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Phytoplankton-zooplankton interactions : data analysis and modelling (with particular reference to Ocean.. 1981

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PHYTOPLANKTON-ZOOPLANKTON INTERACTIONS : DATA ANALYSIS AND MODELLING (WITH PARTICULAR REFERENCE TO OCEAN STATION P (5Q°N,145°S) AND CONTR3LLED ECOSYSTEM EXPERIMENTS ) . By JOHN STANLEY PARSLOK B.Sc., U n i v e r s i t y o f Q u e e n s l a n d , 1974 A THESIS SUBMITTED IN PARTIAL FULFIL MENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (DEPARTMENT OF MATHEMATICS) 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 s ^ u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA J a n u a r y 1981 © John S t a n l e y P a r s l o v , 1981 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e h e a d o f my d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f W#7H£ MflncS The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 D a t e dint, flfrrtl , t<?f>. i i ABSTRACT The anomalous p h y t o p l a n k t o n s e a s o n a l c y c l e i n t h e S u b a r c t i c P a c i f i c h as been a t t r i b u t e d t o g r a z i n g c o n t r o l . In s i m p l e c l a s s i c a l m odels of t h e p h y t o p l a n k t o n - z o o p l a n k t o n i n t e r a c t i o n , g r a z i n g t h r e s h o l d s a r e f o u n d t o be n e c e s s a r y t o o b t a i n t h i s t y p e of c o n t r o l . W e a t h e r s h i p o b s e r v a t i o n s a t O.S.P. a r e a n a l y s e d t o p r o v i d e a b a s i s f o r a more r e a l i s t i c model. P h y t o p l a n k t o n a r e p r e s e n t a t O.S.P. i n a l m o s t u n i f o r m l y low c o n c e n t r a t i o n s ( a b o u t 0.4 mg C h l a . m " 3 ) , have low p h o t o s y n t h e t i c e f f i c i e n c y (<0.5 mg C.mg C h i a " 1 . l y " 1 ) , a d a p t t o s e a s o n a l c h a n g e s i n s o l a r r a d i a t i o n and show most s u r f a c e i n h i b i t i o n i n t h e s p r i n g . A- n u m e r i c a l p r o d u c t i o n model b a s e d on t h e s e r e s u l t s and d r i v e n by p h y s i c a l t i m e s e r i e s from t h e w e a t h e r s h i p s y i e l d s low a n n u a l p r o d u c t i o n l e v e l s compared w i t h p r e v i o u s e s t i m a t e s . P r e d i c t e d p r o d u c t i o n l e v e l s a r e s e n s i t i v e t o t h e c h o i c e o f r e s p i r a t i o n r a t e , and i n t r o d u c t i o n of a r a p i d l i g h t r e s p o n s e or 'Marra' e f f e c t r e s u l t s i n a d o u b l i n g of n e t p r o d u c t i o n . P r e d i c t e d y e a r t o y e a r v a r i a t i o n i s low and might be h i g h e r i f v a r i a t i o n i n S e c c h i d e p t h c o u l d be a c c o u n t e d f o r . In a p h y t o p l a n k t o n - z o o p l a n k t o n ( b i o m a s s ) model b a s e d on t h e p r o d u c t i o n model, g r a z i n g t h r e s h o l d s and o v e r - w i n t e r i n g s t r a t e g i e s a r e b o t h n e c e s s a r y f o r g r a z i n g c o n t r o l . Systems i d e n t i f i c a t i o n t e c h n i q u e s a r e a d a p t e d t o e s t i m a t e p o p u l a t i o n p a r a m e t e r s f o r c o h o r t s o f t h e dom i n a n t g r a z e r s . C o h o r t s t r u c t u r e i s i n t r o d u c e d i n t o t h e p h y t o p l a n k t o n - z o o p l a n k t o n model u s i n g t h e s e e s t i m a t e s . As a r e s u l t , a t t e n t i o n i s s h i f t e d from t h e s p r i n g t o l a t e summer and f a l l where s e n s i t i v i t y and s t a b i l i t y p r o b l e m s a r e a s s o c i a t e d w i t h t h e o v e r - w i n t e r i n g d e p a r t u r e o f t h e dominant g r a z e r s . An a p p r o x i m a t e m a t h e m a t i c a l a n a l y s i s of S t e e l e ' s (1974) n u t r i e n t - p h y t o p l a n k t o n - z o o p l a n k t o n model a l l o w s t h e e x p l a n a t i o n and e l a b o r a t i o n of p r e v i o u s a u t h o r s ' n u m e r i c a l r e s u l t s . S t a b l e c y c l i c s o l u t i o n s a r e shown t o e x i s t under n u t r i e n t l i m i t a t i o n f o r c o n s t a n t m o r t a l i t y r a t e s i n t h e a b s e n c e o f g r a z i n g t h r e s h o l d s . A t t e n t i o n i s f o c u s e d on t h e t r a n s i e n t ( s p r i n g bloom) a p p r o a c h t o t h e n u t r i e n t - l i m i t e d c y c l e and a b r o a d e r ( p h y s i o l o g i c a l and b e h a v i o u r a l ) framework f o r z o o p l a n k t o n r e s p o n s e t o d e c l i n i n g f o o d c o n c e n t r a t i o n s i s p r o p o s e d . Systems i d e n t i f i c a t i o n t e c h n i q u e s a r e a l s o u s e d t o e s t i m a t e z o o p l a n k t o n f e e d i n g and g r o w t h p a r a m e t e r s from CEPEX c o p e p o d t i m e s e r i e s . The e s t i m a t e s a r e compared w i t h l i t e r a t u r e v a l u e s and t h e s t a t i s t i c a l and d e t e r m i n i s t i c l i m i t a t i o n s o f t h e t i m e s e r i e s d i s c u s s e d w i t h a mind t o f u t u r e e x p e r i m e n t s . A n u t r i e n t - p h y t o p l a n k t o n - z o o p l a n k t o n model, b a s e d on t h e p a r a m e t e r e s t i m a t e s , p r o v i d e s a c o n s i s t e n t e x p l a n a t i o n of t h e o b s e r v e d p h y t o p l a n k t o n p e r s i s t e n c e a t low d e n s i t i e s as a s t a b l e n u t r i e n t - l i m i t e d e q u i l i b r i u m . A m a t h e m a t i c a l s o l u t i o n i n t e r m s of B e s s e l f u n c t i o n s i s f o u n d f o r p h y t o p l a n k t o n p o p u l a t i o n s u n d e r g o i n g d i f f u s i o n and s i n k i n g i n t h e c a s e of an e x p o n e n t i a l g r o w t h p r o f i l e . Non- d i m e n s i o n a l i z a t i o n a l l o w s a r e l a t i v e l y c o m p l e t e d i s c u s s i o n o f t h e e f f e c t s o f v a r y i n g p h y s i c a l and b i o l o g i c a l p a r a m e t e r s on p r o f i l e s and p o p u l a t i o n g r o w t h r a t e s . S u b s u r f a c e maxima f o r c o n s t a n t d i f f u s i v i t y and s i n k i n g r a t e , p r e v i o u s l y r e p o r t e d f o r an i d e a l i s e d s t e p - f u n c t i o n g r o w t h p r o f i l e , a r e a l s o o b t a i n e d f o r t h e e x p o n e n t i a l g r o w t h p r o f i l e . S o l u t i o n s t o c o u p l e d n o n - l i n e a r p h y t o p l a n k t o n - n u t r i e n t e q u a t i o n s c o r r e s p o n d i n g t o s u b s u r f a c e maxima o f t h e n u t r i e n t - t r a p t y p e a r e a l s o o b t a i n e d u s i n g b o u n d a r y - l a y e r t e c h n i q u e s . The dependence o f t h e d e p t h , shape and m a g n i t u d e of t h e s e maxima on p a r a m e t e r s i s e x p l o r e d . The a p p r o x i m a t e t h e o r y a g r e e s w e l l w i t h p r e v i o u s l y p u b l i s h e d r e s u l t s f r o m a complex s i m u l a t i o n m o d e l . V CONTENTS ABSTRACT i i LIST OF TABLES i x LIST OF FIGURES x i ACKNOWLEDGEMENTS XX i PREFACE xx i i CHAPTER 1. INTRODUCTION AND ANALYSIS OF SIMPLE GRAZING MODELS. 1.1 G e n e r a l I n t r o d u c t i o n .• 1 1.2 P h y s i c a l Oceanography of the S u b a r c t i c P a c i f i c 2 1.3 B i o l o g y of the S u b a r c t i c P a c i f i c 8 1.4 Simple P h y t o p l a n k t o n - Z o o p l a n k t o n Models f o r the S u b a r c t i c P a c i f i c 14 1.5 Pr e v i e w of C h a p t e r s 2-4 33 CHAPTER 2. QUALITATIVE ANALYSIS OF A COMPLEX SIMULATION MODEL. 2.1 I n t r o d u c t i o n ...38 2.2 Model and A n a l y s i s 40 2.3 S i m u l a t i o n R e s u l t s 47 2.4 C o n c l u s i o n s 59 CHAPTER 3. PHYTOPLANKTON AT O.S.P.: DATA ANALYSIS AND MODELLING. 3.1 I n t r o d u c t i o n 68 3.2 Data A n a l y s i s 68 3.2.1 D e s c r i p t i o n of the Data Set 68 3.2.2 C h l o r o p h y l l Data 69 3.2.3 1 4C Data. 75 3.2.4 N i t r a t e Data 101 3.2.5 N i t r a t e C o n c e n t r a t i o n and P r o d u c t i o n 104 3.3 A P h y t o p l a n k t o n Growth Model I l l 3.3.1 I n t r o d u c t i o n I l l 3.3.2 P h y s i c a l S t r u c t u r e and D r i v i n g V a r i a b l e s 113 3.3.3 B i o l o g i c a l B a s i s f o r the Model 117 3.3.4 S i m u l a t i o n R e s u l t s 124 3.3.5 P r i m a r y P r o d u c t i o n and N i t r a t e D e p l e t i o n 148 CHAPTER 4. HERBIVOROUS ZOOPLANKTON AT O.S.P.: DATA ANALYSIS AND MODELLING. 4.1 Parameter E s t i m a t i o n 150 4.1.1 D e s c r i p t i o n of Data. 150 4.1.2 Review of Parameter E s t i m a t i o n Techniques 151 4.1.3 A p p l i c a t i o n t o O.S.P. Data 156 4.1.4 S t a t i s t i c a l C o n s i d e r a t i o n s 160 4.1.5 R e s u l t s f o r Calanus plumchrus -..161 4.1.6 R e s u l t s f o r Calanus c r i s t a t u s 168 4.1.7 Other S p e c i e s 174 4.1.8 Secondary P r o d u c t i o n E s t i m a t e s 175 4.2 Biomass Model f o r Zoo p l a n k t o n 180 4.2.1 I n t r o d u c t i o n 180 4.2.2 F o r m u l a t i o n of a Biomass G r a z i n g Model 181 4.2.3 C h o i c e of Zoo p l a n k t o n Parameters 183 4.2.4 S i m u l a t i o n R e s u l t s and D i s c u s s i o n 187 4.3 A Cohort Model f o r Zooplankton 198 4.3.1 I n t r o d u c t i o n 198 4.3.2 Model F o r m u l a t i o n 200 4.3.3 Parameters 201 4.3.4 S i m u l a t i o n R e s u l t s and D i s c u s s i o n 202 4.4 C o n c l u s i o n s 224 CHAPTER 5. PARAMETER ESTIMATION AND STABILITY FOR A CEPEX ENCLOSURE. 5.1 I n t r o d u c t i o n 250 5.2 E s t i m a t i o n of Parameters i n a Zoo p l a n k t o n Growth Model.253 5.3 E s t i m a t i o n R e s u l t s 261 5.4 S t a b i l i t y of the P h y t o p l a n k t o n - z o o p l a n k t o n I n t e r a c t i o n . 2 9 8 5.5 C o n c l u s i o n s ; 303 CHAPTER 6. DIFFUSION, SINKING AND GROWTH OF PHYTOPLANKTON. 6.1 I n t r o d u c t i o n 310 6.2 Review of a Simple Model 312 6.3 A More R e a l i s t i c Model 321 6.4 E f f e c t of a Mixed L a y e r 327 6.5 A G e n e r a l N e c e s s a r y C o n d i t i o n For Growth 336 6.6 D i s c u s s i o n 337 CHAPTER 7. MATHEMATICAL ANALYSIS OF DEEP CHLOROPHYLL MAXIMA. 7.1 I n t r o d u c t i o n 342 7.2 A P h y t o p l a n k t o n - N u t r i e n t Model 344 v i i i 7.3 E f f e c t of a Mixed Layer 354 7.4 E f f e c t of N u t r i e n t Dependent S i n k i n g Rates 356 7.5 D i s c u s s i o n 358 CHAPTER 8'. CONCLUDING REMARKS 373 BIBLIOGRAPHY 377 Table I . Parameters used i n S t e e l e ' s Model (2.1) 42 LIST OF TABLES. Table I I . P r e d i c t e d a n n u a l p r i m a r y p r o d u c t i o n a t O.S.P., 1964 t o 1976 128 Table I I I . Monthly means of p r e d i c t e d d a i l y net p r i m a r y p r o d u c t i o n a t O.S.P. u s i n g t h r e e l i g h t a d a p t a t i o n time s c a l e s 134 Table IV. Parameter e s t i m a t e s f o r Calanus plumchrus 163 Table V. Parameter e s t i m a t e s f o r Calanus c r i s t a t u s 169 Table V I . Secondary p r o d u c t i o n e s t i m a t e s 179 Table V I I . Average r e l a t i v e s e a s o n a l abundance of m i c r o z o o p l a n k t o n 233 Table V I I I . F i n a l parameter e s t i m a t e s and SSQ e r r o r s f o r Pseudocalanus 269 Table IX. E x p o n e n t i a l growth r a t e s f o r Calanus 275 Tab l e X. F i n a l parameter e s t i m a t e s and SSQ e r r o r s f o r Calanus 278 T a b l e X I . F i n a l p a r a m e t e r e s t i m a t e s and SSQ e r r o r s f o r P a r a c a l a n u s . x i LIST OF FIGURES. F i g u r e 1. Se a s o n a l c y c l e i n v e r t i c a l s t r u c t u r e a t O.S.P. 5 F i g u r e 2. Schematic diagram of s u r f a c e c u r r e n t s and domains i n the S u b a r c t i c P a c i f i c 6 F i g u r e 3. Se a s o n a l c y c l e i n c h l o r o p h y l l a a t O.S.P. and De p a r t u r e Bay, S t r a i t of G e o r g i a . . . .' 16 F i g u r e 4. Se a s o n a l c y c l e a t O.S.P. i n p r i m a r y p r o d u c t i o n and z o o p l a n k t o n s t a n d i n g s t o c k 17 F i g u r e 5. Phase p l a n e p o r t r a i t s f o r the system 1.2 21 F i g u r e 6. Type I I I f u n c t i o n a l r e sponses 28 F i g u r e 7. Phase p l a n e p o r t r a i t s f o r the system 1.6 29 F i g u r e 8. Phase p l a n e p o r t r a i t s f o r the system 2.3 45 F i g u r e 9. Comparison of g e n e r a t i o n t i m e s from e q u a t i o n 2.4 and those o b t a i n e d i n n u m e r i c a l s o l u t i o n s of 2.2 48 F i g u r e 10. S t a b l e c y c l i c s o l u t i o n s of the system 2.2 f o r GX=0.05 and (a) F=0.4. (b) F=0.2 49 (c) F = 0.6 51 F i g u r e 11. Be h a v i o u r of the system 2.1 f o r GX=0.05 and B=10. w i t h : (a) F=0.4, D=100. ( s t a b l e c y c l e ) ... 52 (b) F=0.2, D=100. ( u n s t a b l e o s c i l l a t i o n s ) . (c) F=0.2, D=175. ( s t a b l e c y c l e ) 53 F i g u r e 12. S i m u l a t i o n of s p r i n g and summer u s i n g the system 2.1 w i t h : (a) B=10., F=0.4, D=100. and GX=0.05 .....58 (b) B=10., F=0.4, D=175. and GX=0.05 60 (c) B=10., F=0.4, D=175. and GX=0.04 61 (d) B=10., F = 0.2, D=250., GX=0.05 and E=0.3 62 F i g u r e 13. S u r f a c e o b s e r v a t i o n s of c h l o r o p h y l l a from the w e a t h e r s h i p s a t O.S.P 70 F i g u r e 14. S u r f a c e c h l o r o p h y l l a ( c r u i s e medians, ann u a l smooth, s e a s o n a l f i t p l u s r e s i d u a l s ) f o r : (a) and (b) 1964-68 72 (c) and (d) 1969-76 73 F i g u r e 15. O b s e r v a t i o n s of C h i b/Chl a and C h i c / C h l a from the w e a t h e r s h i p s a t O.S.P 76 F i g u r e 16. S u r f a c e C h i b/Chl a ( c r u i s e medians, annual smooth, s e a s o n a l f i t p l u s r e s i d u a l s ) 1969-76 77 F i g u r e 17. S u r f a c e C h i c / C h l a ( c r u i s e medians, ann u a l smooth, s e a s o n a l f i t p l u s r e s i d u a l s ) 1969-76 78 F i g u r e 18. S u r f a c e o b s e r v a t i o n s of p r o d u c t i v i t y per u n i t C h i a from the w e a t h e r s h i p s a t O.S.P 80 F i g u r e 19. Frequency h i s t o g r a m s f o r P(0) 81 F i g u r e 20. P(0) ( c r u i s e medians, annua l smooth, s e a s o n a l f i t p l u s r e s i d u a l s ) f o r : (a) and (b) 1964-68 82 (c) and (d) 1969-76 83 F i g u r e 21. S c a t t e r p l o t of P(0) vs I f o r 1964-68 85 F i g u r e 22. S c a t t e r p l o t of A vs I f o r 1964-68 91 F i g u r e 23. Depth p r o f i l e s of P 92 F i g u r e 24. Monthly a v e r a g e s of ex. 94 F i g u r e 25. S c a t t e r p l o t of B vs I. f o r 1964-68 96 F i g u r e 26. S c a t t e r p l o t of P M A X vs I 0 f o r 1964-68 98 F i g u r e 27. Monthly a v e r a g e s of e s t i m a t e s of l i g h t a d a p t a t i o n p a r a m e t e r s : (a) B 99 (b) B s 100 F i g u r e 28. S u r f a c e o b s e r v a t i o n s of n i t r a t e c o n c e n t r a t i o n from the w e a t h e r s h i p s a t O.S.P. 102 x i v F i g u r e 29. S u r f a c e n i t r a t e c o n c e n t r a t i o n s ( c r u i s e medians, annua l smooth, s e a s o n a l f i t p l u s r e s i d u a l s ) 103 F i g u r e 30. N i t r a t e c o n c e n t r a t i o n s from depth p r o f i l e s ( l a y e r a v e r a g e s , annual smooths, s e a s o n a l f i t s p l u s r e s i d u a l s ) 105 (a) 0 - 20m 106 (b) 20 - 40m 107 (c) 40 - 80m 108 (d) 80 - 130m 109 (e) 130 - 200m 110 F i g u r e 31. P(0) vs low s u r f a c e n i t r a t e v a l u e s , May t o O c t o b e r , 1964-68. 112 F i g u r e 32. Time s e r i e s of t o t a l s o l a r r a d i a t i o n , s u r f a c e t e mperature and mixed l a y e r d e pth used t o d r i v e s i m u l a t i o n model 125 F i g u r e 33. P r e d i c t e d d a i l y net p r o d u c t i o n u s i n g s t a n d a r d parameter s e t 127 F i g u r e 34. P r e d i c t e d net p r o d u c t i o n , mixed l a y e r C h i a and mixed l a y e r C:Chl a r a t i o f o r 1976 u s i n g s t a n d a r d parameter s e t and s e a s o n a l l i g h t a d a p t a t i o n 130 F i g u r e 35. As f o r F i g 34, but w i t h ' i n s t a n t a n e o u s ' l i g h t a d a p t a t i o n 131 XV F i g u r e 36. As f o r F i g 34, but w i t h 3-day a d a p t a t i o n 133 F i g u r e 37. P r e d i c t e d C:Chl a r a t i o i n the mixed l a y e r f o r 1976 u s i n g oc = 0.5 and B =2.0 136 F i g u r e 38. P r e d i c t e d d a i l y net p r o d u c t i o n on i n c r e a s i n g y t o 0.1 139 F i g u r e 39. P r e d i c t e d d a i l y net p r o d u c t i o n and mixed l a y e r C:Chl a r a t i o f o r 0c=l.O, B =2.0 w i t h c o n s t r a i n t V^IOO. ..140 F i g u r e 40. P r e d i c t e d d a i l y net p r o d u c t i o n f o r s t a n d a r d parameter s e t w i t h Marra e f f e c t i n t r o d u c e d 142 F i g u r e 41. Observed S e c c h i depths a t O.S.P. vs time of year 143 F i g u r e 42. P r e d i c t e d d a i l y net p r o d u c t i o n u s i n g s t a n d a r d parameter s e t and (a) upper envelope t o S e c c h i d e p t h s . ...144 (b) lower envelope t o S e c c h i depths 145 F i g u r e 43. P r e d i c t e d s e a s o n a l c y c l e i n mixed l a y e r C:Chl a r a t i o u s i n g o p t i m a l i t y c r i t e r i o n 147 F i g u r e 44. P r o j e c t i o n of approximate 95% c o n f i d e n c e r e g i o n s f o r Calanus plumchrus parameter e s t i m a t e s on (a) (9,R T) p l a n e 164 (b) U,6) p l a n e 165 x v i F i g u r e 45. P r o j e c t i o n of a p p r o ximate 95% c o n f i d e n c e r e g i o n s f o r Calanus c r i s t a t u s parameter e s t i m a t e s 171 on (a) (6,R T) p l a n e 172 (b) (y,B) p l a n e 173 F i g u r e 46. R e g r e s s i o n c o e f f i c i e n t s W,- vs c o r r e s p o n d i n g l e n g t h s 1; on l o g - l o g s c a l e 177 F i g u r e 47. P r e d i c t e d mixed l a y e r C h i a and h e r b i v o r e biomass f o r 1976 u s i n g s t a n d a r d parameter s e t 188 F i g u r e 48. E f f e c t of d e c r e a s i n g D on F i g 47 191 F i g u r e 49. E f f e c t of i n t r o d u c i n g o v e r - w i n t e r i n g s t r a t e g y i n F i g 47 194 F i g u r e 50. E f f e c t of r e d u c i n g s p r i n g r e c r u i t m e n t t o 25 mg Cm" 3 on F i g 49 195 F i g u r e 51. P r e d i c t e d C h i a and g r a z e r s t a n d i n g s t o c k f o r biomass model w i t h Type I f u n c t i o n a l response and s p r i n g r e c r u i t m e n t ; 199 F i g u r e 52. P r e d i c t e d C h i a and z o o p l a n k t o n biomass f o r 1976, u s i n g weight t h r e s h o l d s f o r d e p a r t u r e i n c o h o r t model 203 F i g u r e 53. As f o r F i g 52, but w i t h f i x e d r e s i d e n c e t i m e s x v i i and lower growth and m o r t a l i t y r a t e s f o r C. C r i s t a t u s . ..209 F i g u r e 54. E f f e c t of q u a d r u p l i n g s p r i n g r e c r u i t m e n t i n F i g 53 211 F i g u r e 55. E f f e c t of h a l v i n g s p r i n g r e c r u i t m e n t i n F i g 53. 213 F i g u r e 56. E f f e c t of i n c r e a s i n g m o r t a l i t y r a t e s f o r Calanus i n F i g 53 214 F i g u r e 57. E f f e c t of d e c r e a s i n g g r a z i n g parameters CO and D i n F i g 53. 216 F i g u r e 58. E f f e c t of i n c r e a s i n g m e t a b o l i c r a t e s i n F i g 57. 217 F i g u r e 59. P r e d i c t e d mixed l a y e r C h i a and t o t a l z o o p l a n k t o n carbon f o r 1964-76, u s i n g the parameters and f i x e d r e c r u i t m e n t l e v e l s of F i g 53 219 F i g u r e 60. Observed z o o p l a n k t o n wet w e i g h t s (10 day means) a t O.S.P. 1956-78 221 F i g u r e 61. P r e d i c t e d mixed l a y e r C h i a and z o o p l a n k t o n carbon f o r 1964-76 u s i n g c o u p l e d r e c r u i t m e n t 223 F i g u r e 62. Average s e a s o n a l c y c l e s i n (a) obse r v e d x v i i i z o o p l a n k t o n wet w e i g h t s (10 day means, 1956-78) 228 (b) i n g e s t i o n v a r i a b l e VI (10 day means, 1969-78) 229 (c) i n g e s t i o n v a r i a b l e V2 (10 day means, 1969-78) 230 F i g u r e 63. Annual v a r i a t i o n (10 day means) i n (a) i n g e s t i o n v a r i a b l e VI 237 (b) i n g e s t i o n v a r i a b l e V2 238 F i g u r e 64. P h y t o p l a n k t o n carbon (0-20m average) i n CEE5 ..257- F i g u r e -65. J2fj(W), ^(W) f o r WJ-=0.4 jug C, Wj^ =2.0 pq C ...262 F i g u r e 66. F u n c t i o n a l response d a t a f o r P s e u s o c a l a n u s ....264 F i g u r e 67. Observed d e n s i t i e s of Pseudocalanus i n CEE5 ...266 F i g u r e 68. Comparison of p r e d i c t i o n s and o b s e r v a t i o n s f o r Pseudocalanus (a) t r i a l 1 270 (b) t r i a l 6 271 F i g u r e 69. Observed d e n s i t i e s of Calanus i n CEE5 276 F i g u r e 70. Comparison of p r e d i c t i o n s and o b s e r v a t i o n s f o r C a l a n u s . (a) t r i a l 1 279 (b) t r i a l 4 281 (c) t r i a l 6 283 (d) t r i a l 7 284 x i x F i g u r e 71. Observed d e n s i t i e s of P a r a c a l a n u s i n CEE5. ...288 F i g u r e 72. Observed d e n s i t i e s of t o t a l n a u p l i i i n CEE5 ...289 F i g u r e 73. Comparison of p r e d i c t i o n s and o b s e r v a t i o n s f o r P a r a c a l a n u s (a) t r i a l 1 292 (b) t r i a l 2 293 (c) t r i a l 3 295 (d) t r i a l 4 297 F i g u r e 74. P r e d i c t e d p h y t o p l a n k t o n c o n c e n t r a t i o n s i n CEE5 a f t e r day 30 302 F i g u r e 75. An i l l u s t r a t i o n of the e i g e n c o n d i t i o n 6.6. ...316 F i g u r e 76. Contour p l o t of the f u n c t i o n JVioo, Sg ) 317 F i g u r e 77. C h a r a c t e r i s t i c p r o f i l e s p{S) 319 F i g u r e 78. An i l l u s t r a t i o n of the e i g e n c o n d i t i o n 6.9. ...324 F i g u r e 79. Contour p l o t of the f u n c t i o n A.1 (UJ,/3) .325 F i g u r e 80. C h a r a c t e r i s t i c p h y t o p l a n k t o n p r o f i l e f o r c o = 0.7, 13 = 20 328 F i g u r e 81. Contour p l o t of the f u n c t i o n _ A 3 ( / l , i p f o r (a) j3=0 332 XX (b) (3=1 333 (c) /?=3 334 (d) (3 = 10 335 F i g u r e 82. Contour p l o t of S,/(3 vs co,(3 f o r S=0.1 359 F i g u r e 83. C h a r a c t e r i s t i c p r o f i l e s P(S) vs S//5 361 F i g u r e 84. P h y t o p l a n k t o n p r o f i l e s P(<p) vs <p f o r 6 = 1. and 9 = 0.1 365 F i g u r e 85. Contour p l o t s of fT- <pc vs (3,UJ* f o r 5=0.1 and 0=1. and 0.1 366 F i g u r e 86. Contour p l o t s of ^ vs j3,co f o r =w and w4 =5.w 368 F i g u r e 87. Comparison of p h y t o p l a n k t o n p r o f i l e s p r e d i c t e d by complex s i m u l a t i o n model and by a s y m p t o t i c a n a l y s i s . ..372 xx i ACKNOWLEDGEMENTS I w o u l d l i k e t o a c k n o w l e d g e t h e e n c o u r a g e m e n t , s u p p o r t and a d v i c e o f my t h e s i s s u p e r v i s o r , Dr T . R . P a r s o n s , and a l s o t h e o t h e r members o f my t h e s i s c o m m i t t e e f o r h e l p f u l d i s c u s s i o n s . I have t o t h a n k Mr J . D . F u l t o n and Mr O.D.Kennedy o f t h e P a c i f i c B i o l o g i c a l s t a t i o n , Nanaimo, f o r k i n d l y p r o v i d i n g much of t h e raw d a t a u s e d i n t h i s s t u d y and a s s i s t i n g i n i t s i n t e r p r e t a t i o n . The s t u d y of p a r a m e t e r e s t i m a t i o n owes much t o an i n i t i a l , s t i m u l a t i n g c o l l a b o r a t i o n w i t h Dr J.B.L.Matthews and Mr N.C.Sonntag. My s t u d i e s i n Canada were s u p p o r t e d by a p o s t - g r a d u a t e s t u d e n t s h i p from CSIRO, A u s t r a l i a . x x i i PREFACE E c o l o g y has been v a r i o u s l y c a l l e d a s o f t s c i e n c e , an immature s c i e n c e , or even a p r e - s c i e n c e (Kuhn,1970) by comparison w i t h the c l a s s i c a l f i e l d s of p h y s i c s and c h e m i s t r y . The l a t t e r d i s c i p l i n e s have a c q u i r e d a t h e o r e t i c a l c o r e which i s r i g o r o u s l y (and u s u a l l y m a t h e m a t i c a l l y ) f o r m u l a t e d . A m a t h e m a t i c i a n working i n t h e s e f i e l d s i s e s s e n t i a l l y c o n c erned w i t h d educing the i m p l i c a t i o n s of t h i s t h e o r y a c c o r d i n g t o the p r i n c i p l e s of mathematics and l o g i c . He can, f o r the most p a r t , f a c e t h i s t a s k w i t h o u t nervous backward g l a n c e s t o see i f the t h e o r y has j u s t changed. T h i s i s not t o say t h a t the ' r e a l w o r l d ' does not i n t r u d e upon him. The m a t h e m a t i c a l problem u s u a l l y c o r r e s p o n d s t o a q u e s t i o n about the r e a l system i n v o l v i n g experiment and/or o b s e r v a t i o n . Moreover, i n many i n t e r e s t i n g c a s e s , an e x a c t g e n e r a l s o l u t i o n t o the a p p r o p r i a t e m a t h e m a t i c a l problem cannot be found and the d e r i v a t i o n of an a p p r o x i m a t e , t r a c t a b l e problem i s n e c e s s a r y ( a p p l i e d mathematics has been c a l l e d i n p a r t 'the a r t of j u d i c i o u s a p p r o x i m a t i o n ' (Greenspan,1969)). I t i s t h i s p r o c e d u r e which demands of s u c c e s s f u l a p p l i e d m a t h e m a t i c i a n s a f a m i l i a r i t y w i t h the s c i e n c e , or ' p h y s i c a l i n t u i t i o n ' , as w e l l as t e c h n i c a l m a t h e m a t i c a l a b i l i t y . A s o f t s c i e n c e such as e c o l o g y i s d i s t i n g u i s h e d by i t s l a c k of an a c c e p t e d t h e o r e t i c a l c o r e . For any p a r t i c u l a r r e a l system, t h e r e i s no u n i v e r s a l l y a c c e p t e d , m a t h e m a t i c a l l y f o r m u l a t e d model. An a p p l i e d m a t h e m a t i c i a n v e n t u r i n g i n t o such a f i e l d has two c h o i c e s . He can t a k e a p r e v i o u s l y p u b l i s h e d model and a n a l y z e i t s m a t h e m a t i c a l p r o p e r t i e s , p a y i n g l i t t l e a t t e n t i o n t o x x i i i t h e ( s e n s i t i v e ) i s s u e of i t s b i o l o g i c a l a c c u r a c y . A t h e o r y can d e v e l o p w h i c h i s i n c r e a s i n g l y d i v o r c e d f r o m r e a l i t y . T h e r e i s a l a r g e body o f e c o l o g i c a l t h e o r y , o f t e n o f c o n s i d e r a b l e m a t h e m a t i c a l e l e g a n c e , c o n c e r n i n g t r a d i t i o n a l i d e a l i s e d m o d e ls w h i c h do n o t c o r r e s p o n d i n d e t a i l t o any r e a l s y s t e m . T h i s t h e o r y c a n be u s e f u l i n t h e s t u d y o f a p a r t i c u l a r , r e a l s y s t e m ( i t i s u s e d i n t h i s t h e s i s ) , b ut t h e gap between t h e o r y and r e a l i t y may overwhelm t h e f i e l d b i o l o g i s t l a c k i n g f o r m a l m a t h e m a t i c a l t r a i n i n g . The m a t h e m a t i c i a n ' s a l t e r n a t i v e i s t o p l u n g e i n t o t h e b i o l o g i c a l c o n t r o v e r s y and become a m o d e l l e r ; t h a t i s , someone who f o r m u l a t e s h i s own m a t h e m a t i c a l v e r s i o n o f r e a l i t y . W h i l e f a m i l i a r i t y w i t h t h e b i o l o g i c a l s c i e n c e i s a p r e r e q u i s i t e f o r m o d e l l i n g , s o p h i s t i c a t e d m a t h e m a t i c a l a b i l i t y i s n o t . W i t h t h e a d v e n t o f t h e computer, v e r y l a r g e , complex s i m u l a t i o n models can be c o n s t r u c t e d and s t u d i e d w i t h l i t t l e a d v a n c e d m a t h e m a t i c s . One m i g h t wonder t h e n i f b i o l o g i s t s , who p r e s u m a b l y have more f a m i l i a r i t y w i t h t h e i r s c i e n c e , need a p p l i e d m a t h e m a t i c i a n s a t a l l . The answer t o t h i s q u e s t i o n i s y e s , f o r t h e same r e a s o n t h a t c o m p u t e r s have not e l i m i n a t e d a l l m a t h e m a t i c i a n s e x c e p t n u m e r i c a l a n a l y s t s i n t h e p h y s i c a l s c i e n c e s . A s i m u l a t i o n model i s a c l u m s y t o o l w h i c h p r o d u c e s a p a r t i c u l a r outcome f o r a s p e c i f i c s e t o f p a r a m e t e r v a l u e s . P a r t i c u l a r l y i n t h e c a s e o f c o m p l i c a t e d s i m u l a t i o n m o d e l s , i n v o l v i n g many p a r a m e t e r s and i n t e r a c t i o n s , t h e u s u a l c o n s i d e r a t i o n of a s m a l l number o f s i m u l a t i o n s ( s e n s i t i v i t y a n a l y s i s ) c a n g i v e a v e r y m i s l e a d i n g p i c t u r e o f t h e m odel's p r o p e r t i e s . I t i s t h e t a s k o f t h e a p p l i e d m a t h e m a t i c i a n x x i v t o u n c o v e r t h e g e n e r a l d e pendence o f a model's b e h a v i o u r on b o t h p a r a m e t e r v a l u e s and t h e a s s u m p t i o n s w h i c h a r e i m p l i c i t i n i t s m a t h e m a t i c a l s t r u c t u r e . T h i s i s e s p e c i a l l y i m p o r t a n t i n a f i e l d s u c h a s e c o l o g y , where u n c e r t a i n t y r e g a r d i n g b o t h p a r a m e t e r v a l u e s and t h e a p p r o p r i a t e model s t r u c t u r e i s v e r y h i g h . As i n t h e p h y s i c a l s c i e n c e s , e x a c t g e n e r a l s o l u t i o n s t o model e q u a t i o n s a r e n o t u s u a l l y a v a i l a b l e and t e c h n i q u e s o f a p p r o x i m a t i o n and q u a l i t a t i v e a n a l y s i s a r e i m p o r t a n t . Some o f t h e s e c t i o n s of t h i s t h e s i s i n v o l v e t h e a n a l y s i s of s i m u l a t i o n m o d e ls c o n s t r u c t e d by o t h e r s ( C h a p t e r 2 ) , or of s i m p l e i d e a l i s e d m o d e ls ( C h a p t e r s 1 , 6 , 7 ) . O t h e r s i n v o l v e t h e f o r m u l a t i o n and a n a l y s i s of o r i g i n a l models w i t h c o n s i d e r a b l e new b i o l o g i c a l i n p u t ( C h a p t e r 4 ) . In e a c h c a s e , a c l e a r p i c t u r e has been s o u g h t o f t h e r e l a t i o n s h i p between t h e model Vs. m a t h e m a t i c a l o u t p u t ( b e h a v i o u r ) and i n p u t ( s t r u c t u r e and p a r a m e t e r v a l u e s ) , b o t h i n t e r p r e t e d b i o l o g i c a l l y . In an i n t e r e s t i n g d i s c u s s i o n of t h e n o t i o n of i n t e l l i g e n c e , Weizenbaum(1976) o b s e r v e s t h a t b e i n g g i v e n a b l u e p r i n t of t h e human mind, o r o f a complex computer s y s t e m , w h i c h a l l o w e d us t o c o n s t r u c t an e x a c t , w o r k i n g c o p y , would not be a t a l l t h e same t h i n g a s u n d e r s t a n d i n g t h e human mind or t h e c o m p u t e r . The c o n c e p t of ' u n d e r s t a n d i n g ' i s a d i f f i c u l t one t o d e f i n e r i g o r o u s l y and may be a h i g h l y p e r s o n a l one. I s h a r e Weizenbaum's n o t i o n o f u n d e r s t a n d i n g s u f f i c i e n t l y t o a g r e e w i t h h i s s t a t e m e n t and would make t h e same p o i n t a b o u t complex s i m u l a t i o n m o d e l s i n e c o l o g y : an a b i l i t y t o c o n s t r u c t a l a r g e , c o m p l i c a t e d model f r o m component p i e c e s and e v e n t o have i t mimic t h e n a t u r a l s y s t e m p e r f e c t l y (an u n l i k e l y e v e n t ) i s n o t t h e same XXV t h i n g as an u n d e r s t a n d i n g o f t h e e c o s y s t e m . I f i n d t h a t t h e q u a l i t a t i v e , a p p r o x i m a t e a n a l y s e s d e s c r i b e d above and c a r r i e d o ut i n t h i s t h e s i s , by u n c o v e r i n g r e l a t i o n s h i p s between a s s u m p t i o n s and p r e d i c t i o n s a t a s i m p l e , o f t e n g r a p h i c a l l e v e l , p r o v i d e t h a t s e n s e of u n d e r s t a n d i n g w h i c h a complex s i m u l a t i o n model l a c k s . In a d d i t i o n t o t h e c o n s t r u c t i o n and a n a l y s i s o f m o d e l s , a number o f t y p e s of d a t a a n a l y s i s a r e p r e s e n t e d i n t h i s t h e s i s . T h e s e r a n g e from r e l a t i v e l y c o n v e n t i o n a l d e s c r i p t i v e t e c h n i q u e s t o t h e r a t h e r a m b i t i o u s m o d e l - f i t t i n g p r o c e d u r e s of C h a p t e r 5. A l l t e c h n i q u e s of d a t a a n a l y s i s a r e b a s e d on an u n d e r l y i n g s t a t i s t i c a l m o d e l . In many o f t h e c l a s s i c a l t e c h n i q u e s , t h e u n d e r l y i n g model was k e p t d e l i b e r a t e l y s i m p l e , u s u a l l y l i n e a r , t o f a c i l i t a t e t h e c o m p u t a t i o n of p a r a m e t e r e s t i m a t e s and t h e a n a l y s i s o f t h e i r s t a t i s t i c a l p r o p e r t i e s . The d e v e l o p m e n t o f c o m p u t e r s and n u m e r i c a l m o d e l - f i t t i n g a l g o r i t h m s (Benson,1978) now o f f e r s t h e p o s s i b i l i t y . o f e s t i m a t i n g b i o l o g i c a l l y m e a n i n g f u l p a r a m e t e r s i n more complex models from o b s e r v a t i o n s on c o m p l i c a t e d n a t u r a l o r e x p e r i m e n t a l s y s t e m s . Much o f t h e d a t a a n a l y s i s i n t h i s t h e s i s r e p r e s e n t s an e x p l o r a t i o n o f t h e p o t e n t i a l of t h i s a p p r o a c h . 1 CHAPTER 1 INTRODUCTION AND ANALYSIS OF SIMPLE GRAZING MODELS. 1.1 G e n e r a l I n t r o d u c t i o n . The problems d i s c u s s e d i n t h i s t h e s i s have a l l a r i s e n out of an i n i t i a l i n t e r e s t i n the marine p l a n k t o n i c ecosystem of the S u b a r c t i c P a c i f i c . The g e n e r a l i n t r o d u c t i o n g i v e n i n t h i s s e c t i o n i s i n t e n d e d as a b r i e f s u r v e y of the problems t r e a t e d and t h e i r i n t e r r e l a t i o n s h i p s . A d e t a i l e d i n t r o d u c t i o n t o the p h y s i c a l and b i o l o g i c a l oceanography of the S u b a r c t i c P a c i f i c i s g i v e n i n S e c t i o n s 1.2 and 1.3 r e s p e c t i v e l y . Comprehensive i n t r o d u c t i o n s t o the d e r i v e d problems a r e l e f t t o the c o r r e s p o n d i n g c h a p t e r s . At the time t h i s study commenced, i n 1976, m o d e l l i n g of s e a s o n a l c y c l e s i n p l a n k t o n i c n u t r i e n t - p l a n t - h e r b i v o r e systems, p a r t i c u l a r l y the phenomena of s p r i n g and f a l l p h y t o p l a n k t o n blooms, was w e l l e s t a b l i s h e d . A g a i n s t t h i s background of e m p i r i c a l knowledge and t h e o r e t i c a l u n d e r s t a n d i n g , the observ e d s e a s o n a l c y c l e i n the S u b a r c t i c P a c i f i c , showing l i t t l e or no v a r i a t i o n i n p h y t o p l a n k t o n s t a n d i n g s t o c k , s t o o d out as an anomaly. A number of s i m p l e v e r b a l hypotheses had been proposed t o account f o r t h i s c y c l e ( M c A l l i s t e r et a l , 1 9 6 0 ; H e i n r i c h , 1 9 6 2 ) and the time seemed r i p e f o r a more r i g o r o u s e x a m i n a t i o n of the problem v i a m o d e l l i n g , e s p e c i a l l y i n view of the e x i s t e n c e of a l o n g time s e r i e s of b i o l o g i c a l o b s e r v a t i o n s from the w e a t h e r s h i p s at Ocean S t a t i o n 1 1P" ( h e n c e f o r t h a b b r e v i a t e d O.S.P. ) a t 50°N, 145°W i n the S u b a r c t i c P a c i f i c . P r e l i m i n a r y c o n s i d e r a t i o n s , u s i n g some c l a s s i c a l p o p u l a t i o n - 2 i n t e r a c t i o n models (Chapter 1) suggested t h a t g r a z i n g t h r e s h o l d s c o u l d be i m p o r t a n t a t O.S.P. T h i s r e s u l t prompted an approximate q u a l i t a t i v e a n a l y s i s of a complex n u m e r i c a l model ( S t e e l e , 1 9 7 4 ; Landry,1976) whose b e h a v i o u r i n computer s i m u l a t i o n had provoked some d i s c u s s i o n of t h r e s h o l d s . The i n s i g h t s o b t a i n e d from t h i s a n a l y s i s proved u s e f u l i n a more q u a n t i t a t i v e , d e t a i l e d m o d e l l i n g i n v e s t i g a t i o n of ecosystem dynamics a t O.S.P. (Ch a p t e r s 3,4). A n a l y s i s of the time s e r i e s of b i o l o g i c a l o b s e r v a t i o n s from the w e a t h e r s h i p s , conducted as p a r t of t h i s i n v e s t i g a t i o n , i n v o l v e d some n o v e l s t a t i s t i c a l problems, i n c l u d i n g the a d a p t a t i o n of a systems i d e n t i f i c a t i o n t e c h n i q u e f o r e s t i m a t i n g p o p u l a t i o n parameters from z o o p l a n k t o n time s e r i e s ( P a r s l o w et a l ,1979) (Chapter 4) t o overcome i n c o n s i s t e n c i e s i n the .zooplankton time s e r i e s from O.S.P. A v a r i a n t of the parameter e s t i m a t i o n t e c h n i q u e was a p p l i e d t o p h y t o p l a n k t o n and z o o p l a n k t o n time s e r i e s from a c o n t r o l l e d ecosystem e n c l o s u r e s t u d i e d d u r i n g the CEPEX programme (Chapter 5 ) , i n an attempt t o e s t i m a t e f u n c t i o n a l response parameters r e l e v a n t t o b i o l o g i c a l q u e s t i o n s which a r o s e i n the t h e o r e t i c a l a n a l y s i s of c h a p t e r two. Ch a p t e r s 6 and 7 i n v o l v e a t h e o r e t i c a l a n a l y s i s of p h y t o p l a n k t o n p o p u l a t i o n s undergoing growth, d i f f u s i o n , s i n k i n g and n u t r i e n t l i m i t a t i o n . W h i l e o r i g i n a l l y m o t i v a t e d by c o n s i d e r a t i o n of l i g h t - l i m i t e d growth a t O.S.P. , the r e s u l t s proved t o be of more i n t e r e s t i n ca s e s of l o w - t u r b u l e n c e environments and s u b - s u r f a c e c h l o r o p h y l l maxima. 1.2 P h y s i c a l Oceanography of the S u b a r c t i c P a c i f i c . T h i s i n t r o d u c t i o n t o the p h y s i c a l oceanography of the 3 S u b a r c t i c P a c i f i c i s i n t e n d e d as background t o the e c o l o g i c a l models c o n s i d e r e d l a t e r . Two a s p e c t s a r e c o n s i d e r e d h e r e . The s e a s o n a l c y c l e i n water column s t r u c t u r e a t O.S.P. i s im p o r t a n t i n the m o d e l l i n g of p r i m a r y p r o d u c t i o n t h e r e . The l o c a t i o n of O.S.P. r e l a t i v e t o the broad c i r c u l a t i o n p a t t e r n s of the S u b a r c t i c P a c i f i c i s d i s c u s s e d i n an assessment of h o r i z o n t a l a d v e c t i v e e f f e c t s on b i o l o g i c a l p r o c e s s e s . The w e a t h e r s h i p s ' o b s e r v a t i o n s of temperature and s a l i n i t y p r o f i l e s , as w e l l as m e t e o r o l o g i c a l v a r i a b l e s , have p r o v i d e d a b a s i s f o r a number of t h e o r e t i c a l and e m p i r i c a l s t u d i e s of s e a s o n a l c y c l e s a t O.S.P. , p a r t i c u l a r l y the heat budget a s s o c i a t e d w i t h f o r m a t i o n and breakdown of the s e a s o n a l t h e r m o c l i n e (Tabata,1965; Denman,1972). T h i s c y c l e w i l l be p r e s e n t e d q u a n t i t a t i v e l y w i t h a .model of' p r i m a r y p r o d u c t i o n a t O.S.P. i n Chapter 3; a b r i e f d e s c r i p t i o n i s g i v e n h e r e . On the b a s i s of s a l i n i t y , Dodimead e t a l (1963) d i s t i n g u i s h t h r e e permanent zones : an upper zone from 0 t o 100m, a h a l o c l i n e from 100 t o 200m i n which s a l i n i t y i n c r e a s e s from 32.8%<>to 33.8%o, and a lower zone, i n which s a l i n i t y i n c r e a s e s s l o w l y t o 34.4%,at 1000m. The t o p of the h a l o c l i n e c o r r e s p o n d s t o the maximum depth of the s u r f a c e mixed l a y e r , a t t a i n e d i n March a t the. end of the p e r i o d of net heat l o s s t h r o u g h the s u r f a c e . A s e a s o n a l t h e r m o c l i n e i s e s t a b l i s h e d over the p e r i o d of net heat g a i n , from A p r i l t o September, w i t h the mixed l a y e r b e i n g t y p i c a l l y about 30m deep a t the end of t h i s p e r i o d . S u r f a c e t e m p e r a t u r e s i n c r e a s e from about 5°C t o 13°C over t h i s p e r i o d . The s e a s o n a l t h e r m o c l i n e and an a s s o c i a t e d s e a s o n a l h a l o c l i n e a r e eroded over the c o o l i n g p e r i o d , from October t o March, by c o n v e c t i v e o v e r t u r n 4 and s t o r m a c t i v i t y . T h i s s e a s o n a l c y c l e ( F i g 1) i s q u a l i t a t i v e l y c h a r a c t e r i s t i c o f t h e o c e a n i c S u b a r c t i c P a c i f i c a l t h o u g h t h e r e i s some g e o g r a p h i c a l v a r i a t i o n i n t h e m a g n i t u d e and t i m i n g o f t h e c y c l e . Two r e v i e w s of t h e p h y s i c a l o c e a n o g r a p h y of t h e S u b a r c t i c P a c i f i c have been p u b l i s h e d as p a r t o f i n v e s t i g a t i o n s i n t o t h e o c ean e n v i r o n m e n t of salmon by t h e I n t e r n a t i o n a l N o r t h P a c i f i c F i s h e r i e s Commission (Dodimead e t a l ,1963; F a v o r i t e e t a l ,1976). In t h e e a r l i e r r e v i e w , t h e p r i n c i p a l c u r r e n t s y s t e m s and domains were summarized as i n F i g 2. The s o u t h e r n l i m i t o f t h e S u b a r c t i c r e g i o n was d e f i n e d by an a l m o s t v e r t i c a l i s o h a l i n e s u r f a c e of 3 4 % 0 a t a b o u t 42°N l a t i t u d e . I m m e d i a t e l y t o t h e n o r t h a body o f warmer water formed by m i x i n g of t h e K u r o s h i o and O y a s h i o c u r r e n t s and moving, .eastward a t 2 t o 4 n a u t i c a l m i l e s / d a y was i d e n t i f i e d a s t h e t r a n s i t i o n d omain. To t h e n o r t h a g a i n , t h e S u b a r c t i c c u r r e n t was d e s c r i b e d as moving e a s t a t a b o u t 2 n a u t i c a l m i l e s / d a y and c o n s i s t i n g o f t h a t p a r t of t h e O y a s h i o w h i c h does n o t mix w i t h t h e K u r o s h i o . T h e s e e a s t w a r d f l o w i n g c u r r e n t s d i v i d e o f f t h e c o a s t o f N o r t h A m e r i c a , p a r t t r a v e l l i n g s o u t h t o form t h e C a l i f o r n i a c u r r e n t and p a r t f l o w i n g n o r t h w a r d a r o u n d t h e G u l f o f A l a s k a , f o r m i n g a r e l a t i v e l y i n t e n s e b o u n d a r y c u r r e n t ( t h e A l a s k a n S t e a m ) . P a r t of t h e A l a s k a n S t r e a m t u r n s s o u t h t o f o r m t h e A l a s k a n G y r e , w h i l e p a r t moves n o r t h i n t o t h e B e r i n g Sea t o e v e n t u a l l y j o i n t h e E a s t Kamchatka c u r r e n t o r t h e O y a s h i o . The c i r c u l a t i o n t i m e f o r t h e e n t i r e s y s t e m i s g i v e n a s a b o u t 4-6 y e a r s . O.S.P. i s l o c a t e d n o r t h o f t h e t r a n s i t i o n z o n e , on t h e s o u t h - e a s t e r l y edge of t h e A l a s k a n G y r e . The f o l l o w i n g q u o t e i s 0 50 CL CD a 100 net h e a t i n g p\ wind m i x i n g c o n v e c t i v e mix ing seasona l t h e r m o c l i n e — o p i m a r c n t a ' o c l i n e J F M A M J J A S O N D Month F i g u r e 1. S e a s o n a l c y c l e i n v e r t i c a l s t r u c t u r e a t O.S.P. (Ad a p t e d f r o m T a b a t a , 1 9 6 5 ) .  