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Optimal stock size and harvest rate in multistage life history models : Pacific salmon Oncorhynchus spp. Moussalli, Elie Ibrahim 1984

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OPTIMAL STOCK SIZE AND HARVEST RATE IN MULTISTAGE L I F E HISTORY MODELS: PACIFIC SALMON ONCORHYNCHUS SPP. By E L I E IBRAHIM MOUSSALLI M. S c . , A m e r i c a n U n i v e r s i t y Of B e i r u t , L e b a n o n , 1972 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES (De p a r t m e n t of R e s o u r c e Management S c i e n c e ) We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA June 1984 (c) E l i e I b r a h i m M o u s s a l l i , 1984 86 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements fo r an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I further agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head of my department or by h i s or her representatives. I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s for f i n a n c i a l gain s h a l l not be allowed without my written permission. The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6 (3/81) ABSTRACT A simple l i f e history model that analyzes the r e l a t i o n s h i p s between habitat capacity, s u r v i v a l rates, optimal escapement, and harvest rate for P a c i f i c salmon ( Oncorhynchus spp.), . i s i m p l i c i t in nearly a l l considerations of salmon management. This model c o n s i s t s of density dependent events at spawning and rearing, and density independent prespawning mo r t a l i t y , egg-to-fry s u r v i v a l and smolt-to-adult s u r v i v a l . The analysis of t h i s model y i e l d s simple relationships between key parameters and optimal escapement, and optimal harvest rate. Interventions with the salmonid l i f e cycle such as flow regulation, habitat improvement or degradation have simple consequences on optimal escapement and harvest rate. S i m i l a r l y , detectable long term trends in the ocean survival rate w i l l a f f e c t the optimal escapement and harvest rate and thus may be incorporated in management decisions. If there i s a density dependent l i m i t l a t e r in the l i f e h i s t o r y , an increase in the s u r v i v a l rate w i l l r e s u l t in a lower optimum escapement and a higher optimum harvest rate. If there i s no density dependent l i m i t l a t e r in the . l i f e h i s t o r y an increase in s u r v i v a l w i l l result in a higher optimum escapement and a higher optimum harvest rate. I also show that management by escapement requires information on habitat capacity and stock p r o d u c t i v i t y ; whereas management by harvest rate requires only the l a t t e r . This model i s generalized to a multistage form to represent any harvestable population c o n s i s t i n g of successive Beverton and Holt type density dependent stock-recruitment stages. This multistage stock-recruitment model i s expressed r e c u r s i v e l y in terms of the cumulative parameters up to and including any a r b i t r a r y stage. The optimal spawning stock size (escapement) and harvest rate are, analyzed in terms of the parameters of any intermediate stage* The r e s u l t s of th i s extension confirm those of the salmonid model and generalize them. i v TABLE OF CONTENTS ABSTRACT i i LIST OF TABLES v LIST OF FIGURES v i ACKNOWLEDGEMENTS v i i INTRODUCTION 1 CHAPTER ONE LI F E HISTORY MODEL AND ANALYSIS 9 1.1 The g e n e r a l i z e d l i f e h i s t o r y model 9 1.2 D e r i v a t i o n of o p t i m a l escapement and h a r v e s t r a t e 13 1.3 A n a l y s i s of t h e e f f e c t of some key parameters ... 22 CHAPTER TWO EXTENSION OF THE MODEL 44 2.1 I n t r o d u c t i o n 44 2.2 The m u l t i s t a g e s t o c k and r e c r u i t m e n t model 45 2.3 A n a l y s i s of i n t e r m e d i a t e parameters 52 CHAPTER THREE DISCUSSION 64 LITERATURE CITED 69 V LIST OF TABLES T a b l e I S e n s i t i v i t y of o p t i m a l escapement t o parameters of the t h r e e models 39 T a b l e I I S e n s i t i v i t y of o p t i m a l h a r v e s t r a t e t o parameters of the t h r e e models 41 v i LIST OF FIGURES F i g u r e 1.1 A l t e r n a t i v e f u n c t i o n a l r e l a t i o n s h i p s between egg d e p o s i t i o n and f r y p r o d u c t i o n i n P a c i f i c salmon 6 F i g u r e 1.2 B e v e r t o n and H o l t t ype r e c r u i t m e n t f u n c t i o n .... 15 F i g u r e 1.3 R e l a t i o n s h i p between o p t i m a l escapement and parameters of t h e u n c o n s t r a i n e d c u r v i l i n e a r f u n c t i o n ... 24 F i g u r e 1.4 R e l a t i o n s h i p between o p t i m a l escapement and parameters of r e a r i n g c o n s t r a i n e d c u r v i l i n e a r f u n c t i o n . 26 F i g u r e 1.5 R e l a t i o n s h i p between o p t i m a l escapement and parameters of a c o n s t r a i n e d r e c t i l i n e a r f u n c t i o n 28 F i g u r e 1.6 R e l a t i o n s h i p between o p t i m a l h a r v e s t r a t e and parameters of u n c o n s t r a i n e d c u r v i l i n e a r f u n c t i o n 31 F i g u r e 1.7 R e l a t i o n s h i p between o p t i m a l h a r v e s t r a t e and parameters of the c o n s t r a i n e d c u r v i l i n e a r f u n c t i o n 34 F i g u r e 1.8 R e l a t i o n s h i p between o p t i m a l h a r v e s t r a t e and parameters of t h e r e c t i l i n e a r f u n c t i o n 36 F i g u r e 2.1 B e v e r t o n and H o l t t ype s t o c k and r e c r u i t m e n t c u r v e 46 F i g u r e 2.2 R e l a t i o n s h i p between o p t i m a l h a r v e s t r a t e and p r o d u c t i v i t y 55 F i g u r e 2.3 R e l a t i o n s h i p between o p t i m a l s t o c k s i z e and p r o d u c t i v i t y 59 F i g u r e 2.4 R e l a t i o n s h i p between o p t i m a l s t o c k s i z e and c a p a c i t y 62 ACKNOWLEDGEMENTS I s h o u l d l i k e t o thank members of my committee e s p e c i a l l y my c o - s u p e r v i s o r s , Dr. Ray H i l b o r n f o r p o i n t i n g the way out when I was wandering i n b l i n d a l l e y s and Dr. C a r l W a l t e r s f o r h i s i n s i g h t f u l comments and f o r showing me t h e d e r i v a t i o n of the r e c u r s i v e form on pages 50-52. My thanks a r e a l s o due t o Dr. J u s t i n Cook who w i t h a few s h o r t s t e p s c o n f i r m e d and s i m p l i f i e d pages of my a n a l y s i s , and t o P e t e r M i l l i n g t o n f o r h e l p f u l comments he made on an e a r l i e r d r a f t . I a l s o w i s h t o thank my f r i e n d Dr. Thomas B e t l a c h f o r c r i t i c a l d i s c u s s i o n and f o r h i s h o s p i t a l i t y and companionship i n the H i g h S i e r r a s when I needed t o get away from Fungus Land. F i n a l l y , I owe v e r y s p e c i a l thanks t o K a t h l e e n Day. She encouraged and h e l p e d me i n many ways. T h i s work i s d e d i c a t e d t o my p a r e n t s who d i d not have the chance. 1 INTRODUCTION C o n v e n t i o n a l wisdom of salmon management, based on R i c k e r ' s (1954) s t o c k and r e c r u i t m e n t model, c a l l s f o r r e g u l a t i n g the f i s h e r y by a l l o w i n g s u f f i c i e n t escapement t o ensure f u t u r e p r o d u c t i v i t y . Low escapements reduce the l o n g term y i e l d from a s t o c k due t o i n s u f f i c i e n t u t i l i z a t i o n of the r e s o u r c e , but a t h i g h escapements the s u s t a i n a b l e y i e l d i s reduced by c r o w d i n g or compensatory m o r t a l i t y . U s i n g R i c k e r ' s model, a common o b j e c t i v e of management i s t o maximize t h e h a r v e s t a b l e s u r p l u s , and thus t o move the f i s h e r y t o t h a t escapement l e v e l which produces the b e s t b a l a n c e between u t i l i z a t i o n of the r e s o u r c e and reduced s u r v i v a l due t o c r o w d i n g . M a x i m i z i n g the h a r v e s t a b l e s u r p l u s i s u s u a l l y not the s o l e o b j e c t i v e of salmon management. Indeed, b a l a n c i