7 p a r t i c u l a r l y r e l e v a n t f o r m o d e l l i n g the p l a n k t o n i c ecosystem a t O.S.P. a l t h o u g h i t was prompted by c o n s i d e r a t i o n s of s e a s o n a l s a l i n i t y and temperature c y c l e s t h e r e : '. . . the g e o s t r o p h i c f l o w i n the v i c i n i t y of Ocean S t a t i o n "P" i s g e n e r a l l y z o n a l , b u t , more i m p o r t a n t l y , slow (2 m i l e s / d a y ) . Hence, w i t h i n the time p e r i o d c o n s i d e r e d h e r e , ' ( s e a s o n a l ) ' the waters i n the ar e a a r e s u b j e c t e d t o c l i m a t i c c o n d i t i o n s s i m i l a r t o those a t Ocean S t a t i o n "P"; t h u s , they can be r e g a r d e d as h a v i n g r e s i d e d t h e r e . . . ' (Dodimead e t a l ,1963) . The second review ( F a v o r i t e e_t a l ,1976) p r e s e n t s a s i m i l a r p a t t e r n f o r the g e n e r a l c i r c u l a t i o n , w i t h f u r t h e r e l a b o r a t i o n of the c u r r e n t and domain s t r u c t u r e s . There i s , however, a new f e a t u r e p r e s e n t e d which may a f f e c t b i o l o g i c a l m o d e l l i n g and d a t a - i n t e r p r e t a t i o n f o r O.S.P. The dynamic topography f o r J u l y shows a s m a l l c l o c k w i s e gyre i n the s u r f a c e c i r c u l a t i o n o f f the Queen C h a r l o t t e I s l a n d s , e a s t of the n o r t h - s o u t h b r a n c h i n g of the S u b a r c t i c C u r r e n t . A s s o c i a t e d w i t h t h i s gyre i s a ' D i l u t e Domain' of water showing the e f f e c t s of c o a s t a l r u n o f f . The rev i e w i s ambiguous as t o the westward e x t e n t of t h i s domain . C i r c u l a t i o n p a t t e r n s suggested by dynamic topography, i n t e g r a t e d t o t a l w i n d - s t r e s s t r a n s p o r t and n u m e r i c a l model r e s u l t s a l l i n d i c a t e t h a t the c o a s t a l gyre l i e s w e l l t o the e a s t of O.S.P. and t h a t the f l o w a t O.S.P. i s e s s e n t i a l l y as d e s c r i b e d above from the e a r l i e r r e v i e w . However, the D i l u t e Domain d e f i n e d on the b a s i s of the 33%„ i s o h a l i n e c o n t o u r a t 100m extends westward t o a l m o s t 160°W and i n c l u d e s O.S.P. In F i g 41 of F a v o r i t e et. a l (1976), the S u b a r c t i c C u r r e n t i s p o r t r a y e d as d i v i d i n g t o the 8 west of the D i l u t e Domain and O.S.P. The D i l u t e Domain i s d i l u t e by comparison w i t h the h i g h e r s a l i n i t y w a t e r s of the Ridge Domain t o the northwe s t (due t o u p w e l l i n g i n the A l a s k a n G y r e ) , the c o a s t a l u p w e l l i n g domains t o the s o u t h e a s t and the t r a n s i t i o n domain t o the s o u t h . I t i s not c l e a r t o what e x t e n t the lower s a l i n i t y a t O.S.P. may be due t o the a n n u a l e x c e s s of p r e c i p i t a t i o n over e v a p o r a t i o n which o c c u r s t h r o u g h o u t the e a s t e r n S u b a r c t i c (Dodimead e t a l , 1 9 6 3 ) , superimposed on a z o n a l f l o w . I f the f l o w t h r o u g h O.S.P. i s p r e d o m i n a n t l y z o n a l and p a r t of the e a s t w a r d - f l o w i n g S u b a r c t i c c u r r e n t , then as i n d i c a t e d by the quote above, a model of s e a s o n a l changes e i t h e r i n the p h y s i c a l s t r u c t u r e of the water column or i n the p l a n k t o n i c community may r e a s o n a b l y be c o n s t r u c t e d -without t a k i n g l a r g e - s c a l e h o r i z o n t a l a d v e c t i o n i n t o a c c o u n t . The models p r e s e n t e d i n t h i s t h e s i s have been c r e a t e d on t h i s b a s i s . I f O.S.P. r e a l l y l i e s w i t h i n a s m a l l - s c a l e c o a s t a l c i r c u l a t i o n , which seems u n l i k e l y f o r reasons c i t e d above and o t h e r s ( f o r example, the n i t r a t e s e a s o n a l c y c l e a t O.S.P. i s c h a r a c t e r i s t i c of S u b a r c t i c o c e a n i c r a t h e r than c o a s t a l w a t e r s ) , a more e x p l i c i t t r e a t m e n t of a d v e c t i v e e f f e c t s may be r e q u i r e d . 1.3 B i o l o g y of the S u b a r c t i c P a c i f i c . A v e r y c o n s i d e r a b l e l i t e r a t u r e e x i s t s on the b i o l o g y of the S u b a r c t i c P a c i f i c and t h i s b r i e f s u r v ey makes no p r e t e n c e a t an e x h a u s t i v e r e v i e w . The f o l l o w i n g i n f o r m a t i o n i s p r e s e n t e d p a r t l y as an i n t r o d u c t i o n t o the q u e s t i o n s a d d r e s s e d here u s i n g a m o d e l l i n g approach-. I t i s a l s o i n t e n d e d p a r t l y as an ov e r v i e w of the c u r r e n t s t a t e of b i o l o g i c a l knowledge, so t h a t a reader 9 u n f a m i l i a r w i t h t h i s l i t e r a t u r e can p l a c e the models c o n s i d e r e d h e r e , w i t h t h e i r n e c e s s a r y l i m i t a t i o n s and r a t h e r narrow f o c u s , i n b roader p e r s p e c t i v e . Much of the review w i l l be concerned w i t h s t u d i e s a t O.S.P. which i s by f a r the most i n t e n s i v e l y sampled ocean s t a t i o n i n the r e g i o n . R e l e v a n t i n f o r m a t i o n from s u r r o u n d i n g a r e a s w i l l be mentioned where a p p r o p r i a t e . P r o b a b l y the best-known f e a t u r e of the p l a n k t o n i c ecosystem i n the s u b - A r c t i c P a c i f i c i s the absence of a s p r i n g i n c r e a s e i n p h y t o p l a n k t o n abundance. T h i s was r e p o r t e d f o r the B e r i n g Sea over 20 y e a r s ago (Semina,1958), and c o n f i r m e d by the w e a t h e r s h i p o b s e r v a t i o n s a t O.S.P. ( P a r s o n s , 1 9 6 5 ) . D u r i n g a c r u i s e i n J u l y and August of 1959, m a c r o n u t r i e n t s ( n i t r a t e , s i l i c a t e , p h o s p h a t e ) , p h y t o p l a n k t o n ( c h l o r o p h y l l a) and p a r t i c u l a t e o r g a n i c carbon .were measured ( M c A l l i s t e r et a_l ,1960). N u t r i e n t l e v e l s were c o n s i s t e n t l y h i g h (>6 u g a t . l " 1 NO", >16 u g a t . l " 1 S i O P , >1.2 u g a t . l " 1 PO^), a t a time of year when n u t r i e n t d e p l e t i o n might n o r m a l l y be e x p e c t e d . T h i s o b s e r v a t i o n , t o g e t h e r w i t h the obs e r v e d e x p o n e n t i a l i n c r e a s e of p h y t o p l a n k t o n i n b a t c h c u l t u r e t o some 40 tim e s the i n i t i a l c o n c e n t r a t i o n once g r a z e r s were removed, s u p p o r t e d the argument, advanced by Semina (1960), t h a t z o o p l a n k t o n g r a z i n g i s r e s p o n s i b l e f o r the c o n s t a n c y of p h y t o p l a n k t o n c o n c e n t r a t i o n i n the S u b a r c t i c . The s e a s o n a l c y c l e i n the S u b a r c t i c P a c i f i c was c o n t r a s t e d w i t h the ' s p r i n g bloom' c y c l e s of c o a s t a l a r e a s and the N o r t h A t l a n t i c by H e i n r i c h (1962). He a t t r i b u t e d the d i f f e r e n c e t o the l i f e h i s t o r y s t r a t e g i e s of the dominant h e r b i v o r o u s copepods, Calanus plumchrus and Calanus c r i s t a t u s , i n the S u b a r c t i c P a c i f i c . A p r e l i m i n a r y p i c t u r e of z o o p l a n k t o n abundance, c o m p o s i t i o n 10 and v e r t i c a l d i s t r i b u t i o n a t O.S.P. was g i v e n by M c A l l i s t e r ( 1961), based on s u r f a c e t r a w l s and v e r t i c a l h a u l s made from the w e a t h e r s h i p s from 1956 t o 1958. He d e s c r i b e d a w i n t e r minimum i n z o o p l a n k t o n biomass from December t o March and a summer maximum from A p r i l t o J u l y . S u r f a c e t r a w l s were dominated by copepods i n A p r i l and May and by amphipods i n June and J u l y and i n November and December. V e r t i c a l h a u l s (150m t o s u r f a c e ) were p r e d o m i n a n t l y copepods (ca 75%) and ch a e t o g n a t h s (15%) a t a l l t i m e s . Two l a y e r s of maximum abundance of z o o p l a n k t o n were found, above and below the permanent h a l o c l i n e . Zooplankton wet weight i n the t o p 150m ranged from ca 10 mg.nr 3 i n w i n t e r t o 80 mg.m-3 i n May, 1957. Perhaps the s i n g l e most comprehensive study of the b i o l o g y o f . t h e S u b a r c t i c P a c i f i c t o date i s t h a t of L e B r a s s e u r (1969). T h i s s t u d y of p r e d a t o r - p r e y r e l a t i o n s h i p s i n the G u l f of A l a s k a i n c l u d e d a d e t a i l e d a n a l y s i s of n i n e y e a r s (1956-64) of z o o p l a n k t o n d a t a from O.S.P. The l i f e h i s t o r i e s of the dominant z o o p l a n k t o n s p e c i e s were d i s c u s s e d , based on average s e a s o n a l p a t t e r n s of abundance by stage as r e f l e c t e d i n v e r t i c a l h a u l s and s u r f a c e tows. Next t o the two s p e c i e s mentioned above, a t h i r d l a r g e h e r b i v o r o u s copepod, Eu c a l a n u s b u n g i , makes a s i g n i f i c a n t c o n t r i b u t i o n t o biomass, as noted f o r o t h e r l o c a t i o n s i n the S u b a r c t i c P a c i f i c ( H e i n r i c h , 1 9 6 2 ; S e k i g u c h i , 1 9 7 5 ) . S m a l l e r copepods ( Pse u d o c a l a n u s , Calanus p a c i f i c u s , M e t r i d i a p a c i f i c a ) a r e i m p o r t a n t i n f a l l and w i n t e r . A mixed c o l l e c t i o n of p l a n k t o n i c p r i m a r y c a r n i v o r e s , i n c l u d i n g c h a e t o g n a t h s ( S a g i t t a e l e g a n s , E u k r o h n i a hamata) and the trachymedusa A g l a n t h a d i g i t a l e were a l s o d i s c u s s e d . An attempt was made t o e s t i m a t e the 11 s t a n d i n g s t o c k s of f o r a g e organisms (myctophid and s q u i d ) and the annual carbon f l u x t h r ough a s i m p l i f i e d food web l e a d i n g up t o the t e r t i a r y consumers (salmon, b a l e e n whales and p o m f r e t ) . Due t o the n a t u r e of the sa m p l i n g program a t O.S.P. , much more i s known of the h e r b i v o r o u s and c a r n i v o r o u s z o o p l a n k t o n than of the t r o p h i c l e v e l s above or below. For example, the p o p u l a t i o n dynamics of s q u i d , an im p o r t a n t p r i m a r y / s e c o n d a r y c a r n i v o r e , a r e v i r t u a l l y unknown. The s e a s o n a l p a t t e r n i n s p e c i e s c o m p o s i t i o n of p h y t o p l a n k t o n a t O.S.P. i s a l s o c o m p a r a t i v e l y p o o r l y known. W e a t h e r s h i p p h y t o p l a n k t o n samples have not y e t been a n a l y s e d q u a n t i t a t i v e l y . I n f o r m a t i o n on abundance of net p h y t o p l a n k t o n i s a v a i l a b l e from the s h i p s of o p p o r t u n i t y program ( V e n r i c k , 1 9 7 1 ) and from w e a t h e r s h i p m i c r o - z o o p l a n k t o n d a t a ( d i s c u s s e d l a t e r ) . O b s e r v a t i o n s a t O.S.P. i n J u l y , August, 1959 ( M c A l l i s t e r e t a l , 1 9 6 0 ) , d u r i n g the Transpac c r u i s e of 1969 (Parsons,1972) and p r e l i m i n a r y a n a l y s i s of L i n e P s t a t i o n s (R. Waters, p e r s . comm. ) suggest t h a t p h y t o p l a n k t o n a re dominated i n biomass by s m a l l f l a g e l l a t e s , l e s s than 10 pm i n d i a m e t e r . There have been a number of t h e o r e t i c a l a n a l y s e s of p r i m a r y and secondary p r o d u c t i o n a t O.S.P. or i n i t s v i c i n i t y . The s e a s o n a l c y c l e of c h l o r o p h y l l a and p r i m a r y p r o d u c t i o n a t O.S.P. (based on 1 4 C uptake measurements) was d e s c r i b e d by Parsons (1965). In t h a t paper, S v e r d r u p ' s c r i t i c a l depth model (Sverdrup,1953) was used t o e x p l o r e the i n t e r a c t i o n of s e c c h i d e p t h , mixed l a y e r depth and s u r f a c e i r r a d i a n c e . A l l macro- n u t r i e n t s were d e s c r i b e d as n o n - l i m i t i n g t hroughout the G u l f of A l a s k a and an annual p r i m a r y p r o d u c t i o n of ca 60 g Cm" 2 was 12 e s t i m a t e d , based on 1 4 C measurements and the s e a s o n a l d e c r e a s e i n n i t r a t e and phosphate. The c r i t i c a l depth approach was l a t e r expanded t o cov e r the f u l l G u l f of A l a s k a by Parsons and L e B r a s s e u r (1968). The l a r g e - s c a l e p a t t e r n i n the t i m i n g of the s p r i n g i n c r e a s e i n z o o p l a n k t o n s t a n d i n g s t o c k was p r e d i c t e d i n t h i s s t u d y . R e s u l t s of p r i m a r y p r o d u c t i o n s t u d i e s on the Transpac c r u i s e and a s h i p s of o p p o r t u n i t y program conducted from American M a i l L i n e c r u i s e s between S e a t t l e and Yokohama were r e p o r t e d by Parsons and Anderson (1970). A d e p t h - i n t e g r a t e d form of S t e e l e and Menzel's (1962) e q u a t i o n was found t o o v e r e s t i m a t e p r o d u c t i o n on the Transpac c r u i s e . Reducing the p h o t o s y n t h e t i c e f f i c i e n c y parameter from 0.24 t o 0.17 (ug C.ug C h i a ^ . l y " 1 ) i n t h i s e q u a t i o n p r o v i d e d a b e t t e r f i t f o r the Transpac d a t a , but v a l u e s r a n g i n g from 0.07 t o 3.1 (jug C.ug C h i a ^ . l y " 1 ) were n e c e s s a r y t o o b t a i n agreement w i t h d a t a from the AML c r u i s e s . The c o n v e r s i o n of p r i m a r y p r o d u c t i o n t o secondary p r o d u c t i o n a t O.S.P. has been s t u d i e d by M c A l l i s t e r (1969,1972). D a i l y p h y t o p l a n k t o n p r o d u c t i o n ( e s t i m a t e d from 1 4 C measurements), minus e s t i m a t e d dark r e s p i r a t i o n , was assumed t o be i n g e s t e d by z o o p l a n k t o n . Zooplankton r e s p i r a t i o n was c a l c u l a t e d as a f r a c t i o n of z o o p l a n k t o n s t a n d i n g s t o c k . A s s i m i l a t e d r a t i o n minus r e s p i r a t i o n was taken as secondary p r o d u c t i o n . Because of u n c e r t a i n t y i n z o o p l a n k t o n r e s p i r a t i o n r a t e s (eg S t e e l e , 1 9 7 4 ) , e s t i m a t e s of secondary p r o d u c t i o n r e p r e s e n t e d the d i f f e r e n c e of two l a r g e , u n c e r t a i n q u a n t i t i e s and ranged from n e g a t i v e v a l u e s t o a maximum of 23g C.m~ 2.yr _ 1. An e s t i m a t e of 13 g C . i r r 2 . y r ~ l was chosen as most l i k e l y . 13 For the purposes of t h i s t h e s i s , an i n t e r e s t i n g summary of the c u r r e n t s t a t e of b i o l o g i c a l knowledge f o r t h i s l o c a t i o n can be o b t a i n e d by a s s e s s i n g i t s s t r e n g t h s and weaknesses as a b a s i s f o r a d e t a i l e d , r a t i o n a l m e c h a n i s t i c model ( P i a t t e_t a l ,1975) of the ecosystem t h e r e . Whether a model of t h i s k i n d i s a d e s i r a b l e g o a l , p a r t i c u l a r l y i f we wi s h t o make s u c c e s s f u l q u a n t i t a t i v e p r e d i c t i o n s , i s d e b a t a b l e ( P i a t t e t a_l ,1975), but i t does p r o v i d e a u s e f u l way t o s t r u c t u r e an approach t o e x i s t i n g d a t a . There i s a l o n g time s e r i e s of o b s e r v a t i o n s of c h l o r o p h y l l a, 1 4 C p r o d u c t i v i t y and m a c r o n u t r i e n t s a t O.S.P. However, f o r a model of p h y t o p l a n k t o n dynamics, a knowledge of v a r i a t i o n s i n p h y t o p l a n k t o n carbon and c a r b o n r c h l o r o p h y l l r a t i o s , i n the p h o t o s y n t h e s i s vs l i g h t r e l a t i o n s h i p (on s h o r t and l o n g time s c a l e s ) , i n p h y t o p l a n k t o n r e s p i r a t i o n r a t e s and i n l i m i t i n g e f f e c t s of m i c r o n u t r i e n t s , i f any, would a l l be d e s i r a b l e . An attempt w i l l be made t o i n f e r some of the s e i n d i r e c t l y from the a v a i l a b l e o b s e r v a t i o n s i n Chapter 3. I n v e s t i g a t i o n of s i z e and/or s p e c i e s dependent e f f e c t s i n b i o l o g i c a l i n t e r a c t i o n s would r e q u i r e o b s e r v a t i o n s a t a comparable l e v e l of d e t a i l . The l o n g time s e r i e s of 150m v e r t i c a l h a u l s i s the p r i n c i p a l z o o p l a n k t o n d a t a s o u r c e , p r o v i d i n g i n f o r m a t i o n on t o t a l wet we i g h t , s p e c i e s c o m p o s i t i o n and some st a g e and/or s i z e s t r u c t u r e . These d a t a a r e s u p p o r t e d by s t u d i e s on s e a s o n a l and d i u r n a l p a t t e r n s of v e r t i c a l m i g r a t i o n (Vinogradov,1968; F r o s t and McCrone,1974; S e k i g u c h i , 1 9 7 5 ; Marlowe and M i l l e r , 1 9 7 5 ) . Some measurements have been made of g r a z i n g r a t e s , c h e m i c a l c o m p o s i t i o n and r e s p i r a t i o n of the dominant h e r b i v o r e s p r i m a r i l y i n c o a s t a l l o c a t i o n s f a r from O.S.P. (Parsons e t a l ,1969; 14 Ikeda,1972; Taguchi and I s h i i , 1 9 7 2 ; F u l t o n , 1 9 7 3 ; Ikeda,1977). However, i n f o r m a t i o n on the f u n c t i o n a l and n u m e r i c a l responses of thes e dominant copepods i s poor compared w i t h b e t t e r s t u d i e d c o a s t a l s p e c i e s such as Calanus p a c i f i c u s ( P a f f e n h o f f e r , 1 9 7 0 ; Frost,1972,1975) or Pseudocalanus ( P a f f e n h o f f e r and H a r r i s , 1 9 7 6 ) . The average s e a s o n a l p a t t e r n of m i c r o z o o p l a n k t o n ( r e t a i n e d by 44 jam mesh) a t O.S.P. has been r e p o r t e d ( L e B r a s s e u r and Kennedy,1972), but the a u t e c o l o g y of most of these organisms i s v e r y p o o r l y known. A s i m i l a r s t a t e of i g n o r a n c e e x i s t s f o r most of the p r i m a r y c a r n i v o r e s mentioned above, w h i l e even the abundance of f o r a g e organisms such as s q u i d i s p o o r l y known. The study of f o r a g e organisms i s c o m p l i c a t e d by t h e i r v e r y l a r g e d i u r n a l and s e a s o n a l m i g r a t i o n s and l o n g ( m u l t i - y e a r ) g e n e r a t i o n t i m e s . For such l o n g - l i v e d o r g a n i s m s , and of c o u r s e f o r wide- r a n g i n g p r e d a t o r s such as whales, salmon and po m f r e t , a model of c o n d i t i o n s at O.S.P. becomes m e a n i n g l e s s and the l a r g e - s c a l e v a r i a b i l i t y and c i r c u l a t i o n of the S u b a r c t i c P a c i f i c would have t o be m o d e l l e d . 1.4 Simple P h y t o p l a n k t o n - z o o p l a n k t o n Models f o r the S u b a r c t i c Pac i f i c . As the summary of S e c t i o n 1.3 demonstates, an attempt t o c o n s t r u c t a d e t a i l e d , complete ecosystem model f o r O.S.P. , or r a t h e r f o r the G u l f of A l a s k a or the whole S u b a r c t i c P a c i f i c , would be premature, t o say the l e a s t . For m u l t i p l e t r o p h i c l e v e l s and l a r g e h o r i z o n t a l s c a l e s , the s i m p l e r tropho-dynamic arguments of L e B r a s s e u r (1969) and Sanger (1972) a r e more a p p r o p r i a t e a t p r e s e n t . A much narrower range of q u e s t i o n s i s 15 a d d r e s s e d h e r e , c e n t e r i n g on the observed c o n s t a n c y of c h l o r o p h y l l a c o n c e n t r a t i o n s a t O.S.P. These q u e s t i o n s can be ad d r e s s e d u s i n g s i m p l i f i e d s i m u l a t i o n models which a re more commensurate w i t h the p r e s e n t s t a t e of b i o l o g i c a l knowledge. The s e a s o n a l c y c l e of C h i a a t O.S.P. i s p r e s e n t e d i n F i g 3. The c y c l e i n the S t r a i t of G e o r g i a i s a l s o p r e s e n t e d f o r c o n t r a s t . W h i l e the de c r e a s e i n mixed l a y e r depth and i n c r e a s e i n s o l a r r a d i a t i o n a t O.S.P. i n the s p r i n g does r e s u l t i n an i n c r e a s e i n p r i m a r y p r o d u c t i v i t y (Parsons,1965) t h i s i s not r e f l e c t e d i n an i n c r e a s e i n p h y t o p l a n k t o n s t a n d i n g s t o c k (as measured by C h i a ) . I n s t e a d , the biomass (wet weight) of z o o p l a n k t o n above 150m v a r i e s s e a s o n a l l y i n a s i m i l a r manner t o p r i m a r y p r o d u c t i v i t y ( F i g 4 ) . T h i s seems t o be c o n s i s t e n t w i t h the h y p o t h e s i s t h a t z o o p l a n k t o n g r a z i n g r e s u l t s i n a p h y t o p l a n k t o n m o r t a l i t y r a t e which b a l a n c e s p h y t o p l a n k t o n growth throughout the y e a r . T h i s h y p o t h e s i s of g r a z i n g c o n t r o l i n t u r n r a i s e s o t h e r q u e s t i o n s . Why s h o u l d i t o c c u r i n the o c e a n i c S u b a r c t i c P a c i f i c and not i n c o a s t a l a r e a s , nor i n the o c e a n i c N o r t h A t l a n t i c ? Perhaps even more t r o u b l e s o m e , i n view of the r a p i d growth r a t e of p h y t o p l a n k t o n , s e a s o n a l and d a i l y v a r i a t i o n s i n t h i s growth r a t e , and the observed v a r i a b i l i t y i n z o o p l a n k t o n s t a n d i n g s t o c k , i s the t i g h t b a l a n c e r e q u i r e d between g r a z i n g and p h y t o p l a n k t o n growth by t h i s h y p o t h e s i s . The f i r s t q u e s t i o n was answered by H e i n r i c h (1962) as f o l l o w s . The dominant g r a z e r s i n the N o r t h A t l a n t i c and i n many c o a s t a l a r e a s , copepods such as Calanus f i n m a r c h i c u s and Calanus p a c i f i c u s , o v e r - w i n t e r as l a t e c o p e p o d i t e s t a g e s or a d u l t s and 3 0 CO JL 2 0 D Q_ 10 O _o U 0 i " i s 11 /1 M M v O N D M o n t h Figure 3. Seasonal cycle in chlorophyll a at O.S.P. (solid line) and i n Departure Bay, Strait of Georgia (dashed l i n e ) . (Adapted from Parsons, 1965). J F M A M J J A S O N D MONTH Figure 4. Seasonal cycle at O.S.P. i n (a) primary production and (b) zooplankton standing stock, (from McAllister, 1969) . 18 cannot reproduce i n the s p r i n g u n t i l they have accumulated egg t i s s u e by f e e d i n g on adequate p h y t o p l a n k t o n c o n c e n t r a t i o n s . A f u r t h e r p e r i o d i n which n a u p l i i do not f e e d f o l l o w s r e p r o d u c t i o n , so t h a t a s i z e a b l e l a g o c c u r s i n the n u m e r i c a l response of z o o p l a n k t o n t o the s p r i n g i n c r e a s e i n p h y t o p l a n k t o n growth. The dominant copepods i n the S u b a r c t i c , C. plumchrus and C. c r i s t a t u s , reproduce a t depth i n the s p r i n g , u s i n g f a t s t o r e s l a i d down the p r e v i o u s summer. An a c t i v e l y - g r o w i n g p o p u l a t i o n i s t h e r e b y r e c r u i t e d t o the s u r f a c e i n the s p r i n g w i t h o u t any l a g i n response t o i n c r e a s i n g p h y t o p l a n k t o n growth r a t e s . There a r e some c o a s t a l a r e a s such as the S t r a i t of G e o r g i a (Parsons,1965) and the Sea of Japan ( H e i n r i c h , 1 9 6 2 ) where C. plumchrus and/or C. c r i s t a t u s dominate but a p h y t o p l a n k t o n s p r i n g bloom does o c c u r . H e i n r i c h a t t r i b u t e d t h i s t o an e a r l i e r i n c r e a s e i n p h y t o p l a n k t o n growth r a t e i n s t r a t i f i e d c o a s t a l w a t e r s and a consequent f a i l u r e i n t i m i n g of the s p r i n g r e c r u i t m e n t of the s e copepods. W h i l e t h i s argument seems t o o f f e r a s o l u t i o n t o the f i r s t q u e s t i o n (a s u f f i c i e n t l a g i n z o o p l a n k t o n response i n the s p r i n g w i l l c l e a r l y ensure a s p r i n g p h y t o p l a n k t o n bloom), i t does not add r e s s the second q u e s t i o n . In the remainder of t h i s s e c t i o n , t h i s q u e s t i o n of b a l a n c e i s a d d r e s s e d by c o n s i d e r i n g the s t a b i l i t y p r o p e r t i e s of some s i m p l e biomass models of p h y t o p l a n k t o n - z o o p l a n k t o n i n t e r a c t i o n s . These s h o u l d not be regarde d as r e a l i s t i c or p r e d i c t i v e models but r a t h e r as sta t e m e n t s of paradigm, i n the sense of Kuhn(1970). T h e i r c o n s i d e r a t i o n w i l l a l l o w us t o r e l a t e the q u e s t i o n of b a l a n c i n g g r a z i n g l o s s and p h y t o p l a n k t o n growth t o some i m p o r t a n t c u r r e n t 19 i s s u e s i n m a r i n e e c o l o g y . A c l a s s i c a l s t a r t i n g p o i n t f o r s i m p l e m o d e l s of p r e d a t o r - p r e y i n t e r a c t i o n s (as t h e z o o p l a n k t o n - p h y t o p l a n k t o n i n t e r a c t i o n w i l l be r e g a r d e d f o r t h e r e s t o f t h i s s e c t i o n ) i s t h e d i f f e r e n t i a l e q u a t i o n model : x = r . x - a . x . y 1.1a y = e.a.x.y - m.y 1.1b ( L o t k a , 1 9 2 5 ) . In t h i s model, t h e p a r a m e t e r r r e p r e s e n t s an i n t r i n s i c r a t e of g r o w t h of p r e y , x; t h e p a r a m e t e r a i s t h e s u c c e s s f u l e n c o u n t e r r a t e by a s i n g l e p r e d a t o r p e r u n i t p r e y d e n s i t y so t h a t a.x.y i s t h e t o t a l r a t e o f c o n s u m p t i o n o f p r e y by p r e d a t o r s ; e i s t h e e f f i c i e n c y o f c o n v e r s i o n of .consumed p r e y t o p r e d a t o r so t h a t e.a.x.y r e p r e s e n t s t h e c o n s e q u e n t r a t e of i n c r e a s e i n p r e d a t o r , y, w h i l e m i s a c o n s t a n t p e r c a p i t a l o s s r a t e f o r p r e d a t o r s i n t h e a b s e n c e o f f o o d . The o s c i l l a t o r y s o l u t i o n s of s y s t e m 1.1 o r i g i n a l l y a t t r a c t e d some i n t e r e s t , but t h e i r n e u t r a l s t a b i l i t y p r o p e r t i e s and c o n s e q u e n t l a c k o f r o b u s t n e s s under s m a l l s t r u c t u r a l c h a n g e s i n t h e model l e d t o i t s r e p l a c e m e n t . The model c a n be i m p r o v e d by i n t r o d u c i n g f u r t h e r non- l i n e a r i t i e s t o a c c o u n t f o r r e s o u r c e - l i m i t a t i o n o f p r e y and s a t u r a t i o n of p r e d a t o r s ( H o l l i n g , 1 9 5 9 ) . One p o s s i b l e f o r m f o r s u c h a model i s x = r . x . ( l - x / K ) - i M . x . y / ( D + x ) 1.2a y = e . i M . x . y / ( D + x ) - m.y 1.2b 20 H e r e , K i s a c a r r y i n g c a p a c i t y f o r p r e y i n t h e a b s e n c e of p r e d a t o r s , i M i s t h e maximum r a t e o f c o n s u m p t i o n o f p r e y p e r p r e d a t o r and D i s t h e p r e y d e n s i t y a t w h i c h p r e y c o n s u m p t i o n r e a c h e s h a l f i t s maximum v a l u e . The r a t i o i M / D i s c o m p a r a b l e t o t h e p a r a m e t e r a i n e q u a t i o n 1.1. The q u a l i t a t i v e b e h a v i o u r o f t h i s model has seen much d i s c u s s i o n ( f o r a r e v i e w , s e e May,1974). Two t y p e s o f phase p l a n e p o r t r a i t s a r e shown i n F i g 5. The p r e y i s o c l i n e a l w a y s forms a q u a d r a t i c 'hump' and t h e p r e d a t o r i s o c l i n e a v e r t i c a l l i n e . T h e i r i n t e r s e c t i o n ( x , y ) i s a c r i t i c a l p o i n t of t h e s y s t e m and a s i m p l e g e o m e t r i c r u l e d e t e r m i n e s i t s s t a b i l i t y p r o p e r t i e s . When t h e p r e d a t o r i s o c l i n e (x = x) l i e s t o t h e r i g h t of t h e 'hump', ( F i g 5 a ) , (x,y) i s a s y m p t o t i c a l l y s t a b l e and t r a j e c t o r i e s s p i r a l i n t o i t . When t h e p r e d a t o r i s o c l i n e l i e s t o t h e l e f t of t h e hump, t r a j e c t o r i e s s p i r a l o u t w a r d t o a s t a b l e l i m i t c y c l e s o l u t i o n . To r e l a t e t h e s e r e s u l t s t o t h e O.S.P. s e a s o n a l c y c l e , a way must be f o u n d t o i n c o r p o r a t e s e a s o n a l e f f e c t s i n t o t h i s p r e s e n t l y homogeneous mo d e l . The o b v i o u s a p p r o a c h i s t o a l l o w t h e p h y t o p l a n k t o n g r o w th p a r a m e t e r s r and K t o be f u n c t i o n s o f t i m e , t , r e f l e c t i n g t h e s e a s o n a l c y c l e i n p r i m a r y p r o d u c t i v i t y ( F i g 4 ) . The r e s u l t i n g s y s t e m i s no l o n g e r homogeneous and c o n s e q u e n t l y d i f f i c u l t t o t r e a t a n a l y t i c a l l y . Suppose f o r t h e moment t h a t t h e p a r a m e t e r s i n 1.2 a t any p a r t i c u l a r t i m e t a r e s u c h t h a t a s t a b l e n o n - t r i v i a l e q u i l i b r i u m ( x ( t ) , y ( t ) ) e x i s t s f o r t h e c o r r e s p o n d i n g homogeneous s y s t e m . I f t h e t i m e s c a l e f o r a p p r o a c h t o t h i s e q u i l i b r i u m i s s u f f i c i e n t l y f a s t compared w i t h t h e ( s e a s o n a l ) t i m e s c a l e s o v e r w h i c h r and K change, i t seems r e a s o n a b l e t h a t t h e s o l u t i o n t o t h e non-homogeneous s y s t e m , i f i t s t a r t s n e a r 21 C Q) D "O ^ ^ isocl ine predator isocl ine a \ o prey dens i t y x K p r e y dens i ty x Figure 5. Phase plane portraits, for system 1.2. (a) equilibrium stable. (b) equilibrium unstable. 22 ( x ( t ) , y ( t ) ) , w i l l remain c l o s e t o t h i s q u a s i - e q u i l i b r i u m s o l u t i o n . By t h i s s e p a r a t i o n of time s c a l e s (Ludwig e t a l ,1978), a s e a s o n a l c y c l e can be e n v i s a g e d which i s c l o s e l y a p p r o x i m a t e d by the q u a s i - e q u i l i b r i u m s o l u t i o n . Now x ( t ) , y ( t ) are d e t e r m i n e d by e. i M . x / ( D + x ) = m 1.3a y ( t ) = r ( t ) . ( l - x / K ( t ) ) . ( D + x ) / i M 1.3b There a r e two im m e d i a t e l y e n c o u r a g i n g a s p e c t s of these e x p r e s s i o n s . The q u a s i - e q u i l i b r i u m p h y t o p l a n k t o n c o n c e n t r a t i o n i s c o n s t a n t over t i m e , depending o n l y on z o o p l a n k t o n p a r a m e t e r s . The q u a s i - e q u i l i b r i u m z o o p l a n k t o n c o n c e n t r a t i o n i s p r o p o r t i o n a l t o the p h y t o p l a n k t o n growth r a t e . .Phytoplankton and z o o p l a n k t o n s t a n d i n g s t o c k s which f o l l o w e d ( x ( t ) , y ( t ) ) c l o s e l y would behave as observed a t O.S.P. , w i t h p h y t o p l a n k t o n c o n c e n t r a t i o n a p p r o x i m a t e l y c o n s t a n t and z o o p l a n k t o n c o n c e n t r a t i o n v a r y i n g w i t h p r i m a r y p r o d u c t i v i t y . There i s however a s e r i o u s problem w i t h t h i s ' e x p l a n a t i o n ' . The s t a b i l i t y c r i t e r i o n g i v e n above can be w r i t t e n as D > K - 2.x 1.4 K r e p r e s e n t s the s t a n d i n g s t o c k of p h y t o p l a n k t o n a t which r e s o u r c e l i m i t a t i o n causes the growth r a t e t o drop t o z e r o . There a r e two ob v i o u s p o t e n t i a l l y l i m i t i n g r e s o u r c e s f o r p h y t o p l a n k t o n , namely n u t r i e n t s u p p l y and a v a i l a b l e l i g h t . In s e c t i o n 1.3, n u t r i e n t c o n c e n t r a t i o n s a t O.S.P. were d e s c r i b e d as 23 n o n - l i m i t i n g . The v a l u e of K reached i n the c u l t u r e experiment of M c A l l i s t e r e t a l (1960) was about 40 times x. In f a c t , the u s u a l n o n - l i n e a r M i c h a e l i s - M e n t e n r e l a t i o n s h i p between growth and n u t r i e n t c o n c e n t r a t i o n means t h a t any n e g a t i v e feedback on growth r a t e of s m a l l i n c r e a s e s i n x above x w i l l be much s m a l l e r than a v a l u e of x/K of 1/40 i n the l o g i s t i c model would su g g e s t . I n c r e a s e s i n p h y t o p l a n k t o n d e n s i t y can d e c r e a s e p h y t o p l a n k t o n growth r a t e s t h r o u g h s e l f - s h a d i n g . T h i s e f f e c t can be q u a n t i f i e d f o r a homogeneous m i x e d - l a y e r p o p u l a t i o n by employing a r e l a t i o n s h i p between e x t i n c t i o n c o e f f i c i e n t , k, and c h l o r o p h y l l , x, k = k„ + k t .x (adapted from Parsons et a l , 1977), combined w i t h a . p h o t o s y n t h e s i s (P) vs l i g h t ( I ) r e l a t i o n s h i p (eg P= #.I.exp(- I / I MAX)' S t e e l e , 1 9 6 2 ) . I n t e g r a t i n g over depth g i v e s an e x p r e s s i o n f o r growth i n the mixed l a y e r as a f u n c t i o n of mixed l a y e r d e p t h , z M , s u r f a c e l i g h t i n t e n s i t y I 0 , P vs I parameters a and I M A X , the parameters k 0 and k 4 and the C h i a c o n c e n t r a t i o n , x. D i f f e r e n t i a t i n g t h i s e x p r e s s i o n w i t h r e s p e c t t o x and s u b s t i t u t i n g f o r x = x y i e l d s a v a l u e s u i t a b l e f o r i n s e r t i o n as 1/K i n 1.2. For z M =' 30m ( l a t e summer), k 4 = .02 m2.mg C h i a " 1 , (Lorenzen,1980;Megard et a l ,1980), k 0 = .1 n r 1 , and I 0 / I M A x = 2. ( S t e e l e , 1962), t h i s y i e l d s a v a l u e of K, j g h t = 8 mg C h i a . n r 3 or about 20 t i m e s x. T h i s v a l u e a g r e e s w i t h the r e s u l t s of T a kahashi and P a r s o n s ( 1 9 7 2 ) . Both c o n s i d e r a t i o n s of n u t r i e n t l i m i t a t i o n and s e l f - s h a d i n g 24 suggest t h a t ,at O.S.P. ', the p h y t o p l a n k t o n p o p u l a t i o n i s b e i n g m a i n t a i n e d a t c o n c e n t r a t i o n s about o n e - t w e n t i e t h of i t s c a r r y i n g c a p a c i t y or l e s s . A c c o r d i n g t o the c o n d i t i o n 1.4, f o r t h i s e q u i l i b r i u m t o be s t a b l e , D must be about 20 t i m e s x which would imply t h a t z o o p l a n k t o n a t O.S.P. a r e growing and r e p r o d u c i n g a t p h y t o p l a n k t o n c o n c e n t r a t i o n s s u f f i c i e n t t o s u p p l y 1/40 of t h e i r maximum r a t i o n . W h i l e l a r g e v a l u e s of the g r a z i n g h a l f - s a t u r a t i o n c o n s t a n t of t h i s o r d e r have been r e p o r t e d , such a low r e l a t i v e i n g e s t i o n r a t e seems u n l i k e l y t o cov e r even the b a s a l m e t a b o l i c r e q u i r e m e n t s of the z o o p l a n k t o n . Of c o u r s e , the p a r t i c u l a r v a l u e 1/40 can be q u e s t i o n e d as i t depends on the s p e c i f i c form of the g r a z e r s ' f u n c t i o n a l r e s p o n s e , as w e l l as the assumptions u n d e r l y i n g the e s t i m a t e of K. However, the problem i s more fundamental than the s t a b i l i t y c r i t e r i o n a l o n e would s u g g e s t . For example, a p i e c e w i s e l i n e a r , or Type I ( H o l l i n g , 1 9 6 5 ) f u n c t i o n a l response on the p a r t of z o o p l a n k t o n g r a z e r s has r e c e i v e d support ( M u l l i n and Brooks,1975). I f a f u n c t i o n a l response of t h i s type i s s u b s t i t u t e d i n t o 1.2, the e q u i l i b r i u m i s always a s y m p t o t i c a l l y s t a b l e , no matter how s m a l l x/K may be. The l o c a l r a t e of approach t o e q u i l i b r i u m , on the o t h e r hand, i s then g i v e n by r.x/K, and the q u a s i - e q u i l i b r i u m e x p l a n a t i o n depends not o n l y on (x,y) b e i n g s t a b l e , but on the f u r t h e r c o n d i t i o n t h a t approach t o e q u i l i b r i u m o c c u r s on a f a s t time s c a l e compared w i t h changes i n ( x ( t ) , y ( t ) ) . V a l u e s of x/K of o r d e r l / 2 0 t h and a v a l u e of r of 0.2 d a y : 1 , e s t i m a t e d from M c A l l i s t e r et a l (1960), g i v e a c h a r a c t e r i s t i c time of approach t o e q u i l i b r i u m of o r d e r 50 days. T h i s i s not s h o r t compared w i t h the s e a s o n a l time s c a l e of changes i n p h y t o p l a n k t o n growth r a t e and y ( t ) . L a r g e o s c i l l a t i o n s i n p h y t o p l a n k t o n and z o o p l a n k t o n abundance would o c c u r i f s u c h a w e a k l y - s t a b l e s y s t e m was s u b j e c t e d t o s e a s o n a l c y c l e s i n p r o d u c t i v i t y . A l t h o u g h t h e homogeneous s y s t e m w i t h a t y p e I f u n c t i o n a l r e s p o n s e i s s t a b l e , when x/K i s v e r y s m a l l , i t i s l i t t l e b e t t e r t h a n n e u t r a l l y s t a b l e i n t e r m s of i t s a b i l i t y t o t r a c k a s e a s o n a l l y v a r y i n g e q u i l i b r i u m . The s y s t e m 1.2 c a n n o t p r o v i d e a c o n s i s t e n t e x p l a n a t i o n of o b s e r v a t i o n s a t O.S.P. However, t h e q u a s i - e q u i l i b r i u m a p p r o a c h seems p r o m i s i n g and one m i g h t t r y t o p r o c e e d by m o d i f y i n g t h e s y s t e m so as t o overcome t h e s t a b i l i t y t i m e s c a l e p r o b l e m , w h i l e r e t a i n i n g t h e q u a l i t a t i v e b e h a v i o u r of t h e q u a s i - e q u i l i b r i u m s o l u t i o n ( x , y ( t ) ) . One a p p r o a c h - w h i c h d o e s n ' t work i s m e n t i o n e d h e r e f o r i n t e r e s t s a k e . I n t r o d u c i n g a q u a d r a t i c l o s s t e r m f o r p r e d a t o r s i s known t o b r i n g a b o u t a s y m p t o t i c s t a b i l i t y i n s i m p l e p r e d a t o r - p r e y m o d els ( B a z y k i n , 1 9 7 4 ) and h a s been s u g g e s t e d i n o t h e r p h y t o p l a n k t o n - z o o p l a n k t o n models ( L a n d r y , 1 9 7 6 ) . A model of t h i s k i n d , w i t h t y p e I f u n c t i o n a l r e s p o n s e , can be w r i t t e n as x = r . x - a . x . y 1.5a y = e.a.x.y - u . y 2 1.5b f o r x a t sub-maximal r a t i o n l e v e l s . (An i n c r e a s e i n p e r c a p i t a l o s s r a t e (u.y) w i t h p r e d a t o r d e n s i t y c o u l d r e s u l t from i n t r a s p e c i f i c c o m p e t i t i o n f o r r e s o u r c e s o t h e r t h a n f o o d , o r from s w i t c h i n g o r a g g r e g a t i v e r e s p o n s e s i n h i g h e r c a r n i v o r e s . ) T h i s model c a n be r e j e c t e d , w i t h o u t c o n s i d e r i n g i t s s t a b i l i t y 26 p r o p e r t i e s , as the q u a s i - e q u i l i b r i u m s o l u t i o n f o r s e a s o n a l l y v a r y i n g r ( t ) i s y ( t ) = r ( t ) / a x ( t ) = u . y ( t ) / ( e . a ) The q u a s i - e q u i l i b r i u m p h y t o p l a n k t o n c o n c e n t r a t i o n x ( t ) i s not c o n s t a n t but v a r i e s w i t h y ( t ) i n t h i s model. T h i s i s c e r t a i n l y not c o n s i s t e n t w i t h o b s e r v a t i o n s a t O.S.P. Two p o s s i b l e f u n c t i o n a l r esponses have a l r e a d y been c o n s i d e r e d i n 1.2 : a h y p e r b o l i c (Type I I ) and p i e c e w i s e l i n e a r (Type I ) response ( H o l l i n g , 1 9 6 5 ) . W i t h i n the c o n t e x t of s i m p l e biomass models such as 1.2, the s e can be c h a r a c t e r i z e d as d e s t a b i l i z i n g and n e u t r a l l y s t a b l e r e s p e c t i v e l y . A t h i r d form of f u n c t i o n a l response d i s c u s s e d by H o l l i n g (1965) i s the s i g m o i d or Type I I I f u n c t i o n a l r e sponse, i n c o r p o r a t i n g a reduced c l e a r a n c e r a t e by copepods a t low food d e n s i t i e s . T h i s type of f u n c t i o n a l response has a s t a b i l i s i n g e f f e c t i n s i m p l e biomass models. A number of s t u d i e s of p e l a g i c copepods have suggested g r a z i n g r esponses of t h i s t y p e , i n v o l v i n g e i t h e r a c e s s a t i o n of f e e d i n g below some t h r e s h o l d food d e n s i t y ( P a rsons et_ a_l ,1969; F r o s t , 1 9 7 2 ) , or a r e d u c t i o n i n f i l t e r i n g r a t e a t low food d e n s i t i e s ( F r o s t , 1 9 7 5 ) . T h e o r e t i c a l c o n s i d e r a t i o n s suggest t h a t a copepod t r y i n g t o o p t i m i z e energy i n t a k e s h o u l d reduce i t s f i l t e r i n g a c t i v i t y a t low food d e n s i t i e s (Lam and F r o s t , 1 9 7 6 ; S t e e l e and F r o s t , 1 9 7 7 ) . The e f f e c t of a l l o w i n g a type I I I g r a z i n g response can be seen i n a phase p l a n e a n a l y s i s of the system 27 k = r.x - f ( x ) . y 1.6a y = e . f ( x ) . y - m.y 1.6b where the g r a z i n g f u n c t i o n f ( x ) has the t h r e s h o l d form ( F i g 6a) or s i g m o i d form ( F i g 6 b ) . (The p h y t o p l a n k t o n s e l f - l i m i t i n g term (-x/K) has been dropped as i t i s n e g l i g i b l e under the c o n d i t i o n s p r e v a i l i n g a t O.S.P.). Phase p l a n e p o r t r a i t s f o r the system 1.6 a r e g i v e n i n F i g 7. For the case of a g r a z i n g t h r e s h o l d a t x=x„ , the prey i s o c l i n e asymptotes t o the v e r t i c a l l i n e x=x c a t the l e f t , has a minimum a t some p o i n t x*, and a s y m p t o t e s ' t o the l i n e y = r . x / i M as x approaches oo. (The parameter i M r e p r e s e n t s the maximum z o o p l a n k t o n r a t i o n . ) The z o o p l a n k t o n i s o c l i n e i s a v e r t i c a l l i n e as. f o r system 1.2. These i n t e r s e c t a t the e q u i l i b r i u m g i v e n by f (x) = m/(e.a) 1.7a y = r.x.e/m 1.7b By l i n e a r i z i n g i n the neighbourhood of ( x , y ) , i t can be shown t h a t t h i s e q u i l i b r i u m i s a s y m p t o t i c a l l y s t a b l e i f and o n l y i f x < x*; t h a t i s , the z o o p l a n k t o n i s o c l i n e must l i e t o the l e f t of the minimum i n the p h y t o p l a n k t o n i s o c l i n e . G iven t h a t t h i s c o n d i t i o n i s s a t i s f i e d , t h e r e a r e s t i l l two q u a l i t a t i v e l y d i s t i n c t phase p o r t r a i t s . When x i s v e r y l a r g e , the system 1.6 becomes a p p r o x i m a t e l y x = r.x - i M . y 1.8a y = ( e . i M - m ) . y 1.8b 28 a x o p rey dens i t y x p rey dens i t y x Figure 6. Type III functional responses: (a) threshold. (b) sigmoid. 29 0 prey density x F i g u r e 7. Phase p l a n e p o r t r a i t s f o r s y s t e m 1.6. (a) e . l M ~ m > r . (b) e . i ^ - n K r . 30 I f e.i^-m > r , o r , e q u i v a l e n t l y , i f the maximum growth r a t e of z o o p l a n k t o n exceeds t h a t of p h y t o p l a n k t o n , t r a j e c t o r i e s s t a r t i n g from l a r g e x and s m a l l y w i l l always c y c l e back t o low p h y t o p l a n k t o n c o n c e n t r a t i o n s ( F i g 7 a ) . In t h i s sense, i t i s i m p o s s i b l e f o r p h y t o p l a n k t o n t o escape z o o p l a n k t o n c o n t r o l p e r m anently. I f e . i M - m < r , (as seems more l i k e l y ) , the be h a v i o u r of t r a j e c t o r i e s f o r l a r g e x depends on i n i t i a l c o n d i t i o n s , (x, ,y,- ). I f yi i s l a r g e enough, the t r a j e c t o r y w i l l c y c l e , but o t h e r w i s e , x w i l l i n c r e a s e i n d e f i n i t e l y and the t r a j e c t o r y w i l l approach (+00,+ 00). i n the c o r r e s p o n d i n g phase p l a n e p o r t r a i t ( F i g 7b)- t h e r e i s a s e p a r a t r i x which d i v i d e s t r a j e c t o r i e s which c y c l e from those which don't. The phase p l a n e p o r t r a i t s f o r s i g m o i d f u n c t i o n a l responses a r e q u a l i t a t i v e l y s i m i l a r , the p r i n c i p a l d i f f e r e n c e b e i n g t h a t the p h y t o p l a n k t o n i s o c l i n e asymptotes t o x=0. The l o c a l r a t e of approach of t r a j e c t o r i e s t o e q u i l i b r i u m , can be found by l i n e a r i z i n g about ( x , y ) . In f a c t , i f h(x) i s the c l e a r a n c e r a t e of z o o p l a n k t o n ( g i v e n by f ( x ) / x ) , the l o c a l r a t e of approach t o e q u i l i b r i u m i s g i v e n by r . x . h ' ( x ) / h ( x ) The s t a b i l i t y c r i t e r i o n x < x* i s s i m p l y e q u i v a l e n t t o a p o s i t i v e r a t e of approach, x* b e i n g g i v e n by h'(x*)=0. That i s , the e q u i l i b r i u m (x,y) i s l o c a l l y s t a b l e p r o v i d e d the c l e a r a n c e r a t e i n c r e a s e s w i t h x a t x=x. In the case of a t h r e s h o l d , h y p e r b o l i c f u n c t i o n a l response as used by S t e e l e (1974), 31 f ( x ) = i M . ( x - x 0 ) 4 / ( D + ( x - x e ) + ) , and e x p l i c i t f o r m u l a e c a n be w r i t t e n f o r x* and f o r t h e r a t e o f a p p r o a c h : x * - x 0 = /D7X0 r a t e o f a p p r o a c h = r . ( ( x * - x 0 ) 2 - ( x - x „ ) 2 ) / ( ( x - x e ) . D + ( x - x „ ) 2 ) T h a t i s , t h e c l e a r a n c e r a t e i s a maximum a t a d e n s i t y x* w h i c h i s g r e a t e r t h a n t h e t h r e s h o l d by t h e g e o m e t r i c mean of t h e h a l f - s a t u r a t i o n c o n s t a n t , D, and t h e t h r e s h o l d x e . P r o v i d e d t h e e q u i l i b r i u m x does n o t l i e t o o c l o s e t o x*, t h e r a t e of a p p r o a c h t o e q u i l i b r i u m i s of t h e same o r d e r as t h e p h y t o p l a n k t o n g r o w th r a t e , r . I t i s p o s s i b l e t h e n f o r t h e a p p r o a c h t o e q u i l i b r i u m i n 1.6 t o o c c u r on r e l a t i v e l y f a s t t i m e s c a l e s , of t h e same o r d e r as p h y t o p l a n k t o n g r o w t h , and c o n s e q u e n t l y f o r t h e q u a s i - e q u i l i b r i u m a s s u m p t i o n t o be v a l i d f o r s e a s o n a l v a r i a t i o n i n p h y t o p l a n k t o n g r o w t h r a t e , r . Note t h a t t h e q u a s i - e q u i l i b r i u m c y c l e g i v e n by 1.7 p r e d i c t s , as b e f o r e , a c o n s t a n t p h y t o p l a n k t o n s t a n d i n g s t o c k and a z o o p l a n k t o n s t a n d i n g s t o c k w h i c h v a r i e s w i t h p r i m a r y p r o d u c t i o n . The s i m p l e b i o m a s s model 1.6, a c c o r d i n g t o t h i s a p p r o x i m a t e argument b a s e d on s e p a r a t i o n o f t i m e s c a l e s , s h o u l d be c a p a b l e of q u a l i t a t i v e l y r e p r o d u c i n g t h e o b s e r v e d s e a s o n a l c y c l e a t O.S.P. R e f l e c t i o n on t i m e s c a l e s a l l o w s an i n t e r e s t i n g , i n t u i t i v e p e r s p e c t i v e on t h e d e v e l o p m e n t so f a r . The t i m e s c a l e s one t r a d i t i o n a l l y a s s o c i a t e s w i t h p h y t o p l a n k t o n g r o w t h a r e s h o r t and 32 one would expect an agent which r e g u l a t e s p h y t o p l a n k t o n d e n s i t y t i g h t l y t o o p e r a t e on a s i m i l a r time s c a l e . The time s c a l e s one t r a d i t i o n a l l y a s s o c i a t e s w i t h copepod dynamics a r e much l o n g e r and t h i s appears t o be a problem w i t h a g r a z i n g c o n t r o l h y p o t h e s i s a t O.S.P. Where n u t r i e n t s a r e l i m i t i n g , feedback e f f e c t s on p h y t o p l a n k t o n growth can t i g h t l y r e g u l a t e p h y t o p l a n k t o n d e n s i t y , as d i s c u s s e d i n Chapter 2, but t h i s i s a p p a r e n t l y not the case a t O.S.P. The s i g m o i d or t h r e s h o l d f u n c t i o n a l response i n 1.6 e s s e n t i a l l y i n t r o d u c e s a new f a s t time s c a l e , a z o o p l a n k t o n b e h a v i o u r a l time s c a l e , i n t o the problem. T h i s development augments r a t h e r n e a t l y H e i n r i c h ' s ( 1 9 6 2 ) e x p l a n a t i o n of the d i f f e r e n c e s between s e a s o n a l c y c l e s i n the S u b a r c t i c and e l s e w h e r e , based on the l i f e - h i s t o r y s t r a t e g i e s of the dominant g r a z e r s . He a t t r i b u t e d the s p r i n g bloom i n the N o r t h A t l a n t i c t o a d e l a y i n the n u m e r i c a l response of the dominant g r a z e r t o the s p r i n g i n c r e a s e i n p h y t o p l a n k t o n growth and abundance. I f the maximum growth r a t e of p h y t o p l a n k t o n exceeds t h a t of z o o p l a n k t o n biomass, a phase p l a n e p o r t r a i t as i n F i g 7b i s a p p r o p r i a t e . The s p r i n g i n c r e a s e i n p h y t o p l a n k t o n growth r a t e i s e q u i v a l e n t t o a v e r t i c a l s h i f t i n the p h y t o p l a n k t o n i s o c l i n e . A d e l a y i n z o o p l a n k t o n response can then e a s i l y r e s u l t i n the system b e i n g o v e r t a k e n by the s e p a r a t r i x , l e a v i n g the l o c a l s t a b i l i t y domain about (x,y) and e n t e r i n g a r e g i o n i n which p h y t o p l a n k t o n have escaped z o o p l a n k t o n c o n t r o l . A c c o r d i n g t o 1.6, the t r a j e c t o r y w i l l a pproach ( + oo, + oo). In p r a c t i c e , of c o u r s e , n u t r i e n t s a r e d e p l e t e d and the s p r i n g bloom t e r m i n a t e s . H e i n r i c h ( 1 9 6 2 ) a t t r i b u t e d the o c c u r r e n c e of s p r i n g blooms i n 33 c o a s t a l r e g i o n s of the S u b a r c t i c where C. plumchrus i s p r e s e n t t o a f a i l u r e i n t i m i n g of r e c r u i t m e n t . A l t h o u g h the s p r i n g bloom does o c c u r e a r l i e r i n the S t r a i t of G e o r g i a than i n o c e a n i c l o c a t i o n s ( P a r s o n s , 1 9 6 5 ) , the n a u p l i i of C. plumchrus r e a c h the s u r f a c e w a t e r s of the S t r a i t of G e o r g i a i n F e b r u a r y and March ( F u l t o n , 1 9 7 3 ) and i t i s not c l e a r t h a t the t i m i n g i s i n a p p r o p r i a t e . The phase p l a n e p o r t r a i t i n F i g 7b sug g e s t s another e x p l a n a t i o n . In c o a s t a l w a t e r s s u b j e c t t o r u n - o f f , s t r a t i f i c a t i o n and a consequent i n c r e a s e i n p h y t o p l a n k t o n growth r a t e can occur very r a p i d l y . T h i s may r e s u l t i n the system s t a t e ( x ( t ) , y ( t ) ) f a i l i n g t o t r a c k the r a p i d l y s h i f t i n g q u a s i - e q u i l i b r i u m c y c l e ( x , y ( t ) ) and a g a i n b e i n g o v e r t a k e n by the s e p a r a t r i x , r e s u l t i n g i n a s p r i n g bloom. The o b s e r v a t i o n s of F u l t o n (1973) suggest a s i m p l e r and perhaps more c o n v i n c i n g e x p l a n a t i o n : namely, t h a t r e c r u i t m e n t of C. plumchrus i n the s t r a i t f a i l s t o c o i n c i d e i n space, r a t h e r than t i m e , w i t h the s p r i n g bloom. S u c c e s s f u l o v e r - w i n t e r i n g of C. plumchrus o c c u r s o n l y i n water deeper than 300m, which o c c u p i e s o n l y o n e - f o u r t h of the a r e a of the s t r a i t . A r r i v a l of C. plumchrus i n the r e m a i n i n g a r e a s i s presumably d e l a y e d , s u b j e c t t o h o r i z o n t a l a d v e c t i o n . 1.5 Pr e v i e w of C h a p t e r s 2-4. Each of the s i m p l e models c o n s i d e r e d above can be r e g a r d e d as a composite h y p o t h e s i s c o n c e r n i n g t r o p h i c i n t e r a c t i o n s a t O.S.P. There i s c l e a r l y a need f o r independent e x p e r i m e n t a l t e s t s of a s p e c t s of the s e h y p o t h e s e s . For example, the f u n c t i o n a l r esponses of the dominant copepods a t O.S.P. a r e not well-known and the e x i s t e n c e of z e r o or reduced c l e a r a n c e r a t e s 34 a t low p h y t o p l a n k t o n d e n s i t i e s i s an e s s e n t i a l p a r t of the model 1.6. The t h e o r e t i c a l p o s s i b i l i t i e s have h a r d l y been exhausted by the above a n a l y s i s . G i v e n the c o m p a r a t i v e w e a l t h of i n f o r m a t i o n i n the l o n g time s e r i e s of o b s e r v a t i o n s a t O.S.P. , more than rough q u a l i t a t i v e agreement of a model w i t h average s e a s o n a l c y c l e s can be demanded. A q u a n t i t a t i v e comparison of p r e d i c t i o n and o b s e r v a t i o n r e q u i r e s a more c a r e f u l l y c o n s t r u c t e d , more d e t a i l e d model. For example, the s t a t e v a r i a b l e s x and y above have been used r a t h e r l o o s e l y t o r e p r e s e n t p h y t o p l a n k t o n and z o o p l a n k t o n s t a n d i n g s t o c k , a l t h o u g h they have been i m p l i c i t l y i d e n t i f i e d w i t h C h i a i n mg.irr 3 and z o o p l a n k t o n wet weight i n mg.nr 2 r e s p e c t i v e l y . As the p h y t o p l a n k t o n p o p u l a t i o n a t O.S.P. v a r i e s s e a s o n a l l y i n v e r t i c a l d i s t r i b u t i o n and ( h y p o t h e t i c a l l y ) i n C:Chl a r a t i o ( M c A l l i s t e r , 1 9 6 9 ) , the use of C h i a i n mg.m"3 as a s t a t e v a r i a b l e c l e a r l y needs c a r e f u l e x a m i n a t i o n . Zooplankton wet weight i s a rough summary v a r i a b l e , r e p r e s e n t i n g c o n t r i b u t i o n s from c a r n i v o r e s as w e l l as h e r b i v o r e s , a l t h o u g h dominated by h e r b i v o r o u s copepods a t most t i m e s ( L e B r a s s e u r , 1 9 6 9 ) . More d e t a i l e d s i z e and s p e c i e s i n f o r m a t i o n i s a v a i l a b l e f o r O.S.P. and t h i s c e r t a i n l y d e s e r v e s a t t e n t i o n i n view of r e c e n t t h e o r e t i c a l r e s u l t s ( S t e e l e , 1 9 7 4 ; S t e e l e and F r o s t , 1 9 7 7 ) . These problems a r e ad d r e s s e d i n Chapter 3 and Chapter 4. In Chapter 3, an attempt i s made a t a more r e a l i s t i c q u a n t i t a t i v e s i m u l a t i o n model of p h y t o p l a n k t o n growth a t O.S.P. The model i s p a r t l y based on an o r i g i n a l a n a l y s i s of the p h y t o p l a n k t o n d a t a , p a r t i c u l a r l y 1 4 C uptake r a t e s , o b t a i n e d from the w e a t h e r s h i p s . 35 In Chapter 4, a parameter e s t i m a t i o n t e c h n i q u e d e v e l o p e d f o r copepod time s e r i e s i s a p p l i e d t o e s t i m a t e p o p u l a t i o n parameters f o r the dominant h e r b i v o r e s a t O.S.P. Two more e l a b o r a t e models of the p h y t o p l a n k t o n - z o o p l a n k t o n i n t e r a c t i o n a r e c o n s t r u c t e d , u s i n g t h e s e r e s u l t s and those of Chapter 3. Some e f f e c t of h e r b i v o r e s i z e - s t r u c t u r e and s p e c i e s c o m p o s i t i o n i s c o n s i d e r e d i n the second model. The models a r e s t u d i e d u s i n g s i m u l a t i o n and the q u a l i t a t i v e t e c h n i q u e s and r e s u l t s of t h i s c h a p t e r and Chapter 2. Three hypotheses c o n c e r n i n g the p h y t o p l a n k t o n - z o o p l a n k t o n i n t e r a c t i o n a t O.S.P. have been c o n s i d e r e d i n S e c t i o n 1.4. A l l e x p l a i n the s e a s o n a l c y c l e s i n p h y t o p l a n k t o n and z o o p l a n k t o n s t a n d i n g s t o c k as the r e s u l t of the system's a b i l i t y t o c l o s e l y t r a c k a q u a s i - e q u i l i b r i u m s e a s o n a l c y c l e . The hypotheses d i f f e r i n the mechanisms r e s p o n s i b l e f o r the s h o r t term s t a b i l i s i n g feedback n e c e s s a r y f o r t h i s t r a c k i n g of a s e a s o n a l l y s h i f t i n g e q u i l i b r i u m . The model 1.2, i n v o l v i n g r e s o u r c e - l i m i t a t i o n of p h y t o p l a n k t o n growth, i s c e r t a i n l y c a p a b l e of the a p p r o p r i a t e q u a l i t a t i v e b e h a v i o u r , but has been r e j e c t e d on the b a s i s of independent e x p e r i m e n t a l e v i d e n c e . The model 1.5, i n v o l v i n g a q u a d r a t i c l o s s r a t e f o r g r a z e r s , cannot reproduce the o b s e r v e d s e a s o n a l c y c l e . The model 1.6, which assumes a r e d u c t i o n i n g r a z i n g r a t e a t low p h y t o p l a n k t o n d e n s i t i e s , i s c a p a b l e of r e p r o d u c i n g the observed s e a s o n a l c y c l e q u a l i t a t i v e l y . As noted e a r l i e r , the a c t u a l f u n c t i o n a l r e s p o n s e s of g r a z e r s a t O.S.P. a r e not known. The s i m u l a t i o n s c o n s i d e r e d i n Chapter 4 w i l l be p r i m a r i l y based upon the system 1.6 and the assumption of g r a z i n g 36 t h r e s h o l d s . However, t h e c o m p o s i t e h y p o t h e s i s i m p l i c i t i n 1.6 w i l l become m o d i f i e d i n t h e p r o c e s s of model e l a b o r a t i o n . In p a r t i c u l a r , an e x p l i c i t t r e a t m e n t of c o p e p o d l i f e - h i s t o r y s t r a t e g i e s w i l l show t h a t t h e , s p r i n g r e c r u i t m e n t o f n a u p l i i o v e r an e x t e n d e d p e r i o d can s t a b i l i s e t h e p h y t o p l a n k t o n - z o o p l a n k t o n i n t e r a c t i o n and p r e v e n t a s p r i n g bloom, even i n t h e a b s e n c e o f g r a z i n g t h r e s h o l d s . As a r e s u l t , a t t e n t i o n w i l l f o c u s on t h e .problem of m a i n t a i n i n g g r a z i n g c o n t r o l i n t h e summer and f a l l a g a i n s t t h e d e s t a b i l i s i n g e f f e c t of o v e r - w i n t e r i n g d e p a r t u r e by t h e d o m i n a n t c o p e p o d s . A l a c k of model r o b u s t n e s s d u r i n g t h i s p e r i o d and d i s c r e p a n c i e s i n d e t a i l between model p r e d i c t i o n s and o b s e r v a t i o n s w i l l f o r c e a r e - e v a l u a t i o n of t h e g r a z i n g t h r e s h o l d h y p o t h e s i s and a s e a r c h f o r a l t e r n a t i v e s . The p o s s i b i l i t y o f n u t r i e n t l i m i t a t i o n o f p h y t o p l a n k t o n g r o w th i n l a t e summer and f a l l w i l l be r e c o n s i d e r e d and t h e p o t e n t i a l i m p o r t a n c e of s p a t i a l v a r i a t i o n i n m a i n t a i n i n g g r a z i n g c o n t r o l w i l l be d i s c u s s e d . C h a p t e r 2 i s not d i r e c t l y c o n c e r n e d w i t h e v e n t s a t O.S.P. but i s of more g e n e r a l t h e o r e t i c a l i n t e r e s t . The i d e a t h a t f e e d i n g t h r e s h o l d s may be i m p o r t a n t i n p l a n k t o n e c o s y s t e m s i s n o t a new one. In a n u m e r i c a l model of p h y t o p l a n k t o n - z o o p l a n k t o n i n t e r a c t i o n s i n t h e N o r t h Sea, S t e e l e ( 1 9 7 4 ) f o u n d t h a t t h r e s h o l d s were n e c e s s a r y t o o b t a i n r e a l i s t i c b e h a v i o u r . An a l t e r n a t i v e was p r o p o s e d by L a n d r y ( 1 9 7 6 ) , who f o u n d t h a t i n s e r t i n g a q u a d r a t i c l o s s t e r m f o r h e r b i v o r e s i n t o S t e e l e ' s model e l i m i n a t e d t h e need f o r t h r e s h o l d s . B o t h a u t h o r s b a s e d t h e i r t h e o r e t i c a l c o n c l u s i o n s on s i m u l a t i o n r e s u l t s . A q u a l i t a t i v e m a t h e m a t i c a l a n a l y s i s o f t h e s e m o d e l s , b a s e d on s e p a r a t i o n o f t i m e s c a l e s , i s g i v e i n C h a p t e r 2. The a n a l y s i s p r o v i d e s i n s i g h t s i n t o t h e r e s u l t s o f 37 S t e e l e and L a n d r y w h i c h a p p e a r t o have i n t e r e s t i n g i m p l i c a t i o n s f o r m a r i n e e c o s y s t e m models i n g e n e r a l , and t h e m o d e l s c o n s i d e r e d h e r e f o r O.S.P. i n p a r t i c u l a r . 38 CHAPTER 2 QUALITATIVE ANAYLSIS OF A COMPLEX SIMULATION MODEL 2.1 I n t r o d u c t i o n . In a t h e o r e t i c a l t r e a t i s e on the p l a n k t o n i c ecosystem of the N o r t h Sea, S t e e l e (1974) examined the r e l a t i v e importance of i n t e r a c t i o n s between d i f f e r e n t t r o p h i c l e v e l s as ' c o n t r o l mechanisms f o r the whole system'. H i s c o n c l u s i o n s were based on a s i m u l a t i o n model of a s i m p l i f i e d n u t r i e n t - p h y t o p l a n k t o n - c o p e p o d food c h a i n which c o n t i n u e s t o be of t h e o r e t i c a l i n t e r e s t , both as a b a s i s f o r more complex r e a l i s t i c models ( S t e e l e and F r o s t , 1 9 7 7 ) and as a f i r s t s t e p i n c o m p l e x i t y and r e a l i s m above the h i g h l y s i m p l i f i e d p r e d a t o r - p r e y models of the L o t k a - V o l t e r r a type (Lotka,1925;May,1974). S t e e l e ' s c o n c l u s i o n s were based p r i m a r i l y on comparisons of a number of computer s i m u l a t i o n s of the model i n v o l v i n g v a r i o u s a ssumptions c o n c e r n i n g the form and magnitude of t r o p h i c i n t e r a c t i o n s . A key f i n d i n g was t h a t the model c o u l d not p r e d i c t t i m e streams which agreed q u a l i t a t i v e l y w i t h o b s e r v a t i o n s of the N o r t h Sea u n l e s s a t h r e s h o l d - f e e d i n g mechanism, or type I I I f u n c t i o n a l response ( H o l l i n g , 1 9 5 9 ) , was i n v o k e d f o r h e r b i v o r o u s copepods. Landry (1976) o b t a i n e d r e a l i s t i c b e h a v i o u r from S t e e l e ' s model w i t h o u t t h r e s h o l d s by i n t r o d u c i n g a per c a p i t a p r e d a t i o n r a t e on h e r b i v o r e s which i n c r e a s e d i n p r o p o r t i o n t o h e r b i v o r e numbers a t low d e n s i t i e s . T h i s was p a r t l y i n t r o d u c e d by Landry as a s i z e - d e p e n d e n t term ( s m a l l e r copepods b e i n g more abundant) but i t s s i g n i f i c a n c e seemed t o l i e i n the r e s u l t i n g q u a d r a t i c l o s s term f o r h e r b i v o r e s ( S t e e l e , 1 9 7 6 ) ; the 39 i n t r o d u c t i o n of such a l o s s term i s well-known t o produce s t a b l e e q u i l i b r i a i n o t h e r w i s e u n s t a b l e s i m p l e L o t k a - V o l t e r r a models ( B a z y k i n , 1 9 7 4 ) . Both S t e e l e and Landry p r e s e n t e d t h e i r r e s u l t s as a t l e a s t s u g g e s t i v e e v i d e n c e f o r the e x i s t e n c e and importance of t h r e s h o l d s and d e n s i t y - d e p e n d e n t p e r - c a p i t a p r e d a t i o n r a t e s r e s p e c t i v e l y i n r e a l ecosystems. T h i s f o c u s i n g of a t t e n t i o n on p a r t i c u l a r b i o l o g i c a l q u e s t i o n s i s r e c o g n i s e d as a v a l u a b l e p o t e n t i a l c o n t r i b u t i o n of m o d e l l i n g s t u d i e s i n g e n e r a l , and i t appears t o have been s u c c e s s f u l here i n view of the d i s c u s s i o n a r o u s e d c o n c e r n i n g the e x p e r i m e n t a l o b s e r v a t i o n of f e e d i n g t h r e s h o l d s ( M u l l i n et a l ,1975; F r o s t , 1 9 7 5 ) . However, any argument t h a t some a s p e c t of a model i s n e c e s s a r y i f r e a l i s t i c r e s u l t s a r e t o be o b t a i n e d must, always be viewed w i t h c a u t i o n . One reason f o r t h i s , common t o a l l m o d e l l i n g s t u d i e s , i s t h a t c o n c e p t u a l e l a b o r a t i o n of a model i s always p o s s i b l e (as i n the case of Landry v e r s u s S t e e l e ) , so t h a t the n e c e s s i t y of any p a r t i c u l a r concept can never be e s t a b l i s h e d . A second reason a p p l i e s p a r t i c u l a r l y t o s t u d i e s such as S t e e l e ' s and L andry's which a r e based on computer s i m u l a t i o n . G iven the l a r g e u n c e r t a i n t y i n most e c o l o g i c a l p a r a m e t e r s , i t i s not c l e a r t h a t a model h a v i n g a p r e s c r i b e d f u n c t i o n a l form s h o u l d be judged t o be i n c a p a b l e of s a t i s f y i n g a s e t of q u a l i t a t i v e c r i t e r i a on the b a s i s of a s m a l l number of" n u m e r i c a l s o l u t i o n s u s i n g p a r t i c u l a r parameter v a l u e s . T h i s c r i t i c i s m can be a d d r e s s e d by the t h e o r e t i c i a n , a t l e a s t i n p r i n c i p l e , as q u a l i t a t i v e m a t h e m a t i c a l a n a l y s i s of the model can p r o v i d e i n f o r m a t i o n about the b e h a v i o u r of s o l u t i o n s over r e g i o n s i n 40 p a r a m e t e r s p a c e , r a t h e r t h a n a t p o i n t s w i t h i n them. A n a l y s i s o f t h i s t y p e c a n a l s o l e a d t o a b e t t e r u n d e r s t a n d i n g o f t h e b e h a v i o u r o f t h e model and i t s dependence on p a r a m e t e r s and t h e r e b y a l l o w u s e f u l b i o l o g i c a l i n s i g h t s . A d d i t i o n a l m o t i v a t i o n f o r a q u a l i t a t i v e a n a l y s i s of S t e e l e ' s model i s p r o v i d e d by t h e s t u d y o f s i m p l e L o t k a - V o l t e r r a models i n C h a p t e r 1 w h i c h s u g g e s t e d t h a t g r a z i n g t h r e s h o l d s a r e i m p o r t a n t i n t h e S u b a r c t i c P a c i f i c e c o s y s t e m . W h i l e t h i s i s s u p e r f i c i a l l y c o n s i s t e n t w i t h S t e e l e ' s c o n c l u s i o n f o r t h e N o r t h Sea, a p u z z l i n g d i s c r e p a n c y e x i s t s . I t was n e c e s s a r y t o i n v o k e t h r e s h o l d s i n s i m p l e m o d els of t h e S u b a r c t i c P a c i f i c o n l y b e c a u s e p h y t o p l a n k t o n t h e r e a r e n o t n u t r i e n t - l i m i t e d . In t h e N o r t h Sea s i m u l a t i o n m o d e l, p h y t o p l a n k t o n a r e n u t r i e n t - l i m i t e d t h r o u g h o u t much of t h e summer, and i t i s not c l e a r why t h r e s h o l d s s h o u l d be needed f o r s t a b i l i t y . An e x p l a n a t i o n o f t h i s d i s c r e p a n c y i s so u g h t h e r e t h r o u g h t h e q u a l i t a t i v e a n a l y s i s o f S t e e l e ' s m o d e l . 2.2 Model and A n a l y s i s The model d e v e l o p e d by S t e e l e (1974) and u s e d , w i t h c e r t a i n a l t e r a t i o n s , by L a n d r y ( 1 9 7 6 ) i s : R = V.(RO-R) + U. ( E . ( P - P l )/(D+P) + F).Z.W 0- 7 - A.R.P/(B+R) 2.1a ( r a t e o f change of n u t r i e n t e q u a l s m i x i n g t h r o u g h t h e r m o c l i n e + z o o p l a n k t o n e x c r e t i o n - p h y t o p l a n k t o n u p t a k e ) P = A.R.P/(B+R) - V.P - C.Z.W 0 7.(P-P1)/(D+P) 2.1b ( r a t e of change of p h y t o p l a n k t o n = g r o w t h - m i x i n g - g r a z i n g ) 41 W = ( (0.7.C-E) . (P-P1)/(D+P) - F).W 0- 7 2.1c ( r a t e o f i n d i v i d u a l g r o w t h = n e t a s s i m i l a t i o n - a c t i v e and b a s a l m e t a b o l i s m ) Z = -GW.(W-Wl).(Z-Z1)/(H+Z.W) - GX.Z 2 . I d ( z o o p l a n k t o n m o r t a l i t y r a t e = n o n l i n e a r , w e i g h t - d e p e n d e n t t e r m + 1 i n e a r t e r m ) . When c o p e p o d s r e a c h a d u l t w e i g h t , W2, growth i s d i v e r t e d t o r e p r o d u c t i v e s t o r e , S, f o r a p e r i o d of J d a y s , a f t e r w h i c h ZO n a u p l i a r e r e l e a s e d , ZO b e i n g g i v e n by ZO = X.S/WO • 2.1e where X i s e f f i c i e n c y and WO t h e i n i t i a l n a u p l i a r w e i g h t . The n o t a t i o n h e r e ( T a b l e I) f o l l o w s t h a t o f S t e e l e ( 1 9 7 4 ) , e x c e p t t h a t GW i s u s e d i n 2 . I d t o a l l o w G t o be r e s e r v e d f o r z o o p l a n k t o n b i o m a s s . The r e a d e r i s r e f e r r e d t o S t e e l e f o r a d e t a i l e d d e r i v a t i o n of t h e mo d e l . The s y s t e m 2.1 r e p r e s e n t s a s e t o f f o u r s i m u l t a n e o u s , n o n l i n e a r , d i f f e r e n t i a l e q u a t i o n s i n v o l v i n g t h r e s h o l d s and t h e d i s c o n t i n u o u s r e c r u i t m e n t o f n a u p l i i and t h e r e i s l i t t l e p o i n t i n l o o k i n g f o r n o n - t r i v i a l s o l u t i o n s i n c l o s e d f o r m . A s e r i e s o f s i m p l i f i c a t i o n s and a p p r o x i m a t i o n s i s employed h e r e t o o b t a i n some u n d e r s t a n d i n g of t h e b e h a v i o u r of t h e model o v e r c o r r e s p o n d i n g r e g i o n s i n p a r a m e t e r s p a c e . T h e s e a p p r o x i m a t e r e s u l t s a r e t h e n c h e c k e d and e x t e n d e d by computer s i m u l a t i o n . T h e . b a s i c s t e p i n t h i s q u a l i t a t i v e a n a l y s i s i s t h e r e c o g n i t i o n of 42 Table I. Parameters used i n Steele's model (2.1). R...nutrient concentration (carbon equivalent) i n mixed layer. RO...nutrient concentration (carbon equivalent) below mixed layer. V...mixing rate through thermocline. U...fraction of excreted nutrient recycled. P...phytoplankton carbon concentration i n mixed layer. A. ..maximum phytoplankton growth rate (day ^ ) . B. ..half-saturation constant (carbon equivalent) for nutrient- dependent growth. Z...zooplankton density (#/l). W...zooplankton weight (ug C/ind). C. ..fixes maximum zooplankton ingestion rate. Pi..threshold for zooplankton grazing on phytoplankton carbon. D. ..fixes zooplankton grazing rate above PI. E. ..fixes component of metabolic rate proportional to ingestion. F. ..fixes basal metabolic rate. GW. .maximum of weight and density-dependent mortality rate. Wl..weight threshold for mortality. Zl..number threshold for mortality. H...'half-saturation' constant for mortality. GX..constant mortality rate. ZO. . i n i t i a l number of nauplii i n cohort. WO..initial naupliar weight. W2..adult weight. S...reproductive store. J...period over which reproductive store accumulates. 43 two a s p e c t s of the s e a s o n a l b e h a v i o u r s t u d i e d by S t e e l e and Landry; namely, the t r a n s i e n t response t o h i g h i n i t i a l n u t r i e n t c o n c e n t r a t i o n s (the s p r i n g bloom), and the approach t o a s t a b l e c y c l i c p a t t e r n i n the n u t r i e n t - l i m i t e d p e r i o d which f o l l o w s . The l a t t e r i s more l i k e l y t o be amenable t o q u a l i t a t i v e a n a l y s i s and i s t r e a t e d here f i r s t . The a n a l y s i s proceeds t h r o u g h the r e c o g n i t i o n of t h r e e d i s t i n c t time s c a l e s i n the model under n u t r i e n t l i m i t a t i o n . (For an i n s t r u c t i v e example of the use of m u l t i p l e time s c a l e s i n the a n a l y s i s of a c o m p l i c a t e d e c o l o g i c a l model, see Ludwig,Jones and H o l l i n g ( 1 9 7 8 ) . ) A time s c a l e f o r n u t r i e n t t u r n o v e r can be o b t a i n e d by d i v i d i n g the source term, V.RO, by the h a l f - s a t u r a t i o n c o n s t a n t f o r n u t r i e n t u p t a k e , B. For S t e e l e ' s v a l u e s of V ( 0 . 0 1 , d a y 1 ) and RO (760 pg C ( e q ) . ! - 1 ) , V.RO e q u a l s 7.6. S t e e l e used a r a t h e r h i g h v a l u e of B (96 p g C ( e q ) . l _ 1 a c c o r d i n g t o L a n d r y ) . Recent chemostat r e s u l t s , ' c o m b i n e d w i t h o b s e r v a t i o n s of v e r y low n u t r i e n t c o n c e n t r a t i o n s i n the oceans, suggest t h a t h a l f - s a t u r a t i o n c o n s t a n t s f o r growth s h o u l d be s m a l l e r than t h i s , of o r d e r 0.1 )jg a t N . l " 1 or a p p r o x i m a t e l y 10 jug C ( e q ) . ! - 1 (McCarthy and Goldman,1978). T h i s r e s u l t s i n a time s c a l e f o r n u t r i e n t t u r n o v e r of o r d e r 1 day, much s h o r t e r than t h a t of p h y t o p l a n k t o n (maximum growth r a t e 0.2 d a y 1 , y i e l d i n g a time s c a l e of 5 days) or z o o p l a n k t o n . We proceed t h e r e f o r e by t r e a t i n g R as a f a s t v a r i a b l e ; t h a t i s , by assuming t h a t n u t r i e n t c o n c e n t r a t i o n a d j u s t s r a p i d l y t o changes i n o t h e r s t a t e v a r i a b l e s so t h a t R R(P,Z,W), where R makes the r i g h t - h a n d s i d e of e q u a t i o n 2.1a z e r o . S u b s t i t u t i n g R = R i n e q u a t i o n 2.1b g i v e s 44 P = V.RO - V.P - Z.W0-.7 . ( (C-U.E) .f (P) - U.F) , 2.2a where f ( P ) s t a n d s f o r (P-P1)/(D+P), and the term V.R has been n e g l e c t e d s i n c e R i s assumed t o be of o r d e r B or a p p r o x i m a t e l y one-hundredth RO. When combined w i t h e q u a t i o n s 2.1 c,d,e, e q u a t i o n 2.2a forms a system ( 2 . 2 ) , from which n u t r i e n t s have been e l i m i n a t e d . The second and t h i r d time s c a l e s can be d i s t i n g u i s h e d p r o v i d e d the copepod m o r t a l i t y r a t e i s low enough t h a t Z changes s l o w l y compared w i t h p o t e n t i a l growth r a t e s of P and W. Then Z can be t r e a t e d as a slow v a r i a b l e and the b e h a v i o u r of P and W c o n s i d e r e d w i t h Z f i x e d . The system P = V.RO - V.P - Z.W0-7 .( (C-U.E) . f ( P ) - U.F) 2.3a W = ( (0.7.C-E) .f (P) - F).W°-7 2.3b has the phase p l a n e p o r t r a i t shown i n F i g 8. The n o n - t r i v i a l e q u i l i b r i u m s o l u t i o n (P,W(Z)) of 2.3 i s s t a b l e p r o v i d e d f ' ( P ) i s p o s i t i v e , a c o n d i t i o n which i s always s a t i s f i e d r e g a r d l e s s of the v a l u e of P I . Then, a c c o r d i n g t o the s l o w - v a r i a b l e a p p r o x i m a t i o n , as Z ( t ) d e c r e a s e s t h r o u g h m o r t a l i t y , P ( t ) and W(t) s h o u l d t r a c k the q u a s i - e q u i l i b r i u m s o l u t i o n ( P , W ( Z ( t ) ) ) . A c c o r d i n g t o 2.3, P and Z ( t ) . W 0 , 7 are both c o n s t a n t . I t f o l l o w s t h a t a c o h o r t w i l l r e a c h a d u l t w e i g h t , W2, from an i n i t i a l w e i g h t , WO, when the d e n s i t y has dropped by a f a c t o r (WO/W2) 0 7. For example, i f the per c a p i t a m o r t a l i t y i s c o n s t a n t (GW = 0, GX f 0 ) , Z ( t ) = ZO.exp(-GX.t) and, a l l o w i n g f o r the i n c u b a t i o n p e r i o d J , the g e n e r a t i o n time i s g i v e n by F i g u r e 8. Phase p l a n e p o r t r a i t f o r t h e s y s t e m 2.3- 46 T = J + 0.7.1n(W2/WO)/GX. 2.4 So f a r , o n l y the growth of a s i n g l e c o h o r t has been d e a l t w i t h but an approximate t r e a t m e n t of r e p r o d u c t i o n i s a l s o p o s s i b l e . D u r i n g the p e r i o d of J days over which r e p r o d u c t i v e s t o r e s a r e acc u m u l a t e d , W i s f i x e d a t a d u l t weight W2, so t h a t e q u a t i o n 2.3b and the e q u i l i b r i u m v a l u e P a r e not r e l e v a n t . I t i s c o n s i s t e n t w i t h the s l o w - v a r i a b l e a p p r o x i m a t i o n t o assume t h a t P i s a p p r o x i m a t e l y e q u a l t o P ( Z ) , where P makes P = 0 i n 2.3a f o r W = W2. S u b s t i t u t i n g P = P(Z) i n the e q u a t i o n f o r S and i n t e g r a t i n g g i v e s , a f t e r J days: S = SO - Z.W° 7 . S I where Z.W07 i s e v a l u a t e d a t the b e g i n n i n g of the r e p r o d u c t i v e p e r i o d and 50 = V.RO.J.(0.7.C-E)/(C-E.U) , 51 = F.C.(1.-0.7.U) . ( l . - e x p ( - G X . J ) ) / ((C-E.U).GX) . But a c c o r d i n g t o the approximate t r e a t m e n t of growth, Z.W0-7 = Z03.W00-7 where ZO 3 i s the i n i t i a l s i z e of the g t h c o h o r t . U s i n g e q u a t i o n 2.1e, i t f o l l o w s t h a t Z 0 9 + 1 = S0.X/W0 - Z0 9.S1.X/W0°- 3 . 2.5 T h i s c o n s t i t u t e s a d i f f e r e n c e e q u a t i o n f o r ZO 3. I f the c o e f f i c i e n t of ZO 9 i n e q u a t i o n 2.5 i s l e s s than 1 i n magnitude, the e q u a t i o n has a s t a b l e c o n s t a n t s o l u t i o n ZO* and, a c c o r d i n g t o 47 t h i s a p p r o x i m a t e t h e o r y , t h e r e i s a c o r r e s p o n d i n g s t a b l e c y c l i c s o l u t i o n o f t h e p h y t o p l a n k t o n - z o o p l a n k t o n s y s t e m 2.2. I f t h e c o e f f i c i e n t i s g r e a t e r t h a n 1 i n m a g n i t u d e , ZO i s u n s t a b l e and a s t a b l e c y c l i c s o l u t i o n t o 2.2 h a v i n g c o n s t a n t a m p l i t u d e c a n n o t be e x p e c t e d . T h i s c o e f f i c i e n t i s p r o p o r t i o n a l t o t h e b a s a l m e t a b o l i c r a t e , F. 2.3 S i m u l a t i o n R e s u l t s . The a p p r o x i m a t e t h e o r y p r e d i c t s a s t a b l e c y c l i c s o l u t i o n t o S t e e l e ' s model under c o n d i t i o n s of n u t r i e n t - l i m i t a t i o n w i t h o u t any need f o r t h r e s h o l d s o r q u a d r a t i c p r e d a t i o n t e r m s . T h i s i s i n k e e p i n g w i t h t h e r e s u l t s of C h a p t e r 1 but somewhat s u r p r i s i n g i n view of S t e e l e ' s s i m u l a t i o n r e s u l t s . The p r e d i c t i o n has been t e s t e d by computer s i m u l a t i o n of b o t h t h e s i m p l i f i e d p h y t o p l a n k t o n - z o o p l a n k t o n model 2.2 and t h e f u l l model 2.1 f o r t h e s i m p l e s t c a s e of no t h r e s h o l d s , c o n s t a n t m o r t a l i t y r a t e and f i x e d m e t a b o l i c r a t e (P1=E=GW=0). The f i r s t s e t of s i m u l a t i o n s were o b t a i n e d by f i x i n g t h e m e t a b o l i c r a t e F and v a r y i n g t h e m o r t a l i t y r a t e GX i n t h e s i m p l i f i e d model 2.2. F o r F = 0.4 and GX r a n g i n g from 0.02 t o 0.06 d a y " 1 , s t a b l e c y c l i c s o l u t i o n s were a p p r o a c h e d w i t h g e n e r a t i o n t i m e d e c r e a s i n g a s GX i n c r e a s e d , i n q u a l i t a t i v e a greement w i t h t h e s l o w - v a r i a b l e t h e o r y . However, q u a n t i t a t i v e a greement between s i m u l a t e d g e n e r a t i o n t i m e s and thos,e p r e d i c t e d a c c o r d i n g t o 2.4 i s n o t p a r t i c u l a r l y good ( F i g 9 ) . P a r t , o f t h e e x p l a n a t i o n f o r t h e p o o r agreement can be seen i n F i g 10a, where a s i m u l a t e d c y c l e i s p o r t r a y e d . The p h y t o p l a n k t o n d e n s i t y i s f a r from b e i n g c o n s t a n t o v e r t h e p e r i o d o f c o p e p o d g r o w t h , due p a r t l y 48 F i g u r e 9. C o m p a r i s o n o f g e n e r a t i o n t i m e s p r e d i c t e d by e q u a t i o n 2.4 ( s o l i d l i n e ) and t h o s e o b t a i n e d i n n u m e r i c a l s o l u t i o n s o f t h e s y s t e m 2.2 ( d o t s ) . 49 300 200 U O) 100 r15 50 100 150 200 TIME (days) 250 300 350 400 400n 3004 200A U CJ) 3. 100H 150 200 250 TIME (days) 300 350 400 F i g u r e 10. S t a b l e c y c l i c s o l u t i o n s o f the system 2.2 f o r : (a) F=0.4, GX=0.05; (b) F=0.2, GX=0.05. 50 t o the l i m i t a t i o n s of the s l o w - v a r i a b l e a p p r o x i m a t i o n and p a r t l y t o the p e r t u r b a t i o n imposed by the r e l e a s e of n a u p l i i a t the end of each r e p r o d u c t i v e p e r i o d . In s p i t e of t h i s , f u r t h e r q u a l i t a t i v e agreement between the a p proximate t h e o r y and s i m u l a t i o n was found. When F i s d e c r e a s e d t o 0.2, the c y c l e p e r i o d i s not a f f e c t e d , b u t , as shown i n F i g 10b, the s i m u l a t e d c y c l e i n v o l v e s lower P and h i g h e r ZO, as p r e d i c t e d by the s l o w - v a r i a b l e t h e o r y . A l s o , i n c r e a s i n g F t o 0.6 i n the s i m u l a t i o n r e s u l t s i n an approach t o a r a t h e r c u r i o u s c y c l e i n v o l v i n g a l t e r n a t e l y h i g h and low n a u p l i a r r e c r u i t m e n t s ( F i g 1 0 c ) . T h i s b e h a v i o u r c o r r e s p o n d s i n the a p p r o ximate t h e o r y t o a s t a b l e s o l u t i o n of p e r i o d 2 i n the d i f f e r e n c e e q u a t i o n 2.5, ZO* h a v i n g been d e s t a b i l i z e d by i n c r e a s i n g F. The phenomenon of p e r i o d i c and a p e r i o d i c s o l u t i o n s t o d i f f e r e n c e e q u a t i o n s i n s i m p l e e c o l o g i c a l models has a r o u s e d c o n s i d e r a b l e i n t e r e s t (eg May,1975). The v a l i d i t y of u s i n g the f a s t - v a r i a b l e a ssumption t o e l i m i n a t e n u t r i e n t s has been t e s t e d by computer s i m u l a t i o n of S t e e l e ' s f u l l model 2.1. For F = 0.4, GX = 0.05 day" 1 and low B (10 jug C ( e q ) . ! " 1 ) , n u m e r i c a l s o l u t i o n s of the f u l l model approach a s t a b l e c y c l i c s o l u t i o n ( F i g 1 1 a ) , which i s almost i d e n t i c a l t o t h a t o b t a i n e d from system 2.2 ( F i g 10a). Throughout the c y c l e , R remains low (of o r d e r B) and the f a s t - v a r i a b l e a p p r o x i m a t i o n i s q u i t e a c c u r a t e . However, i t can be v i o l a t e d i n a number of ways. For example, i f F i s reduced t o 0.2, w i t h o t h e r parameters unchanged, l a r g e , d i v e r g i n g o s c i l l a t i o n s i n R, P and Z appear ( F i g l i b ) . The e x p l a n a t i o n f o r t h i s can be found i n the n u t r i e n t e q u a t i o n 2.1a and the f a c t t h a t , f o r F=0.2, the c y c l i c s o l u t i o n .51 TIME (days) Figure 10c. Stable cyclic solution of the system 2.2 for F=0.6, GX=0.05. O 50 100 150 200 250 300 350 400 TIME (days) Figure 11a. Stable cyclic solution of the system 2.1 for F=0.4, GX=0.05 and B=10. 