n g t h e needs of v a r i o u s u s e r groups i n the f a c e of u n c e r t a i n t y i s what c o n f r o n t s managers most o f t e n . M u l t i - A t t r i b u t e U t i l i t y A n a l y s i s (MUA) as f o r m u l a t e d by Keeney and R a i f f a (1976) and a p p l i e d by Keeney (1977), H i l b o r n and W a l t e r s (1977), and Walker e t a l . (1983), i s a more a p p r o p r i a t e t o o l f o r the complex r e a l i t i e s of salmon management than a s i m p l e m a x i m i z a t i o n of t o t a l c a t c h . However, a l l such management e x e r c i s e s a r e based on some t h e o r e t i c a l f u n c t i o n a l form, w i t h compensatory m o r t a l i t i y a t h i g h p o p u l a t i o n d e n s i t i e s ( P a u l i k 1973; G u l l a n d 1974; C u s h i n g 1973a, b ) . C o n c l u s i o n s about r e t u r n s t o escapement a r e reached ex p o s t  f a c t o by a n a l y s i s of h i s t o r i c a l d a t a , and a r e assumed t o h o l d on the a v e r a g e . M o r t a l i t y d u r i n g a " c r i t i c a l p e r i o d " f o l l o w i n g y o l k sac e x h a u s t i o n , was sug g e s t e d by H j o r t (1914) as an 2 e x p l a n a t i o n f o r year c l a s s s t r e n g t h and f l u c t u a t i o n s of N o r t h e r n European f i s h e r i e s . A l t e r n a t i v e t o t h e c r i t i c a l p e r i o d h y p o t h e s i s a r e the c o n s t a n t s u r v i v a l and i n c r e a s i n g s u r v i v a l h y p o t h e s e s , which c l a i m t h a t m o r t a l i t y a t i n t e r m e d i a t e l i f e h i s t o r y s t a g e s i s c u m u l a t i v e r a t h e r than d i s c o n t i n u o u s (Marr 1956; May 1974). However, f a c t o r s i n f l u e n c i n g t h e s e s t a g e s a r e v e r y complex and t h e r e f o r e g e n e r a l l y not e x p l i c i t l y i n c o r p o r a t e d i n p o p u l a t i o n models commonly used i n f i s h e r i e s management. In two commonly a p p l i e d models, namely the s t o c k and r e c r u i t m e n t models put f o r w a r d by R i c k e r (1954) and B e v e r t o n and H o l t (1957), the c u m u l a t i v e e f f e c t s of a l l s t a g e s i n the l i f e h i s t o r y of a p o p u l a t i o n a r e subsumed i n two pa r a m e t e r s . One of th e s e parameters r e p r e s e n t s s t o c k p r o d u c t i v i t y , the o t h e r the c a r r y i n g c a p a c i t y . These models d e s c r i b e a r e l a t i o n s h i p between a spawning s t o c k and the c o r r e s p o n d i n g r e c r u i t m e n t , or p r o d u c t i o n , from t h i s p a r e n t s t o c k . The dome shaped R i c k e r c u r v e d e s c r i b e s p o p u l a t i o n s w i t h low r e c r u i t m e n t a t h i g h s t o c k l e v e l s , whereas the B e v e r t o n and H o l t c u r v e i s a p p l i c a b l e f o r p o p u l a t i o n s r e a c h i n g some a s y m p t o t i c r e c r u i t m e n t beyond some s t o c k s i z e . The R i c k e r c u r v e i m p l i e s g e o m e t r i c a l l y i n c r e a s i n g d e n s i t y dependence over a range of s t o c k s i z e s . The B e v e r t o n and H o l t c u r v e i m p l i e s an a r i t h m e t i c a l l y p r o g r e s s i v e r e d u c t i o n i n r e c r u i t m e n t r a t e as s t o c k d e n s i t y i n c r e a s e s . With o n l y two para m e t e r s t o e s t i m a t e , t h e s e models have become a v e r y u s e f u l t o o l f o r f i s h e r i e s management. Where i n f o r m a t i o n about i n t e r m e d i a t e l i f e h i s t o r y s t a g e s i s a v a i l a b l e , t h e i r u s e f u l n e s s may be i n c r e a s e d by i n c o r p o r a t i n g such i n f o r m a t i o n . 