53 400 r20 100 150 200 TIME (days) KI 300 r15 c r <D U cn 3 CC t_ o D_ 200 100 0 50 . 100 150 200 250 TIME (DAYS) 300 350 400 Figure ll b , c . Behaviour of the system 2.1 for F=0.2, GX=0.05 and B=10 with: (b) D=100. (unstable oscillations) and (c) D=175. (stable cycle). 54 p r e d i c t e d under the f a s t - v a r i a b l e a p p r o x i m a t i o n ( F i g 10b) i n v o l v e s low p h y t o p l a n k t o n c o n c e n t r a t i o n s . N u t r i e n t uptake by the p h y t o p l a n k t o n i s n o n - l i n e a r i n R w i t h a maximum v a l u e of A.P. Thus, i f P i s too s m a l l , p h y t o p l a n k t o n uptake cannot match the m i x i n g i n p u t V.RO and the e q u i l i b r i u m s o l u t i o n R(P,Z) s w i t c h e s from a low v a l u e (of o r d e r B) de t e r m i n e d by p h y t o p l a n k t o n u p t a k e , t o a h i g h v a l u e (of o r d e r RO) de t e r m i n e d by m i x i n g l o s s e s . The f a s t - v a r i a b l e assumption f o r n u t r i e n t s i s of co u r s e no l o n g e r v a l i d and, i n the computer s i m u l a t i o n , the i n s t a b i l i t y a s s o c i a t e d w i t h the p h y t o p l a n k t o n - z o o p l a n k t o n i n t e r a c t i o n appears t o dominate. T h i s e x p l a n a t i o n suggests a number of ways t o r e c o v e r s t a b l e c y c l e s f o r low F. A s u f f i c i e n t d e c r e a s e i n the r a t e of n u t r i e n t i n p u t , V.RO, or i n c r e a s e i n maximum growth r a t e , A, w i l l p r e v e n t n u t r i e n t b u i l d - u p . Changes can a l s o be made i n o t h e r parameters which i n c r e a s e P, compensating f o r the decrease i n F. For example, P i s s c a l e d by the g r a z i n g h a i f - s a t u r a t i o n c o n s t a n t , D. When D i s i n c r e a s e d from 100 t o 175 ug C . l " 1 , (the v a l u e used by L a n d r y ) , the s o l u t i o n of the f u l l model f o r F=0.2 ( F i g 11c) approaches the s t a b l e c y c l i c s o l u t i o n p r e d i c t e d by the f a s t - v a r i a b l e model 2.2 ( F i g 10b). An o b v i o u s way t o v i o l a t e the f a s t - v a r i a b l e assumption i s t o i n c r e a s e B. I f S t e e l e ' s v a l u e f o r B of 96 ug C ( e q ) . l _ 1 i s adopted, the time s c a l e f o r changes i n n u t r i e n t becomes comparable t o t h a t of changes i n p h y t o p l a n k t o n and the p r e c e d i n g a pproximate t r e a t m e n t cannot be a p p l i e d . In f a c t , f o r F=0.4,GX = 0.05 and D = 100 ug C . l " 1 , r a p i d l y d i v e r g i n g o s c i l l a t i o n s i n P and R a r e o b t a i n e d i n n u m e r i c a l s o l u t i o n s on i n c r e a s i n g B t o 96. 55 A s t a b l e c y c l i c s o l u t i o n c an be r e c o v e r e d by i n c r e a s i n g D t o 175 ^jg C . l " 1 , t h i s s o l u t i o n b e i n g c h a r a c t e r i s e d by c o n s i s t e n t l y h i g h P and low R/B. A n o t h e r a p p r o a c h w h i c h a p p e a r s t o be u s e f u l i n i n t e r p r e t i n g t h e s e n u m e r i c a l r e s u l t s i s t o compare them w i t h a n a l y t i c r e s u l t s f r o m a more m a t h e m a t i c a l l y t r a c t a b l e model o b t a i n e d by d r o p p i n g t h e c o p e p o d age s t r u c t u r e : R = V.RO - A.R.P/(B+R) + U.F.G 2.6a P = A.R.P/(B+R) - G . C . f ( P ) 2.6b G = 0.7.G.C.f(P) - m.G 2.6c where G i s c o p e p o d b i o m a s s . T h i s s y s t e m p o s s e s s e s a n o n - t r i v i a l e q u i l i b r i u m s o l u t i o n (R,P,G) d e t e r m i n e d by f ( P ) = m/(0.7.C) G = 0.7.V.R0/(m-0.7.U.F) A.R.P/(B+R) = V.R0.m/(m-0.7.U.F). A T a y l o r - s e r i e s e x p a n s i o n a b o u t t h i s e q u i l i b r i u m y i e l d s t h e l i n e a r s y s t e m / • \ SR SP SG G. I 8R SP SG where the c o n s t a n t m a t r i x GV has t r a c e 56 I c- = -A.B.P/(B+R) + G . C . ( f ( P ) / P - f ' ( P ) ) . 2.7 I t i s w e l l - k n o w n t h a t a n e c e s s a r y c o n d i t i o n f o r t h e l o c a l A A > s t a b i l i t y o f (R,P,G) i s t h a t 2. c,, be n e g a t i v e . In t h e a b s e n c e of t h r e s h o l d s (PI = 0 ) , t h e s e c o n d t e r m i n 2.7 i s p o s i t i v e ; t h i s i s t h e d e s t a b i l i z i n g e f f e c t o f t h e t y p e II f u n c t i o n a l r e s p o n s e ( H o l l i n g , 1 9 6 5 ) . The f i r s t t e r m i s n e g a t i v e and r e p r e s e n t s t h e s t a b i l i z i n g e f f e c t of n u t r i e n t l i m i t a t i o n . I f B i s i n c r e a s e d w i t h t h e o t h e r p a r a m e t e r s i n 2.6 h e l d c o n s t a n t , t h e f i r s t t erm d e c r e a s e s l i k e 1/B w h i l e t h e s e c o n d t e r m r e m a i n s c o n s t a n t ; t h u s i n c r e a s i n g B t e n d s t o d e s t a b i l i z e t h e s y s t e m . I f D i s i n c r e a s e d and t h e o t h e r p a r a m e t e r s h e l d c o n s t a n t , i t c a n a l s o be shown t h a t t h e s e c o n d t e r m d e c r e a s e s l i k e 1/D w h i l e t h e f i r s t t e r m i n c r e a s e s i n m a g n i t u d e , so t h a t i n c r e a s i n g D t e n d s t o s t a b i l i z e t h e s y s t e m . T h i s i s c o n s i s t e n t w i t h t h e s i m u l a t i o n r e s u l t s r e p o r t e d above f o r S t e e l e ' s m o d e l . T h i s c o n c l u d e s t h e a n a l y s i s of t h e n u t r i e n t l i m i t e d p e r i o d . B e f o r e p r o c e e d i n g , i t seems a d v i s a b l e t o summarise t h e r e s u l t s o b t a i n e d so f a r . By use of a s e r i e s of a p p r o x i m a t i o n s b a s e d on t h e r e c o g n i t i o n of f a s t and s l o w t i m e s c a l e s , s t a b l e c y c l i c s o l u t i o n s t o S t e e l e ' s model have been p r e d i c t e d i n t h e a b s e n c e of t h r e s h o l d s and f o r c o n s t a n t p e r c a p i t a m o r t a l i t y r a t e and f i x e d m e t a b o l i c r a t e . Computer s i m u l a t i o n has shown t h a t t h e s e s o l u t i o n s do e x i s t but a r e r e s t r i c t e d t o a r e g i o n i n p a r a m e t e r s p a c e f o r w h i c h t h e s t a b i l i z i n g e f f e c t s o f n u t r i e n t l i m i t a t i o n o u t w e i g h t h e d e s t a b i l i z i n g e f f e c t s o f t h e s a t u r a t i n g t y p e II f u n c t i o n a l r e s p o n s e of t h e c o p e p o d s . Of c o u r s e , S t e e l e demanded of h i s model t h a t i t r e p r o d u c e 57 the q u a l i t a t i v e f e a t u r e s c h a r a c t e r i s t i c of the observed s e a s o n a l c y c l e i n the N o r t h Sea, s t a r t i n g from a p r e s c r i b e d s e t of i n i t i a l c o n d i t i o n s c o r r e s p o n d i n g t o the i n i t i a t i o n of the s p r i n g bloom. The e x i s t e n c e of a s t a b l e c y c l i c s o l u t i o n f o r a p a r t i c u l a r s e t of parameters i n no way guarantees t h a t the t r a n s i e n t approach t o t h i s c y c l e from the p r e s c r i b e d i n i t i a l c o n d i t i o n s w i l l s a t i s f y S t e e l e ' s c r i t e r i a . In f a c t , i f the v a l u e s B = 10 pg C ( e q ) . ! " 1 , F = 0.4, D = 100 pg C . l " 1 and GX = 0.05 d a y " 1 , c o r r e s p o n d i n g to the s t a b l e c y c l e of F i g 11a, are used t o g e t h e r w i t h S t e e l e ' s i n i t i a l c o n d i t i o n s , the r e s u l t i n g s i m u l a t i o n i s most u n r e a l i s t i c ( F i g 12a). D u r i n g the s p r i n g bloom, a l a r g e c o h o r t of copepods i s produced. I n d i v i d u a l copepods i n i t i a l l y see h i g h p h y t o p l a n k t o n c o n c e n t r a t i o n s and grow r a p i d l y , q u i c k l y p r o d u c i n g a v e r y h i g h g r a z i n g p r e s s u r e on the p h y t o p l a n k t o n . The p h y t o p l a n k t o n c o n c e n t r a t i o n d e c l i n e s q u i c k l y and d u r i n g the p r o l o n g e d p e r i o d of low P which f o l l o w s , the copepods s t a r v e i n an i m p r e s s i v e but r a t h e r u n r e a l i s t i c manner, the c o n s t a n t m e t a b o l i c r a t e r e d u c i n g t h e i r weight t o z e r o by day 54. To p r e v e n t t h i s from happening, i t i s o b v i o u s l y n e c e s s a r y t o reduce t h i s b u r s t of i n t e n s e g r a z i n g p r e s s u r e . One way t o do t h i s i s t o i n t r o d u c e a p r e d a t i o n term which r e s u l t s i n h i g h copepod m o r t a l i t y r a t e s when copepod numbers a r e h i g h , t h e r e b y r a p i d l y r e d u c i n g the s i z e of t h i s l a r g e second c o h o r t . T h i s appears t o be the p r i m a r y r o l e of Landry's q u a d r a t i c p r e d a t i o n term; any s t a b i l i z i n g e f f e c t of t h i s term i n the l a t e r n u t r i e n t - l i m i t e d regime i s of secondary i m p o r t a n c e . Another a l t e r n a t i v e i s t o s e t a maximum l i m i t on c o h o r t s i z e i n the manner of S t e e l e and M u l l i n (1977), who made ZO a h y p e r b o l i c f u n c t i o n of 58 F i g u r e 12a. A t t e m p t e d s i m u l a t i o n o f s p r i n g b l o o m u s i n g t h e s y s t e m 2.1 w i t h B=10, F=0.4, D=100 and GX=0.05. 59 r e p r o d u c t i v e s t o r e , S. A t h i r d a l t e r n a t i v e i s t o i n c r e a s e t h e h a l f - s a t u r a t i o n c o n s t a n t f o r g r a z i n g , t h e r e b y r e d u c i n g g r a z i n g p r e s s u r e a t low p h y t o p l a n k t o n d e n s i t i e s . T h i s has a l r e a d y been done by L a n d r y , who i n c r e a s e d D from 100 t o 175 jag C . l - 1 . I f t h e same change i s made t o t h e s i m u l a t i o n d e s c r i b e d a bove, t h e r e s u l t i s much im p r o v e d ( F i g 1 2 b ) . P h y t o p l a n k t o n d e p l e t i o n i s l e s s p r o n o u n c e d and of s h o r t e r d u r a t i o n and a r a p i d a p p r o a c h t o t h e s t a b l e c y c l i c s o l u t i o n o c c u r s . U n f o r t u n a t e l y , w e i g h t l o s s f o l l o w i n g t h e s p r i n g bloom i s s t i l l u n r e a l i s t i c a l l y h i g h , i n d i v i d u a l s b e i n g r e d u c e d from more t h a n 3 pq C t o l e s s t h a n 0.35 pq C by s t a r v a t i o n . I t i s p o s s i b l e t o do much b e t t e r t h a n t h i s w i t h o u t a d o p t i n g t h r e s h o l d s o r L a n d r y ' s p r e d a t i o n t e r m . In F i g 12c, t h e r e s u l t s o f a s i m u l a t i o n u s i n g F=0.5,GX=0.04, B=10 and D=175 a r e p r e s e n t e d . The p e r i o d o f p h y t o p l a n k t o n d e p l e t i o n i s r e d u c e d t o t h e e x t e n t t h a t c o p e p o d w e i g h t l o s s f o l l o w i n g t h e s p r i n g bloom i s o n l y a b o u t 50%. The f i n a l s i m u l a t i o n ( F i g 12d) u s e s an even l a r g e r v a l u e of D (250 pq C . l " 1 ) and L a n d r y ' s c o m b i n a t i o n i n g e s t i o n - d e p e n d e n t and c o n s t a n t m e t a b o l i c r a t e (E=0.3, F=0.2). The r e s u l t i n g c o p e p o d w e i g h t l o s s f o l l o w i n g t h e s p r i n g bloom i s l i m i t e d t o l e s s t h a n 25%. 2.4 C o n e l u s i o n s . • P e r h a p s t h e most s t r i k i n g r e s u l t of t h i s r e v i e w o f S t e e l e ' s s i m u l a t i o n model i s t h a t ' r e a l i s t i c ' i m i t a t i o n s o f t h e s e a s o n a l p l a n k t o n c y c l e i n t h e N o r t h Sea can be o b t a i n e d f r o m t h e model w i t h o u t i n v o k i n g t h r e s h o l d s i n h e r b i v o r e g r a z i n g o r q u a d r a t i c p r e d a t i o n t e r m s . T h i s o f c o u r s e does n o t e s t a b l i s h t h a t F i g u r e 12b. S i m u l a t i o n o f s p r i n g and summer w i t h p a r a m e t e r s u s e d f o r F i g 12a e x c e p t D=175. F i g u r e 12c. As f o r F i g 12b except GX=0.04. TIME (days) F i g u r e 12 d. S i m u l a t i o n o f s p r i n g and summer w i t h F=0.2, E=0.3, GX=0.05 and D=250. 63 t h r e s h o l d s or q u a d r a t i c p r e d a t i o n a r e n o t p r e s e n t and i m p o r t a n t i n t h e N o r t h Sea e c o s y s t e m , a l t h o u g h i t weakens any argument f o r t h e i r e x i s t e n c e b a s e d on t h e i r n e c e s s i t y i n S t e e l e ' s m o d e l . In f a c t , t h e r e a p p e a r s t o be b e t t e r e x p e r i m e n t a l e v i d e n c e now f o r t h e e x i s t e n c e of t h r e s h o l d - t y p e phenomena, a t l e a s t i n some s p e c i e s o f c o p e p o d s ( F r o s t , 1 9 7 5 ) , and t h e a s s u m p t i o n o f a c o n s t a n t p e r c a p i t a m o r t a l i t y r a t e f o r c o p e p o d s i s c e r t a i n l y no more s a t i s f a c t o r y a way of t r u n c a t i n g t h e f o o d c h a i n t h a n u s i n g L a n d r y ' s p r e d a t i o n t e r m . The p r e c e d i n g r e s u l t s do i l l u s t r a t e t h e a d v a n t a g e s o f t h e q u a l i t a t i v e a n a l y s i s of complex e c o l o g i c a l m o d e l s , w h e r e v e r t h i s i s p o s s i b l e . The a n a l y s i s c o n d u c t e d h e r e , a l t h o u g h a p p r o x i m a t e and i n c o m p l e t e , l e d t o t h e d i s c o v e r y o f s t a b l e c y c l i c s o l u t i o n s i n t h e n u t r i e n t - l i m i t e d r e g i m e i n t h e a b s e n c e of t h r e s h o l d s and q u a d r a t i c m o r t a l i t y . T h e s e e x i s t o v e r a l i m i t e d r e g i o n of p a r a m e t e r s p a c e and m i g h t not have been f o u n d a t a l l by s i m p l y t r y i n g v a r i o u s p a r a m e t e r c o m b i n a t i o n s i n computer s i m u l a t i o n s . The i n t e r a c t i o n s between p a r a m e t e r s w h i c h were r e v e a l e d by t h e q u a l i t a t i v e a n a l y s i s t e s t i f y t o t h e i n a d e q u a c y o f s t a n d a r d s e n s i t i v i t y a n a l y s e s i n w h i c h p a r a m e t e r s a r e v a r i e d one a t a t i m e . F o r example, i f t h e h a l f - s a t u r a t i o n c o n s t a n t f o r g r a z i n g i s l a r g e enough, s t a b l e c y c l i c s o l u t i o n s e x i s t o v e r a wide r a n g e of v a l u e s of B, a t l e a s t 0 < B < 100 jug C ( e q ) . ! ' 1 . However, i f D i s l o w e r , s t a b l e s o l u t i o n s o n l y e x i s t f o r B s m a l l and i t c o u l d be c o n c l u d e d t h a t t h e v a l u e o f B i n n a t u r e i s c r i t i c a l . S t e e l e ' s s i m u l a t i o n r e s u l t s were i n s e n s i t i v e t o c h a n g e s i n B, but t h r e s h o l d s i n g r a z i n g and p r e d a t i o n were a l r e a d y i n f o r c e . He d i d n o t o b t a i n r e a s o n a b l e r e s u l t s when t h e t h r e s h o l d s were 64 o m i t t e d due p a r t l y t o t h e v a l u e s assumed f o r B and D. The i n s i g h t i n t o S t e e l e ' s model o b t a i n e d h e r e s u g g e s t s a b r o a d e r a p p r o a c h t o t h e b i o l o g i c a l q u e s t i o n of h e r b i v o r e r e s p o n s e t o low f o o d c o n c e n t r a t i o n s , i n c l u d i n g t h e m a t t e r o f t h r e s h o l d s . The major p r o b l e m e n c o u n t e r e d h e r e i n o b t a i n i n g a r e a l i s t i c s e a s o n a l p a t t e r n from S t e e l e ' s model i n v o l v e s t h e t r a n s i e n t r e s p o n s e r a t h e r t h a n s t a b i l i t y ; of t h e r e g i o n o f p a r a m e t e r s p a c e a l l o w i n g s t a b l e c y c l i c s o l u t i o n s , o n l y a s m a l l s u b - r e g i o n a l l o w s a r e a l i s t i c a p p r o a c h t o t h i s c y c l e from i n i t i a l c o n d i t i o n s o f h i g h n u t r i e n t . As was e x p l a i n e d e a r l i e r , t h e d i f f i c u l t y l i e s i n t h e u n r e a l i s t i c w e i g h t l o s s e s w h i c h can o c c u r i n t h e p o s t - b l o o m p e r i o d o f p h y t o p l a n k t o n d e p l e t i o n due t o t h e a s s u m p t i o n o f a h i g h b a s a l m e t a b o l i c r a t e . S t e e l e a v o i d e d t h i s d i f f i c u l t y i n h i s s t a n d a r d s i m u l a t i o n by a s s u m i n g a t h r e s h o l d i n g r a z i n g ( t h e r e b y p r e v e n t i n g extreme d e p l e t i o n o f p h y t o p l a n k t o n ) and a m e t a b o l i c r a t e e n t i r e l y p r o p o r t i o n a l t o i n g e s t i o n ( z e r o b a s a l m e t a b o l i s m ) , so t h a t c o p e p o d w e i g h t l o s s was i m p o s s i b l e . (When a n o n - z e r o b a s a l m e t a b o l i c r a t e was a d o p t e d i n one of S t e e l e ' s t r i a l s i m u l a t i o n s , c o p e p o d w e i g h t l o s s d i d o c c u r and t h e r e s u l t was c o n s e q u e n t l y d i s m i s s e d as u n r e a l i s t i c ) . In a l a t e r , more d e t a i l e d a n a l y s i s of t h e e n e r g y budget and o p t i m a l f e e d i n g r e s p o n s e o f c o p e p o d s , S t e e l e and F r o s t (1977) c o n c l u d e d t h a t t h e r e i s an i m p o r t a n t component o f m e t a b o l i s m p r o p o r t i o n a l t o f i l t e r i n g r a t e , not i n g e s t i o n r a t e . However, t h e o p t i m a l f e e d i n g r e s p o n s e t h e y o b t a i n e d i s r o u g h l y c o n s i s t e n t w i t h S t e e l e ' s (1974) a s s u m p t i o n as b o t h i n v o l v e r e d u c e d f i l t e r i n g r a t e a t low f o o d d e n s i t i e s and n o n - n e g a t i v e g r o w t h a t a l l f o o d d e n s i t i e s . 65 S t e e l e and F r o s t ' s argument does imply t h a t an organism which does not behave o p t i m a l l y , and i n s t e a d m a i n t a i n s a h i g h f i l t e r i n g r a t e a t low food d e n s i t i e s , w i l l be s u b j e c t t h e r e t o a h i g h m e t a b o l i c c o s t and r a p i d l o s s of body w e i g h t . T h i s i s e q u i v a l e n t t o an assumption of no t h r e s h o l d and a h i g h b a s a l m e t a bolism i n S t e e l e ' s (1974) model. O b v i o u s l y , t h i s weight l o s s , which w i l l be e s p e c i a l l y s e v e r e f o r s m a l l i n d i v i d u a l s a c c o r d i n g t o the W0-7 m e t a b o l i c law, cannot c o n t i n u e i n d e f i n i t e l y . There appear t o be two extreme outcomes p o s s i b l e . One p o s s i b i l i t y i s t h a t , as i t s body c o n d i t i o n d e t e r i o r a t e s , the organism's f i l t e r i n g r a t e and c o n s e q u e n t l y i t s m e t a b o l i c r a t e w i l l d e c r e a s e . P r o v i d e d t h i s o c c u r s on a short-enough time s c a l e (most l i k e l y f o r s m a l l o r g a n i s m s ) , t h i s p h y s i o l o g i c a l response w i l l have the same e f f e c t i n , t h e s i m u l a t i o n model as an ' o p t i m a l ' b e h a v i o u r a l response or a t h r e s h o l d . T h i s p o s s i b i l i t y may have been e n v i s a g e d by S t e e l e (1974), when he mentioned t h a t responses t o food c o n c e n t r a t i o n s might occur on s e v e r a l time s c a l e s . An im p o r t a n t p o i n t i s t h a t the p h y s i o l o g i c a l response would not be observ e d i n s h o r t - t e r m g r a z i n g e x p e r i m e n t s of the type commonly conducted over 24 hours or l e s s u s i n g l a r g e , l a t e - s t a g e copepods c o l l e c t e d i n the f i e l d ( M a r s h a l l , 1 9 7 3 ) . E f f e c t s of food c o n c e n t r a t i o n and d u r a t i o n (up t o 5 d a y s ) ' o f p r e - c o n d i t i o n i n g on f u n c t i o n a l response parameters have been r e p o r t e d f o r f r e s h - w a t e r c r u s t a c e a n s (Buckingham,1978). The second p o s s i b i l i t y i s t h a t the i n d i v i d u a l c o n t i n u e s t o f i l t e r and l o s e weight a t maximum r a t e s u n t i l weight l o s s exceeds some maximum p e r m i s s a b l e amount, when the organi s m d i e s . In the h i g h l y s i m p l i f i e d s i n g l e c o h o r t v e r s i o n of S t e e l e ' s model, the 66 e n t i r e c o h o r t w i l l d i e s i m u l t a n e o u s l y . In a more r e a l i s t i c m u l t i - c o h o r t v e r s i o n , as p r e s e n t e d by Landry (1976), or one i n v o l v i n g d i f f e r e n c e s i n growth r a t e between i n d i v i d u a l s , as r e p o r t e d even i n l a b o r a t o r y c u l t u r e s under homogeneous c o n d i t i o n s ( P a f f e n h o f f e r and H a r r i s , 1 9 7 6 ) , s t a r v a t i o n m o r t a l i t y w i l l l e a d t o a d e c r e a s e i n g r a z i n g p r e s s u r e on the p h y t o p l a n k t o n and, a t l e a s t f o r s m a l l z o o p l a n k t o n , may have a s i m i l a r q u a l i t a t i v e e f f e c t t o a g r a z i n g t h r e s h o l d . Thus a l l t h r e e t y p e s of response t o low food d e n s i t i e s (the s h o r t - t e r m b e h a v i o u r a l r e s p o n s e , the l o n g e r - t e r m p h y s i o l o g i c a l response and s t a r v a t i o n m o r t a l i t y ) c o u l d ensure r e a s o n a b l e t r a n s i e n t b e h a v i o u r i n S t e e l e ' s model. The a c t u a l r e s p o n s e , which p r o b a b l y l i e s somewhere between these extremes, may be i m p o r t a n t i n o t h e r c i r c u m s t a n c e s , s u c h as the S u b a r c t i c P a c i f i c , and i t s e x p e r i m e n t a l d e f i n i t i o n appears t o be a c h a l l e n g i n g and v a l u a b l e g o a l . Most of the c u r r e n t e x p e r i m e n t a l approaches a d d r e s s o n l y one a s p e c t . The s h o r t - t e r m g r a z i n g e x p e r i m e n t s r e f e r r e d t o above can c o n s i d e r o n l y the b e h a v i o u r a l r e s ponse. There have been a number of l a b o r a t o r y s t u d i e s of r e s p i r a t i o n r a t e s and weight l o s s i n s t a r v e d i n d i v i d u a l s (Reeve et aj. ,1970; Ikeda,1977). A g a i n , most of the s t u d i e s have been conducted on l a t e - s t a g e i n d i v i d u a l s , and any p h y s i o l o g i c a l e f f e c t of weight l o s s on f i l t e r i n g r a t e cannot be measured i n s t a r v e d i n d i v i d u a l s , w h i l e measurements of r e s p i r a t i o n r a t e s a r e open t o the problems of i n t e r p r e t a t i o n " d i s c u s s e d by S t e e l e ( 1 9 7 4 ) . I t i s i n t e r e s t i n g t o note t h a t C h e c k l e y (1980) has r e p o r t e d mean s u r v i v a l t i m e s f o r P a r a c a l a n u s parvus females w i t h o u t food t o be 4-5 days so t h a t s t a r v a t i o n m o r t a l i t y c o u l d be a s i g n i f i c a n t f a c t o r , a t l e a s t f o r 67 s m a l l copepods w i t h o u t f a t s t o r e s . I t i s o b v i o u s l y not an i m p o r t a n t c o n s i d e r a t i o n f o r the l a t e s t a g e s of l a r g e r s p e c i e s such as C a lanus plumchrus which b u i l d up e x t e n s i v e f a t s t o r e s ( F u l t o n , 1 9 7 3 ) . The e x p e r i m e n t a l p r o c e d u r e s most l i k e l y t o r e s o l v e t h i s q u e s t i o n appear t o be those r e p o r t e d by P a f f e n h o f f e r (1970), and r e v i e w e d by P a f f e n h o f f e r and H a r r i s ( 1 9 7 9 ) , f o r s t u d y i n g growth, m o r t a l i t y and c l e a r a n c e r a t e s over the e n t i r e copepod l i f e c y c l e i n l a b o r a t o r y c u l t u r e s . I n c r e a s e d m o r t a l i t y has been r e p o r t e d f o r Calanus p a c i f i c u s and Temora l o n g i c o r n u s a t low food d e n s i t i e s (about 20 jug C . l " 1 ) , a l t h o u g h these food d e n s i t i e s were s t i l l s u f f i c i e n t t o a l l o w growth a t c l o s e t o maximum r a t e s ( P a f f e n h o f f e r , 1 9 7 0 ; P a f f e n h o f f e r and H a r r i s , 1 9 7 6 ) . I t would c e r t a i n l y be i n t e r e s t i n g t o see t h e s e e x p e r i m e n t s r e p e a t e d a t even lower food d e n s i t i e s . 68 CHAPTER 3 PHYTOPLANKTON AT O.S.P.: DATA ANALYSIS AND MODELLING 3.1 I n t r o d u c t i o n The aim i n t h i s c h a p t e r i s t o d e v e l o p a r e a l i s t i c model of p h y t o p l a n k t o n growth a t O.S.P. The model i s based p r i m a r i l y on an a n a l y s i s of data c o l l e c t e d from the w e a t h e r s h i p s a t O.S.P. and the l e v e l of d e t a i l c o n s i d e r e d has been chosen t o match these o b s e r v a t i o n s . The u n d e r l y i n g s t r u c t u r e of the model i s adapted from the l i t e r a t u r e , as a r e c e r t a i n parameter v a l u e s which cannot be i n f e r r e d from the d a t a . The model i s used o n l y t o p r e d i c t net p r i m a r y p r o d u c t i o n from a f i x e d s t a n d i n g s t o c k i n t h i s c h a p t e r , but i s c o n s t r u c t e d i n such a way t h a t i t can e a s i l y be extended t o i n c l u d e g r a z e r s i n Chapter 4, where the q u a l i t a t i v e hypotheses r a i s e d i n Chapter 1 w i l l be a d d r e s s e d q u a n t i t a t i v e l y . 3.2 Data A n a l y s i s 3.2.1 D e s c r i p t i o n of the Data S e t . We a t h e r s h i p o b s e r v a t i o n s r e l a t e d t o p h y t o p l a n k t o n f o r the y e a r s 1964 t o 1976 have been p u b l i s h e d i n m a n u s c r i p t form (Stephens,1966; Stephens,1968; Stephens,1970; Stephens,1977). V a r i a b l e s measured i n c l u d e • S e c c h i d e p t h , c h l o r o p h y l l a, 1 4 C uptake r a t e s and n i t r a t e c o n c e n t r a t i o n . C h l o r o p h y l l s b and c were measured s t a r t i n g i n 1969. The d a t a r e c o r d c o n s i s t s of a p p r o x i m a t e l y 200 depth p r o f i l e s and an a d d i t i o n a l 600 s u r f a c e o b s e r v a t i o n s , a l t h o u g h not a l l v a r i a b l e s were n e c e s s a r i l y measured on a l l days or a t a l l d e p t h s . There a r e c e r t a i n p e c u l i a r i t i e s t o the s a m p l i n g regime which 69 can be seen i n the s c a t t e r p l o t of s u r f a c e c h l o r o p h y l l c o n c e n t r a t i o n s g i v e n i n F i g 13. From 1964 t o 1968, o b s e r v a t i o n s were made from o n l y one of the two w e a t h e r s h i p s , r e s u l t i n g i n a s e r i e s of 6 week gaps i n the time s e r i e s . W h i l e on s t a t i o n , s u r f a c e samples were c o l l e c t e d e v e r y second day, more or l e s s r e g u l a r l y . A f t e r 1968, o b s e r v a t i o n s were made from both w e a t h e r s h i p s , but the sampl i n g f r e q u e n c y dropped t o one t o s i x o b s e r v a t i o n s per 6 week c r u i s e . U s u a l l y one or two p r o f i l e s were taken on each c r u i s e . 3.2.2 C h l o r o p h y l l Data. The c o n c e n t r a t i o n of c h l o r o p h y l l a has been r o u t i n e l y measured from the w e a t h e r s h i p a t O.S.P. and i s the p r i n c i p a l i n d i c a t o r of p h y t o p l a n k t o n s t a n d i n g s t o c k . The s u r f a c e v a l u e s p l o t t e d i n F i g 13 a r e g e n e r a l l y r e p r e s e n t a t i v e of the mixed l a y e r d e s c r i b e d i n Chapter 1. I t i s c l e a r from F i g 13 t h a t c h l o r o p h y l l a a t O.S.P. tends t o be u n i f o r m l y low: l e s s than 1 ug C h i a . l " 1 i n a l l but a few o b s e r v a t i o n s . T h i s f e a t u r e of the S u b a r c t i c P a c i f i c has been remarked upon by many r e s e a r c h e r s (Semina,1958; H e i n r i c h , 1 9 6 2 ; Parsons,1965) and i s i n sharp c o n t r a s t t o the c l a s s i c s p r i n g bloom seen i n o t h e r temperate p l a n k t o n i c ecosystems such as the S t r a i t of G e o r g i a ( P a r s o n s , 1 9 6 5 ) , the No r t h Sea ( S t e e l e , 1 9 7 4 ) and the N o r t h A t l a n t i c ( C ushing,1959). E x c e p t i o n s t o the r u l e of c o n s t a n c y can be seen i n F i g 13: s c a t t e r e d h i g h v a l u e s , from 1 t o 4 ug C h i a . l " 1 were obser v e d i n 1964,1965,1969,1972 and .1975. In 1964,1965 and 1969, thes e h i g h v a l u e s o c c u r i n s m a l l groups, a p a t t e r n c o n s i s t e n t w i t h a s h o r t - l i v e d bloom (of low i n t e n s i t y compared w i t h c o a s t a l blooms (eg P a r s o n s , 1 9 6 5 ) ) , or w i t h the a d v e c t i o n of a 'patch' p a s t the 2 70 CO X C D CE I E C J 0 2 > • • 0 2 0 2 0 1961 1 9 6 2 1 9 6 3 1 9 6 4 * • + t * * 1 9 6 5 1 9 6 6 1 9 6 7 . 1 9 6 8 • * -* • * * • • • • + * * * . v. *** • 1 9 6 9 1970 1971 ^ d s 1 9 7 2 * . . • •** • ^ ** • A A A « * » * * * * * * * * »• » • » » » » ** 1 9 7 3 1 9 7 4 . 1 9 7 5 1 9 7 6 Figure 13. Surface observations of chlorophyll a from the weatherships at O.S.P. (Triangles denote high values, given to nearest integer by d i g i t above.) 71 s t a t i o n . The h i g h o b s e r v a t i o n s i n 1975 are i s o l a t e d but the s a m p l i n g f r e q u e n c y i s low t h e r e , so t h a t o n l y one o b s e r v a t i o n might be e x p e c t e d t o c o i n c i d e w i t h a s h o r t - l i v e d bloom. The i m p l i c a t i o n s of t h e s e h i g h o b s e r v a t i o n s f o r t h e o r i e s of p h y t o p l a n k t o n r e g u l a t i o n i n the S u b a r c t i c P a c i f i c w i l l be d i s c u s s e d l a t e r . Any s e a s o n a l or a n n u a l p a t t e r n i n t h e i r o c c u r r e n c e would be of i n t e r e s t , but i t i s c l e a r from F i g 13 t h a t the s a m p l i n g i n t e n s i t y i s not s u f f i c i e n t t o r e v e a l such a p a t t e r n . H i g h v a l u e s o c c u r i n June,1964, J u l y and August,1965, March and September,1969, A p r i l and December,1972 and J a n u a r y , June, J u l y and October,1975, r e v e a l i n g no c o n s i s t e n t s e a s o n a l p a t t e r n . Gaps i n the time s e r i e s are such t h a t h i g h v a l u e s c o u l d have e a s i l y been missed i n o t h e r months or y e a r s . Any c o n s i s t e n t l o w - a m p l i t u d e s e a s o n a l p a t t e r n i n the background C h i a v a l u e s c o u l d a l s o be of i n t e r e s t . Annual and s e a s o n a l c o n t r i b u t i o n s t o v a r i a b i l i t y i n a time s e r i e s can be d i s t i n g u i s h e d by p e r f o r m i n g a r u n n i n g a n n u a l average and l o o k i n g f o r s e a s o n a l v a r i a t i o n i n the r e s i d u a l s ( K e n d a l l , 1 9 7 3 ) . ( The word 'annual' i s used h e r e , f o l l o w i n g K e n d a l l (1973), t o denote year t o year v a r i a t i o n s . These may be r e f e r r e d t o as i n t e r - a nnual v a r i a t i o n s by some a u t h o r s . The s e a s o n a l p a t t e r n s d i s c u s s e d here a r e s h o r t e r term f l u c t u a t i o n s of 12 month p e r i o d . ) Two problems p r e v e n t such a s t r a i g h t - f o r w a r d approach t o these d a t a . One i s t h a t the o c c a s i o n a l h i g h v a l u e s c o n t r i b u t e o v e r w h e l m i n g l y t o the v a r i a n c e and w i l l dominate any r e s u l t i n g p a t t e r n . T h i s has been overcome here by l o o k i n g f o r a s e a s o n a l p a t t e r n i n the s i x week c r u i s e medians, p l o t t e d i n F i g 14a,c. The second problem i s t h a t the presence of i r r e g u l a r gaps i n the 1964 1 1965 : 1966 ! 1967 ' 1968 1.2 0 "3? E L CD G O CD C ? J : F 1 M 1 fl 1 M 1 J ' J 1 fl 1 5 1 0 1 N 1 D -3 Figure 14. Surface c h l o r o p h y l l a (mg.m ): (a) Cruise medians and annual smooth for 1964-68. (b) Seasonal f i t plus r e s i d u a l s f o r 1964-68. ho ~ 1 J .' F 1 M ' f l 1 M ' J ' J ' H ' S ' 0 1 N 1 D 1 _3 Figure 14. Surface chlorophyll a (mg.m ): (c) Cruise medians and annual smooth for 1969-76. (d) Seasonal f i t plus residuals for 1969-76. 74 t i m e s e r i e s w i l l l e a d t o c o n f u s i o n of a n n u a l and s e a s o n a l c o n t r i b u t i o n s t o v a r i a t i o n . T h i s i s a s i m i l a r p r o b l e m t o t h e two-way a n a l y s i s of v a r i a n c e w i t h empty b l o c k s . A s o l u t i o n t o th e l a t t e r p r o b l e m , s u g g e s t e d by M o s t e l l e r and T u k e y ( 1 9 7 7 ) , i s t o i t e r a t e . The p r o c e d u r e t h e y d e s c r i b e has been a d a p t e d t o t h e t i m e s e r i e s p r o b l e m as f o l l o w s . An a n n u a l s m o o t h i n g (moving window o f w i d t h one y e a r ) i s a p p l i e d t o t h e t i m e s e r i e s and t h e f i r s t t h r e e s e a s o n a l h a r m o n i c s ( p e r i o d one y e a r , o n e - h a l f y e a r and o n e - t h i r d y e a r ) a r e l e a s t - s q u a r e s f i t t e d t o t h e r e s i d u a l s . The r e s i d u a l s f r o m t h e s e a s o n a l f i t a r e t h e n added back t o t h e a n n u a l smooth and r e - s m o o t h e d ; t h e r e s u l t i n g r e s i d u a l s a r e added back t o t h e s e a s o n a l f i t and t h e s e a s o n a l h a r m o n i c s a r e r e f i t t e d . T h i s p r o c e d u r e i s r e p e a t e d u n t i l no f u r t h e r change o c c u r s i n a n n u a l smooth o r s e a s o n a l h a r m o n i c s . (The same f i n a l r e s u l t s a r e of c o u r s e o b t a i n e d i f t h e s e a s o n a l f i t i s a p p l i e d f i r s t ) . The p r o c e d u r e i s e q u i v a l e n t t o a s s u m i n g a model f o r o b s e r v a t i o n s o f th e f o r m : Y ( t ) = Y A ( t ) + Y s ( t ) + e ( t ) where Y ^ ( t ) r e p r e s e n t s an a n n u a l e f f e c t , Y s ( t ) a s e a s o n a l e f f e c t ( r e p e a t e d e a c h y e a r ) and e ( t ) a r e s i d u a l e f f e c t , and c h o o s i n g a n n u a l and s e a s o n a l components so a s t o m i n i m i z e t h e sum of s q u a r e s o f u n e x p l a i n e d r e s i d u a l s . In t h e c a s e of C h i a, t h e f i n a l a n n u a l and s e a s o n a l components a c c o u n t e d f o r 19% a n d 12% r e s p e c t i v e l y o f t h e o r i g i n a l SSQ a b o u t t h e mean. B e c a u s e of t h e gaps i n t h e t i m e s e r i e s , no 75 attempt has been made t o t e s t f o r s i g n i f i c a n c e of the s e components. The e s t i m a t e d s t a n d a r d d e v i a t i o n of r e s i d u a l s i s 0.12 ug C h i a . l " 1 compared w i t h a s e a s o n a l c y c l e a m p l i t u d e of 0.2 ug C h i a . l " 1 . T h i s c y c l e a r i s e s almost e n t i r e l y from the second h a l f of the time s e r i e s . When the proce d u r e i s a p p l i e d t o the p e r i o d s 1964-1968 and 1969-1976 s e p a r a t e l y , the s e a s o n a l c y c l e f o r the f i r s t p e r i o d has much lower a m p l i t u d e (0.04 ug C h i a . l " 1 ) and e x p l a i n s o n l y 1.5% of the SSQ about the mean ( F i g 1 4 ) . C o n c e n t r a t i o n s of C h i b and C h i c, measured a f t e r 1968, p o t e n t i a l l y c o n t a i n i n f o r m a t i o n about the taxonomic c o m p o s i t i o n of the p h y t o p l a n k t o n . S c a t t e r p l o t s of the r a t i o s C h i b/Chl a and C h i c / C h l a a r e g i v e n i n F i g 15. Seaso n a l and annual p a t t e r n s were sought among c r u i s e medians of t h e s e r a t i o s u s i n g the p r o c e d u r e a l r e a d y d e s c r i b e d f o r C h i a. The s e a s o n a l ' c y c l e ' found f o r the C h i b/Chl a r a t i o i s of v e r y s m a l l a m p l i t u d e , e x p l a i n i n g o n l y 4% of the SSQ about the mean. The annual smooth e x p l a i n e d 44% of the SSQ and v a r i e d from a low of 0.28 i n 1969 t o a h i g h of 0.83 i n 1975 ( F i g 1 6 ) . The s e a s o n a l c y c l e o b t a i n e d f o r C h i c / C h l a ( F i g 17) e x p l a i n e d 11% of the SSQ about the mean, w h i l e the annua l smooth e x p l a i n e d 30%. The average v a l u e of C h i c / C h l a was 1.6, compared w i t h 0.6 f o r C h i b/Chl a, both r a t i o s showing an i n c r e a s i n g t r e n d throughout the p e r i o d of o b s e r v a t i o n s . 3.2.3 Data. P r i m a r y p r o d u c t i v i t y has been r o u t i n e l y measured a t O.S.P. s i n c e 1964. I n c u b a t i o n s were u s u a l l y c o nducted from noon t o 1800 h ( l o c a l t i m e ) and average uptake r a t e s r e p o r t e d as jug C . l ^ . h r " 1 ) . Uptake r a t e s per u n i t C h i a have been c a l c u l a t e d 76 c r •IE C J CO 2 0 2 5 (a) • • * • r. * ' f • »•* # * * i *' .*V ... <«: ** t • .-• i * 0 1.969 1970 3 1971 1 9 7 2 3__ 1 9 7 3 1 9 7 4 1975 1 9 7 6 5 C E _ J • E C J \ C J _ l I E C J 0 5 0 • . * * V * * , * .»» • » • a) • • • * * 1 9 6 9 1970 1971 1 9 7 2 • * • • * • * - • * *• * t » * 1 9 7 3 1 9 7 4 1 9 7 5 1 9 7 6 F i g u r e 15. O b s e r v a t i o n s o f (a) C h i b / C h l a and (b) C h i c / C h l a f r o m t h e w e a t h e r s h i p s a t O.S.P. ( T r i a n g l e s as i n F i g 13.) Figure 16. Surface Chi b/Chl a: (a) Cruise medians and annual smooth for 1969-76. (b) Seasonal f i t plus residuals for 1969-76. Figure 17. Surface Chi c/Chl a: (a) Cruise medians and annual smooth for 1969-76. (b) Seasonal f i t plus residuals f o r 1969-76. 79 u s i n g the c o i n c i d e n t measurements of C h i a and w i l l be r e f e r r e d t o here by the symbol P. V a l u e s of t h i s r a t i o f o r a l l s u r f a c e samples, denoted by P(0) (ug C.ug C h i a _ 1 . h r _ 1 ) , are p l o t t e d i n F i g 18. I t can be seen t h e r e t h a t the s c a t t e r i n P(0) i n c r e a s e s markedly a f t e r 1968. A number of anomalous v a l u e s ( f o r example, 21 pg C.pg C h i a ^ . h r " 1 on 13th December, 1970, which saw a d a i l y t o t a l of 20 l y s o l a r r a d i a t i o n ) a l s o appear i n the second h a l f of the time s e r i e s . H i s t o g r a m s of the f r e q u e n c y d i s t r i b u t i o n of P ( 0 ) , s q u a r e - r o o t s c a l e d t o reduce skewness, f o r the p e r i o d s 1964-1968 and 1969-1976 are p l o t t e d i n F i g 19. W h i l e the h i g h v a l u e s a r e most i m p r e s s i v e i n the second h a l f of F i g 18, the h i s t o g r a m r e v e a l s a l a r g e number of v e r y low v a l u e s from 1969 on, and a tendency f o r the g a u s s i a n - s h a p e d d i s t r i b u t i o n f o r 1964-1968 t o become almost u n i f o r m i n 1969-1976. S e a s o n a l and a n n u a l p a t t e r n s based on c r u i s e medians of P(0) were e x t r a c t e d u s i n g the p r o c e d u r e d e s c r i b e d above f o r c h l o r o p h y l l . The s e a s o n a l p a t t e r n f o r 1964-1968 i s as e x p e c t e d w i t h low v a l u e s from December t o March and a broad peak from June t o August ( F i g 20a,b). T h i s c y c l e e x p l a i n s 47% of the SSQ about the mean, w i t h the a n n u a l smooth e x p l a i n i n g an a d d i t i o n a l 24%. The s e a s o n a l c y c l e o b t a i n e d f o r 1969-1976 ( F i g 20c,d) e x p l a i n s o n l y 20% of the SSQ about the mean, and the h i g h s t a n d a r d d e v i a t i o n of r e s i d u a l s (0.71) i s not s u r p r i s i n g i n view of the i n c r e a s e d s c a t t e r noted above. The m i d - w i n t e r peak i n the s e a s o n a l c y c l e o b t a i n e d f o r t h i s p e r i o d may w e l l be an a r t i f a c t . The s u r f a c e o b s e r v a t i o n s of 1 4 C uptake r a t e s from 1969 t o 1976 a r e not i n c l u d e d i n the r e m a i n i n g a n a l y s i s because of the d i s c r e p a n c i e s a l r e a d y n o t e d , and o t h e r s d i s c u s s e d l a t e r . 80 I E c r _ j I E C J C D 21 C J C D CD 0 5 0 0 0 * • • • * f * ; * A 1961 1 9 6 2 1 9 6 3 1 9 6 4 • • * + * * * ? * V * A V * * + * • • * • 1 9 6 5 1 9 6 6 " R R ?fi 7 ?1 1 9 6 7 1 9 6 8 fi ' T • * • * • * * * * * * » * . •** •* * » * A • • A A • .*• * • • + • • A ** • * * • • • • • *** » - * ^ • 1 9 6 9 1970 1971 1 9 7 2 7 RR * * * * • *** • * • U * * » * * * * A A • * • * * . » . 1 9 7 3 1 9 7 4 1 9 7 5 1 9 7 6 F i g u r e 18. Surface observations of p r o d u c t i v i t y per u n i t C h i a from the weatherships at O.S.P. ( T r i a n g l e s as i n F i g 13.) 81 Figure 19. Frequency histograms for P(0) (square-root scaled) : (a) 1964-68, (b) 1969-76. 0 . 2 > — CJ LU Z3 a L U c r ^ 0 . 1 L U CE LU CT > — C J L U C E _J L U cr. 0 0 0 . 2 a L U c r ^ 0 . H L U 0 0 (a) 0 . 2 i .0 2 . 0 4 . 0 (b) 0 . 2 1 .0 . 2 . 0 . 4 . 0 P ( 0 ) (MG C/MG CHL fl.HR) CO O Q O 1 9 6 4 1 1 9 6 5 ' 1 9 6 6 ' 3 9 6 7 R 1 9 6 8 (b) . CD JSZT ~ — CD CD Nv o D J 1 F 1 M ' fl ' M 1 J ' J 1 fl 1 S l-0 1 N 1 D A 0 Figure 20. . P(0) (mg C.mg Chi a~ 1.hr~ 1) (a) Cruise medians and annual smooth for 1964-68. (b) Seasonal f i t plus residuals for 1964-68.  84 As n o t e d i n Chapter 1, m a c r o - n u t r i e n t s a r e abundant a t O.S.P. and p r i m a r y p r o d u c t i o n i s a p p a r e n t l y l i g h t - l i m i t e d . T o t a l d a i l y s o l a r r a d i a t i o n i s measured on the w e a t h e r s h i p s and r e p o r t e d i n the Monthly R a d i a t i o n Summary. P h o t o s y n t h e t i c a l l y a c t i v e r a d i a t i o n at the s u r f a c e has been c a l c u l a t e d f o l l o w i n g Parsons and Anderson (1970) as the minimum of obser v e d t o t a l r a d i a t i o n and o n e - h a l f the t h e o r e t i c a l c l e a r sky r a d i a t i o n f o r the same day, and i s denoted here by I e . A p l o t of P(0) a g a i n s t I c ( F i g 21) su g g e s t s some r e l a t i o n s h i p between P(0) and I 0 d e s p i t e the l a r g e s c a t t e r . A l i n e a r r e g r e s s i o n of P(0) on I 0 g i v e s P(0) =• 0.81 + 0.003.1 0 , the s l o p e b e i n g s i g n i f i c a n t l y d i f f e r e n t from z e r o a t the 5% l e v e l , a l t h o u g h the r e g r e s s i o n e x p l a i n s o n l y 14.6% of the t o t a l SSQ about the mean. Both P(0) and I 0 t e n d t o be h i g h i n w i n t e r and low i n summer so t h a t i t i s not c l e a r how t h i s s t a t i s t i c a l r e l a t i o n s h i p s h o u l d be i n t e r p r e t e d . A t tempts t o f i n d a c o n s i s t e n t r e l a t i o n s h i p between P(0) and Ia on time s c a l e s d i s t i n c t from t h i s s e a s o n a l c o - v a r i a t i o n were not s u c c e s s f u l . R e g r e s s i o n s of P(0) on I 0 w i t h i n c r u i s e s y i e l d e d o n l y two out of 18 s l o p e s s i g n i f i c a n t l y d i f f e r e n t from z e r o a t the 10% l e v e l , w i t h 12 out of 18 s l o p e s b e i n g n e g a t i v e . S i m i l a r r e s u l t s were o b t a i n e d f o r r e g r e s s i o n s of P(0) on I 0 by month. T h i s f a i l u r e t o f i n d a c o n s i s t e n t s h o r t - t e r m e f f e c t of I G on P(0) may be p a r t l y due t o the l a r g e s c a t t e r i n o b s e r v a t i o n s , but i t can a l s o be ex p e c t e d on t h e o r e t i c a l grounds i n the absence of  86 any s c a t t e r . W h i l e p h y t o p l a n k t o n g r o w t h o v e r t h e mixed l a y e r a s a whole may be l i g h t - l i m i t e d , p h o t o - s a t u r a t i o n o r even p h o t o - i n h i b i t i o n c a n be e x p e c t e d a t t h e s u r f a c e . The i n f o r m a t i o n a v a i l a b l e i n d e p t h p r o f i l e s from O.S.P. can p o t e n t i a l l y c o n t r i b u t e much more t o an u n d e r s t a n d i n g of t h e l i g h t - d e p e n d e n c e of p h y t o p l a n k t o n p r o d u c t i o n t h e r e . The p r o f i l e s t a k e n a t O.S.P. c o n s i s t of sam p l e s a t between s i x and t w e l v e d e p t h s . The 1 4 C u p t a k e measurements were made a t t h e upper s i x d e p t h s , w h i c h , w i t h a few e x c e p t i o n s , b e l o n g e d t o one of 3 s t a n d a r d s e t s : (0,2,5,12,25,50m), (0,3,7,15,25,60m) and (0,6,10,25,33,75m). Uptake r a t e s were measured u s i n g s i m u l a t e d i n - s i t u i n c u b a t i o n s on deck under a s t a n d a r d s e t of mesh l i g h t s c r e e n s . The S e c c h i d e p t h (and p r e s u m a b l y t h e e x t i n c t i o n c o e f f i c e n t ) a t O.S.P. v a r i e s a s much as t w o - f o l d o r more from summer t o w i n t e r ( P a r s o n s , 1 9 6 5 ) . C o n s e q u e n t l y , t h e l i g h t i n t e n s i t y a t t h e i n c u b a t e d sample ( u n d e r a s t a n d a r d l i g h t s c r e e n ) need n o t c o i n c i d e w i t h t h e l i g h t i n t e n s i t y a t t h e ( s t a n d a r d ) d e p t h f r o m w h i c h t h e sample was t a k e n . In f a c t , t h e i r r a t i o c a n v a r y w i d e l y d e p e n d i n g on t h e e x t i n c t i o n c o e f f i c i e n t a t t h e t i m e . T h i s means t h a t d i r e c t d e p t h - i n t e g r a t i o n of p r o f i l e s t o o b t a i n t o t a l p r o d u c t i o n under 1 m2 i s n o t v a l i d . The p r o f i l e s c a n s t i l l be u s e d t o examine t h e r e l a t i o n s h i p between p h o t o s y n t h e s i s and l i g h t i n t e n s i t y as f o l l o w s . The t r a n s m i s s i o n c o e f f i c i e n t s f o r t h e s t a n d a r d l i g h t s c r e e n s , and t h e s u r f a c e l i g h t i n t e n s i t y , a r e known. The p r o f i l e c a n t h e n be r e g a r d e d a s a s i n g l e P vs I e x p e r i m e n t , d i s r e g a r d i n g t h e s t a t e d d e p t h s e x c e p t as an i n d e x t o t h e c o r r e s p o n d i n g s c r e e n . T h i s r e q u i r e s t h e s t r o n g a s s u m p t i o n t h a t t h e p h y t o p l a n k t o n p o p u l a t i o n 87 i s homogeneous w i t h d e p t h , or a t l e a s t t h a t the r e l a t i o n s h i p between p r o d u c t i o n per u n i t C h i a and l i g h t i n t e n s i t y i s independent of dep t h . T h i s seems p l a u s i b l e i n l a t e - f a l l , w i n t e r and s p r i n g when the mixed l a y e r extends below the deepest 1 4 C sample. However, i n the l a t e summer and e a r l y f a l l , when the s e a s o n a l t h e r m o c l i n e i s s h a l l o w e s t , a d i s t i n c t shade-adapted p o p u l a t i o n can be ex p e c t e d below the t h e r m o c l i n e , and r e s u l t s o b t a i n e d under the assumption of v e r t i c a l homogeneity must be regarde d w i t h s u s p i c i o n . Each p r o f i l e from O.S.P. has been used here t o e s t i m a t e parameters i n a f o r m u l a f o r the l i g h t - d e p e n d e n c e of p h o t o s y n t h e s i s . There a r e a number of form u l a e of t h i s type i n the l i t e r a t u r e ( f o r a r e v i e w , see J a s s b y and P i a t t , 1 9 7 6 ) , r a n g i n g from the s i m p l e t o the complex (eg Banni s t e r , 1979.) . A s i m p l e v e r s i o n of t h a t proposed by S t e e l e (1962): P = oc.i . e x p ( - I / I M A X ) 3.1 has been used h e r e . T h i s e q u a t i o n has been s a t i s f a c t o r i l y f i t t e d t o o t h e r s e t s of f i e l d o b s e r v a t i o n s i n v o l v i n g p h o t o - i n h i b i t i o n (Hameedi,1977; Chan,pers comm). Other fo r m u l a e have been found t o g i v e a s u p e r i o r d e s c r i p t i o n a t s u b - o p t i m a l l i g h t i n t e n s i t i e s ( J a s s b y and P i a t t , 1 9 7 6 ) , but t h e s e r e q u i r e t h r e e parameters t o account f o r p h o t o - i n h i b i t i o n , and f i t t i n g more than two parameters t o the s i x o b s e r v a t i o n s i n each p r o f i l e seems unwise, to say the l e a s t . The t h e o r y proposed by S t e e l e ( 1 9 6 2 ) f o r the s e a s o n a l v a r i a t i o n of parameters i n e q u a t i o n 3.1 and t h e i r i n t e r p r e t a t i o n i n terms of l i g h t a d a p t a t i o n and cha n g i n g 88 c a r b o n : c h l o r o p h y l l r a t i o s w i l l a l s o prove u s e f u l . E q u a t i o n 3.1 i s used here t o p r e d i c t d a i l y uptake r a t e s on the b a s i s of d a i l y r a d i a t i o n t o t a l s . The e q u a t i o n was f i r s t p r oposed f o r i n s t a n t a n e o u s r a t e s , but i t was shown t o be a p p l i c a b l e t o d a i l y t o t a l s on the b a s i s of an i d e a l i s e d d i u r n a l v a r i a t i o n i n l i g h t i n t e n s i t y ( S t e e l e , 1 9 6 2 ) . As noted e a r l i e r , i n c u b a t i o n s a t O.S.P. were u s u a l l y made from 1200 t o 1800 hr l o c a l time and average 1*C uptake r a t e s per hour over the •in c u b a t i o n p e r i o d were r e p o r t e d . These have been m u l t i p l i e d by 12 t o g i v e a crude e s t i m a t e of d a i l y uptake. Where the i n c u b a t i o n p e r i o d i n c l u d e s dusk, t h i s p rocedure i s most r e a s o n a b l e . Where i t does n o t , d a i l y uptake i s u n d e r e s t i m a t e d by a s m a l l undetermined amount. Other s o u r c e s of e r r o r , such as d i u r n a l v a r i a t i o n i n p h o t o s y n t h e t i c a c t i v i t y and p a r t i c u l a r l y a f t e r n o o n d e p r e s s i o n ( Y e n t s c h and R y t h e r , 1 9 5 7 ; M c A l l i s t e r , 1 9 6 3 ) may be more i m p o r t a n t . To f i t e q u a t i o n 3.1 t o the p r o f i l e s , i t was r e w r i t t e n as Pj = 0C.I o . t r , . e x p ( - I D . t r ; / I M A X ) where P; i s the uptake r a t e per u n i t C h i a a t depth i , I 0 the s u r f a c e l i g h t i n t e n s i t y and t r ; the t r a n s m i s s i o n c o e f f i c i e n t of the f i l t e r used f o r depth i . The q u a n t i t i e s A = <x.I0 and B = I0/IM>»X were e s t i m a t e d f o r each p r o f i l e and t h e i r dependence on I 0 c o n s i d e r e d l a t e r . I f e r r o r s i n P; were l o g - n o r m a l l y d i s t r i b u t e d , the s t a t i s t i c a l model c o u l d be w r i t t e n as : l n ( P ; ) = l n ( A ) + l n ( t r j ) - B . t r ; + e; (e ; i . i . d . N(0,a 2) ) 89 r e s u l t i n g i n a c o n v e n i e n t l i n e a r r e g r e s s i o n f o r l n ( A ) and B. T h i s was t r i e d but i t was f o u n d t h a t deep (low l i g h t i n t e n s i t y ) o b s e r v a t i o n s t h e n c o n t r i b u t e d d i s p r o p o r t i o n a t e l y t o t h e SSQ o f r e s i d u a l s . I n s p e c t i o n o f c u r v e s f i t t e d t h r o u g h i n d i v i d u a l p r o f i l e s r e v e a l e d a t e n d e n c y t o i g n o r e s h a l l o w o b s e r v a t i o n s e n t i r e l y . The o b s e r v a t i o n e r r o r does not d e c r e a s e q u i c k l y enough w i t h t h e mean t o be l o g - n o r m a l . When a n o n - l i n e a r l e a s t - s q u a r e s f i t t o t h e raw d a t a was t r i e d , i t was f o u n d t h a t s h a l l o w ( h i g h l i g h t i n t e n s i t y ) o b s e r v a t i o n s made a d i s p r o p o r t i o n a t e l y l a r g e c o n t r i b u t i o n t o t h e SSQ o f r e s i d u a l s , and, e s p e c i a l l y where s u r f a c e i n h i b i t i o n was e x a g g e r a t e d , deep o b s e r v a t i o n s were i g n o r e d . A t h i r d a t t e m p t was made, a g a i n u s i n g t h e n o n - l i n e a r l e a s t - s q u a r e s a p p r o a c h , b u t f i r s t s q u a r e - r o o t s c a l i n g t h e o r i g i n a l P; . T h i s s c a l i n g i s u s e d t o n o r m a l i z e o b s e r v a t i o n s from a P o i s s o n d i s t r i b u t i o n ( B a r n e s , 1 9 5 2 ) , but was u s e d h e r e as an e m p i r i c a l measure w h i c h r e s u l t e d i n e q u a l w e i g h t b e i n g a t t a c h e d t o s h a l l o w and deep o b s e r v a t i o n s ( c f F i g 1 9 a ) . As a c h e c k on p a r a m e t e r e s t i m a t e s o b t a i n e d u s i n g S t e e l e ' s e q u a t i o n ( 1 ) , an ad-hoc p r o c e d u r e was a l s o u s e d . A s t r a i g h t l i n e was f i t t e d t h r o u g h t h e t h r e e d e e p e s t o b s e r v a t i o n s and t h e o r i g i n t o e s t i m a t e A. O b s e r v a t i o n s a t s u c c e s s i v e l y s h a l l o w e r d e p t h s were t h e n i n c l u d e d , b u t u n l e s s t h i s r e s u l t e d i n a h i g h e r e s t i m a t e o f A, t h e o r i g i n a l e s t i m a t e was r e t a i n e d . The h i g h e s t o b s e r v e d P ; , d e n o t e d by P M A X , was a l s o n o t e d f o r e a c h p r o f i l e . The p a r a m e t e r oc r e p r e s e n t s a p h o t o s y n t h e t i c e f f i c i e n c y , h a v i n g u n i t s o f mg C.mg C h i a _ 1 . l y " 1 , and has been measured on many o c c a s i o n s . E a r l y v a l u e s a r e g i v e n i n terms of l u x - h r r a t h e r 90 than l y and c o n v e r s i o n can be p r o b l e m a t i c a l . S t e e l e ( 1 9 6 2 ) proposed a u n i v e r s a l v a l u e of 4.10" 4 mg C.mg C h i a ^ . l u x - h r " 1 or .48 mg C.mg C h i a ^ . l y 1 . Steeman N i e l s e n ( 1 9 7 8 ) suggests t h a t a v a l u e of 0.5 mg C.mg C h l a ^ . l y 1 may be assumed f o r most p h y t o p l a n k t o n . On the o t h e r hand, a t h e o r e t i c a l maximum of 2 mg C.mg C h i a ^ . l y 1 i s proposed by B a n n i s t e r ( 1 9 7 4 ) , and h i g h v a l u e s , g r e a t e r than lmg C.mg C h i a ^ . l y 1 , have been r e p o r t e d f o r l a b o r a t o r y p o p u l a t i o n s (Chan,1978), and i n the f i e l d (Hameedi,1977) . A p l o t of A ( e s t i m a t e d u s i n g the ad hoc p r o c e d u r e ) v e r s u s I f o r the p e r i o d 1964-1968 i s g i v e n i n F i g 22. The l i n e drawn i s the b e s t f i t t o a s t r a i g h t l i n e t h r o u g h the o r i g i n ; i t has a s l o p e c o r r e s p o n d i n g t o ot = 0.25 mg C.mg C h i a " 1 . l y 1 . The e s t i m a t e s of A o b t a i n e d u s i n g e q u a t i o n 3.1 y i e l d e d a s i m i l a r p l o t , w i t h od=0.34 mg C.mg C h i a'1.ly'1. Because of the c o n s i d e r a b l e s c a t t e r , the 95% c o n f i d e n c e l i m i t s f o r each v a l u e i n c l u d e s the o t h e r . An e q u i v a l e n t p l o t f o r the p e r i o d 1969-1976 showed no A r e l a t i o n s h i p a t a l l between A and I„ . The reason f o r t h i s q u i c k l y became apparent on i n s p e c t i o n of i n d i v i d u a l p r o f i l e s . In F i g 23, 10 randomly s e l e c t e d p r o f i l e s of P; v e r s u s depth a r e g i v e n f o r t h e p e r i o d s 1964-1968 and 1969-1976. Stephens(1977) has e x p r e s s e d r e s e r v a t i o n s c o n c e r n i n g some of the 1 4 C o b s e r v a t i o n s a f t e r 1969. In view of the i n c r e a s e d s c a t t e r and a l t e r e d f r e q u e n c y d i s t r i b u t i o n of s u r f a c e v a l u e s r e p o r t e d e a r l i e r , and the i n c o h e r e n t p a t t e r n of P w i t h depth as shown i n F i g 23, a l l 1 4 C o b s e r v a t i o n s a f t e r 1969 have been i g n o r e d and the r e m a i n i n g d i s c u s s i o n a p p l i e s o n l y t o the p e r i o d 1964-68. T h i s 91 F i g u r e 22. S c a t t e r p l o t of A v s I f o r 1964-68. ( L i n e drawn i s l e a s t - s q u a r e s f i t t h r o u g h o r i g i n . ) 92 ioo i i I r I (b) Figure 23. Depth profiles of P for (a) 1964-68 and (b) 1969-76. (Straight-line segments connect observations.) 93 reduces the number of p r o f i l e s from 200 t o 45 and s e v e r e l y r e s t r i c t s the u s e f u l n e s s of the d a t a s e t , p a r t i c u l a r l y f o r examining s e a s o n a l p a t t e r n s and a n n u a l a n o m a l i e s . Rather than p l o t % vs I 0 and e s t i m a t e a s i n g l e mean v a l u e of cx , an e s t i m a t e <X = A / I 0 can be c a l c u l a t e d f o r each p r o f i l e . A A p l o t of monthly means of oc ( F i g 24) suggests a s t r o n g s e a s o n a l p a t t e r n i n oc w i t h a maximum i n w i n t e r and minimum i n summer. In l a b o r a t o r y e x p e r i m e n t s , cx has been found t o v a r y w i t h p h y t o p l a n k t o n c o m p o s i t i o n and w i t h e n v i r o n m e n t a l c o n d i t i o n s such as t e m p e r a t u r e , n u t r i e n t s and l i g h t q u a l i t y ( P a rsons e_t a l , 1977). The low v a l u e s i n August and September must be viewed w i t h some s u s p i c i o n as the t h e r m o c l i n e i s s h a l l o w a t t h i s time and the assumption of v e r t i c a l homogeneity of the p h y t o p l a n k t o n may be v i o l a t e d . Even d i s c o u n t i n g these v e r y low v a l u e s , the r e m a i n i n g monthly means ar e l e s s than or e q u a l t o Steeman N i e l s e n ' s v a l u e of 0.5 mg C.mg C h i a ~ x . l y _ 1 which i s i n t u r n low compared w i t h o t h e r r e c e n t o b s e r v a t i o n s mentioned above. The i n t e r p r e t a t i o n of e s t i m a t e s of B i s not as A s t r a i g h t f o r w a r d . I f I M A x was c o n s t a n t , a p l o t of B vs I s h o u l d show a l i n e a r r e l a t i o n . However, Stee'le(1962) quotes a number of s t u d i e s i n which I M A x was found t o c o - v a r y w i t h I c so t h a t MAX l a y between 2 and 3. S t e e l e and B a i r d ( 1 9 6 1 ) found lower v a l u e s of t h i s r a t i o , between 1 and 2, i n the N o r t h Sea i n summer. In a t h e o r e t i c a l d i s c u s s i o n , S t e e l e ( 1 9 6 2 ) a t t r i b u t e d t h i s c o - v a r i a t i o n t o l i g h t a d a p t a t i o n by p h y t o p l a n k t o n t o s e a s o n a l l i g h t i n t e n s i t y t h r ough changes i n the c a r b o n : c h l o r o p h y l l r a t i o , and used a f i x e d v a l u e f o r I„/I M Ax of 2. F i g u r e 24. M o n t h l y a v e r a g e s o f OL. ( L i n e f i t t e d b y eye.) 95 Estimates of B are p l o t t e d a g a i n s t I 0 i n F i g 25. While there i s no suggestion of a l i n e a r r e l a t i o n between B and I 0 , as would be expected i f I M A x was const a n t , the s c a t t e r i s ra t h e r l a r g e to d e s c r i b e B as constant. The average value of B, 1.2, l i e s i n the range suggested by S t e e l e and Baird(1961) and i s lower than values r e p o r t e d in other r e f e r e n c e s quoted i n St e e l e ( 1 9 6 2 ) . Theory suggests that the s c a t t e r i n F i g 25 might be e x p l a i n e d by a more c a r e f u l c o n s i d e r a t i o n of the time-course of l i g h t a d a p t a t i o n . I t i s u n l i k e l y that phytoplankton can maintain a constant r a t i o of I 0 / I M A X , s i n c e t h i s r e q u i r e s instantaneous a d a p t a t i o n to changing l i g h t i n t e n s i t y . C e l l s c o u l d adapt to seasonal l i g h t i n t e n s i t i e s (Steele,1962) through some long-term averaging response. A seasonal average l i g h t i n t e n s i t y , I s ( t ) , was c a l c u l a t e d using a running average with a window of width 20 days. Estimates of the seasonal a d a p t a t i o n parameter, B s, given A A by B s = I S / I ^ x = B . I S / I 0 , showed no l e s s - s c a t t e r than estimates of B i t s e l f . L i g h t a d a p t a t i o n through changes i n C:Chl a r a t i o s has been r e p o r t e d to occur on time s c a l e s of a few days (Steeman N i e l s e n and Park,1964). A simple model f o r l i g h t - a d a p t a t i o n on these time s c a l e s would allow the parameter I M A ) < to vary as a l i n e a r combination of r a d i a t i o n on the preceding days. Such a model was f i t t e d to the p r o f i l e r e s u l t s as a l i n e a r r e g r e s s i o n i n the form: 1/B(t) = W / I 0 ( t ) = b 0 + b 4 , I 0 ( t - l ) / I 0 (t) + bs . I 0 (t-2)/l 0 (t) + b 3 . I 0 (t-3)/l 0 (t) , • • • • • • • • • • • • • • LO • • • • • • • o -Q— Q—Q- • • mm • LLTLTJ U -B-0 P . R . R . (LY/DRY) 360 Figure 25. S c a t t e r p l o t o f B v s I Q f o r 1964-68. 97 but t h i s d i d not produce a s i g n i f i c a n t r e d u c t i o n i n the v a r i a n c e of 1/B. In F i g 26, a p l o t of P M A X v e r s u s I 0 i s g i v e n . Under i n s t a n t a n e o u s a d a p t a t i o n ( I 0 / l M A x c o n s t a n t ) , P̂AX s h o u l d i n c r e a s e l i n e a r l y w i t h I 0 . The l i n e i n F i g 26 r e p r e s e n t s the best f i t t o a s t r a i g h t l i n e t h r o u g h the o r i g i n and has a s l o p e of 0.01 ± 0.001, c o r r e s p o n d i n g t o an average v a l u e f o r I 0 / I M A x between 0.9 and 1.2, c o n s i s t e n t w i t h the average v a l u e o b t a i n e d by f i t t i n g e q u a t i o n 3.1 t o p r o f i l e s . I t i s i n t e r e s t i n g t h a t the p o i n t s i n F i g 26 a t low l i g h t i n t e n s i t i e s ( l e s s than 120 l y / d a y ) tend t o l i e above t h i s l i n e t h r ough the o r i g i n . T h i s i s c o n s i s t e n t w i t h the e x i s t e n c e of a lower l i m i t t o l i g h t a d a p t a t i o n ; t h a t i s , a minimum v a l u e of I M A X , and a c c o r d i n g t o S t e e l e ( 1 9 6 2 ) a minimum C:Chl a - r a t i o , which i s enc o u n t e r e d a t the s e l i g h t i n t e n s i t i e s . The r e s u l t s o b t a i n e d from a n a l y s i s of 1 4 C p r o f i l e s a t O.S.P. can t h e r e f o r e be summarised as f o l l o w s . P h o t o s y n t h e t i c e f f i c i e n c y oc i s low a t O.S.P. , w i t h a s e a s o n a l c y c l e r a n g i n g from about 0.45 mg C.mg C h i a ^ . l y " 1 from October t o A p r i l t o 0.3 mg C.mg C h i a ^ . l y " 1 or l e s s from May t o September. There i s r e a s o n a b l e e v i d e n c e of a d a p t a t i o n t o l i g h t i n t e n s i t y w i t h an i n c r e a s e i n I M A X and P M A X w i t h i n c r e a s i n g I„ , a l t h o u g h the d a t a a r e not s u f f i c i e n t t o d i s t i n g u i s h amongst ' i n s t a n t a n e o u s ' , s h o r t - term or l o n g - t e r m a d a p t a t i o n t o changes i n s o l a r i r r a d i a n c e . Some of the s c a t t e r i n F i g 25 can be e x p l a i n e d by a l l o w i n g A the r a t i o I M A x / I o t o v a r y s e a s o n a l l y . Monthly averages of B and A Bs a r e p l o t t e d i n F i g 27. Both show the same s e a s o n a l t r e n d . The h i g h v a l u e s i n F e b r u a r y t o May agree r o u g h l y w i t h S t e e l e ' s f i g u r e of 2.0. T h i s i s a p e r i o d of deep mixed l a y e r s and  99 1. 0. J \ A A A A A ? A? jjj A J F M A M J J A S O N D M O N T H gure 27a. Monthly averages of estimates of light adaptation parameter B. (Line f i t t e d by eye.) J F MA M J J A S ON D MONTH Figure 27b. Monthly averages of estimates of light adaptation parameter B g. (Line f i t t e d by eye.) 101 i n c r e a s i n g l i g h t i n t e n s i t y . Throughout the remainder of the y e a r , monthly averages of B a r e low and c o r r e s p o n d t o maximum growth a t or near the s u r f a c e . The v e r y low v a l u e s i n August and September may a g a i n be due t o the breakdown of the assumption of v e r t i c a l homogeneity. In view of the low number of p r o f i l e s c o n s i d e r e d and the u n e x p l a i n e d s c a t t e r i n e s t i m a t e s of OL and B, t h e s e s e a s o n a l c y c l e s i n oc and B must be r e g a r d e d as b e i n g suggested r a t h e r than c o n f i r m e d by the d a t a . The h i g h s c a t t e r , t o g e t h e r w i t h the s i x week gaps i n the s a m p l i n g regime d u r i n g 1964 t o 1968, has a l s o p r e v e n t e d a s e a r c h f o r annual v a r i a t i o n i n t h e s e p a r a m e t e r s . The l o s s of the 1 4 C d a t a from 1969-1976 has been a c o n s i d e r a b l e h a n d i c a p i n t h e s e r e s p e c t s . 3.2.4 N i t r a t e Data. N i t r a t e c o n c e n t r a t i o n s i n b o t h s u r f a c e and deep samples were r e c o r d e d a f t e r 1966. The f r e q u e n c y of o b s e r v a t i o n s i s low d u r i n g 1966-1968 and i n c r e a s e s s u b s t a n t i a l l y t h e r e a f t e r ( F i g 2 8 ) . S e a s o n a l and a n n u a l summaries were c a l c u l a t e d as f o r o t h e r v a r i a b l e s and are g i v e n i n F i g 29. The annual smooth and s e a s o n a l c y c l e remove 42% and 37% r e s p e c t i v e l y of the t o t a l SSQ about the mean. The s t a n d a r d d e v i a t i o n of r e s i d u a l s i s 2.1 jug a t . l " 1 compared w i t h a s e a s o n a l c y c l e r a n g i n g from 7 t o 15 ug a t . l " 1 . In a d d i t i o n , average n i t r a t e v a l u e s i n the depth ranges 0- 20,20-40,40-80, 80-130 and 130-200 m were c a l c u l a t e d f o r each p r o f i l e and the annual smooth and average s e a s o n a l c y c l e e x t r a c t e d ( u s i n g i n d i v i d u a l p r o f i l e s r a t h e r than c r u i s e medians) f o r each depth range. The r e s u l t i n g s e a s o n a l c y c l e s and the 102 20 oo X X c r CD 21 LU h— c r cr: 0 20 0 20 0 • . . , • *• * 1965 1 9 6 6 1 9 6 7 ?? 1 9 6 8 • * * *• * • . * *• '• ' ' ** . * * . • • * * 1 9 6 9 1970 1971 1 9 7 2 r * i f * * • ** • • « * i ** # * *» •» '•• •.• . v." **• 1 9 7 3 1 9 7 4 1 9 7 5 1 9 7 6 F i g u r e 28. S u r f a c e o b s e r v a t i o n s o f n i t r a t e c o n c e n t r a t i o n f r o m t h e w e a t h e r s h i p s a t O.S.P.  104 a n n u a l smooths are p l o t t e d i n F i g 30. The s e a s o n a l c y c l e s f o r 0- 20m and 20-40m are both s i m i l a r t o t h a t o b t a i n e d f o r s u r f a c e samples and a l l can be e x p l a i n e d as a r e s u l t of uptake by p h y t o p l a n k t o n d u r i n g s p r i n g and summer, w i t h a minimum o c c u r r i n g about the time of maximum s t a b i l i s a t i o n of the water column. The f i r s t change i n s e a s o n a l c y c l e w i t h depth comes a t 40-80m and i s q u i t e i n t e r e s t i n g . A s m a l l d e c l i n e i n n i t r a t e c o n c e n t r a t i o n o c c u r s i n s p r i n g and e a r l y summer, but as the t h e r m o c l i n e s h a l l o w s t o above 40m, an i n c r e a s e i n n i t r a t e c o n c e n t r a t i o n o c c u r s . T h i s i s presumably due t o a c o m b i n a t i o n of reduced n i t r a t e uptake ( p h y t o p l a n k t o n a t the s e depths a r e no l o n g e r mixed upward i n t o r e g i o n s of h i g h l i g h t i n t e n s i t y ) , v e r t i c a l m i x i n g from below and p o s s i b l y r e c y c l i n g . The s e a s o n a l c y c l e s f o r the t h r e e upper depth ranges e x p l a i n 41%, 33% and 8% r e s p e c t i v e l y of the t o t a l SSQ about the means. The s e a s o n a l c y c l e a t 80-130m e x p l a i n s o n l y 6% of the SSQ; the p r i n c i p a l i n t e r p r e t a b l e f e a t u r e i n the c y c l e i s the minimum i n Fe b r u a r y and March, the time of maximum mixed l a y e r d e p t h , when n u t r i e n t d e p l e t e d water from the p r e v i o u s summer i s mixed t o thes e d e p t h s . The s e a s o n a l c y c l e at-130-200m e x p l a i n s o n l y 3% of the t o t a l SSQ w h i l e the an n u a l smooth e x p l a i n s 49%. The c o n s i d e r a b l e v a r i a t i o n i n n i t r a t e c o n c e n t r a t i o n a t t h e s e depths i s not of a s e a s o n a l n a t u r e and presumably r e f l e c t s the passage of l a r g e water masses w i t h d i f f e r e n t n u t r i e n t h i s t o r i e s p a s t the s t a t i o n . 3.2.5 N i t r a t e C o n c e n t r a t i o n and P r o d u c t i o n . The average s e a s o n a l c y c l e i n n i t r a t e c o n c e n t r a t i o n a t the s u r f a c e has a minimum of 7 jug a t . l - 1 but lower v a l u e s of 2.7, 1.9 105 Figure 30. Ni t r a t e concentrations (mg at.m J) from depth p r o f i l e s . Layer averages and annual smooths, seasonal f i t s plus residuals. (a) 0 - 20m. (b) 20 - 40m. (c) 40 - 80m. (d) 80 - 130m. (e) 130 - 200m. 1966 1967 1 1968 ' 1969 ' 1970 ' 1971 1 1972 1 1973 1 1974 ' 1975 ' 1976 Figure 30a.     I l l and 1.1 jug a t . l " 1 were obser v e d t o depths of 50 t o 75 m. V a l u e s r e p o r t e d from n u t r i e n t - d e p l e t e d waters a r e u s u a l l y l e s s than 1 y j g a t . l " 1 (McCarthy and Goldman,1978) and even the lower v a l u e s from O.S.P. would not n o r m a l l y be c o n s i d e r e d t o l i m i t p h y t o p l a n k t o n growth. The p o s s i b i l i t y remains t h a t p h y t o p l a n k t o n at O.S.P. have u n u s u a l l y h i g h h a l f - s a t u r a t i o n c o n s t a n t s f o r growth and t h i s would a f f e c t g r e a t l y the t r e a t m e n t of p h y t o p l a n k t o n - z o o p l a n k t o n i n t e r a c t i o n s . The time s e r i e s a v a i l a b l e here a re not s u i t e d t o t e s t i n g f o r n u t r i e n t dependence of p h o t o s y n t h e s i s . However, i n the absence of e x p e r i m e n t a l e v i d e n c e c o n c e r n i n g n u t r i e n t uptake k i n e t i c s at O.S.P. , a check was made f o r u n u s u a l l y low v a l u e s of P(0) a t these t i m e s -of low n u t r i e n t c o n c e n t r a t i o n . To a v o i d c o n f u s i o n w i t h e f f e c t of l i g h t i n t e n s i t y , v a l u e s of P(0) over the p e r i o d May t o October were p l o t t e d a g a i n s t n i t r a t e c o n c e n t r a t i o n . There i s no e v i d e n c e of any dec r e a s e i n P(0) a t low n i t r a t e c o n c e n t r a t i o n s ( F i g 3 1 ) . A g a i n , these r e s u l t s must be taken as s u g g e s t i v e o n l y . The s c a t t e r i n F i g 31 i s l a r g e enough t o mask any s l i g h t downward t r e n d a t low n i t r a t e c o n c e n t r a t i o n s so t h a t v a l u e s of a h a l f - s a t u r a t i o n c o n s t a n t between 0.1 and 2 jjg a t . l " 1 would p r o b a b l y not be d i s t i n g u i s h e d . A second r e s e r v a t i o n i s that,- i f c e l l s responded t o n u t r i e n t l i m i t a t i o n by i n c r e a s i n g t h e i r C:Chl a r a t i o ( S t e e l e , 1 9 6 2 ; A n t i a et a l ,1963), a de c r e a s e i n growth r a t e might not be r e f l e c t e d i n d e c r e a s e d P ( 0 ) . 3.3 A P h y t o p l a n k t o n Growth Model. 3.3.1 I n t r o d u c t i o n . The p r e c e d i n g d a t a a n a l y s i s , t o g e t h e r w i t h i n f o r m a t i o n from 0 5 10 15 NITRATE (MG RT/MXX3) Figure 31. P(0) vs low surface nitrate values, May to October, 1964-68. 113 the l i t e r a t u r e where n e c e s s a r y , w i l l now be used t o c o n s t r u c t a model of p h y t o p l a n k t o n growth a t O.S.P. which can be d r i v e n by time s e r i e s of o b s e r v a t i o n s of p h y s i c a l v a r i a b l e s . A s i m p l e study of the s e a s o n a l c y c l e of net p r i m a r y p r o d u c t i o n a t O.S.P. has appeared p r e v i o u s l y ( M c A l l i s t e r , 1 9 6 9 ) , but a more s o p h i s t i c a t e d approach, f o l l o w i n g the example of S t e e l e ( 1 9 7 4 ) and Jamart et a l ( 1 9 7 7 ) , i s i n t e n d e d h e r e . 3.3.2 P h y s i c a l S t r u c t u r e and D r i v i n g V a r i a b l e s . Based on the arguments p r e s e n t e d i n Chapter 1, p h y t o p l a n k t o n s t a t e v a r i a b l e s w i l l be t r e a t e d as f u n c t i o n s of time t and depth z but not h o r i z o n t a l p o s i t i o n ; t h a t i s , a water column 1 metre square w i l l be m o d e l l e d . The v e r t i c a l d i s t r i b u t i o n of p h y t o p l a n k t o n w i l l be de t e r m i n e d by the depth-dependent p r o c e s s e s of growth and g r a z i n g as w e l l as v e r t i c a l m i x i n g . The s e a s o n a l changes i n the p h y s i c a l s t r u c t u r e of the water column are reviewed here as background t o the m o d e l l i n g of v e r t i c a l m i x i n g of p h y t o p l a n k t o n . The mixed l a y e r i s deepest ( g r e a t e r than 100m) i n March, a t the end of the p e r i o d of net heat l o s s . The f o r m a t i o n and maintenance of the s e a s o n a l t h e r m o c l i n e i n subsequent months o c c u r s t h r o u g h the a l t e r n a t i o n of p e r i o d s of calm, sunny weather which produce s h a l l o w , t r a n s i e n t t h e r m o c l i n e s , w i t h p e r i o d s of h i g h wind a c t i o n d u r i n g which t h e s e t r a n s i e n t s t r u c t u r e s a r e mixed downward t o s t r e n g t h e n the s e a s o n a l t h e r m o c l i n e (Denman,1972). The l a t t e r s h a l l o w s t h r o u g h the p e r i o d of net heat g a i n t o rea c h a minimum of about 30m i n August t o September. I t i s then eroded d u r i n g the p e r i o d of net heat l o s s , b o t h by c o n v e c t i o n a l o v e r t u r n and by wind m i x i n g . 114 T h e r e a r e two c o n v e n t i o n a l ways t o r e p r e s e n t v e r t i c a l m i x i n g i n a p h y t o p l a n k t o n m o d e l . I t c a n be r e p r e s e n t e d as a d i f f u s i o n p r o c e s s , a s i n J a m a r t e t a_l ( 1 9 7 7 ) , w i t h d i f f u s i o n r a t e s d e p e n d e n t on d e p t h and t i m e , h i g h r a t e s b e i n g assumed above t h e t h e r m o c l i n e and l o w e r r a t e s w i t h i n and below i t . The r e s u l t i n g p a r t i a l d i f f e r e n t i a l e q u a t i o n f o r p h y t o p l a n k t o n c o n c e n t r a t i o n as a f u n c t i o n o f d e p t h and t i m e can be s o l v e d n u m e r i c a l l y . A s i m p l e r a p p r o a c h i s t o r e g a r d t h e m i x e d l a y e r above t h e t h e r m o c l i n e as a u n i f o r m o r w e l l - m i x e d r e g i o n , i n w h i c h p h y s i c a l m i x i n g overwhelms d e p t h - v a r y i n g b i o l o g i c a l p r o c e s s e s (eg S t e e l e , 1 9 7 4 ) . I t i s c l e a r from t h e above r e v i e w t h a t n e i t h e r a p p r o a c h i s e n t i r e l y r e a l i s t i c . The a l t e r n a t i o n o f c a l m and r o u g h p e r i o d s and t h e f o r m a t i o n and breakdown o f t r a n s i e n t t h e r m o c l i n e s mean t h a t b o t h a s e a s o n a l l y - v a r y i n g d i f f u s i o n r a t e and a u n i f o r m mixed l a y e r c a n o n l y be r e g a r d e d as a p p r o x i m a t i o n s . , w h i c h may be s a t i s f a c t o r y on l o n g e r t i m e s c a l e s , but a r e c e r t a i n l y i n a c c u r a t e on a day t o day b a s i s . I t a l s o seems u n l i k e l y t h a t t h e p r o c e s s of c o n v e c t i v e o v e r t u r n i n f a l l o r w i n t e r w i l l be a c c u r a t e l y r e p r e s e n t e d by a s i m p l e d i f f u s i o n p r o c e s s . As t h e r e i s l i t t l e r e a s o n t o b e l i e v e t h a t t h e d i f f u s i o n r e p r e s e n t a t i o n i s more a c c u r a t e , t h e s i m p l e r u n i f o r m m i x e d - l a y e r model i s u s e d h e r e . (The f a c t t h a t a l m o s t a l l C h i a p r o f i l e s f r o m O.S.P. show no s i g n i f i c a n t v e r t i c a l v a r i a t i o n w i t h i n t h e m i x e d l a y e r can be t a k e n a s e m p i r i c a l s u p p o r t f o r t h e s i m p l e r a p p r o a c h ) . The t o p 150m of t h e w ater column i s r e p r e s e n t e d i n t h e m o d e l . T h i s i n c l u d e s a m i x e d l a y e r o f d e p t h z M ( t ) w i t h i n w h i c h p h y t o p l a n k t o n c o n c e n t r a t i o n w i l l be assumed t o be i n d e p e n d e n t o f 115 d e p t h . S i n c e z M v a r i e s over t i m e , the c o n c e n t r a t i o n below the mixed l a y e r as a f u n c t i o n of depth must be m o d e l l e d . In the model, p h y t o p l a n k t o n c o n c e n t r a t i o n C ( z , t ) i s r e p r e s e n t e d by the v a l u e s C ( z i , t ) , i=1...30, i n 5m t h i c k l a y e r s . (The nominal depths z; a r e a r b i t r a r i l y t a ken a t the m i d p o i n t s of t h e i r r e s p e c t i v e l a y e r s . ) The mixed l a y e r depth i s g i v e n t o the n e a r e s t 5m and c o n c e n t r a t i o n s C ( z , , t ) f o r z-, l e s s than z M ( t ) are s e t e q u a l a t each d a i l y s t e p . When z M i n c r e a s e s from one day t o the n e x t , the l a y e r s above the new mixed l a y e r depth are averaged. In view of the slow changes i n s a l i n i t y and n u t r i e n t c o n c e n t r a t i o n s below the s e a s o n a l t h e r m o c l i n e , i t can be assumed t h a t m i x i n g r a t e s t h e r e a r e v e r y low. For s i m p l i c i t y , i t i s assumed here t h a t l o c a l depth-dependent p r o c e s s e s of growth and g r a z i n g dominate the e f f e c t s of m i x i n g below the t h e r m o c l i n e ; the. 5m l a y e r s t h e r e are l e f t unconnected. The p h y s i c a l d r i v i n g v a r i a b l e s a r e mixed l a y e r d e p t h , t e m p e r a t u r e as a f u n c t i o n of d e p t h , d a i l y s o l a r r a d i a t i o n and s e c c h i d e p t h . These a r e a v a i l a b l e as time s e r i e s (at v a r y i n g l e v e l s of completeness) from w e a t h e r s h i p o b s e r v a t i o n s . The mixed l a y e r d epth has been d e t e r m i n e d from STD p r o f i l e s , p r o v i d e d by Environment Canada on magnetic d a t a t a p e , i n the f o l l o w i n g manner. The p o t e n t i a l energy r e q u i r e d t o mix the water column down t o depth z, P E ( z ) , was c a l c u l a t e d a s : A v a l u e of PE(z) of 0.02 j . c n r 2 was chosen t o d e f i n e the mixed l a y e r d epth z M . T h i s i s comparable t o the m i x i n g energy p r o v i d e d PE(z) 116 by a t y p i c a l wind storm (Denman,1973) or the d e c r e a s e i n PE which would r e s u l t from s e v e r a l calm sunny days. By u s i n g t h i s c r i t e r i o n , t r a n s i e n t s h a l l o w t h e r m o c l i n e s c r e a t e d d u r i n g s h o r t p e r i o d s of calm were i g n o r e d and a smoother time s e r i e s f o r mixed l a y e r depth r e s u l t e d . I f p r o f i l e s had been a v a i l a b l e f o r each day, t h e s e t r a n s i e n t t h e r m o c l i n e s c o u l d have been t r e a t e d as s h a l l o w mixed l a y e r s but i t was n e c e s s a r y t o i n t e r p o l a t e over gaps of s e v e r a l days and o c c a s i o n a l l y s e v e r a l weeks. Under these c i r c u m s t a n c e s , the use of the above c r i t e r i o n was judged l e s s l i k e l y t o r e s u l t i n e r r o n e o u s i n t e r p o l a t i o n . I t i s c o n s i s t e n t w i t h the use of the u n i f o r m l y - m i x e d l a y e r approach as an a p p r o x i m a t i o n v a l i d on time s c a l e s of s e v e r a l days or l o n g e r . The temperature i n the mixed l a y e r was taken as the s u r f a c e temperature i n STD p r o f i l e s . To a v o i d s t o r i n g and reading-, l a r g e q u a n t i t i e s of temperature p r o f i l e d a t a on each s i m u l a t i o n r u n , the t e m p e r a t u r e below the mixed l a y e r was assumed t o drop o f f e x p o n e n t i a l l y towards 5 C w i t h a decay c o n s t a n t of 20m, chosen a f t e r i n s p e c t i o n of temperature p r o f i l e s i n one y e a r . T h i s r a t h e r crude r e p r e s e n t a t i o n was c o n s i d e r e d s u f f i c i e n t as almost a l l p h y t o p l a n k t o n p r o d u c t i o n a t O.S.P. o c c u r s w i t h i n the mixed l a y e r and the e f f e c t s of s m a l l e r r o r s i n temperature-dependent growth r a t e s below the mixed l a y e r a r e i n s i g n i f i c a n t . D a i l y s o l a r r a d i a t i o n was taken from the Monthly R a d i a t i o n Summaries and p h o t o s y n t h e t i c a l l y a c t i v e r a d i a t i o n , I„ , was c a l c u l a t e d as d e s c r i b e d i n the a n a l y s i s of 1 4 C p r o f i l e s . Gaps i n the time s e r i e s were f i l l e d by the 1964 t o 1976 means f o r t h a t day of the y e a r . The d r i v i n g v a r i a b l e w i t h the l e a s t complete time s e r i e s i s 117 s e c c h i d e p t h , SD, which was used t o determine the background e x t i n c t i o n c o e f f i c i e n t f o r l i g h t a c c o r d i n g t o the f o r m u l a : k 8 = 1.7/SD (Parsons et a l ,1977). The o b s e r v a t i o n s suggest a s e a s o n a l c y c l e w i t h a minimum i n summer and maximum i n w i n t e r , as w e l l as c o n s i d e r a b l e year t o year v a r i a t i o n . As o b s e r v a t i o n s a r e not a v a i l a b l e f o r a l l y e a r s , an average s e a s o n a l c y c l e was e x t r a c t e d and used f o r a l l y e a r s : SD = 14.4 + 2.8 SIN(2. 77 . ( t + 41)/365) 3.2 3.3.3 B i o l o g i c a l B a s i s f o r the Model. The p h y t o p l a n k t o n biomass i n the model i s d e f i n e d as p l a n t carbon (pg C I - 1 ) . S t a n d i n g s t o c k i s observed as C h i a and comparison of p r e d i c t i o n s w i t h o b s e r v a t i o n s w i l l r e q u i r e a knowledge of the c a r b o n r c h l o r o p h y l l r a t i o . T h i s i s n e c e s s a r y i n any case as our a n a l y s i s of 1 * C uptake r a t e s a l l o w s us t o p r e d i c t carbon produced per u n i t C h i a, whereas a dynamic model r e q u i r e s a growth r a t e i n d a y s " 1 . The s i g n i f i c a n c e of t h i s problem was emphasized by E p p l e y ( 1 9 7 2 ) , but i t remains u n s o l v e d due t o the l a c k of a r e l i a b l e method f o r measuring l i v i n g p l a n t carbon i n the f i e l d . In the absence of d i r e c t o b s e r v a t i o n s of C:Chl a r a t i o s , a t h e o r e t i c a l framework which w i l l a l l o w t h e i r p r e d i c t i o n i s n e c e s s a r y . The t h e o r y used here was p r e s e n t e d by S t e e l e ( 1 9 6 2 ) and depends on t h r e e assumptions : (1) At low l i g h t i n t e n s i t i e s , p h o t o s y n t h e s i s i s p r o p o r t i o n a l t o C h i a, w i t h some c o n s t a n t p h o t o s y n t h e t i c e f f i c i e n c y <x which i s not s u b j e c t t o change th r o u g h l i g h t a d a p t a t i o n . 118 (2) The maximum r a t e o f c a r b o n f i x a t i o n p e r c e l l i s g o v e r n e d by t h e d a r k r e a c t i o n and, more s p e c i f i c a l l y , by t h e enzyme c o n t e n t w h i c h i s assumed p r o p o r t i o n a l t o p l a n t c a r b o n . T h i s i m p l i e s t h e e x i s t e n c e of a maximum gr o w t h r a t e , u H ( h r - 1 ) w h i c h i s t e m p e r a t u r e - d e p e n d e n t but d o e s n o t change t h r o u g h l i g h t a d a p t a t i o n . The d a r k r e s p i r a t i o n r a t e i s assumed t o be a f i x e d p e r c e n t a g e of u w . ( N u t r i e n t s w i l l be assumed t o be n o n - l i m i t i n g a t O.S.P. ). (3) L i g h t a d a p t a t i o n o c c u r s t h r o u g h c h a n g e s i n t h e C : C h l a r a t i o . The t h e o r y was r e c o g n i z e d t o be a s i m p l i f i c a t i o n by S t e e l e ( 1 9 6 2 ) a t t h e t i m e o f i t s p r o p o s a l . More i s now known of t h e s t r u c t u r e and f u n c t i o n of t h e p h o t o s y n t h e t i c a p p a r a t u s i n r e l a t i o n t o l i g h t a d a p t a t i o n . Examples i n c l u d e t h e s p e c i a l r o l e o f t h e P-700 m o l e c u l e i n t h e l i g h t c o l l e c t i n g , a p p a r a t u s (Steeman N i e l s e n , 1 9 7 4 ) and t h e s i g n i f i c a n c e o f d a r k - r e a c t i o n enzymes s u c h as RuDP c a r b o x y l a s e ( B e a r d a l l and M o r r i s , 1 9 7 6 ) . However, t h e s e m o l e c u l a r s t u d i e s have y e t t o g i v e r i s e t o a c o h e r e n t t h e o r y of l i g h t a d a p t a t i o n a t t h e c e l l u l a r o r p o p u l a t i o n l e v e l . R e c e n t s u p p o r t f o r a s p e c t s of S t e e l e ' s t h e o r y (Myers and Graham,1971; B a n n i s t e r , 1 9 7 4 ; Steeman N i e l s e n , 1 9 7 4 ) has e n c o u r a g e d i t s use h e r e . The l i g h t - d e p e n d e n c e of g r o w t h i s b a s e d on t h e r e l a t i o n s h i p P = a.l . e x p ( - I / l M A X ) 3.1 u s e d i n t h e a n a l y s i s of 1 4 C p r o f i l e s . The d a i l y n e t g r o w t h r a t e o f p h y t o p l a n k t o n e x p o s e d t o d a i l y r a d i a t i o n I i s t h e n g i v e n by p = oi.l . e x p ( - I / I M A ) < )/V - d 3.3 119 where V i s the C:Chl a r a t i o and d i s the r e s p i r a t i o n r a t e ( d a y " 1 ) . The maximum d a i l y growth r a t e i s g i v e n by UMAX = a . I M A X / ( V . e ) - d A c c o r d i n g t o assumption ( 2 ) , a maximum (temperature-dependent) h o u r l y growth r a t e u H ( T ) can be d e f i n e d , so t h a t ( X . I M A X / ( V . e ) = u H ( T ) . D L + d 3.4 where DL i s d a y l e n g t h i n h o u r s . R e s p i r a t i o n i s assumed t o be a f i x e d p r o p o r t i o n , v, of pn(T); t h a t i s , d = V . U H ( T ) . 2 4 3.5 E q u a t i o n s 3.4 and 3.5 can be r e g a r d e d as d e f i n i n g I M A x r g i v e n u H ( T ) and V, or as d e f i n i n g V, g i v e n I M A X and j u H ( T ) . L i g h t i n t e n s i t y i s assumed t o f a l l o f f e x p o n e n t i a l l y w i t h d e p t h , so t h a t I ( z ) = I 0 e x p ( - / k ( z ' ) . d z ' ) , the e x t i n c t i o n c o e f f i c i e n t k ( z ) b e i n g d e f i n e d by k ( z ) = k B + 0.02.C(z)/V(z) (Lorenzen,1980, Megard e t a l ,1980). The c o n v e n t i o n a l method f o r c a l c u l a t i n g the average growth r a t e over a mixed l a y e r of depth z M ( i n which k i s c o n s t a n t ) i s t o i n t e g r a t e 3.3 v e r t i c a l l y t o o b t a i n : ju = o c . I M A X . ( e x p ( - I M / I M A x ) " e x p ( - I 0 / I , ^ ) ) / ( k . z M . V ) - d 3.6 120 where I M = I 0 e x p ( - k . z M ) i s the l i g h t i n t e n s i t y a t the bottom of the mixed l a y e r . U s i n g e q u a t i o n s 3.4 and 3.5, t h i s can be r e w r i t t e n a s : jj = ju H(T) .DL.e. (1 + 24. V /DL) . (exp ( - I M / I M A X ) - e x p ( - I e / I M A X ) ) / ( k , z M ) - d 3.7 Some r e c e n t l a b o r a t o r y and f i e l d e v i d e n c e p r e s e n t e d by Marra(1978 a,b) s u g g e s t s t h a t t h i s approach may not be c o r r e c t . Marra found t h a t l i g h t - s a t u r a t i o n and p h o t o - i n h i b i t i o n occur over time s c a l e s of t e n s of minutes t o hours and t h a t p h o t o s y n t h e t i c r a t e s over s h o r t exposure p e r i o d s may i n c r e a s e l i n e a r l y w i t h l i g h t i n t e n s i t y up t o v e r y h i g h l e v e l s . I t f o l l o w s t h a t , p r o v i d e d c i r c u l a t i o n t i m e s i n t h e mixed l a y e r a r e s h o r t enough, p h y t o p l a n k t o n i n the ocean may not e x p e r i e n c e p h o t o - i n h i b i t i o n or even p h o t o - s a t u r a t i o n , a l t h o u g h i n - s i t u i n c u b a t i o n s a t f i x e d depths w i l l show these e f f e c t s . An a c c u r a t e assessment of the Marra e f f e c t a t O.S.P. would r e q u i r e much more i n f o r m a t i o n about c i r c u l a t i o n i n the mixed l a y e r on s h o r t time s c a l e s , and about p h y t o p l a n k t o n p h y s i o l o g y , than i s c u r r e n t l y a v a i l a b l e . U l t i m a t e l y , the assessment of t r u e i n - s i t u p h y t o p l a n k t o n p r o d u c t i o n i n the ocean may r e q u i r e t h a t the same a t t e n t i o n be p a i d t o s h o r t - t e r m f l u c t u a t i o n s i n ambient l i g h t i n t e n s i t y as i s c u r r e n t l y b e i n g p a i d t o f l u c t u a t i o n s i n n u t r i e n t a v a i l a b i l i t y ( T u r p i n , 1 9 8 0 ; T u r p i n , H a r r i s o n and P a r s l o w , 1 9 8 0 ) . As a crude a p p r a i s a l of the importance of the Marra e f f e c t , the average growth r a t e i n the mixed l a y e r can be d e t e r m i n e d as a f u n c t i o n of the average l i g h t i n t e n s i t y t h e r e : 121 u = « . I . e x p ( - I / I M A X ) / V - d 3.8 where I = I 0 . ( l - e x p ( - k . z M ) ) / ( k . z M ) . T h i s almost c e r t a i n l y e x a g g e r a t e s the importance of the Marra e f f e c t a s , f o r a l l but v e r y s h a l l o w mixed l a y e r s , I / I M A x i s much l e s s than 1 and e q u a t i o n 3.8 r e p r e s e n t s a l i n e a r response t o l i g h t i n t e n s i t y , as might be e x p e c t e d f o r c o n t i n u o u s v e r y r a p i d m i x i n g of c e l l s t h r o u g h t h e mixed l a y e r . Below the mixed l a y e r , where d a i l y v e r t i c a l movements of p h y t o p l a n k t o n c e l l s a r e assumed t o be v e r y s m a l l , the l o c a l growth r a t e a t depth z i s g i v e n by p ( z ) = cX.l (z) .exp(-I ( z ) / I M A X ) / V - d 3.9 In the n u m e r i c a l model, t h i s e x p r e s s i o n has been i n t e g r a t e d over each 5m l a y e r t o g i v e ju(Zj ) = oc. ( e x p ( - I ( Z j + 2 . 5 ) / I M A X ) - e x p ( - I (z,-2 . 5 ) / l M A X ) ) / (5.k.V) - d 3.10 U s i n g the above e x p r e s s i o n s , p h y t o p l a n k t o n growth r a t e s t h r o u g h o u t the water column can now be p r e d i c t e d , g i v e n d a y l e n g t h , I 0 , mixed l a y e r d epth and t e m p e r a t u r e , and s e c c h i d e p t h . In a d d i t i o n , the parameters ot,ju H(T), and e i t h e r I M A X or V must be s p e c i f i e d . The temperature dependence of p H was r e v i e w e d by E p p l e y ( 1 9 7 2 ) who gave as an upper bound 122 l o g l o ( j J M A X ) = 0.0275.T - 0.070 3.11 c o r r e s p o n d i n g t o an i n c r e a s e i n growth r a t e per 10°C (Q 1 0 ) of 1.88X. Growth r a t e s f o r p a r t i c u l a r s p e c i e s g e n e r a l l y l i e below t h i s c u r v e and f a l l s h a r p l y away from i t a t s u f f i c i e n t l y h i g h t e m p e r a t u r e s . Taxanomic d i f f e r e n c e s i n J U h were c l e a r l y shown by Chan(1978), who found t h a t maximum growth r a t e s of diatoms were a p p r o x i m a t e l y t w i c e as l a r g e as those of d i n o f l a g e l l a t e s a t 20 C. These d i f f e r e n c e s can be a l l o w e d f o r by r e t a i n i n g the c o e f f i c i e n t of temperature i n e q u a t i o n 3.11 and v a r y i n g the i n t e r c e p t . The s e a s o n a l p a t t e r n of p h y t o p l a n k t o n s p e c i e s c o m p o s i t i o n a t O.S.P. i s not well-known. An e x t e n s i v e s e t of o b s e r v a t i o n s of l a r g e r diatoms and d i n o f l a g e l l a t e s has been accumulated as p a r t of r e g u l a r m i c r o z o o p l a n k t o n s a m p l i n g from the w e a t h e r s h i p s ( d a t a p r o v i d e d by Kennedy and F u l t o n ) . However, p r e l i m i n a r y a n a l y s i s of more c a r e f u l l y c o l l e c t e d and p r e s e r v e d p h y t o p l a n k t o n samples has suggested 'that the p r i m a r y p r o d u c e r s a r e p r e d o m i n a t e l y s m a l l ( l e s s than 10 ju) f l a g e l l a t e s t h r o u g h o u t the year (R. W a t e r s , p e r s . Comm.). The growth of marine n a n o f l a g e l l a t e s as a f u n c t i o n of l i g h t i n t e n s i t y has not been as w e l l s t u d i e d as t h a t of diatoms and d i n o f l a g e l l a t e s . C o c c o l i t h o p h o r i d s have been r e p o r t e d as the dominant p h y t o p l a n k t e r a t O.S.P. on c e r t a i n o c c a s i o n s ( M c A l l i s t e r , 1 9 6 1 ; I s h i m a r u and Nemoto,1977). A maximum growth r a t e of 2 d i v i s i o n s / d a y was r e p o r t e d by Paasche(1967) f o r C o c c o l i t h u s h u x l e y i under a 16H:8H l i g h t . d a r k c y c l e a t 20°C. T h i s c o r r e s p o n d s t o l o g 1 0 (ju H) = 0.0275.T - 1.65 3.12 123 The p h y t o p l a n k t o n r e s p i r a t i o n r a t e i n the model i s a c o n s t a n t f r a c t i o n v of ^ u H ( T ) . The v a l u e of V w i l l be shown t o be c r i t i c a l i n d e t e r m i n i n g net p h y t o p l a n k t o n growth r a t e s , p a r t i c u l a r l y f o r deep mixed l a y e r s . An o f t - q u o t e d v a l u e of 0.1 was suggested by R y t h e r ( 1 9 5 6 ) , a l t h o u g h he noted the l a r g e range i n r e p o r t e d v a l u e s , and observed t h a t a lower v a l u e seemed t o be needed t o e x p l a i n p h y t o p l a n k t o n p e r s i s t e n c e i n deep mixed l a y e r s i n the ocean. There have been many measurements of v s i n c e , but l i t t l e more consensus has r e s u l t e d . I t has r e c e n t l y been r e p o r t e d as between 0.03 and 0.07 by H a r r i s and P i c c i n i n ( 1 9 7 7 ) and as c o n s i s t e n t l y g r e a t e r than 0.1 by B u r r i s ( 1 9 7 7 ) . In s t a n d a r d s i m u l a t i o n s , the s e a s o n a l p a t t e r n of <X r e p r e s e n t e d by the s o l i d l i n e i n F i g 24 w i l l be used. The s e a s o n a l p a t t e r n f o r B g i v e n by the s o l i d l i n e i n F i g 27 w i l l be used t o determine I M„ X f o r the mixed l a y e r , s u b j e c t t o the c o n s t r a i n t t h a t the c o r r e s p o n d i n g v a l u e of V a c c o r d i n g t o e q u a t i o n s 3.4 and 3.5 must be g r e a t e r than 20. Below the mixed l a y e r , V i s s e t e q u a l t o 20 t o r e p r e s e n t a shade-adapted p o p u l a t i o n . The use of these e s t i m a t e s i n e q u a t i o n 3.3 h i n g e s on an i n t e r p r e t a t i o n of the 1 4 C d a t a which d e s e r v e s some d i s c u s s i o n . The parameters oc and I M A x i n e q u a t i o n 3.3 d e t e r m i n e g r o s s p h o t o s y n t h e s i s , and e q u a t i o n 3.1 has been f i t t e d t o w e a t h e r s h i p o b s e r v a t i o n s as i f the 1*C t e c h n i q u e measured g r o s s p h o t o s y n t h e s i s . The i n t e r p r e t a t i o n of 1*C uptake r a t e s has seen much c o n t r o v e r s y : i t i s g e n e r a l l y a c c e p t e d t o measure something between net and g r o s s p h o t o s y n t h e s i s (Parsons e t a l ,1977). At low l i g h t i n t e n s i t i e s , e s p e c i a l l y near or below the compensation d e p t h , the 1*C uptake r a t e must be c l o s e t o g r o s s p h o t o s y n t h e s i s , 124 as n e g a t i v e 1 4 C uptake r a t e s a r e not t e c h n i c a l l y p o s s i b l e . Much of the sampled depth range a t O.S.P. c o r r e s p o n d s t o such low l i g h t i n t e n s i t i e s , e s p e c i a l l y i n w i n t e r . The i n t e r p r e t a t i o n of 1 4 C uptake r a t e s as g r o s s r a t h e r than net p h o t o s y n t h e s i s i s then l e s s l i k e l y t o i n v o l v e major e r r o r . I f an i n t e r p r e t a t i o n as net carbon uptake i s p r e f e r r e d , the e s t i m a t e s of oc and I ^ x used i n e q u a t i o n 3.3 can be reg a r d e d as p r e d i c t i n g net d a y l i g h t p r o d u c t i o n , w i t h a reduced v a l u e of v c o r r e s p o n d i n g t o n o c t u r n a l r e s p i r a t i o n o n l y ( c f M c A l l i s t e r , 1 9 6 9 ) . 3.3.4 S i m u l a t i o n R e s u l t s . The development so f a r a l l o w s the p r e d i c t i o n of p h y t o p l a n k t o n growth r a t e s over t i m e . The p r e d i c t i o n of p h y t o p l a n k t o n s t a n d i n g s t o c k over time r e q u i r e s t h a t z o o p l a n k t o n g r a z i n g be m o d e l l e d , and t h i s w i l l be done i n the f o l l o w i n g c h a p t e r . In t h i s c h a p t e r , p h y t o p l a n k t o n p r o d u c t i o n w i l l be p r e d i c t e d on the b a s i s of the ob s e r v e d p h y t o p l a n k t o n s t a n d i n g s t o c k . I t w i l l be assumed t h a t g r a z i n g i s s u f f i c i e n t t o keep s t a n d i n g s t o c k i n the mixed l a y e r a t 0.4 ug C h i a . l " 1 , the mean observed c o n c e n t r a t i o n . S t a n d i n g s t o c k below the mixed l a y e r w i l l not be a l l o w e d t o exceed t h i s c o n c e n t r a t i o n and w i l l be a l l o w e d t o f a l l below i t where r e s p i r a t i o n exceeds p h o t o s y n t h e s i s . In F i g 32, the time s e r i e s of p h y s i c a l d r i v i n g v a r i a b l e s f o r the y e a r s 1964 t o 1976 are p l o t t e d . These were used t o p r e d i c t p r i m a r y p r o d u c t i o n on the f i x e d s t a n d i n g s t o c k b a s i s , u s i n g the s e a s o n a l p a t t e r n s f o r oc and B ( F i g 24,27) d i s c u s s e d above, e q u a t i o n 3.12 f o r ju H(T) and a v a l u e f o r v of 0.05. One of the noteworthy r e s u l t s of t h i s and subsequent s i m u l a t i o n s i s the l a c k 125 Figure 32. Time series of total solar radiation, surface temperature and mixed layer depth used to drive simulation model. YEAR 126 of y e a r t o y e a r v a r i a t i o n i n p r e d i c t e d p h y t o p l a n k t o n p r o d u c t i o n ( F i g 33 and T a b l e I I ) . T h i s f e a t u r e w i l l be o f i n t e r e s t i n C h a p t e r 4; f o r t h e r e m a i n d e r o f t h i s c h a p t e r , a t t e n t i o n w i l l be f o c u s e d on s e a s o n a l p a t t e r n s and r e s u l t s p r e s e n t e d f o r one y e a r , 1976, t o a v o i d r e p e t i t i o n . In t h e a n a l y s i s o f 1 4 C p r o f i l e s , i t was f o u n d t h a t t h e t i m e s c a l e o f a d a p t a t i o n t o c h a n g i n g l i g h t i n t e n s i t i e s c o u l d n o t be i n f e r r e d from t h e d a t a . The e f f e c t s of d i f f e r e n t a d a p t a t i o n t i m e s c a l e s on p r e d i c t e d p r i m a r y p r o d u c t i o n can now be c o n s i d e r e d u s i n g t h e m o d e l . T h r e e c a s e s w i l l be c o n s i d e r e d : I 0 ( t ) / B ( i n s t a n t a n e o u s a d a p t a t i o n ) , ( I 0 (t-l-)+I„ (t-2)+I„ ( t - 3 ) )/(3.B) ( s h o r t t i m e s c a l e ) I s ( t ) / B ( s e a s o n a l t i m e s c a l e ) . B e f o r e c o n s i d e r i n g t h e f i r s t two c a s e s , t h e c o n c e p t o f f i x i n g p h y t o p l a n k t o n s t a n d i n g s t o c k must be c o n s i d e r e d more c a r e f u l l y . In t h e f i r s t s i m u l a t i o n , l i g h t a d a p t a t i o n on a s e a s o n a l t i m e s c a l e was assumed and i t f o l l o w s f r o m e q u a t i o n s 3.4 and 3.5 t h a t V ( t ) =< a . I s / ( p»(T).DL.(1 + 24.V/DL).e.B) 3.13 so t h a t b o t h V and p h y t o p l a n k t o n c a r b o n c o n c e n t r a t i o n i n t h e mixed l a y e r , (C = V . 0 . 4 ) , v a r y s m o o t h l y o v e r t i m e . I f i n s t a n t a n e o u s l i g h t a d a p t a t i o n i s assumed ( c a s e 1 ) , V ( t ) , g i v e n by r e p l a c i n g I s i n e q u a t i o n 3.13 by I Q , w i l l u n d e r g o v e r y l a r g e f l u c t u a t i o n s f r o m day t o day, as I 0 d o e s . F i x i n g t h e C h i a c o n c e n t r a t i o n a t 0.4 ^ g C h i a . l ' 1 w i l l c a u s e t h e c a r b o n (1) l M A X < t ) = (2) I M A X ( t ) = (3) I M A X ( t ) = Figure 33. Predicted d a i l y net production using standard parameter, set (see t e x t ) . 128 Table II _2 Predicted annual primary production (gC.m ) at O.S.P. for 1964 to. 1-976. ( Details of simulation provided i n text.) Year Gross (Daylight) Net 1964 35.2 20.6 1965 35.5 21.2 1966 35.2. 19.1 1967 35.6 20.5 1968 35.4 21.5 1969 35.6 20.7 1970 35.2 19.8 1971 35.1 18.7 1972 35.3 20.0 1973 35.6 19.2 1974 36.2 21.6 1975 34.6 17.9 1976 34.9 19.6 129 c o n c e n t r a t i o n t o f l u c t u a t e i n t h e same manner. A l t h o u g h t h e s e s i m u l a t i o n s a r e not e x p l i c i t l y c o n c e r n e d w i t h c o n s e r v a t i o n o f c a r b o n , s u c h r a p i d f l u c t u a t i o n s i n p l a n t c a r b o n a r e n o t r e a l i s t i c , e s p e c i a l l y i n v i e w o f t h e g e n e r a l l y low c a r b o n g r o w t h r a t e s p r e d i c t e d . I f V i s t o v a r y r a p i d l y , i t seems more r e a s o n a b l e t h a t t h i s s h o u l d o c c u r t h r o u g h c h a n g e s i n C h i a c o n c e n t r a t i o n . T h i s can be a c c o m p l i s h e d w h i l e k e e p i n g a v e r a g e C h i a c o n c e n t r a t i o n s n e a r 0.4 ug C h i a . l " 1 as f o l l o w s : (a) F i x a s e a s o n a l l y v a r y i n g c a r b o n c o n c e n t r a t i o n , C s ( t ) , i n t h e m i x e d l a y e r by C s ( t ) = V s ( t ) .0.4 3.14 where V s ( t ) i s g i v e n by e q u a t i o n 3.13. (b) F i x t h e a c t u a l C : C h l a r a t i o , V, u s i n g one of t h e s h o r t t i m e s c a l e m o d e l s ; t h e C h i a c o n c e n t r a t i o n i s t h e n g i v e n by C h i a ( t ) = C s ( t ) / V ( t ) . In F i g 34, t h e n e t d a i l y p r i m a r y p r o d u c t i o n p r e d i c t e d f o r 1976 i n t h e f i r s t s i m u l a t i o n i s p l o t t e d a g a i n on an expanded t i m e s c a l e . The d a i l y v a l u e s of V and C h i a c o n c e n t r a t i o n i n t h e m i x e d l a y e r a r e a l s o p l o t t e d . L i g h t a d a p t a t i o n i n t h i s s i m u l a t i o n o c c u r r e d on a s e a s o n a l t i m e s c a l e , so t h a t V (= V s ( t ) ) v a r i e s s m o o t h l y w h i l e C h i a i s c o n s t a n t a t 0.4 j u g . l - 1 . The c o r r e s p o n d i n g o u t p u t f o r i n s t a n t a n e o u s a d a p t a t i o n i s p l o t t e d i n F i g 35. H e r e , V f l u c t u a t e s m a r k e d l y w i t h d a i l y c h a n g e s i n I Q and so does C h i a c o n c e n t r a t i o n . On t h e o t h e r hand, s h o r t - t e r m f l u c t u a t i o n s i n n e t p r i m a r y p r o d u c t i o n a r e much r e d u c e d . T h e r e i s a s i m p l e e x p l a n a t i o n f o r t h i s . The dominant c o n t r i b u t i o n t o 130 150 J F M R M J J R S ON D MONTH ' Figure 34. Predicted net production, mixed layer Chi a and mixed layer C:Chl a ratio for 1976 using standard parameter set and seasonal light adaptation. 131 1 5 0 C J OH 1 1 1 1 1—'—i ; r 1 1 1 J F M fl M J J A S. O N D MONTH Figure 35. As for Fig . 34,. but with 'instantaneous' light adaptation. 132 net p r i m a r y p r o d u c t i o n i n the water column comes from the mixed l a y e r . U s i n g e q u a t i o n 3.7, d a i l y p r i m a r y p r o d u c t i o n i n the mixed l a y e r i s g i v e n by p . C s . z M = C s . p H ( T ) . D L . e . ( 1 + 2 4 . v / D L ) . ( e x p ( - I M / l w ^ ) - e x p ( - I D / W ) )/k - 2 4 . p H ( T ) .V.C S , z M 3.15 As T and z M v a r y smoothly over t i m e , the major c o n t r i b u t o r t o d a i l y f l u c t u a t i o n s i n p r o d u c t i o n i s the exponent I O / I M A X • F O R s e a s o n a l a d a p t a t i o n , I W A X v a r i e s w i t h I S and t h i s exponent f l u c t u a t e s from day t o day w i t h I Q . For i n s t a n t a n e o u s a d a p t a t i o n , I M A X v a r i e s w i t h I 0 and the exponent does not f l u c t u a t e from day t o day. For the i n t e r m e d i a t e case ( 2 ) , V and C h i a show reduced s h o r t - t e r m f l u c t u a t i o n as e x p e c t e d ( F i g 36). The c h o i c e of the a d a p t a t i o n time s c a l e t h e r e f o r e d e t e r m i n e s the s i z e of s h o r t - t e r m f l u c t u a t i o n s i n V, C h i a and d a i l y p r i m a r y p r o d u c t i o n . I t has v e r y l i t t l e e f f e c t on average p r i m a r y p r o d u c t i o n l e v e l s , as shown i n T a b l e I I I . Comparison of the p r e d i c t e d time s e r i e s f o r C h i a i n F i g 35 w i t h o b s e r v a t i o n s i n F i g 13 shows t h a t i n s t a n t a n e o u s a d a p t a t i o n r e s u l t s i n much l a r g e r day t o day f l u c t u a t i o n s i n C h i a than were obse r v e d d u r i n g the i n t e n s i v e s a m p l i n g p e r i o d s from 1964 t o 1968. The s e a s o n a l v a r i a t i o n i n V i s s m a l l compared w i t h the f i v e - f o l d v a r i a t i o n i n I s from w i n t e r t o summer. T h i s v a r i a t i o n i n I s i s p a r t l y compensated f o r by the s e a s o n a l v a r i a t i o n i n pH(T) and DL. The p r e d i c t e d v a l u e s of V a r e r e a s o n a b l e f o r f l a g e l l a t e s a l t h o u g h they would be c o n s i d e r e d l a r g e f o r d i a t o m s , where C:Chl a r a t i o s of 20 or l e s s can be e x p e c t e d under low l i g h t 1 5 0 C J 0 ~i 1 1 1 1 1 1 1 1 1—1—i 1 — J F M R M J J R S Q N D MONTH Figure 36. As for Fig 34, but with 3-day adaptation. 134 Table III _2 Monthly means of predicted daily net primary production (mgC.m ) at O.S.P. using taree light adaptation time scales. Seasonal 3-day Instantaneous Month January 3. "5; 6. February 26. 26. 26. March 51. 53. 54. A p r i l 73. 75. 77. May 77. 75. 76. June 112. 111. 111. July 99. 100. 102. August 77. 78. 80. September 57. 58. 61. October 47. 47. 49. November 22. 18. 19. December 7. 6. 7. 135 c o n d i t i o n s . I t i s i n t e r e s t i n g t h a t t h e s e a s o n a l v a r i a t i o n i n and B, b a s e d on a n a l y s i s of t h e 1*C p r o f i l e s , a l s o r e d u c e s t h e s e a s o n a l v a r i a t i o n i n V. The p r e d i c t e d C : C h l a r a t i o f o r 1976 under t h e a s s u m p t i o n of c o n s t a n t oc (0.5 mg C.mg C h i a ^ . l y " 1 ) and c o n s t a n t B (2.0) ( F i g 37) shows a much g r e a t e r p e r c e n t a g e i n c r e a s e f r o m w i n t e r t o summer. T h i s s u g g e s t s t h a t t h e s e a s o n a l p a t t e r n i n oc and B may be p a r t l y a r e s u l t o f c o n s t r a i n t s on a t t a i n a b l e C : C h l a r a t i o s i n O.S.P. p h y t o p l a n k t o n . The n e t a n n u a l p r o d u c t i o n t o t a l s i n T a b l e I I a r e r a t h e r low compared w i t h p r e v i o u s e s t i m a t e s f o r t h i s r e g i o n (Sanger,1972; M c A l l i s t e r , 1 9 6 9 ) . M c A l l i s t e r (1969) gave upper and l o w e r bounds of 47 g C m - 2 and 31 g C m " 2 f o r n e t p r i m a r y p r o d u c t i o n , and 61 g C m " 2 and 48 g C m " 2 r e s p e c t i v e l y f o r d a y l i g h t p r o d u c t i o n . The upper bounds were o b t a i n e d by u s i n g a c o r r e c t i o n f a c t o r o f 1.25 t o a l l o w f o r e f f e c t s of e n c l o s u r e o f samples o v e r 10 t o 20 hr p e r i o d s . H i s a n n u a l c y c l e o f d a i l y p r o d u c t i o n ( F i g 4) has a peak e x c e e d i n g 300 mg C m " 2 . d a y " 1 , whereas t h e peak i n F i g 34 does n o t r e a c h h a l f t h i s t o t a l . The p o s s i b l e s o u r c e s o f t h i s d i s c r e p a n c y become c l e a r e r when e q u a t i o n 3.14 f o r net d a i l y p r o d u c t i o n i n t h e m i x e d l a y e r i s r e w r i t t e n a s : P r o d = 0.4.«.I S.(l-exp ( - B)-24.v . z M.k/(e.(DL+24.V) ) / ( B.k) 3.16 ( T h i s i s f o r t h e c a s e o f i n s t a n t a n e o u s a d a p t a t i o n . The a p p r o x i m a t i o n exp ( - I M / I M A X ) 1 has been made). Note t h a t g r o s s ( o r d a y l i g h t , M c A l l i s t e r ) p r o d u c t i o n depends o n l y on t h e p a r a m e t e r s oc and B w h i c h were e s t i m a t e d f r o m 1 4 C p r o f i l e s , and t h e e x t i n c t i o n c o e f f i c i e n t . Of t h e p a r a m e t e r s t a k e n from t h e 1 0 0 CE I £ 5 0 ~ CJ 0 i 1 1 1— ' i 1 1 — : — i 1 1 1 J F M fl M.J J fl S 0 .N D MONTH Figure 37. Predicted C:Chl a ratio i n the mixed layer for 1976 using OC =0.5 and B„ = 2.0. 137 l i t e r a t u r e , y d e t e r m i n e s r e s p i r a t i o n but j u H ( T ) ' does n o t a p p e a r a t a l l . T h i s i s not s u r p r i s i n g , s i n c e under t h e c o n s t a n t s t a n d i n g s t o c k a s s u m p t i o n , t h e model i s p r e d i c t i n g p r o d u c t i o n p e r u n i t C h i a, p r e c i s e l y what i s m easured i n t h e f i e l d . I t f o l l o w s t h a t t h e d i f f e r e n c e between M c A l l i s t e r ' s e s t i m a t e f o r a n n u a l d a y l i g h t p r o d u c t i o n o f 48 g C m - 2 and t h e e s t i m a t e p r o d u c e d h e r e of a b o u t 35 g C m " 2 c a n n o t be a t t r i b u t e d t o t h e a s s u m p t i o n o f d i f f e r e n t p a r a m e t e r v a l u e s f r o m t h e l i t e r a t u r e , b ut must be due t o t h e use o f d i f f e r e n t w e a t h e r s h i p d a t a , o r d i f f e r e n t t e c h n i q u e s of d a t a a n a l y s i s . The d i r e c t d e p t h i n t e g r a t i o n o f x 4 C p r o f i l e s f r o m t h e w e a t h e r s h i p t o o b t a i n d a i l y p r o d u c t i o n f o r t h e w a t e r column, u s e d by M c A l l i s t e r , was c r i t i c i s e d e a r l i e r . However, t h i s was now done f o r t h e 1964-1968 d a t a t o see i f i t r e s u l t e d i n h i g h e r p r o d u c t i o n e s t i m a t e s t h a n t h e p a r a m e t e r e s t i m a t i o n and s i m u l a t i o n ° a p p r o a c h . The r e s u l t i n g a n n u a l d a y l i g h t p r o d u c t i o n e s t i m a t e was 32 g C m - 2 , a l i t t l e - l o w e r t h a n t h a t o b t a i n e d t h r o u g h p a r a m e t e r e s t i m a t i o n , and about 2/3 o f M c A l l i s t e r ' s l o w e r v a l u e . T h e r e a r e o t h e r p o s s i b l e c a u s e s o f t h e d i s a g r e e m e n t . M c A l l i s t e r u s e d an e m p i r i c a l l y o b t a i n e d r e l a t i o n s h i p between d e p t h - i n t e g r a t e d p r o d u c t i o n and t h e p r o d u c t o f s u r f a c e p r o d u c t i o n and s u r f a c e l i g h t i n t e n s i t y t o a l l o w t h e i n c l u s i o n of s u r f a c e o b s e r v a t i o n s i n t h e a n n u a l p r o d u c t i o n e s t i m a t e . He a l s o u s e d o b s e r v a t i o n s from an e a r l i e r t i m e p e r i o d , w h i c h were b a s e d on d i f f e r e n t i n c u b a t i o n t e c h n i q u e s . G i v e n t h e low e s t i m a t e o f g r o s s o r d a y l i g h t p r o d u c t i o n , t h e r e s p i r a t i o n t e r m i s r e s p o n s i b l e f o r t h e v e r y low n e t a n n u a l p r o d u c t i o n i n T a b l e I I , r e d u c i n g g r o s s p r o d u c t i o n a l m o s t by h a l f . The c h o i c e of y i s c l e a r l y c r i t i c a l i n t h e p r e d i c t i o n o f n e t 138 p r i m a r y p r o d u c t i o n . For example, i n c r e a s i n g y t o 0.10 (th e v a l u e chosen by Ry t h e r ( 1 9 5 6 ) ) , reduces net annual p r o d u c t i o n a l m o s t t o z e r o and r e s u l t s i n n e g a t i v e d a i l y p r o d u c t i o n over h a l f of the year ( F i g 3 8 ) . I t i s c l e a r t h a t f o r o c e a n i c environments w i t h deep mixed l a y e r s , such as O.S.P., a b e t t e r u n d e r s t a n d i n g of p h y t o p l a n k t o n r e s p i r a t i o n i s e s s e n t i a l t o the a c c u r a t e e s t i m a t i o n of net p r i m a r y p r o d u c t i o n . L a b o r a t o r y r e p o r t s of p a r t i a l c e l l shutdown and/or low p o s i t i v e net p r o d u c t i o n a t ve r y low l i g h t i n t e n s i t i e s (Smayda and M i t c h e l l - l n n e s , 1 9 7 4 ) a r e i n t e r e s t i n g i n t h i s r e g a r d . A g a i n the time s c a l e s of f l u c t u a t i o n s i n ambient l i g h t i n t e n s i t y and of the p h y s i o l o g i c a l response of c e l l s i n the mixed l a y e r may be of c r i t i c a l i m p o r t a n c e . The i n c l u s i o n of l o n g time s c a l e a d a p t a t i o n such as the f o r m a t i o n of r e s t i n g s p o r e s i n v o l v e d i n the l i f e c y c l e s of some p h y t o p l a n k t e r s c o u l d a l s o reduce p r e d i c t e d l o s s r a t e s i n w i n t e r . I n s p e c t i o n of e q u a t i o n 3.16 shows t h a t a s i m p l e way t o i n c r e a s e a n n u a l p r o d u c t i o n t o M c A l l i s t e r ' s l e v e l s i s t o i n c r e a s e oc. The s e a s o n a l p a t t e r n s of net d a i l y p r i m a r y p r o d u c t i o n and C:Chl a r a t i o s p r e d i c t e d by the model f o r <X = 1.0 and B = 2.0, w i t h the a d d i t i o n a l c o n s t r a i n t t h a t V be l e s s than 100, a r e p l o t t e d i n F i g 39. Annual g r o s s p r o d u c t i o n i s 67 g Cm" 2 and net p r o d u c t i o n i s 45 g Cm" 2. A v a l u e of cx as l a r g e as 1.0 i s not su p p o r t e d by the O.S.P. da t a but i s c e r t a i n l y w i t h i n the range of v a l u e s r e p o r t e d from o t h e r f i e l d or l a b o r a t o r y s t u d i e s , as d i s c u s s e d above. Another way t o i n c r e a s e p r o d u c t i o n i s t o i n t r o d u c e the Marra e f f e c t by u s i n g e q u a t i o n 3.8. Wi t h the s e a s o n a l c y c l e s of oc and B o b t a i n e d from p r o f i l e a n a l y s i s and used e a r l i e r , p r e d i c t e d 139 1 5 0 — 9 0 -! 1 1—~i i 1 1 1 1 1 1 1—-I J F M fl M J J S 0 N D MONTH Figure 38. Predicted daily net production on increasing V to 0.1. 140 Figure 3 9 . Predicted d a i l y net production and mixed l a y e r C:Chl a r a t i o f o r oC =1.0, B =2.0 with c o n s t r a i n t s V 100. ' 141 g r o s s and n e t a n n u a l p r o d u c t i o n i n c r e a s e s t o 54 and 39 g C m - 2 r e s p e c t i v e l y . The d o u b l i n g i n n e t p r i m a r y p r o d u c t i o n i s s t r i k i n g ( F i g 40) and i s due t o t h e s u b t r a c t i o n of a c o n s t a n t r e s p i r a t i o n l o s s f r o m an i n c r e a s e d g r o s s p r o d u c t i o n l e v e l . As d i s c u s s e d e a r l i e r , e q u a t i o n 3.8 a l m o s t c e r t a i n l y e x a g g e r a t e s t h e i m p o r t a n c e of t h e M a r r a e f f e c t , b ut on t h e b a s i s of t h e s e r e s u l t s f u r t h e r t h e o r e t i c a l and e x p e r i m e n t a l i n v e s t i g a t i o n a p p e a r s w a r r a n t e d . The e x t i n c t i o n c o e f f i c i e n t , k, a l s o d i r e c t l y a f f e c t s p r o d u c t i o n e s t i m a t e s ( e q u a t i o n 3.15). The v a l u e s of k i n t h e above s i m u l a t i o n s a r e b a s e d on t h e a v e r a g e s e a s o n a l c y c l e of S e c c h i d e p t h ( e q u a t i o n 3 . 2 ) . (The e f f e c t of o b s e r v e d low C h i a l e v e l s on t h e e x t i n c t i o n c o e f f i c i e n t i s n e g l i g i b l e . ) The s e c c h i d e p t h i s t h e one p h y s i c a l d r i v i n g v a r i a b l e w h i c h was n o t o b s e r v e d i n s u f f i c i e n t d e n s i t y t o p r o v i d e an i n t e r p o l a t e d t i m e s e r i e s f o r e a c h y e a r and c o n s e q u e n t l y r e p r e s e n t s a s o u r c e of a n n u a l v a r i a b i l i t y not i n c l u d e d i n F i g 33 or T a b l e I I . Upper and lower e n v e l o p e s t o a p l o t of S e c c h i d e p t h a g a i n s t t i m e of y e a r ( F i g 41) were u s e d t o g e n e r a t e s e a s o n a l c y c l e s of p r e d i c t e d n e t d a i l y p r o d u c t i o n , c o r r e s p o n d i n g t o extreme s e a s o n a l c y c l e s of e x t i n c t i o n c o e f f i c i e n t s ( F i g 4 2 ) . E s t i m a t e s o f g r o s s and n e t a n n u a l p r i m a r y p r o d u c t i o n were 61 g C m - 2 and 35 g C m " 2 r e s p e c t i v e l y f o r maximum S e c c h i d e p t h s (minimum e x t i n c t i o n c o e f f i c i e n t s ) and 25 g C m " 2 and 10 g C m - 2 r e s p e c t i v e l y f o r • minimum S e c c h i d e p t h s . T h i s more t h a n t h r e e - f o l d v a r i a t i o n i n a n n u a l n e t p r o d u c t i o n i s i m p r e s s i v e , a l t h o u g h a n n u a l c y c l e s i n S e c c h i d e p t h need not match t h e e x t r e m e s o b t a i n e d by u s i n g e n v e l o p e s f o r a l l y e a r s . The c a u s e s of o b s e r v e d s e a s o n a l and a n n u a l v a r i a t i o n s i n 142 270 J ' F ' M ' R ' M ' J ' J ' R ' S "O 'N 'D MONTH F i g u r e 40. P r e d i c t e d d a i l y net p r o d u c t i o n f o r standard parameter set w i t h Marra e f f e c t i n t r o d u c e d . 143 3 0 2 5 - < J > a <x>® 5 - . • • 0 i i i i 1 1 r~—i 1 r 1 1 — J F M fl M J J fl S O ' N D MONTH Figure 41. Observed Secchi depths at O.S.P. vs time of year, ^before 1964, D 1964 to 1976. Solid lines represent upper and lower evelopes and seasonal f i t to observations after 1964. 144 2 7 0 - 3 0 ~! 1 n i 1 — : — i 1 r 1 1 ~i 1 1 J F M R M J J R S 0 N D MONTH Figure 42a. Predicted d a i l y net production using standard parameter set and upper envelope to Secchi depths. 145 150 J F M M J J . R S O N D MONTH Figure 42b. Predicted d a i l y net production using standard parameter set and lower envelope to Secchi depths. 146 S e c c h i depth i s . not c l e a r . The c o n c e n t r a t i o n of C h i a i s g e n e r a l l y i n s u f f i c i e n t t o account f o r observed S e c c h i depths and i n any case does not show c o r r e s p o n d i n g s e a s o n a l or annual v a r i a t i o n s . M c A l l i s t e r e t a l ( 1 9 6 1 ) r e p o r t e d 100-200 mg C m - 3 of d e t r i t a l m a t e r i a l i n the s u r f a c e w a t e r s a t O.S.P. i n J u l y - A u g u s t , some f o u r t o e i g h t t i m e s the e s t i m a t e d l i v i n g p l a n t c a r b o n . Recent o b s e r v a t i o n s i n d i c a t e a s e a s o n a l c y c l e i n d e t r i t a l carbon w i t h a minimum i n w i n t e r and maximum i n summer, ( K . I s e k i and C.S.Wong,pers. comm.), c o n s i s t e n t w i t h the n o t i o n t h a t changes i n the e x t i n c t i o n c o e f f i c i e n t may be due t o changes i n d e t r i t u s l e v e l s . R a ther than f i x i n g V or I M A X e m p i r i c a l l y i n e q u a t i o n 3.4, i t i s i n t e r e s t i n g t o c o n s i d e r u s i n g an o p t i m a l i t y p r i n c i p l e f o r p h y t o p l a n k t o n by demanding t h a t l i g h t a d a p t a t i o n of p h y t o p l a n k t o n s h o u l d p r o c e e d so as t o maximise the average growth r a t e i n the mixed l a y e r . T a k i n g oc and as g i v e n , i t i s easy t o show t h a t i n both e q u a t i o n 3.6 and e q u a t i o n 3.8, p i s maximised when V = CX . I / (e . u H . DL . (1 . + 24 . V/DL) ) . When V was c a l c u l a t e d i n t h i s way, u s i n g the ob s e r v e d s e a s o n a l c y c l e f o r ex, i t d i d not exceed the minimum v a l u e of 20 a l l o w e d i n the model ( F i g 43a). Even f o r tx=1.0, the r e s u l t i n g s e a s o n a l c y c l e of V was much lower than t h a t o b t a i n e d u s i n g the v a l u e s f o r cX and B e s t i m a t e d from p r o f i l e s ( F i g 43b). - A p p a r e n t l y , p h y t o p l a n k t o n a t O.S.P. a r e not adapted so as t o maximise growth r a t e s i n the mixed l a y e r . As some marine p h y t o p l a n k t o n a re known t o a c h i e v e C:Chl a r a t i o s of 20 or lower (eg Chan,1978), o t h e r 147 1 0 0 1 0 0 0 " T — I r — — i r — — i 1 1 r — — I 1 1 1 J F M R M J J H S ,0 ,N D ' MONTH F i g u r e 43. P r e d i c t e d s e a s o n a l c y c l e i n mixed l a y e r C : C h l a r a t i o u s i n g o p t i m a l i t y c r i t e r i o n and c o n s t r a i n t V ^ 20 f o r (a) e s t i m a t e d OC . . (b) c*:=i.o. 148 s e l e c t i v e p r e s s u r e s not c o n s i d e r e d i n the growth model, such as g r a z i n g , must p r e v e n t t h e i r dominance a t O.S.P. 3.3.5 P r i m a r y P r o d u c t i o n and N i t r a t e D e p l e t i o n . The amount of c o n f i d e n c e p l a c e d on the i n i t i a l s i m u l a t i o n of p r i m a r y p r o d u c t i o n a t O.S.P. ( F i g 33) depends on a number of f a c t o r s . I f the 1 4 C p r o f i l e s f o r 1964-1968 a r e a c c e p t e d as a c c u r a t e measures of p r o d u c t i o n , and t h i s p e r i o d i s c h a r a c t e r i s t i c of the s i m u l a t i o n p e r i o d , 1964-1976, then the e s t i m a t e d g r o s s p r o d u c t i o n may be t a k e n as a c c u r a t e t o w i t h i n the s t a n d a r d e r r o r i n OC : a p p r o x i m a t e l y +30%. As noted above, e s t i m a t e s of net p r i m a r y p r o d u c t i o n depend c r i t i c a l l y on V, and v a l u e s between 0 and 26 g Cm" 2 can be o b t a i n e d u s i n g the e s t i m a t e d oc and B w i t h o u t d e p a r t i n g from l i t e r a t u r e ranges f o r V. I f the 1*C o b s e r v a t i o n s a r e d i s c o u n t e d , g r o s s p r o d u c t i o n can e a s i l y be d o u b l e d w i t h o u t assuming v a l u e s f o r ot and B o u t s i d e l i t e r a t u r e r anges. The n i t r a t e p r o f i l e s o f f e r the p o s s i b i l i t y f o r an independent check on p r i m a r y p r o d u c t i o n , a l t h o u g h o t h e r problems of i n t e r p r e t a t i o n a r e i n v o l v e d . A lower l i m i t t o g r o s s a n n u a l p r o d u c t i o n can be o b t a i n e d by t a k i n g the average a n n u a l d e c r e a s e of 6.4 ug a t NO .m~3 i n the t o p 40m ( F i g 30) and assuming a C:N r a t i o of 7 mg.rng"1 t o g i v e 25 g Cm" 2. T h i s e s t i m a t e does not a l l o w f o r growth below 40m, v e r t i c a l m i x i n g of n i t r a t e from below 40m or n i t r a t e r e c y c l i n g . A s l i g h t l y h i g h e r e s t i m a t e can be o b t a i n e d by summing a l l d e c r e a s e s i n the average s e a s o n a l c y c l e f o r 40 t o 80 m ( F i g 3 0 ) . The r e s u l t i n g f i g u r e of 28 g Cm" 2 i n c l u d e s p r o d u c t i o n down t o 80m and a t l e a s t some e f f e c t of v e r t i c a l m i x i n g and/or n i t r a t e r e c y c l i n g as d i s c u s s e d e a r l i e r . 149 The C:N r a t i o can p o t e n t i a l l y v a r y from 3 t o 15 (Banse,1974) a l t h o u g h h i g h e r v a l u e s a r e most l i k e l y t o occur under n i t r o g e n l i m i t a t i o n . The n i t r a t e time s e r i e s has not been used t o l o o k a t a n n u a l v a r i a t i o n s i n p r i m a r y p r o d u c t i o n , as the l a r g e - s c a l e f l u c t u a t i o n s i n deep-water n i t r a t e c o n c e n t r a t i o n s ( F i g 3 0 ) , presumably c o r r e s p o n d i n g t o a d v e c t i o n of d i f f e r e n t water masses p a s t the s t a t i o n , p r e v e n t an i n t e r p r e t a t i o n of n i t r a t e d e c r e a s e s i n any p a r t i c u l a r year as p h y t o p l a n k t o n uptake. (These l o n g - t e r m e f f e c t s have been removed i n the average s e a s o n a l c y c l e s ) . The e s t i m a t e s of average a n n u a l p r o d u c t i o n based on n i t r a t e d e c r e a s e s ar e u n d e r e s t i m a t e s by an unknown amount which depends on m i x i n g and r e c y c l i n g r a t e s . As they a r e lower than the p r e d i c t e d annual p r o d u c t i o n (Table I I ) , they a l l o w o n l y the weak c o n c l u s i o n t h a t t h e r e i s no e v i d e n c e i n the n i t r a t e o b s e r v a t i o n s f o r h i g h e r p r o d u c t i o n than p r e d i c t e d i n the s t a n d a r d s i m u l a t i o n ( F i g 33). 150 CHAPTER 4 HERBIVOROUS ZOOPLANKTON AT O.S.P.: DATA ANALYSIS AND MODELLING. 4.1 Parameter E s t imat i o n . 4.1.1 D e s c r i p t i o n of Data. The time s e r i e s of o b s e r v a t i o n s of z o o p l a n k t o n from the O.S.P. w e a t h e r s h i p s extends from 1956 t o the p r e s e n t and c o n s t i t u t e s one of the most e x t e n s i v e open ocean data s e t s of i t s t y p e . F o l l o w i n g an e a r l y r e p o r t on v a r i a t i o n w i t h season and depth of z o o p l a n k t o n c o m p o s i t i o n and abundance ( M c A l l i s t e r , 1 9 6 1 ) , two summaries of the time s e r i e s have appeared i n m a n u s c r i p t form ( L e B r a s s e u r , 1 9 6 5 ; F u l t o n , 1 9 7 8 ) . The data have a l s o been used as a b a s i s f o r trophodynamic s t u d i e s ( L e B r a s s e u r , 1 9 6 9 ; M c A l l i s t e r , 1 9 6 9 , 1 9 7 2 ) . The abundance d a t a a n a l y s e d here come from 150 m v e r t i c a l h a u l s u s i n g a 350 jam n e t . Q u a l i t a t i v e d a t a from s u r f a c e tows w i l l not be c o n s i d e r e d . Changes i n the d e s i g n of the 350 p net over the s a m p l i n g p e r i o d a r e d i s c u s s e d by F u l t o n ( 1 9 7 8 ) : e a r l i e r abundance e s t i m a t e s based on t h e NORPAC net have been a d j u s t e d here t o match the c u r r e n t l y - u s e d SCOR net u s i n g F u l t o n ' s c o r r e c t i o n f a c t o r . Wet w e i g h t s of 150m v e r t i c a l h a u l samples were r e c o r d e d t h r o u g h o u t t h e sa m p l i n g p e r i o d . These sample wet w e i g h t s , t o g e t h e r w i t h e s t i m a t e d wet w e i g h t s of f o u r major taxanomic groups : copepods, c h a e t o g n a t h s , e u p h a u s i i d s and amphipods, were r e p o r t e d as time s e r i e s by L e B r a s s e u r ( 1 9 6 5 ) f o r the p e r i o d 1956 t o 1964. Average s e a s o n a l c y c l e s and abundance f o r 28 im p o r t a n t s p e c i e s d u r i n g t h i s p e r i o d were p r e s e n t e d and d i s c u s s e d by 151 LeBrasseur(1965,1969) . Sample a n a l y s i s changed a f t e r 1965 and the summary s t a t i s t i c s p r e s e n t e d by -Fulton(1978) c o n s i s t of sample wet weight and #ind/m 3 i n each of 5 taxanomic groups which i n c l u d e L e B r a s s e u r ' s 4 groups and the trachymedusae, A g l a n t h a . S t a r t i n g i n 1969, samples were a n a l y s e d i n d e t a i l w i t h most organisms b e i n g i d e n t i f i e d t o s p e c i e s l e v e l and a s s i g n e d t o s i z e c l a s s e s . These d a t a a re s t o r e d on magnetic tape and a p r e l i m i n a r y a n a l y s i s has been made i n c o - o p e r a t i o n w i t h J . F u l t o n . The da t a a re s u f f i c i e n t l y comprehensive t o a l l o w L e B r a s s e u r ' s average s e a s o n a l c y c l e s t o be r e f i n e d , and annual v a r i a t i o n s i n t h e s e c y c l e s t o be c o n s i d e r e d . T h i s a n a l y s i s w i l l be r e p o r t e d e l s e w h e r e . The d a t a are used i n t h i s t h e s i s i n two r a t h e r s p e c i a l i s e d r o l e s . In t h i s s e c t i o n , a parameter e s t i m a t i o n t e c h n i q u e d e v e l o p e d f o r a n a l y s i n g copepod c o h o r t s i s adapted and a p p l i e d t o the s i z e - s t r u c t u r e d abundance d a t a f o r the dominant h e r b i v o r o u s copepods a t O.S.P. In the remainder of the .chapter, the da t a w i l l be used i n the c o n s t r u c t i o n and study of a model of the p h y t o p l a n k t o n - z o o p l a n k t o n i n t e r a c t i o n a t O.S.P. 4.1.2 Review of Parameter E s t i m a t i o n T e c h n i q u e s . As background t o the parameter e s t i m a t i o n f o r O.S.P., the r e s u l t s of P a r s l o w et. a l (1979) and Sonntag and Parslow(1980) a r e re v i e w e d . In t h e s e p a p e r s , methods f o r o b t a i n i n g p o p u l a t i o n parameters from time s e r i e s of d e n s i t i e s i n l i f e h i s t o r y s t a g e s u s i n g a t e c h n i q u e of systems i d e n t i f i c a t i o n were e x p l o r e d . The u n d e r l y i n g t e c h n i q u e has the f o l l o w i n g g e n e r a l s t r u c t u r e . G i v e n a s e t of o b s e r v a t i o n s Y ( t ; ) , i = 1,...,N, and a model which A a l l o w s p r e d i c t i o n s Y ( t ; , p ) as a f u n c t i o n of time and parameters 152 p r t h a t s e t o f p a r a m e t e r s p i s sought w h i c h m i n i m i z e s t h e sum of s q u a r e d e r r o r s (SSQ = I ( Y ( t ; ) - Y ( t j ) ) 2 ) . i In P a r s l o w e t a_l ( 1 9 7 9 ) , t h e o b s e r v a t i o n s c o n s i s t e d o f d e n s i t i e s S j ( t ) , j = 1,...,M i n M s t a g e s ( o r g r o u p s of s t a g e s ) and t h e p a r a m e t e r s c o n s i s t e d o f t h e r e s i d e n c e t i m e s T j and m o r t a l i t y r a t e s 0j i n t h e s e s t a g e s ( o r g r o u p s of s t a g e s ) . A model i s t h e n r e q u i r e d w h i c h p r e d i c t s Sj ( t ; ) g i v e n a s e t of p a r a m e t e r s , T and 6 . U n d e r l y i n g t h e s t a g e - s t r u c t u r e of t h e p o p u l a t i o n i s t h e a g e - s t r u c t u r e model w h i c h has t h e form d_Z(a,t) + d_Z(a,t) = - e ( a ) . Z ( a , t ) , where Z ( a , t ) i s t h e p o p u l a t i o n d e n s i t y of age a a t t i m e t and 0(a) i s t h e p e r c a p i t a m o r t a l i t y r a t e f o r i n d i v i d u a l s o f age a T h i s must be c o u p l e d w i t h a f o r m u l a f o r mapping Z ( a , t ) i n t o S j ( t ) . I f a l l i n d i v i d u a l s s p end t h e same t i m e T,- i n s t a g e j , k Z ( a , t ) . d a . Sj ( t ) k--i K More c o m p l i c a t e d f o r m u l a e a r e r e q u i r e d i f i n d i v i d u a l r e s i d e n c e t i m e s a r e a l l o w e d t o v a r y a b o u t a mean v a l u e , Ij . The p r o b l e m i n f i t t i n g s u c h an a g e - s t r u c t u r e model t o o b s e r v a t i o n s i s t h a t t h e i n i t i a l c o n d i t i o n s Z ( a , 0 ) and t h e b oundary c o n d i t i o n s Z ( 0 , t ) a r e r e q u i r e d , but a r e n o t d i r e c t l y a v a i l a b l e . V a r i o u s s i m p l i f y i n g a s s u m p t i o n s can be made t o overcome t h i s d i f f i c u l t y and t h e s e r e s u l t i n a number of s i m p l e r models w h i c h p r e d i c t S j ( t ) . F o u r s u c h m o d els were c o n s i d e r e d by P a r s l o w e t a l ( 1 9 7 9 ) , and t h e i r p e r f o r m a n c e s i n t h e p a r a m e t e r e s t i m a t i o n p r o c e d u r e compared. 153 Any model w h i c h p r e d i c t s Sj ( t ) c an be t h o u g h t o f as a compartment model, w i t h Sj ( t ) b e i n g t h e compartment c o n t e n t s . T h e r e i s a f l u x i n , R j ( t ) , w h i c h r e p r e s e n t s r e c r u i t m e n t from t h e p r e v i o u s s t a g e ( o r from r e p r o d u c t i o n ) and two f l u x e s o u t , one b e i n g r e c r u i t m e n t t o t h e n e x t s t a g e , R j + ) ( t ) , and t h e o t h e r b e i n g m o r t a l i t y . Thus Sj = R j ( t ) - Rj + ) ( t ) - 6 j . S j ( t ) The s i m p l e r models d i f f e r i n t h e way t h e R j ' s a r e c a l c u l a t e d . The l a g model i s e s s e n t i a l l y an a g e - s t r u c t u r e model w i t h f i x e d r e s i d e n c e t i m e s . R e c r u i t m e n t r a t e s i n t o and out of s t a g e j a r e t h e n c o n n e c t e d by t h e r e l a - t i o n s h i p R j + | ( t ) = Rj ( t - T,-) . e x p ( - Bj.'Cj) The b o u n d a r y c o n d i t i o n p r o b l e m a p p e a r s when we t r y t o c a l c u l a t e r e c r u i t m e n t i n t o t h e f i r s t s t a g e , R j ( t ) . T h i s c a n n o t be done e x a c t l y b u t an a p p r o x i m a t e r e c r u i t m e n t r a t e i n t o t h e s e c o n d s t a g e c a n be c a l c u l a t e d i f t h e r e s i d e n c e t i m e i n t h e f i r s t s t a g e i s s h o r t compared w i t h t h e t i m e s c a l e s o f c h a n g e s i n S ^ t ) . Then t h e l i n e a r t r a n s f e r r a t e a p p r o x i m a t i o n R 2 ( t ) = p . S i ( t ) c a n be u s e d , where t h e o b s e r v e d d e n s i t i e s S j ( t ) a r e u s e d t o c o n s t r u c t a d r i v i n g v a r i a b l e . C l e a r l y , i f o b s e r v a t i o n s s t a r t a t t=0, R j ( t ) i s o n l y c a l c u l a b l e a f t e r t = £ T k , so t h a t some o b s e r v a t i o n s i n l a t e r s t a g e s have t o be d i s c a r d e d . I f t h e r e s i d e n c e t i m e s i n a l l s t a g e s a r e assumed t o be s h o r t compared w i t h t i m e s c a l e s o f c h a n g e s i n d e n s i t i e s , t h e l i n e a r 154 t r a n s f e r r a t e a s s u m p t i o n : R j + 1 ( t ) = P j .Sj ( t ) can be made f o r a l l s t a g e s , w i t h S 1 ( t ) a g a i n b e i n g u s e d as a d r i v i n g v a r i a b l e . The r e s u l t i n g ' l i n e a r t r a n s f e r ' model i s s i m p l e (and l i n e a r ) and a l l o b s e r v a t i o n s c an be u s e d . A r a t h e r d i f f e r e n t s o l u t i o n t o t h e b o u n d a r y c o n d i t i o n p r o b l e m was p r o p o s e d by M a n l y ( 1 9 7 4 ) f o r t i m e s e r i e s r e p r e s e n t i n g d i s t i n c t c o h o r t s . M a n l y assumed t h a t r e c r u i t m e n t i n t o e a c h s t a g e f o l l o w s a g a u s s i a n d i s t r i b u t i o n o v e r t i m e . R a t h e r t h a n p r e d i c t S-. ( t ) , t h i s model p r e d i c t s T S k ( t ) by a s s u m i n g a common J im- m o r t a l i t y r a t e f o r t h e s e s t a g e s . E s t i m a t e s o f t h e s i z e , mean t i m e and s p r e a d o f r e c r u i t m e n t a r e o b t a i n e d f o r e a c h s t a g e . A l t h o u g h s t a g e - d e p e n d e n t m o r t a l i t y r a t e s c an be o b t a i n e d , t h e model i s n o t i n t e r n a l l y c o n s i s t e n t i f m o r t a l i t y r a t e s v a r y o v e r s t a g e s . The l a g - M a n l y model u s e s M a n l y ' s a s s u m p t i o n of g a u s s i a n r e c r u i t m e n t i n t o t h e f i r s t s t a g e and t h e l a g - m o d e l ' s r e l a t i o n s h i p between r e c r u i t m e n t i n t o and o u t o f e a c h s t a g e . I t a l l o w s t h e c o n s i s t e n t e s t i m a t i o n o f s t a g e - d e p e n d e n t m o r t a l i t y r a t e s , under t h e a s s u m p t i o n t h a t i n d i v i d u a l r e s i d e n c e t i m e s i n e a c h s t a g e do not v a r y . The p e r f o r m a n c e o f t h e p a r a m e t e r e s t i m a t i o n t e c h n i q u e u s i n g e a c h o f t h e s e models i n t u r n was c o n s i d e r e d f o r d a t a p r o d u c e d by a more complex s i m u l a t i o n model d e s i g n e d t o p r o b e t h e weaknesses of e a c h of t h e m o d e l s . I t was f o u n d t h a t t h e l i n e a r t r a n s f e r model was m a r k e d l y i n f e r i o r t o t h e o t h e r s under a l m o s t a l l 155 c o n d i t i o n s . The o t h e r t h r e e models p r o v i d e d good e s t i m a t e s of r e s i d e n c e t i m e s f o r a range of s a m p l i n g i n t e r v a l s and degrees of stage a g g r e g a t i o n . The la g - M a n l y model performed b e s t , w i t h e s t i m a t i o n e r r o r s of a few p e r c e n t a t most. E s t i m a t i o n of r e s i d e n c e t i m e s by the l a g and la g - M a n l y models was r e l a t i v e l y i n s e n s i t i v e t o l o g - n o r m a l o b s e r v a t i o n e r r o r i n the d a t a , w i t h average e r r o r s l e s s than 25% f o r c o e f f i c i e n t s of v a r i a t i o n up t o 0.4 i n the o b s e r v a t i o n e r r o r s . The e s t i m a t i o n of m o r t a l i t y r a t e s was much l e s s s a t i s f a c t o r y . R e l i a b l e e s t i m a t e s of m o r t a l i t y r a t e i n each stage were not o b t a i n e d f o r any model. The la g - M a n l y model d i d produce r e l i a b l e e s t i m a t e s of average m o r t a l i t y r a t e s i n aggr e g a t e d s t a g e s f o r s m a l l s a m p l i n g i n t e r v a l s under z e r o or low o b s e r v a t i o n e r r o r . When a p p l i e d t o a s e t of CEPEX data r e p r e s e n t i n g Pseudocalanus minutus, u n s a t i s f a c t o r y r e s u l t s were o b t a i n e d f o r a l l models. I t was not c l e a r whether t h i s was due t o u n e x p e c t e d l y h i g h o b s e r v a t i o n e r r o r s i n the d a t a , or t o a v i o l a t i o n of the models' assumption t h a t p o p u l a t i o n parameters a r e c o n s t a n t over t i m e . E s t i m a t e s of secondary p r o d u c t i o n , i n c l u d i n g m o r t a l i t y , can be o b t a i n e d as a by-product of the parameter e s t i m a t i o n p r o c e d u r e , once average i n d i v i d u a l w e i g h t s i n each stage a re s p e c i f i e d . E s t i m a t e s g e n e r a t e d i n t h i s way have been compared w i t h t h e s e o b t a i n e d u s i n g a c l a s s i c a l t e c h n i q u e (Winberg,1968) f o r b o t h s i m u l a t e d d a t a and a d i f f e r e n t s e t of CEPEX d a t a r e p r e s e n t i n g a P a r a c a l a n u s parvus p o p u l a t i o n . E s t i m a t e s of secondary p r o d u c t i o n f o r s i m u l a t e d d a t a were found t o be much 156 more r e l i a b l e than m o r t a l i t y r a t e e s t i m a t e s . The l a t t e r a r e c a l c u l a t e d e s s e n t i a l l y as d i f f e r e n c e s between e s t i m a t e s of t o t a l r e c r u i t m e n t i n t o s u c c e s s i v e s t a g e s , w h i l e secondary p r o d u c t i o n r e p r e s e n t s a weig h t e d average of the s e r e c r u i t m e n t s . 4.1.3 A p p l i c a t i o n t o O.S.P. Data. A l t h o u g h these t e c h n i q u e s were d e v e l o p e d f o r time s e r i e s of d e n s i t i e s i n l i f e h i s t o r y s t a g e s , they a r e e q u a l l y a p p l i c a b l e t o time s e r i e s of d e n s i t i e s i n s i z e c l a s s e s . However, t h e i r a p p l i c a t i o n t o the da t a from O.S.P. has p r e s e n t e d a number of problems due t o the l i f e h i s t o r y s t r a t e g i e s of the dominant copepods, the sample methods employed, and the d e t a i l s of sample a n a l y s i s . The l i f e - h i s t o r y s t r a t e g i e s of Calanus plumchrus and Calanus c r i s t a t u s were d e s c r i b e d i n Chapter 1. Both s p e c i e s a r e r e c r u i t e d as n a u p l i i i n the s u r f a c e waters i n l a t e w i n t e r or e a r l y s p r i n g . I n d i v i d u a l s f e e d and grow i n the s u r f a c e waters u n t i l t h ey r e a c h stage V and have accumulated a l i p i d r e s e r v e . These copepods then l e a v e the s u r f a c e waters and o v e r - w i n t e r below 200m. R e p r o d u c t i o n t a k e s p l a c e a t depth i n the case of Calanus plumchrus ( F u l t o n , 1 9 7 3 ) and n a u p l i i a r e r e c r u i t e d t o s u r f a c e waters the f o l l o w i n g s p r i n g . There i s some s u g g e s t i o n i n the O.S.P. da t a t h a t C. c r i s t a t u s a d u l t s may appear b r i e f l y i n the s u r f a c e waters i n s p r i n g . These c y c l e s a r e q u i t e c o m p a t i b l e w i t h the e s t i m a t i o n t e c h n i q u e s d i s c u s s e d above, as they r e s u l t i n a d i s t i n c t a n n u a l c o h o r t . However, the v e r t i c a l h a u l s a t O.S.P. sample the t o p 150m o n l y so t h a t the d e p a r t u r e of o v e r - w i n t e r i n g copepodids from s u r f a c e w a t e r s appears i n the time s e r i e s as a d e c l i n e which can 157 p o t e n t i a l l y be c o n f u s e d w i t h m o r t a l i t y . T h i s w i l l b a d l y d i s t o r t e s t i m a t e s of m o r t a l i t y r a t e u s i n g the Manly model, u n l e s s a l l o b s e r v a t i o n s taken a f t e r the d e p a r t u r e commences a r e d i s c a r d e d . Even assuming t h e r e was some a p r i o r i way t o choose t h i s t i m e , t h i s would r e s u l t i n an u n a c c e p t a b l e l o s s of i n f o r m a t i o n f o r broad c o h o r t s of the type o b s e r v e d a t O.S.P. O v e r - w i n t e r i n g can be s i m p l y r e p r e s e n t e d i n the l a g - M a n l y model as r e c r u i t m e n t out of the l a r g e s t o b s e r v e d s i z e - c l a s s i n t o a n o ther which i s not ob s e r v e d . T h i s does not e l i m i n a t e the p o t e n t i a l c o n f u s i o n between o v e r - w i n t e r i n g and m o r t a l i t y however, which w i l l appear l a t e r as u n c e r t a i n t y i n r e c r u i t m e n t and m o r t a l i t y r a t e e s t i m a t e s . There i s a l s o a problem i n t h e i n t e r p r e t a t i o n of r e c r u i t m e n t i n the s p r i n g . The s m a l l e s t s i z e c l a s s e s i d e n t i f i e d as C. plumchus or C. c r i s t a t u s have a nominal l e n g t h of 1mm. ( N a u p l i a r s t a g e s a r e not i d e n t i f i e d and are p r o b a b l y p o o r l y r e t a i n e d by the 350 jam n e t ) . I t i s assumed here t h a t a s i z e t h r e s h o l d e x i s t s , due e i t h e r t o net a p e r t u r e s i z e or sample a n a l y s i s t e c h n i q u e s , below which a l l i n d i v i d u a l s a r e i g n o r e d , and above which a l l a r e sampled w i t h c o n s t a n t e f f i c i e n c y . E s t i m a t e s of r e c r u i t m e n t produced here t h e r e f o r e a p p l y t o r e c r u i t m e n t a c r o s s t h i s s i z e t h r e s h o l d . I t i s p o s s i b l e t h a t s i z e c l a s s e s above t h i s t h r e s h o l d a r e sampled w i t h v a r i a b l e e f f i c i e n c y , or t h a t some i n d i v i d u a l s a r e r e c r u i t e d i n t o the t o p 150m at s i z e s above the t h r e s h o l d . To the e x t e n t t h a t e i t h e r o c c u r s , the e s t i m a t e s produced here may be d i s t o r t e d . E s t i m a t e s of secondary p r o d u c t i o n produced here do not i n c l u d e m o r t a l i t y of s m a l l i n d i v i d u a l s but t h i s u s u a l l y makes a n e g l i g i b l e c o n t r i b u t i o n t o t o t a l secondary p r o d u c t i o n (Sonntag and P a r s l o w , 1 9 8 0 ) . 1 158 The above problems can be handl e d by s i m p l e a d j u s t m e n t s of the l a g or lag-Manly models. A t h i r d problem has r e q u i r e d the c r e a t i o n of a new e s t i m a t i o n model. The a n a l y s i s of O.S.P. z o o p l a n k t o n samples has i n v o l v e d a l a r g e number of t e c h n i c i a n s over s e v e r a l y e a r s . W h i l e c a r e has been taken t o ensure c o n s i s t e n c y of c o u n t i n g and s p e c i e s i d e n t i f i c a t i o n ( F u l t o n , 1 9 7 8 ) , the s i z e c l a s s e s used have v a r i e d amongst t e c h n i c i a n s and over t i m e . As a r e s u l t , s e v e r a l d i f f e r e n t s e t s of s i z e c l a s s e s , i n v o l v i n g from one t o f i v e d i f f e r e n t d i v i s i o n s , can be found even w i t h i n one y e a r ' s d a t a f o r one s p e c i e s . Any attempt t o f o r c e these d a t a i n t o a s t a n d a r d s e t of s i z e - c l a s s e s i n o r d e r t o a p p l y the l a g or la g - M a n l y models w i l l i n v o l v e much l o s s of i n f o r m a t i o n , t h rough the b l u r r i n g of some s i z e c l a s s e s and the d i s c a r d i n g of p o o r l y r e s o l v e d samples. An a l t e r n a t e approach has been dev e l o p e d which a v o i d s any l o s s of i n f o r m a t i o n i n u s i n g the a v a i l a b l e d a t a . Rather than assume a s e t of f i x e d r e s i d e n c e t i m e s Tj f o r a unique s e t of s i z e c l a s s e s , a c o n t i n u o u s r e l a t i o n s h i p between age and l e n g t h , a = A ( l ) , i s assumed and p a r a m e t r i s e d . The assumption of g a u s s i a n r e c r u i t m e n t , t o g e t h e r w i t h the a g e - s t r u c t u r e model, a l l o w the p r e d i c t i o n of an a g e - d i s t r i b u t i o n Z ( a , t ) over t i m e , which can then be t r a n s l a t e d i n t o a s i z e - d i s t r i b u t i o n Z ' ( l , t ) . For each sample, i n t e g r a t i o n of the p r e d i c t e d s i z e - d i s t r i b u t i o n between a p p r o p r i a t e l i m i t s p r o v i d e s p r e d i c t i o n s of d e n s i t i e s i n s i z e c l a s s e s which e x a c t l y match the s i z e c l a s s e s used i n the a n a l y s i s of t h a t sample. A power law i s chosen f o r A ( l ) : a = a 1 . ( l l f - l X ) 159 where 1 R i s the t h r e s h o l d s i z e below which i n d i v i d u a l s a r e not sampled. The parameters a± and y a r e e s t i m a t e d . Note t h a t i f weight i s p r o p o r t i o n a l t o l 3 and growth (dW/dt) p r o p o r t i o n a l t o l 2 , ^ = 1. I f growth i s a s m a l l e r power of l e n g t h (eg S t e e l e and F r o s t ( 1 9 7 7 ) suggest i n g e s t i o n may be p r o p o r t i o n a l t o l e n g t h ) , X i s l a r g e r than 1. As e x p o n e n t i a l growth i s approached, y approaches z e r o . The m o r t a l i t y r a t e law e ( a ) , or e q u i v a l e n t l y 0 ( 1 ) , must a l s o be p a r a m e t r i z e d . W h i l e some dependence of m o r t a l i t y r a t e on s i z e might be e x p e c t e d (eg Cush i n g , 1 9 7 6 ) , s c a t t e r i n the d a t a , the l a c k of s i z e - c l a s s r e s o l u t i o n and c o n f u s i o n due t o o v e r - w i n t e r i n g d i s c o u r a g e d any attempt t o go beyond a c o n s t a n t m o r t a l i t y r a t e . The e x a c t parameter e s t i m a t i o n p r o c e d u r e i s t h e r e f o r e as f o l l o w s . G i v e n i n i t i a l guesses a t t o t a l r e c r u i t m e n t , R T, spread of r e c r u i t m e n t <T, mean time of r e c r u i t m e n t jj and m o r t a l i t y r a t e e, the a g e - d i s t r i b u t i o n Z ( a , t ) i s g i v e n by Z ( a , t ) = R T . e x p ( - ( t - a - p ) 2 / ( 2 . a 2 ) - ©.a) / (7277.(7) Gi v e n v a l u e s f o r ai and #, the l e n g t h - d i s t r i b u t i o n can be deduced. The l e n g t h s a t r e c r u i t m e n t , 1 R , and a t o v e r - w i n t e r i n g , 1 F , a r e f i x e d f o r each s p e c i e s based on minimum and maximum ob s e r v e d l e n g t h s . On each day, g i v e n a s e t of M s i z e c l a s s e s w i t h nominal s i z e s 1 A ,1 2 , . . . , 1 M , a c o r r e s p o n d i n g s e t of s i z e c l a s s l i m i t s i s d e f i n e d by U = IR 160 l j = ( l ] + l > , )/2 j = l , . . . , M - l IM = I F r'J P r e d i c t e d d e n s i t i e s S j ( t ) = J Z ' ( l , t ) . d l can then be compared w i t h o b s e r v e d d e n s i t i e s s j ( t ) and the parameters c o r r e s p o n d i n g t o minimum SSQ e r r o r s o b t a i n e d u s i n g the Marquardt a l g o r i t h m (Marquardt,1963). 4.1.4 S t a t i s t i c a l C o n s i d e r a t i o n s . In view of the l a c k of comparable e s t i m a t e s of the parameters sought h e r e , an attempt was made t o p r o v i d e a t l e a s t an i n d i c a t i o n of s t a t i s t i c a l u n c e r t a i n t y i n e s t i m a t e d p a r a m e t e r s . As a p r e l i m i n a r y s t e p , the n a t u r e of v a r i a b i l i t y i n the z o o p l a n k t o n samples from O.S.P. was i n v e s t i g a t e d . A n a l y s i s of 10 r e p l i c a t e samples taken w i t h i n a s h o r t time p e r i o d (3 hours) on 10th June, 1978, i n d i c a t e d t h a t the v a r i a n c e i n s p e c i e s c o u n t s i n c r e a s e d w i t h the mean a t low d e n s i t i e s ( P o i s s o n ) and w i t h the mean squared a t h i g h d e n s i t i e s ( l o g - n o r m a l ) w i t h an a s y m p t o t i c c o e f f i c i e n t of v a r i a t i o n of 0.2. F o l l o w i n g Barnes (1952), the n o r m a l i s i n g t r a n s f o r m a t i o n Y' = s i n h " 1 (/3.JI)//3 w i t h j8= .2, was employed and the SSQ e r r o r s between t r a n s f o r m e d p r e d i c t i o n s and o b s e r v a t i o n s was m i n i m i z e d . ( E s t i m a t e s i n i t i a l l y o b t a i n e d w i t h o u t t r a n s f o r m i n g the da t a showed g r e a t v a r i a b i l i t y from year t o year and i n s p e c t i o n of p r e d i c t e d timestreams showed a tendency t o f i t one or two anomalous h i g h o b s e r v a t i o n s c l o s e l y and i g n o r e the r e s t . ) 161 The.exact s t a t i s t i c a l d i s t r i b u t i o n of e s t i m a t e s o b t a i n e d through n o n - l i n e a r , l e a s t - s q u a r e s t e c h n i q u e s can not g e n e r a l l y be found. However, an approximate d i s t r i b u t i o n can be o b t a i n e d by a p p r o x i m a t i n g the n o n - l i n e a r f u n c t i o n of parameters by a l i n e a r t h i s a p p r o x i m a t i o n , i f the s t a t i s t i c a l model i s Y,- = f j (p) + £\ e v a l u a t e d a t p (Benson,1978). The a c c u r a c y of the a p p r o x i m a t i o n c l e a r l y depends on the r e l a t i v e magnitude of the l i k e l y e r r o r i n p and the s c a l e of n o n - l i n e a r i t i e s i n i. 4.1.5 R e s u l t s f o r Calanus plumchrus . I t was p o s s i b l e t o a p p l y the parameter e s t i m a t i o n t e c h n i q u e s t o s i z e - s t r u c t u r e d d a t a f o r C. plumchrus i n the y e a r s 1969,1970,1973,1975,1976 and 1977. In 1971 and 1972, s i z e - s t r u c t u r e i n f o r m a t i o n was e i t h e r not p r e s e n t or i n s u f f i c i e n t t o a l l o w parameter e s t i m a t i o n . In 1974, t r a n s i t i o n water appeared i n mid-summer and t h i s , combined w i t h a 6-week d a t a gap, made i t i m p o s s i b l e t o f i t a c o h o r t model s e n s i b l y . S i x parameters were e s t i m a t e d i n each y e a r . These were t o t a l r e c r u i t m e n t R T, mean r e c r u i t m e n t time ju, s t a n d a r d d e v i a t i o n of r e c r u i t m e n t cr , i n d i v i d u a l r e s i d e n c e time i n samples T, age- l e n g t h power y and m o r t a l i t y r a t e ©. The t o t a l r e s i d e n c e t i m e , T, e s t i m a t e d i n p l a c e of the c o n s t a n t a t , i s g i v e n by f u n c t i o n i n the neighborhood of the parameter e s t i m a t e . Under . . . £• i i d N ( 0 , ( 7 e 2 ) , P* i s the t r u e parameter s e t and p i s the l e a s t - s q u a r e s e s t i m a t e , p -p* has a m u l t i v a r i a t e normal d i s t r i b u t i o n w i t h c o v a r i a n c e m a t r i x -i 162 These s i x parameters c o u l d be d i v i d e d i n t o two groups based on the p r o p e r t i e s of the approximate c o v a r i a n c e m a t r i c e s . One s e t c o n s i s t e d of u, 0 and T. These v a r i a b l e s , a l l r e l a t e d t o t i m i n g , were not h i g h l y c o r r e l a t e d w i t h one another or w i t h members of the second s e t , c o n s i s t i n g of R T, 0 and Jf. E s t i m a t e s f o r t h i s second s e t were v e r y h i g h l y c o r r e l a t e d w i t h one a n o t h e r , c o r r e l a t i o n c o e f f i c i e n t s of 0.98 b e i n g common. There was e s s e n t i a l l y one degree of freedom i n v o l v i n g these parameters which c o u l d not be r e s o l v e d by the e s t i m a t i o n t e c h n i q u e from the s e time s e r i e s . The parameter e s t i m a t e s a r e g i v e n i n T a b l e IV. The 95% c o n f i d e n c e i n t e r v a l s g i v e n i n t h a t t a b l e f o r u, <J and T a r e a c t u a l l y the l i m i t s of a 95% c o n f i d e n c e r e g i o n f o r a l l 6 parameters d e t e r m i n e d by F^„.ff s t a t i s t i c s . . Because of the h i g h c o r r e l a t i o n amongst R T,e and y, i n d i v i d u a l c o n f i d e n c e i n t e r v a l s a r e u n i n f o r m a t i v e and i n s t e a d p r o j e c t i o n s of the 95% c o n f i d e n c e r e g i o n s on the (R T,e) and (0, y ) p l a n e s have been p r e s e n t e d i n F i g 44. Note t h a t , because of the c o n f u s i o n of ©, R T and y, i t became n e c e s s a r y t o c o n s t r a i n 9 t o be n o n - n e g a t i v e i n some y e a r s . In f a c t , as z e r o m o r t a l i t y r a t e s a r e u n l i k e l y i n n a t u r e , the c o n s t r a i n t 6 ^ 0.005 day" 1 was imposed and c o n f i d e n c e r e g i o n s f o r (R T,9) and ( 6 , ) are t r u n c a t e d a c c o r d i n g l y . The mean r e c r u i t m e n t time o c c u r s i n A p r i l and e a r l y May w i t h a p p a r e n t l y e a r l i e r r e c r u i t m e n t i n 1975 and 1976. The s t a n d a r d d e v i a t i o n of r e c r u i t m e n t i n t o the s m a l l e s t s i z e c l a s s i s f a i r l y c o n s t a n t a t about 30 days, w i t h a l a r g e r s p r e a d i n 1975. Residence t i m e s above 150m a r e re m a r k a b l y c o n s i s t e n t from y e a r - 163 T a b l e IV. P a r a m e t e r e s t i m a t e s f o r C a l a n u s p l u m c h r u s . Y e a r (days) CT(days) T ( d a y s ) R T ( i n d . m ~ 3 ) e ( d a y _ 1 ) 1969 121.±27. 27.± 9. 56.±34. 50. 0.005 1.4 1970 126.±21. 31.± 5. 58.±23. 238. 0.005* 1.0 1973 112.±18. 24.± 7. 66.±31. 318. 0.040 2.2 1975 93.±30. 48.±12. 65.±65. 72. 0.005* 0.45 1976 96.±29. 34.± 8. 64.±37. 141. 0.005* 1.0 1977 105.±10. 31.± 4. "66. ±20. 498. 0.027 1.6 * These v a l u e s were f r o z e n t o a l l o w c o n v e r g e n c e . 164 165 Figure 44b. Projection of approximate 95% confidence regions for Calanus plumchrus parameter estimates on (y,0) plane. 166 t o - y e a r c o n s i d e r i n g the l a r g e c o n f i d e n c e i n t e r v a l s . The sampled p o p u l a t i o n of C. plumchrus above 150m a t O.S.P. can t h e r e f o r e be c h a r a c t e r i z e d as a broad c o h o r t , r e c r u i t e d over 2 months or more from March t o May, of i n d i v i d u a l s which spend about 60 days i n the s u r f a c e l a y e r and l e a v e d u r i n g June and J u l y . A d e t a i l e d account of the l i f e h i s t o r y of C. plumchrus i n the S t r a i t of G e o r g i a was g i v e n by F u l t o n ( 1 9 7 3 ) . In t h a t s t u d y , deep v e r t i c a l h a u l s a l l o w e d the f u l l l i f e c y c l e , i n c l u d i n g m a t u r a t i o n and r e p r o d u c t i o n , t o be f o l l o w e d . F u l t o n r e p o r t e d t h a t maximum egg p r o d u c t i o n o c c u r r e d on March 5th and maximum abundance of n a u p l i i i n s u r f a c e waters on March 1 7 t h . The peak abundance of c o p e p o d i t e stage V o c c u r r e d 62 days l a t e r , on May 18 t h . H i s schematic diagram s u g g e s t s t h a t a broad c o h o r t , e x c e e d i n g 2 months i n w i d t h , i s r e c r u i t e d i n t o the t o p 150m d u r i n g F e b r u a r y , March and A p r i l and d e p a r t s as stage V cop e p o d i d s i n May, June and J u l y . Comparison of h i s r e s u l t s w i t h the e s t i m a t e s produced here i s d i f f i c u l t as n a u p l i i a r e not sampled a t O.S.P. and our r e c r u i t m e n t e s t i m a t e s r e f e r t o e a r l y c o p e p o d i d s . A l s o , depending on the importance of m o r t a l i t y , h i s peak t o peak times may not be comparable t o the mean r e c r u i t m e n t and r e s i d e n c e times e s t i m a t e d here u s i n g a dynamic model. I g n o r i n g the second problem, i t i s c l e a r t h a t r e c r u i t m e n t of n a u p l i i i n the S t r a i t of G e o r g i a p r e c e d e s r e c r u i t m e n t of e a r l y c o pepodids a t O.S.P. which i s not s u r p r i s i n g . In f a c t , i t might be e x p e c t e d t h a t r e c r u i t m e n t of n a u p l i i would o c c u r l a t e r a t O.S.P. as w e l l , as the s p r i n g i n c r e a s e i n p r i m a r y p r o d u c t i o n o c c u r s e a r l i e r i n the S t r a i t of G e o r g i a ( P a r s o n s , 1 9 6 5 ) . The p e r i o d of 62 days between peak 167 n a u p l i a r and peak CV abundance i n the S t r a i t of G e o r g i a i s the same as the e s t i m a t e d r e s i d e n c e t i m e s from e a r l y c o p e p o d i t e t o stage V a t O.S.P. I f t h e s e a r e t e c h n i c a l l y comparable, growth i s s l o w e r a t O.S.P. which would h a r d l y be s u r p r i s i n g , i n view of the much lower p h y t o p l a n k t o n s t a n d i n g s t o c k t h e r e . The l a r g e u n c e r t a i n t y i n R T , 0 and V i s u n f o r t u n a t e , as a knowledge of m o r t a l i t y r a t e s would be of g r e a t i n t e r e s t . The u n c e r t a i n t y a r i s e s from a c o m b i n a t i o n of the c o n f u s i o n of m o r t a l i t y and o v e r - w i n t e r i n g d e p a r t u r e , o b s e r v a t i o n e r r o r s , a r e l a t i v e l y s m a l l number of c o a r s e s i z e c l a s s e s and the broad c o h o r t w i d t h . For example, i n c r e a s i n g the m o r t a l i t y r a t e i n the model would n o r m a l l y have the e f f e c t of d e c r e a s i n g the number of i n d i v i d u a l s i n l a r g e r s i z e c l a s s e s . However, because of the broad c o h o r t w i d t h , t h i s e f f e c t can be c o u n t e r a c t e d by i n c r e a s i n g t o t a l r e c r u i t m e n t and i n c r e a s i n g X , which i n c r e a s e s the r e l a t i v e d u r a t i o n and c o n s e q u e n t l y the r e l a t i v e abundance, of l a r g e r s i z e c l a s s e s . Note t h a t i n 1973 and 1977, the optimum parameter s e t o c c u r s f o r p o s i t i v e & w h i l e i n the o t h e r y e a r s , 0 must be c o n s t r a i n e d t o be p o s i t i v e . However, the c o n f i d e n c e r e g i o n s i n the ( 0 ,V ) p l a n e show s i m i l a r o r i e n t a t i o n and p o s i t i o n , so t h a t some o v e r l a p o c c u r s amongst a l l y e a r s . A range f o r 0 of 0.0 t o 0.1 day" 1 i s i m p l i e d by the c o n f i d e n c e r e g i o n s . Much s m a l l e r c o n f i d e n c e r e g i o n s can be o b t a i n e d f o r R T and e by f i x i n g the growth law parameter Y. For V = 1, low v a l u e s of R T, r a n g i n g from between 200 and 450 ind/m 3 i n 1977 t o between 25 and 60 ind/m 3 i n 1969, a r e e s t i m a t e d . H i g h e r v a l u e s of R T and 6 a r e o b t a i n e d f o r Y= 2. 168 4.1.5 R e s u l t s f o r C. c r i s t a t u s . Parameter e s t i m a t e s c o u l d be o b t a i n e d f o r the y e a r s 1969,1970,1974,1975,1976 and 1977. The y e a r s 1971 and 1972 were a g a i n o m i t t e d f o r l a c k of s i z e - s t r u c t u r e i n f o r m a t i o n . As C. c r i s t a t u s i s p r e s e n t l a t e r i n the year than C. plumchrus, a f a l l i n t r u s i o n of t r a n s i t i o n water i n 1973 i n t e r r u p t e d the time s e r i e s f o r t h i s s p e c i e s . The t e c h n i q u e was a p p l i e d t o 1974 but the e a r l y presence of t r a n s i t i o n water c l e a r l y d i s t o r t s parameter e s t i m a t e s . In g e n e r a l , the C. c r i s t a t u s time s e r i e s were not as ' w e l l - behaved' as those f o r C. plumchrus and e s t i m a t e s show g r e a t e r u n c e r t a i n t y . E s t i m a t e s of u, <T and T are g i v e n i n Ta b l e V. There i s c l e a r l y much l e s s c o n s i s t e n c y from year t o year i n mean r e c r u i t m e n t time f o r t h i s s p e c i e s , w i t h u r a n g i n g from day 67 i n 1977 t o day 142 i n 1970. (The v e r y l a t e r e c r u i t m e n t i n 1974 i s due t o t r a n s i t i o n water i n t r u s i o n . ) D i s c o u n t i n g 1974, the spre a d of r e c r u i t m e n t i s s i m i l a r t o t h a t f o r C. plumchrus, r a n g i n g from 20 days i n 1977 t o 42 days i n 1970. Great v a r i a b i l i t y i s shown i n r e s i d e n c e t i m e s , w i t h T r a n g i n g . f r o m 83 and 89 days i n 1976 and 1969, t o 213 days i n 1977. No r e a l l y c o n s i s t e n t p i c t u r e of the l i f e h i s t o r y of C. c r i s t a t u s emerges from t h i s a n a l y s i s . E x c l u d i n g 1974, i t can be s a i d t h a t a broad c o h o r t i s r e c r u i t e d over a p p r o x i m a t e l y a 2 month p e r i o d a t some time between F e b r u a r y and June and l e a v e s a t some time between June and Oc t o b e r . In view of the l a r g e a pproximate c o n f i d e n c e i n t e r v a l s , t h i s apparent a n n u a l v a r i a b i l i t y may be a s t a t i s t i c a l a r t i f a c t . E s p e c i a l l y w i t h r e g a r d t o the v a r i a t i o n i n r e s i d e n c e t i m e , i t i s i n t e r e s t i n g t h a t 169 T a b l e V. P a r a m e t e r e s t i m a t e s f o r C a l a n u s c r i s t a t u s . Y e a r yt/(days) <7(days) r ( d a y s ) T -3 R ( i n d . m ) e ( d a y ~ 1 ) 1969 89.±45. 24.±17. 89.± 61. 9.5 0.008 * 0.5 1970 142.±25. 42.±13. 119.±100. 97. 0.034 1.4 1974 188.±29. 18.± 9. 53.± 39. 16. 0.005* 3.0 1975 100.±35. 34.±19. 134.±171. 34. 0.005* 0.9 1976 99.±22. 34.± 8. 83.± 41. 118. 0.020 * 0.5 1977 67.±27. 20.±15. 213.± 84. 12. 0.027 1.7 * These v a l u e s were f r o z e n t o a l l o w c o n v e r g e n c e . 170 o b s e r v a t i o n s of C. c r i s t a t u s i n l a t e summer and f a l l a r e o f t e n v e r y low except f o r r a r e , h i g h o b s e r v a t i o n s . T h i s may be due t o s p a t i a l a g g r e g a t i o n , as l a t e s t a g e s of C. c r i s t a t u s a r e l a r g e (7- 8mm) and may show a c t i v e swarming b e h a v i o u r (Nemoto,1957). • D i u r n a l v e r t i c a l m i g r a t i o n by t h e s e l a t e s t a g e s has a l s o been r e p o r t e d (Marlowe and M i l l e r , 1 9 7 5 ) and i t i s q u i t e p o s s i b l e t h a t they a r e p o o r l y sampled by the 150m d a y l i g h t v e r t i c a l h a u l s . P r o j e c t i o n s of the 95% c o n f i d e n c e r e g i o n s on the (© ,Y ) and ( R T , 0 ) p l a n e s are g i v e n i n F i g 45. As f o r C. plumchrus, h i g h c o r r e l a t i o n s were found between R T and 0 i n a l l y e a r s . However, y was not h i g h l y c o r r e l a t e d w i t h R T or G i n 1970,1975 and.1977. In t h e s e y e a r s , numbers of l a r g e (ca 8mm) i n d i v i d u a l s appeared b e f o r e day 100, c o i n c i d e n t w i t h or e a r l - i e r than s m a l l i n d i v i d u a l s . These were re g a r d e d as p a r t of a s e p a r a t e c o h o r t , and a c o h o r t s t a r t i n g w i t h s m a l l s i z e c l a s s e s was d i s t i n g u i s h e d by eye. The r e s u l t i n g time s e r i e s showed a c l e a r e r l a g i n the appearance of l a r g e r s i z e c l a s s e s and t h i s would be e x p e c t e d t o reduce c o n f u s i o n between y and 0 . To o b t a i n convergence, i t was n e c e s s a r y t o f r e e z e V ( a t 0.5) i n 1969 and 1976, and t o f r e e z e O (at 0.005 d a y " 1 ) i n 1974. The s i m u l t a n e o u s appearance of s m a l l and i n t e r m e d i a t e s i z e d i n d i v i d u a l s f o l l o w i n g a t r a n s i t i o n water i n t r u s i o n i n 1974 r e s u l t e d i n an anomalously h i g h e s t i m a t e of #. O t h e r w i s e , low e s t i m a t e s of ^, i n the range 0.5 t o 2.0, were o b t a i n e d . A g a i n , the c o n f i d e n c e r e g i o n s encompass a wide range of ©, from 0.005 day " 1 t o 0.07 d a y " 1 , e x c l u d i n g 1974. F i x i n g the v a l u e of X r e s u l t s i n s m a l l e r ranges i n 1969 and 1976, but has l i t t l e e f f e c t i n t hose y e a r s where ( e,y ) c o r r e l a t i o n s a r e low. The e s t i m a t e s 171 Figure 45. Projection of approximate 95% confidence regions for Calanus cristatus parameter X estimates on (a) (6,R ) plane and (b) plane. 172 Figure 45a. 173 174 of r e c r u i t m e n t i n 1970 and 1976 a r e a p p r o x i m a t e l y 10 t i m e s those f o r 1969 and 1977. E s t i m a t e s of r e c r u i t m e n t f o r C. c r i s t a t u s a r e low compared w i t h those f o r C. plumchrus , except i n 1976. 4.1.7 Other spec i e s . The next major c o n t r i b u t o r t o h e r b i v o r e biomass a t O.S.P. i s a l s o a l a r g e copepod, E u c a l a n u s b u n g i . The l i f e h i s t o r y of t h i s s p e c i e s i s not as c l e a r from o b s e r v a t i o n s . I n d i v i d u a l s t y p i c a l l y appear i n the 150m v e r t i c a l h a u l s i n i n t e r m e d i a t e (2.5 t o 5 mm) s i z e c l a s s e s i n March. There i s a f a i r l y c l e a r p r o g r e s s i o n through t o the l a r g e s t (6.5 t o 7 mm) s i z e c l a s s e s by June or J u l y . In J u l y or August, l a r g e numbers of s m a l l (1.5 t o 2.5mm) i n d i v i d u a l s appear, and the l a r g e i n d i v i d u a l s d i s a p p e a r . Growth o c c u r s t h r o u g h t o i n t e r m e d i a t e s i z e c l a s s e s (3 t o 5 mm) but these i n d i v i d u a l s d i s a p p e a r by the end of O c t o b e r . These o b s e r v a t i o n s a r e m o s t l y c o n s i s t e n t w i t h a l i f e h i s t o r y f o r E u c a l a n u s which i n v o l v e s o v e r - w i n t e r i n g below 150m as e a r l y c o p e p o d i t e s t a g e s , r a t h e r slow growth i n the e u p h o t i c zone t o mature by June or J u l y and p r o d u c t i o n of a new g e n e r a t i o n which grows t o o v e r - w i n t e r i n g s i z e by l a t e f a l l . T h i s p i c t u r e i s c o m p l i c a t e d by the appearance of s m a l l numbers of s m a l l (1.5 t o 2.5 mm) i n d i v i d u a l s i n s p r i n g . Whether some i n d i v i d u a l s o v e r - w i n t e r a t t h i s s i z e , or whether some r e p r o d u c t i o n t a k e s p l a c e below 150m i n the s p r i n g , i s not c l e a r . A s p r i n g b r e e d i n g p e r i o d has been r e p o r t e d f o r E u c a l a n u s i n c o a s t a l w a t e r s (Krause and L e w i s , 1979). In any c a s e , i t was not p o s s i b l e t o a p p l y the parameter e s t i m a t i o n t e c h n i q u e s t o E u c a l a n u s d a t a . The growth of Euc a l a n u s from e a r l y t o l a t e c o p e p o d i t e s t a g e s i s i n t e r r u p t e d by the o v e r - 175 w i n t e r i n g p e r i o d . Both h a l v e s of the growth p e r i o d u s u a l l y i n v o l v e o n l y 2 s i z e c l a s s e s , so t h e r e i s a l a c k of s i z e c l a s s i n f o r m a t i o n . Another problem i s t h a t i n d i v i d u a l s may o v e r - w i n t e r a t a range of l e n g t h s , so t h a t o v e r - w i n t e r i n g d e p a r t u r e and s p r i n g r e c r u i t m e n t may a f f e c t a number of s i z e c l a s s e s . Numbers of E u c a l a n u s a re t y p i c a l l y low (5 - 15 ind/m 3) and i t s c o n t r i b u t i o n t o s t a n d i n g s t o c k and secondary p r o d u c t i o n i s p r o b a b l y s m a l l compared w i t h t h a t of C. c r i s t a t u s and C. plumchrus. Other h e r b i v o r o u s copepods a t O.S.P. a r e of s m a l l ( O i t h o n a , P s e u d o c a l a n u s ) or i n t e r m e d i a t e (Calanus p a c i f i c u s , M e t r i d i a p a c i f i c a ) s i z e and r a r e l y c o n t r i b u t e s i g n i f i c a n t l y t o s t a n d i n g s t o c k by w e i g h t . T h e i r p o p u l a t i o n parameters would s t i l l be of i n t e r e s t , of c o u r s e , but da t a f o r thes e s p e c i e s were not.adequate f o r parameter e s t i m a t i o n . P a r t of the problem i s t h a t t h e i r g e n e r a t i o n time i s s h o r t e r compared w i t h s a m p l i n g i n t e r v a l s but the main reason i s t h a t the s i z e c l a s s e s used a r e too c o a r s e , so t h a t i n d i v i d u a l s a r e o f t e n a s s i g n e d o n l y t o one of two s i z e c l a s s e s . In the case of O i t h o n a and Ps e u d o c a l a n u s , , o n l y the l a r g e s t i n d i v i d u a l s a r e r e t a i n e d by the 350 um mesh n e t . 4.1.8 Secondary p r o d u c t i o n e s t i m a t e s . E s t i m a t e s of secondary p r o d u c t i o n can be o b t a i n e d u s i n g the above t e c h n i q u e , once a w e i g h t - l e n g t h r e l a t i o n s h i p i s p r e s c r i b e d . The weight or carbon c o n t e n t of i n d i v i d u a l copepods from O.S.P. has not been measured d i r e c t l y and l i t e r a t u r e v a l u e s were i n i t i a l l y used t o d e f i n e a w e i g h t - l e n g t h r e l a t i o n s h i p . An average wet weight of 4 mg was r e p o r t e d by F u l t o n ( 1 9 7 3 ) f o r 4.5 mm C. plumchrus stage V. I f i t i s assumed t h a t W i s p r o p o r t i o n a l t o l 3 , t h i s o b s e r v a t i o n c o r r e s p o n d s t o W = .044.1 3. F u l t o n 176 (undated) g i v e s a g e n e r a l w e i g h t - l e n g t h r e l a t i o n s h i p f o r copepod p o p u l a t i o n s i n the S t r a i t of G e o r g i a : W = .068. 1 2" 4 5 which y i e l d s a wet weight of 2.7 mg f o r a 4.5 mm C. plumchrus. Taguchi and I s h i i (1970) r e p o r t wet w e i g h t s r a n g i n g from 2.21 t o 4.55 mg/ind and l e n g t h s of 5 and 4.5 mm r e s p e c t i v e l y f o r C. plumchrus s t a g e V, r e s u l t i n g i n c o e f f i c i e n t s of .02 t o 0.05 mg.mnr 3. Use of any of the s e formulae w i t h O.S.P. s i z e - s t r u c t u r e d d a t a r e s u l t e d i n severe o v e r - e s t i m a t i o n of sample wet w e i g h t s . An attempt was made t o d e r i v e a l e n g t h - w e i g h t r e l a t i o n s h i p a p p r o p r i a t e t o O.S.P. from the s i z e - s t r u c t u r e d a t a as f o l l o w s . For each sample, a l l the copepod d a t a was r e v i e w e d , and the number of i n d i v i d u a l s c ounted i n each s i z e c l a s s added t o one of 9 1mm 'super' s i z e c l a s s e s a c c o r d i n g t o i t s nominal s i z e . A m u l t i p l e l i n e a r r e g r e s s i o n of sample wet w e i g h t s on numbers N? i n each of these s u p e r - s i z e c l a s s e s was then performed : t h a t i s , the c o e f f i c i e n t s Wj i n w = I N , . W J were e s t i m a t e d . An o b v i o u s c r i t i c i s m i s t h a t W* i n c l u d e s wet w e i g h t s of organisms o t h e r than copepods : w h i l e samples h a v i n g i n t e r m e d i a t e t o h i g h wet w e i g h t s a r e p r e d o m i n a n t l y (90%) copepods, the c o e f f i c i e n t s Wj may ten d t o be s l i g h t l y h i g h . These c o e f f i c i e n t s a r e i n t e r p r e t e d as average w e i g h t s f o r each 1mm s i z e c l a s s and a r e p l o t t e d a g a i n s t l e n g t h on a l o g - l o g s c a l e i n F i g 46. A p a r t from anomalous r e s u l t s from two,, almost empty, s i z e c l a s s e s , the c o e f f i c i e n t s f a l l c l o s e t o a l i n e h a v i n g the s l o p e 2.45 suggested by F u l t o n (undated) and an i n t e r c e p t 177 -4 -5 - 6 -7 -8 - 9 + t t -10 0 0 5 10 15 l n ( l i ) 20 2 5 Figure 46. Regression coefficients w\ (±1 standard error) vs corresponding lengths 1^ on log-log scale. Line corresponds to W = 0.033.1 6 mm and 7 icm size classes. 2.45 , xgnores 178 c o r r e s p o n d i n g t o W = . 033 . 1 2 4 5 . The c o e f f i c i e n t 0.033 i s h a l f the v a l u e found by F u l t o n f o r the S t r a i t of G e o r g i a . G i v e n the c o a r s e n a t u r e of the s i z e c l a s s e s used f o r the O.S.P. d a t a and the s e n s i t i v i t y of t h i s c o e f f i c i e n t t o any b i a s i n a s s i g n i n g l e n g t h s , i t may be r a s h t o c o n c l u d e t h a t O.S.P. copepods a r e t h i n n e r than t h e i r c o a s t a l c o u s i n s on the b a s i s of t h i s r e s u l t . W h i l e the lower food c o n c e n t r a t i o n s and tem p e r a t u r e s a t O.S.P. might r e a s o n a b l y be e x p e c t e d t o r e s u l t i n lower growth r a t e s (Parsons e t a l ,1977), i t i s not c l e a r t h a t lower body we i g h t s f o r a g i v e n body l e n g t h s h o u l d r e s u l t . In any c a s e , the s t a t i s t i c a l r e l a t i o n s h i p i s used here i n the e s t i m a t i o n of secondary p r o d u c t i o n based on O.S.P. d a t a . E s t i m a t e s of secondary p r o d u c t i o n and 95% C.I. f o r thes e e s t i m a t e s a r e g i v e n i n Table V I . The u n c e r t a i n t y i n the s e e s t i m a t e s i s c l e a r l y much l e s s than u n c e r t a i n t y i n m o r t a l i t y and r e c r u i t m e n t p a r a m e t e r s , as found f o r s i m u l a t e d d a t a by Sonntag and P a r s l o w ( 1 9 8 0 ) . Large c o n f i d e n c e l i m i t s a r e o b t a i n e d f o r 1975 and 1976 f o r C. plumchrus, but the l i m i t s a r e s m a l l enough i n o t h e r y e a r s t o c l e a r l y d i s t i n g u i s h a year of low p r o d u c t i o n such as 1969 from y e a r s of h i g h p r o d u c t i o n such as 1970 or 1977. Note t h a t , w h i l e h i g h e s t r e c r u i t m e n t e s t i m a t e s f o r C. plumchrus were o b t a i n e d f o r 1973 and 1977, h i g h e r m o r t a l i t y r a t e s i n the s e y e a r s reduced e s t i m a t e d secondary p r o d u c t i o n t o 4 t h and 2nd rank r e s p e c t i v e l y . E s t i m a t e s of secondary p r o d u c t i o n f o r C. c r i s t a t u s showed a s i m i l a r range t o those f o r C. plumchrus. There i s no d i s c e r n i b l e p a t t e r n i n the a b s o l u t e or r e l a t i v e c o n t r i b u t i o n s of the two s p e c i e s t o t o t a l secondary p r o d u c t i o n . Both a r e u n u s a l l y low i n Table VI. -2 Secondary production estimates (g wet wt.m ) Year C_. plumchrus £. c r i s t a t u s T o t a l . 1969 26±10 21+9 47±13 1970 119±45 56+25 165+51 1973 57±17 — -- 1974 -- 51±18 1975 35±35 20+11 55±37 1976 69±54 146±61 215±81 1977 114±22 15+3 129±22 180 1969 and 1975 and both a r e h i g h i n 1970 and 1976. In 1977, C. plumchrus i s v e r y h i g h and C. c r i s t a t u s v e r y low. W h i l e s m a l l e r s p e c i e s a r e not i n c l u d e d , t h e s e l a r g e copepods ar e g e n e r a l l y assumed t o dominate secondary p r o d u c t i o n and i t i s i n t e r e s t i n g t o t r e a t t h e i r combined t o t a l s as a c o n s e r v a t i v e e s t i m a t e of t o t a l secondary p r o d u c t i o n . There i s c o n s i d e r a b l e (>4 f o l d ) v a r i a t i o n over the 5 y e a r s f o r which t o t a l s a r e a v a i l a b l e . I f the s e e s t i m a t e s a r e c o n v e r t e d t o carbon u s i n g a v a l u e of .05 f o r the carbon:wet weight r a t i o , ( P a r sons e t a l ,1977), e s t i m a t e s r a n g i n g from 2.4 t o 10.7 g C.m^.yr" 1 are o b t a i n e d . For comparison, trophodynamic c o n s i d e r a t i o n s l e d M c A l l i s t e r ( 1 9 6 9 ) t o a 'most l i k e l y ' e s t i m a t e of 13 g C.m- 2.yr _ 1, w i t h a minimum v a l u e of 2 g C . i r r 2 . y r _ 1 , depending on assumed r e s p i r a t i o n r a t e s . 4.2 Biomass Model f o r Z o o p l a n k t o n . 4.2.1 I n t r o d u c t i o n . In Chapter 1, s i m p l e biomass models of the p h y t o p l a n k t o n - z o o p l a n k t o n i n t e r a c t i o n a t O.S.P. were c o n s i d e r e d . A model i n v o l v i n g g r a z i n g t h r e s h o l d s appeared t o be c a p a b l e of m i m i c k i n g the s e a s o n a l c y c l e a t O.S.P.. i n a q u a l i t a t i v e sense. U s i n g the r e s u l t s of the da t a a n a l y s i s of C h a p t e r s 3 and 4, an attempt w i l l be made t o a s s i g n v a l u e s t o t h e parameters i n a model of t h i s k i n d , and compare p r e d i c t i o n s and o b s e r v a t i o n s q u a n t i t a t i v e l y . The q u a n t i t a t i v e models c o n s i d e r e d here a r e n u m e r i c a l models, based on the p h y t o p l a n k t o n p r o d u c t i o n model of Chapter 3, so as t o a l l o w a r e a l i s t i c t r e a t m e n t of p h y t o p l a n k t o n c o m p o s i t i o n and v e r t i c a l d i s t r i b u t i o n . The l i m i t a t i o n s of computer s i m u l a t i o n 181 and s e n s i t i v i t y a n a l y s i s when p a r a m e t e r s a r e u n c e r t a i n were d i s c u s s e d i n C h a p t e r 2. To overcome t h e s e l i m i t a t i o n s a s f a r as p o s s i b l e , t h e a p p r o x i m a t e a n a l y t i c a l r e s u l t s o f C h a p t e r 1 and C h a p t e r 2 have been u s e d t o d i r e c t and i n t e r p r e t s i m u l a t i o n o f t h e m o d els w h e r e v e r p o s s i b l e . 4.2.2 F o r m u l a t i o n of a Biomass G r a z i n g M o d e l . In t h e f i r s t model c o n s i d e r e d , h e r b i v o r o u s z o o p l a n k t o n a r e r e p r e s e n t e d by a s i n g l e b i o m a s s v a r i a b l e , G (mg C m " 2 ) , as i n C h a p t e r 1. P h y t o p l a n k t o n c a r b o n i n g e s t e d p e r day, F T , i s c o n v e r t e d t o z o o p l a n k t o n b i o m a s s w i t h e f f i c i e n c y e. A c o n s t a n t l o s s r a t e , m ( d a y " 1 ) c a n be r e g a r d e d as a c o m b i n a t i o n of b a s a l m e t a b o l i s m and m o r t a l i t y i n u n s p e c i f i e d p r o p o r t i o n . Then G = e . F T - m.G 4.1 The p h y t o p l a n k t o n c a r b o n i n e a c h 5m l a y e r c h a n g e s a c c o r d i n g t o a gr o w t h r a t e c a l c u l a t e d a s i n C h a p t e r 3 and t h e c a l c u l a t e d g r a z i n g l o s s . The g r a z i n g l o s s i n t h e model i s b a s e d upon t h e f u n c t i o n a l r e s p o n s e of C h a p t e r 1 but t h e v a r i a t i o n o f p h y t o p l a n k t o n c o n c e n t r a t i o n w i t h d e p t h must now be t a k e n i n t o a c c o u n t . Not o n l y must t h e d i s t r i b u t i o n of z o o p l a n k t o n g r a z i n g a c t i v i t y w i t h d e p t h be c o n s i d e r e d , b u t , as t h e n o n - l i n e a r f u n c t i o n a l r e s p o n s e i t s e l f r e p r e s e n t s a f a s t - v a r i a b l e a p p r o x i m a t i o n ( H o l l i n g , 1 9 5 9 ) , t h e r e l a t i v e t i m e s c a l e s of z o o p l a n k t o n movement and s a t i a t i o n o f f e e d i n g must be c o n s i d e r e d . I f t h e d i s t r i b u t i o n o f z o o p l a n k t o n i s d e n o t e d by q ( z ) , two extr e m e a s s u m p t i o n s w h i c h c o u l d be u s e d i n t h i s model a r e : 182 ( i ) z o o p l a n k t o n are on average u n i f o r m l y d i s t r i b u t e d t hroughout the t o p 150m so q ( z ) = 1/150. ( i i ) z o o p l a n k t o n a r e on average d i s t r i b u t e d i n p r o p o r t i o n t o - " S O p h y t o p l a n k t o n carbon so q ( z ) = C ( z ) / / C ( z ' ) d z ' . Jo The average d i s t r i b u t i o n q ( z ) g i v e s no i n f o r m a t i o n about the v e r t i c a l movement of i n d i v i d u a l z o o p l a n k t e r s . For any g i v e n q ( z ) , two extreme assumptions c o n c e r n i n g the r e l a t i v e time s c a l e s of v e r t i c a l movement and s a t i a t i o n a r e p o s s i b l e : (a) Zooplankton v e r t i c a l movements oc c u r on time s c a l e s much f a s t e r than those of s a t i a t i o n . Each z o o p l a n k t e r then sees the average p h y t o p l a n k t o n c o n c e n t r a t i o n (weighted by q ( z ) ) , so f a r as the f u n c t i o n a l response i s c o n c e r n e d . I f the f u n c t i o n a l response a t c o n s t a n t food l e v e l s i s g i v e n by f ( C ) , the amount i n g e s t e d i n _ _ r'so the water column i s g i v e n by f ( C ) , where C = / C ( z ) . q ( z ) d z . (b) Zooplankton v e r t i c a l movement o c c u r s on time s c a l e s much slower than those of s a t i a t i o n . The amount i n g e s t e d i n the water —* r'so column i s then g i v e n by f ( C ) = / f ( C ( z ) ) . q ( z ) d z . Jo The dominant g r a z e r s a t O.S.P. ar e l a r g e copepods, presumably c a p a b l e of extended v e r t i c a l movement. As the p h y t o p l a n k t o n c o n c e n t r a t i o n i s always low, i t seems u n l i k e l y t h a t they would f e e d a t the v e r y low c o n c e n t r a t i o n s below the mixed l a y e r . In f a c t , as a g r a z i n g t h r e s h o l d v a l u e CO w i l l be c o n s i d e r e d i n the model, i t seems u n l i k e l y t h a t copepods would spend time a t depths where C ( z ) < CO. T h e r e f o r e , q ( z ) has been set p r o p o r t i o n a l t o ( C ( z ) - C 0 ) + . W h i l e the dominant h e r b i v o r o u s copepods a r e presumably c a p a b l e of v e r t i c a l movement, o n l y the l a t e s t a g e Calanus c r i s t a t u s c o pepodids were o b s e r v e d t o undergo e x t e n s i v e d a i l y 183 v e r t i c a l m i g r a t i o n (Marlowe and M i l l e r , 1 9 7 5 ) . For i n d i v i d u a l s u n d e r g o i n g d i u r n a l v e r t i c a l m i g r a t i o n , the extreme assumption (a) would be a p p r o p r i a t e i f s a t u r a t i o n o c c u r r e d on a l o n g time s c a l e (eg g u t-emptying t i m e ) . For organisms which do not undergo d i u r n a l m i g r a t i o n , or become s a t u r a t e d on s h o r t e r time s c a l e s , a s sumption (b) i s more l i k e l y t o be a p p r o p r i a t e and i t has been adopted h e r e , so t h a t the g r a z i n g l o s s r a t e a t depth z i s g i v e n by f ( C ( z ) ) . q ( z ) . G 4.2.3 C h o i c e of Zooplankton P a r a m e t e r s . The parameters which must be s p e c i f i e d i n t h i s s i m p l e biomass model of z o o p l a n k t o n a r e the f u n c t i o n a l response p a r a m e t e r s i M , D and CO, the growth e f f i c i e n c y e and the combined r e s p i r a t i o n and m o r t a l i t y r a t e parameter, m. I t would be u n r e a l i s t i c t o expect t o s p e c i f y t h e s e parameters p r e c i s e l y f o r a number of r e a s o n s . The r e p r e s e n t a t i o n of h e r b i v o r e s by a s i n g l e biomass v a r i a b l e i s i t s e l f a crude a p p r o x i m a t i o n : the g r a z e r s a t O.S.P. a r e composed of i n d i v i d u a l s c o v e r i n g a range of s p e c i e s and s i z e s , and these parameters can be e x p e c t e d t o v a r y a c c o r d i n g l y ( S t e e l e and F r o s t , 1 9 7 7 ; F r o s t , 1 9 7 9 ) . The parameter v a l u e s i n s e r t e d i n the model must i n some sense be r e g a r d e d as approximate a v e r a g e s . These parameters have y e t t o be measured a t O.S.P. f o r a range of s i z e c l a s s e s of the dominant h e r b i v o r e s . Some have been measured f o r the same s p e c i e s i n c o a s t a l l o c a t i o n s (Parsons e t a_l ,1969; Taguchi and I s h i i , 1 9 7 2 ; Ikeda,1972; F r o s t , 1 9 7 9 ) but the e x t r a p o l a t i o n of t h e s e r e s u l t s t o the v e r y d i f f e r e n t food c o n d i t i o n s a t O.S.P. i s not s t r a i g h t f o r w a r d (Buckingham,1978). V a l u e s o b t a i n e d f o r o t h e r s p e c i e s of g r a z e r s under low food 184 c o n d i t i o n s may prove more a p p r o p r i a t e . In a d d i t i o n , as d i s c u s s e d by o t h e r a u t h o r s ( S t e e l e , 1 9 7 4 ; Buckingham,1978) t h e r e i s s t i l l c o n t r o v e r s y a t t a c h e d t o the t e c h n i q u e s f o r measuring t h e s e parameters and the i n t e r p r e t a t i o n of r e s u l t s . The d i s c u s s i o n g i v e n here i s not an e x h a u s t i v e r e v i e w by any means, but i s i n t e n d e d t o g i v e an i n d i c a t i o n of the range of parameter v a l u e s r e p o r t e d i n the l i t e r a t u r e . The r a t i o of growth t o food i n g e s t e d i s known as the g r o s s growth e f f i c i e n c y , K . A range of 10 - 40% f o r t h i s parameter i n marine z o o p l a n k t o n i s g i v e n by P arsons et. al (1977). Maximum v a l u e s of abut 30% were r e p o r t e d by P a f f e n h o f f e r (1976) f o r C. p a c i f i c u s i n l a b o r a t o r y c u l t u r e s , w h i l e s i m i l a r e x p e r i m e n t s y i e l d e d v a l u e s of about 18% f o r Pseudocalanus e l o n g a t u s ( P a f f e n h o f f e r and H a r r i s , 1 9 7 6 ) . The g r o s s growth e f f i c i e n c y i m p l i e d by 4.1 v a r i e s w i t h i n g e s t i o n and depends on the p r o p o r t i o n of m which i s re g a r d e d as b a s a l m e t a b o l i s m . S t e e l e (1974) found t h a t the problems i n i n t e r p r e t a t i o n of r e s p i r a t i o n e x p e r i m e n t s make i t d i f f i c u l t t o a s s e s s the r e l a t i v e importance of a b a s a l m e t a b o l i c r a t e and one which i s p r o p o r t i o n a l t o i n g e s t i o n i n copepods. In a s e r i e s of c a r e f u l e x p e r i m e n t s , Ikeda (1977) found r e s p i r a t i o n r a t e s of 1- 2%/day f o r s t a r v e d C. plumchrus ( s t a g e V) w i t h r a t e s up t o 3 t i m e s h i g h e r f o r a c t i v e l y f e e d i n g i n d i v i d u a l s . R e s p i r a t i o n r a t e s f o r s t a r v e d P a r a c a l a n u s ( w i t h a p p r o x i m a t e l y 1/100 the body weight of C. plumchrus) were much h i g h e r , about 9-18%/day. (The s i z e dependence of r e s p i r a t i o n r a t e i n copepods i s g e n e r a l l y a c c e p t e d (Parsons e t a l ,1977; S t e e l e , 1 9 7 4 ) . ) Taguchi and I s h i i (1972) r e p o r t e d r e s p i r a t i o n r a t e s of 5%/day f o r C. c r i s t a t u s and 9%/day 185 f o r C. plumchrus under c o n d i t i o n s of a c t i v e f e e d i n g . Given the l a r g e s i z e of the dominant h e r b i v o r e s a t O.S.P., a low b a s a l m e t a b o l i c r a t e i n the range 1-5%/day seems r e a s o n a b l e . The e f f e c t i v e g r o s s growth e f f i c i e n c y w i l l be lower than e by an amount dependent on the r a t i o of r a t i o n t o body w e i g h t . G i v e n the low b a s a l metabolism suggested above, a v a l u e f o r e of .5 w i l l r e s u l t i n g r o s s growth e f f i c i e n c e s of .4 or l e s s a t the low food c o n d i t i o n s found a t O.S.P. The parameter i M r e p r e s e n t s maximum r a t i o n as a f r a c t i o n of body w e i g h t . Ranges of 40-60%/day f o r s m a l l e r copepods and 10- 20%/day f o r l a r g e r copepods were found by Parsons et. a l (1967). V a l u e s of 20-60%/day were found by Parsons et a_l (1969) f o r C. plumchrus f e e d i n g on n a t u r a l p h y t o p l a n k t o n assemblages i n . t h e S t r a i t of G e o r g i a . P a f f e n h o f f e r and H a r r i s (1976) found v e r y h i g h r a t e s f o r Pseudocalanus i n c u l t u r e (ca 150% body weight/day) and P a f f e n h o f f e r (1970) r e p o r t e d v a l u e s f o r C. p a c i f i c u s of ca 100%/day f o r stage V copepodids and 300%/day f o r stage V n a u p l i i . Apparent maximum i n g e s t i o n r a t e s f o r stage V C. c r i s t a t u s r e p o r t e d by Taguchi and I s h i i (1972) were v e r y low, ca 5%/day. Ag a i n the v a l u e chosen must r e p r e s e n t an average a c r o s s s i z e c l a s s e s . A v a l u e f o r i M of .5 or l e s s may be r e a s o n a b l e f o r l a t e c o p e p o d i t e s t a g e s of the l a r g e copepods, but a v a l u e of 1.0 may be needed i f i n g e s t i o n by n a u p l i i and e a r l y c opepodids i s i n c l u d e d . T h r e s h o l d s i n z o o p l a n k t o n f e e d i n g on n a t u r a l assemblages from the S t r a i t of G e o r g i a were found by Parsons e t a l (1967) t o range from 50-190 ug C . l " 1 w i t h v a l u e s of D ( h a l f - s a t u r a t i o n c o n s t a n t ) i n a s i m i l a r range. F r o s t (1972) f i t t e d h i s 186 o b s e r v a t i o n s on C. p a c i f i c u s w i t h a type I f u n c t i o n a l response and half-maximum r a t i o n was o b t a i n e d a t food c o n c e n t r a t i o n s r a n g i n g from 50 t o 150 jug C . l " 1 , depending, on food p a r t i c l e s i z e . I n a l a t e r s t u d y , F r o s t (1975) found t h a t t h e c l e a r a n c e r a t e was reduced a t food c o n c e n t r a t i o n s y i e l d i n g l e s s than 15% of maximum r a t i o n , c o r r e s p o n d i n g t o 15-45 ug C . l " 1 . A l l these r e s u l t s were based on r e l a t i v e l y s h o r t - t e r m i n c u b a t i o n s of i n d i v u a l s c a p t u r e d from the f i e l d . In a s e r i e s of l o n g - t e r m c u l t u r i n g e x p e r i m e n t s , no h i n t of t h r e s h o l d s down t o about 25 ug C . l " 1 was found f o r C. pac i f i c u s ( P a f f e n h o f f e r , 1 9 7 0 ) or Pseudocalanus e l o n g a t u s ( P a f f e n h o f f e r and H a r r i s , 1 9 7 6 ) . T h e i r r e s u l t s s uggested a h a l f - s a t u r a t i o n c o n s t a n t f o r Pseudocalanus i n g e s t i o n of 25 t o 50 pg C . l " 1 . The c o n c l u s i o n s of Buckingham (1978) and Mayzaud and P o u l e t ( 1 9 7 8 ) c o n c e r n i n g l o n g - t e r m a d a p t a t i o n t o food c o n c e n t r a t i o n may e x p l a i n the d i f f e r e n c e s between the s e r e s u l t s and the s h o r t - t e r m i n c u b a t i o n s r e p o r t e d above. A d a p t a t i o n t o low food c o n c e n t r a t i o n s may a l s o e x p l a i n the h i g h c l e a r a n c e r a t e s ( f o r example, over 200 ml/day f o r a 10 pg dry weight P s e u d ocalanus ) r e p o r t e d by P a f f e n h o f f e r (1970) and P a f f e n h o f f e r and H a r r i s (1976). C l e a r a n c e r a t e s , or e q u i v a l e n t l y the h a l f - s a t u r a t i o n c o n s t a n t D and t h r e s h o l d CO, are well-known t o depend on the s i z e and v a l u e of food p a r t i c l e s (Parsons e_t a l ,1967; F r o s t , 1 9 7 5 ; S t e e l e and F r o s t , 1 9 7 7 ) . G i v e n the s m a l l s i z e of the p h y t o p l a n k t o n c e l l s and t h e l a r g e s i z e of the dominant h e r b i v o r e s a t O.S.P., low c l e a r a n c e r a t e s or h i g h v a l u e s of CO and D might be e x p e c t e d t h e r e . However, F r o s t (1978) has r e p o r t e d t h a t C. plumchrus feeds e f f i c i e n t l y on s m a l l c e l l s , due t o an u n u s a l l y 187 s m a l l i n t e r - s e t u l e s p a c i n g f o r a c o p e p o d o f i-ts s i z e . The f a c t t h a t t h e s e z o o p l a n k t o n do grow and r e p r o d u c e a t t h e low f o o d l e v e l s o b s e r v e d a t O.S.P. s u g g e s t s t h a t t h e h a l f - s a t u r a t i o n c o n s t a n t , and c e r t a i n l y g r a z i n g t h r e s h o l d s , must be r e l a t i v e l y low. E s t i m a t e s of m o r t a l i t y r a t e a t O.S.P. were o b t a i n e d by a n a l y s i s o f t i m e - s e r i e s of s i z e - c l a s s a b u n d a n c e s i n S e c t i o n 4.1. V a r i o u s p r o b l e m s r e s u l t e d i n a h i g h u n c e r t a i n t y i n t h e s e e s t i m a t e s , b u t r e l a t i v e l y low e s t i m a t e s , i n t h e ra n g e 0.0 t o 0.05 d a y " 1 , were o b t a i n e d . 4.2.4 S i m u l a t i o n R e s u l t s and D i s c u s s i o n . The f i r s t s i m u l a t i o n i s b a s e d on t h e p h y t o p l a n k t o n g r o w t h p a r a m e t e r s u s e d i n F i g 33 and t h e f o l l o w i n g z o o p l a n k t o n p a r a m e t e r v a l u e s : i M = 1.0 d a y " 1 , D = 40 ug C . l " 1 , CO = 20 pq C . l " 1 , e = .5, m = .05 d a y " 1 . T h i s s i m u l a t i o n was run o v e r t h e p e r i o d 1964 t o 1976, but a t t e n t i o n w i l l be f o c u s e d f i r s t on t h e f o r m of t h e s e a s o n a l c y c l e . The c y l e of C h i a i n t h e mixed l a y e r and h e r b i v o r e s t a n d i n g s t o c k i n t h e t o p 150m i n 1976 i s g i v e n i n F i g 47; w h i l e t h e a m p l i t u d e o f p e a k s v a r i e d somewhat from y e a r t o y e a r , a s d i s c u s s e d l a t e r , t h i s i s t y p i c a l of t h e q u a l i t a t i v e v a r i a t i o n i n a l l y e a r s . I t i s i m m e d i a t e l y c l e a r t h a t t h e s i m u l a t i o n r e s u l t s a r e n o t q u a l i t a t i v e l y c o n s i s t e n t e i t h e r w i t h t h e o b s e r v a t i o n s or t h e a p p r o x i m a t e a n a l y s i s o f a s i m i l a r model i n C h a p t e r 1. I n s t e a d o f p h y t o p l a n k t o n r e m a i n i n g c o n s t a n t , a s m a l l bloom o f a b o u t 20 d a y s d u r a t i o n o c c u r s i n A p r i l and a s h a r p z o o p l a n k t o n peak f o l l o w s i n May. A f t e r J u n e , t h e c h l o r o p h y l l c o n c e n t r a t i o n s e t t l e s down a t c a .5 mg C h i a/m 3 and z o o p l a n k t o n s t a n d i n g s t o c k t h e n v a r i e s a p p r o x i m a t e l y w i t h p r i m a r y p r o d u c t i o n . Figure 47. Predicted mixed layer Chi a and herbivore biomass for 1976 using standard parameter set (see text). 189 I t may seem p o s s i b l e t h a t t h e added c o m p l e x i t y of t h e p h y t o p l a n k t o n p r o d u c t i o n model, w h i c h t a k e s d e p t h - d i s t r i b u t i o n and v a r y i n g c a r b o n : C h i a c o m p o s i t i o n i n t o a c c o u n t , has r e n d e r e d t h e a p p r o x i m a t e a n a l y s i s of C h a p t e r 1 i r r e l e v a n t . T h i s i s n o t t h e c a s e and, i n f a c t , t h e s i m u l a t i o n model c a n be c l o s e l y a p p r o x i m a t e d by a model as s i m p l e as t h a t c o n s i d e r e d i n C h a p t e r 1. T h i s i s p o s s i b l e b e c a u s e , as n o t e d i n C h a p t e r 3, p h y t o p l a n k t o n p r o d u c t i o n below t h e m i x e d l a y e r i s a l w a y s v e r y low and p h y t o p l a n k t o n c a r b o n f a l l s o f f q u i c k l y below t h e mixed l a y e r , so t h a t a l m o s t a l l z o o p l a n k t o n f e e d i n g t a k e s p l a c e i n t h e m i x e d l a y e r , where p h y t o p l a n k t o n can be r e p r e s e n t e d by a s i n g l e c a r b o n c o n c e n t r a t i o n , C ( t ) . I f z M ( t ) i s t h e mixed l a y e r d e p t h , t h e c o n c e n t r a t i o n of z o o p l a n k t o n i n t h e mixed l a y e r -is g i v e n by G M ( t ) = G ( t ) / z M ( t ) . I f r ( t ) is....the p h y t o p l a n k t o n g r o w t h r a t e i n t h e mixed l a y e r c a l c u l a t e d as i n C h a p t e r 3, t h e n , on i g n o r i n g e v e n t s below t h e m i x e d l a y e r , t h e s i m u l a t i o n model c a n be r e p l a c e d by C = r ( t ) . C - G M . f ( C ) 4.2a G M = ( e . f ( C ) - m ) . G M 4.2b (Terms r e s u l t i n g f r o m c h a n g e s i n m i x e d l a y e r d e p t h have been n e g l e c t e d as s m a l l on t h e g r o u n d s t h a t c h a n g e s i n mixed l a y e r d e p t h o c c u r r e l a t i v e l y s l o w l y ) . The a p p r o x i m a t e s t a b i l i t y a n a l y s i s o f C h a p t e r 1 a p p l i e s t o t h i s model : p r o v i d e d s e a s o n a l c h a n g e s i n r ( t ) and z M ( t ) o c c u r s l o w l y , t h e s y s t e m s t a t e s h o u l d t r a c k t h e q u a s i - e q u i l i b r i u m c y c l e g i v e n by 190 f ( C ) = m/e = c o n s t a n t G = r ( t ) . C . z M ( t ) . e / m The p h y t o p l a n k t o n c o n c e n t r a t i o n s h o u l d be c o n s t a n t , w h i l e z o o p l a n k t o n s t a n d i n g s t o c k s h o u l d v a r y w i t h p r i m a r y p r o d u c t i o n i n t h e water c o l u m n . T h i s ' p r e d i c t i o n ' depends on t h e s l o w - v a r i a b l e a p p r o x i m a t i o n f o r r ( t ) and z M ( t ) and t h e s i m p l e s t c o n c l u s i o n f r o m F i g 47 i s t h a t t h i s a p p r o x i m a t i o n has b r o k e n down d u r i n g t h e s p r i n g . The s p r i n g i n c r e a s e i n p r o d u c t i o n a p p e a r s t o have o c c u r r e d t o o q u i c k l y f o r z o o p l a n k t o n t o r e s p o n d , r e s u l t i n g i n a t r a n s i e n t d e p a r t u r e f r o m t h e q u a s i - e q u i l i b r i u m c y c l e , w h i c h i s r e t u r n e d t o i n t h e s e c o n d h a l f of t h e y e a r . The r a t e of a p p r o a c h of t h e s y s t e m t o e q u i l i b r i u m c an be i n c r e a s e d by d e c r e a s i n g D and t h i s m i g ht be e x p e c t e d t o r e s u l t i n a r e d u c t i o n i n t h e s p r i n g bloom. The p a r a m e t e r c o m b i n a t i o n CO = 22, D = 20 c o r r e s p o n d s t o t h e same v a l u e o f C a s C0=20, D=40 and doe s r e s u l t i n a much s m a l l e r s p r i n g bloom i n 1976 ( F i g 4 8 ) . However, t h e r e s u l t i n g C h i a peak' i n M a r c h - A p r i l i s s t i l l n o t c o n s i s t e n t w i t h t h e a v e r a g e o b s e r v e d c y c l e ( F i g 14) and t h e z o o p l a n k t o n b i o m a s s s t i l l h a s a v e r y s h a r p and u n r e a l i s t i c e a r l y peak. In f a c t , t h e b a s i c p r o b l e m h e r e l i e s n o t w i t h t h e a b i l i t y o f t h e s y s t e m t o t r a c k t h e q u a s i - e q u i l i b r i u m c y c l e a f t e r t h e s p r i n g p r o d u c t i o n has begun, b ut w i t h t h e v e r y low and o f t e n s l i g h t l y n e g a t i v e n e t p r i m a r y p r o d u c t i o n i n t h e p r e c e d i n g w i n t e r . D u r i n g t h i s p e r i o d , t h e q u a s i - e q u i l i b r i u m d i s a p p e a r s , p h y t o p l a n k t o n s t a n d i n g s t o c k f a l l s below t h e t h r e s h o l d l e v e l and z o o p l a n k t o n s t a n d i n g s t o c k d e c l i n e s e x p o n e n t i a l l y t o v e r y low v a l u e s by Figure 48. Effect of decreasing D on Fig 47. 192 March. When p o s i t i v e p r i m a r y p r o d u c t i o n i s resumed, the system l i e s f a r from the q u a s i - e q u i l i b r i u m c y c l e and a t r a n s i e n t p h y t o p l a n k t o n bloom i s i n e v i t a b l e . Note t h a t w h i l e the i n i t i a l c o n d i t i o n s i n each year a r e c r i t i c a l t o the model's b e h a v i o u r i n the s p r i n g , the c l o s e approach of the system t o the q u a s i - e q u i l i b r i u m c y c l e i n the second h a l f of each year means t h a t i n i t i a l c o n d i t i o n s i n the f o l l o w i n g year a re d e t e r m i n e d e n t i r e l y by the model's parameters and p h y s i c a l d r i v i n g v a r i a b l e s . In p a r t i c u l a r , i n l o n g - t e r m s i m u l a t i o n s , o n l y the f i r s t y e a r i s a f f e c t e d by the c h o i c e of i n i t i a l c o n d i t i o n s . S i m i l a r l y , t o c o n s i d e r the s e a s o n a l c y c l e i n 1976, i t i s s u f f i c i e n t t o s i m u l a t e o n l y 1975 and 1976. The problem a r i s i n g from the d e c l i n e i n z o o p l a n k t o n s t a n d i n g s t o c k i n w i n t e r i m m e d i a t e l y b r i n g s t o mind the l i f e h i s t o r y s t r a t e g y of the dominant copepods a t O.S.P., d i s c u s s e d i n Chapter 1. A l l t h r e e of the major s p e c i e s o v e r w i n t e r below 150m and e i t h e r t h e y , or t h e i r o f f s p r i n g , r e t u r n t o the mixed l a y e r i n the s p r i n g . The ob v i o u s advantage of t h i s b e h a v i o u r t o the copepods i s t h a t w i n t e r l o s s e s t h r o u g h m o r t a l i t y and r e s p i r a t i o n a r e reduced. I t i s p r e c i s e l y t h e s e l o s s e s which a r e r e s p o n s i b l e f o r the s p r i n g bloom i n t h i s s i m p l e model and the r e c r u i t m e n t of z o o p l a n k t o n biomass t o the s u r f a c e l a y e r i n the s p r i n g s h o u l d p r e v e n t t h i s bloom. R e c r u i t m e n t has been i n t r o d u c e d i n t o the model as a d a i l y biomass i n p u t , n o r m a l l y d i s t r i b u t e d over time w i t h mean u, s t a n d a r d d e v i a t i o n cr and t o t a l i n t e g r a t e d r e c r u i t m e n t , G R . The parameter e s t i m a t i o n r e s u l t s of s e c t i o n 4.1 c o r r e s p o n d t o cr between 20 and 40 days, JJ between 70 and 110 days and G R between 193 40 and 400 mg Cm'2. The d e p a r t u r e of i n d i v i d u a l s f o r o v e r - w i n t e r i n g i s r e p r e s e n t e d by an a d d i t i o n a l l o s s r a t e , mw. I f a l l t h r e e dominant s p e c i e s a r e c o n s i d e r e d , t h e d e p a r t u r e p e r i o d e x t e n d s from June t o O c t o b e r . The p a r a m e t e r v a l u e s of F i g 47, t o g e t h e r w i t h (7=30, ju=90, G R =200 mg C m " 2 and m w = .05 d a y " 1 ' f o r day 170 t o 300, r e s u l t e d i n t h e p r e d i c t e d t i m e s e r i e s of C h i a and h e r b i v o r e b i o m a s s g i v e n i n F i g 49. The s p r i n g bloom has e s s e n t i a l l y d i s a p p e a r e d , a l t h o u g h s m a l l f l u c t u a t i o n s i n C h i a due t o s m a l l c h a n g e s i n p h y t o p l a n k t o n c a r b o n and v a r i a t i o n s i n c a r b o n : C h l a r a t i o a r e p r e s e n t . The p r o n o u n c e d peak i n z o o p l a n k t o n b i o m a s s i n May has d i s a p p e a r e d , but t h e i n c r e a s e i n z o o p l a n k t o n b i o m a s s i s e a r l y compared w i t h o b s e r v a t i o n s a t O.S.P. Biomass d e c l i n e s m a r k e d l y as o v e r - w i n t e r i n g commences i n June and s t a b i l i z e s a t ab o u t h a l f t h e l e v e l i n F i g 47 o v e r t h e d e p a r t u r e p e r i o d . The q u a s i - e q u i l i b r i u m b i o m a s s d u r i n g t h i s p e r i o d i s g i v e n by G ( t ) = r ( t ) . C . z M ( t ) . e / ( m + m w ) o r j u s t h a l f of t h e v a l u e p r e d i c t e d w i t h o u t o v e r - w i n t e r i n g . The s e a s o n a l c y l e was f o u n d t o be i n s e n s i t i v e t o c h a n g e s i n t h e t i m i n g and s p r e a d of r e c r u i t m e n t w i t h i n t h e r a n g e s g i v e n a b o v e . As G R i s d e c r e a s e d , t h e s p r i n g bloom s t a r t s t o r e a p p e a r g r a d u a l l y , a l t h o u g h even a low r e c r u i t m e n t l e v e l , G R = 25 mg C m " 2 , r e s u l t s i n a marked r e d u c t i o n ( F i g 50) i n t h e c h l o r o p h y l l and z o o p l a n k t o n p e a k s o f F i g 47. The p r e d i c t e d p h y t o p l a n k t o n s t a n d i n g s t o c k v a r i e s between 0.3 and 0.6 mg C h i a.m'3 i n t h e mixed l a y e r . The q u a s i - e q u i l i b r i u m p h y t o p l a n k t o n c o n c e n t r a t i o n i s d e t e r m i n e d by z o o p l a n k t o n p a r a m e t e r s : 1 .5 J F M fl M . J J fl S 0 N D TH Figure 49. E f f e c t of introducing over-wintering strategy i n F i g 47. 1 .5 cn >K Figure 50. Effect of reducing spring recruitment to -3 25 mg Cm on .Fig 49. 196 C = CO + m.D/(e.i M-m) The peak z o o p l a n k t o n biomass i s about 1100 mg Cm" 2. Assuming a carbon : wet weight r a t i o of 20:1, t h i s r e p r e s e n t s 20 g wet weight.m" 2, or a p p r o x i m a t e l y 150 mg wet wt.m" 3, c o r r e s p o n d i n g t o an i n t e r m e d i a t e year a t O.S.P. Zooplankton s t a n d i n g s t o c k i s g i v e n a p p r o x i m a t e l y by e/m t i m e s p r i m a r y p r o d u c t i o n . The u n c e r t a i n t y i n parameter v a l u e s d i s c u s s e d above i s such t h a t i t would be m e a n i n g l e s s t o c l a i m more than t h a t the magnitudes of observe d s t a n d i n g s t o c k s a r e c o n s i s t e n t w i t h the v e r y broad range of p o s s i b l e parameter v a l u e s . At t h i s p o i n t , the d i s c u s s i o n of p h y t o p l a n k t o n r e g u l a t i o n a t O.S.P. may appear t o have gone almost f u l l c i r c l e . I t s t a r t e d i n Chapter 1 w i t h H e i n r i c h ' s c o n t e n t i o n t h a t the l i f e h i s t o r i e s of the dominant g r a z e r s a t O.S.P. were r e s p o n s i b l e . C o n s i d e r a t i o n of s i m p l e , q u a l i t a t i v e models suggested t h a t the l i f e h i s t o r i e s were not a s u f f i c i e n t e x p l a n a t i o n and a s i m p l e biomass model i n c o r p o r a t i n g t h r e s h o l d s and n e g l e c t i n g l i f e h i s t o r i e s appeared at a q u a l i t a t i v e l e v e l t o be c a p a b l e of e x p l a i n i n g the o b s e r v a t i o n s . In the s i m u l a t i o n model i n t r o d u c e d h e r e , both l i f e h i s t o r y s t r a t e g i e s and t h r e s h o l d s appear t o be n e c e s s a r y . So f a r , the s p r i n g r e c r u i t m e n t of z o o p l a n k t o n biomass has been d i s c u s s e d as i f i t were a s t e p change i n the system s t a t e which b r i n g s the system c l o s e r t o q u a s i - e q u i l i b r i u m b e f o r e p r i m a r y p r o d u c t i o n i n c r e a s e s . In the s i m u l a t i o n model, r e c r u i t m e n t of h e r b i v o r e s t a k e s p l a c e over an extended p e r i o d and may a l l o w a phenomenon, r e p o r t e d by Brauer and Soudack (1979) as a p r e d a t o r - s t o c k i n g e f f e c t , t o o c c u r . For s i m p l i c i t y , c o n s i d e r a s i m p l e n e u t r a l l y s t a b l e L o t k a - V o l t e r r a type model w i t h a c o n s t a n t 197 3 7 r e c r u i t m e n t r a t e f o r h e r b i v o r e s added : C = r.C - a.C.G G = e.a.C.G - m.G + R The n o n - t r i v i a l e q u i l i b r i u m , g i v e n by C = (m-R.a/r)/(e.a) G = r/a can e a s i l y be shown t o be a s y m p t o t i c a l l y s t a b l e ( p r o v i d e d of c o u r s e C i s n o n - n e g a t i v e ) . That i s , d u r i n g the p e r i o d of z o o p l a n k t o n r e c r u i t m e n t i n the s p r i n g , the r e c r u i t m e n t i t s e l f c o n t r i b u t e s t o the s t a b i l i t y of the p h y t o p l a n k t o n - z o o p l a n k t o n i n t e r a c t i o n . I t i s p o s s i b l e t h a t a f e e d i n g t h r e s h o l d i s not n e c e s s a r y t o ensure t h a t the system t r a c k s the q u a s i - e q u i l i b r i u m c y c l e d u r i n g t h i s p e r i o d of i n c r e a s i n g p r i m a r y p r o d u c t i o n . I f the f e e d i n g t h r e s h o l d s h o u l d t u r n out t o be u nnecessary w i t h r e c r u i t m e n t i n c l u d e d , the d i s c u s s i o n would indeed have come f u l l c i r c l e . T h i s was t e s t e d under the most f a v o u r a b l e c o n d i t i o n s i n the s i m u l a t i o n model by r e p l a c i n g the t h r e s h o l d f u n c t i o n a l response w i t h a type I f u n c t i o n a l response w i t h an i d e n t i c a l maximum c l e a r a n c e r a t e . The r e s u l t i n g s p r i n g bloom i s r e l a t i v e l y s m a l l and somewhat d e l a y e d , but a second l a r g e r peak i n C h i a d e v e l o p s as o v e r - w i n t e r i n g d e p a r t u r e commences(Fig 51) . The C h i a and h e r b i v o r e biomass time s e r i e s show l i t t l e resemblance t o the ' o b s e r v a t i o n s . Both l i f e h i s t o r i e s , and a c o n t i n u i n g s t a b i l i s i n g 198 e f f e c t a s p r o v i d e d by a g r a z i n g t h r e s h o l d , a p p e a r t o be n e c e s s a r y t o r e p r o d u c e t h e o b s e r v e d c o n s t a n c y of p h y t o p l a n k t o n s t a n d i n g s t o c k . 4.3 A C o h o r t Model f o r Z o o p l a n k t o n . 4.3.1 I n t r o d u c t i o n . T h e r e a r e a number of r e a s o n s t o c o n s i d e r a more r e a l i s t i c model o f h e r b i v o r e d y n a m i c s a t O.S.P., i n v o l v i n g some s p e c i e s and s i z e s t r u c t