3 The most common a p p l i c a t i o n of R i c k e r ' s s t o c k and r e c r u i t m e n t model, however, lumps t o g e t h e r i n t e r m e d i a t e s t e p s i n the p r o d u c t i o n p r o c e s s and r e l i e s on the p r i n c i p l e of f a l l i n g m a r g i n a l p r o d u c t i v i t y w i t h o u t f o c u s i n g on the d e t a i l s of s u c c e s s i v e s t e p s i n t h e l i f e h i s t o r y of t h e p o p u l a t i o n i n q u e s t i o n . Management by s t o c k and r e c r u i t m e n t o b v i a t e s the e x p e n s i v e s t u d y of c o m p l i c a t e d i n t e r m e d i a t e s t e p s because the p r i m a r y t o o l of management i s escapement; i t s o b j e c t i v e i s maximum s u s t a i n a b l e y i e l d . However, as such management i s based on average r e c r u i t s t o a g i v e n l e v e l of escapement, s t o c k and r e c r u i t m e n t a n a l y s i s does not p r e s c r i b e escapement i n the f a c e of c h a n g i n g h a b i t a t q u a l i t y or o t h e r p r o d u c t i o n f a c t o r s . I f , f o r example, an observed t emperature regime and d i s c h a r g e r a t e of a salmon b e a r i n g r i v e r causes some s i g n i f i c a n t prespawning m o r t a l i t y o r reduces smolt s u r v i v a l (Hartman e t a l . 1982), and the f l o w regime i s such t h a t a change i n prespawning m o r t a l i t y i s e x p e c t e d , how does the d e s i r e d escapement change? S t o c k and r e c r u i t m e n t w i l l o n l y p r o v i d e an answer based on p a s t d a t a ; i t w i l l not p r o v i d e a method t o i n c o r p o r a t e new changes i n l i f e h i s t o r y e v e n t s . Oregon coho ( Oncorhynchus k i s u t c h ) (ODFW 1982), p r o v i d e a n o t h e r example. From 1961 t o 1971 p r o d u c t i o n of a d u l t s c o r r e l a t e d w i t h i n c r e a s i n g h a t c h e r y smolt r e l e a s e s . T h i s was f o l l o w e d by a p e r i o d of f l u c t u a t i n g r e t u r n s , i n s p i t e of i n c r e a s i n g smolt r e l e a s e s . R e t u r n s peaked i n 1976 a t 4.1 m i l l i o n a d u l t s , c r a s h e d t o a r e c o r d low of 1.1 m i l l i o n i n the f o l l o w i n g y e a r , and remained around t h e l a t t e r l e v e l by 1981 as 4 r e l e a s e s s u r p a s s e d 60 m i l l i o n s m o l t s . These o b s e r v a t i o n s were c o n t r a r y t o the a c c e p t e d n o t i o n t h a t t h e r e i s no d e n s i t y dependence i n the o c e a n i c phase of t h e coho l i f e c y c l e (Peterman 1978). C l a r k and M c C a r l (1983) show t h a t two e x p l a n a t o r y f a c t o r s , smolt r e l e a s e s and l a g g e d u p w e l l i n g (a p r o x y f o r n e a r s h o r e food a v a i l a b i l i t y ) p r o v i d e the b e s t e x p l a n a t i o n f o r v a r i a t i o n i n coho r e t u r n s . Suppose t h a t ocean s u r v i v a l f o r a g i v e n c o h o r t of r e l e a s e d s m o l t s c o u l d be p r e d i c t e d from o b s e r v a t i o n s of n e a r s h o r e o c e a n o g r a p h i c p a t t e r n s , such as the cube of t h e wind-speed inde x d e v e l o p e d by Bakun (1973, 1975) and a p l i e d by L a s k e r (1981) f o r the n o r t h e r n anchovy ( E n q r a u l i s  mordax ; how c o u l d t h i s i n f o r m a t i o n be used t o d e t e r m i n e the a p p r o p r i a t e l e v e l of c a t c h and escapement f o r t h a t brood y e a r ? A g a i n , s t o c k and r e c r u i t m e n t p r e d i c t s average r e t u r n s t o a g i v e n s t o c k s i z e based on h i s t o r i c a l r e c o r d s ; i t does not a l l o w f o r t h e use of e x p e c t e d s u r v i v a l r a t e s . The above examples p o i n t towards the need f o r t o o l s t h a t use l i f e h i s t o r y d a t a , when a v a i l a b l e , t o d e t e r m i n e h a r v e s t i n g p o l i c i e s . S e v e r a l workers ( R i c k e r 1954, 1958; Neave 1958, W i c k e t t 1958, 1962; and Thomas 1975) r e p o r t e d t h a t an upper l i m i t t o the average y i e l d of f r y i s approached as t h e d e n s i t y of spawners reaches an optimum l e v e l . M c N e i l (1964) d e r i v e d a r e l a t i o n s h i p f o r spawning s u c c e s s as a f u n c t i o n of spawner d e n s i t y which i s based on B e v e r t o n and H o l t ' s (1957) a s y m p t o t i c r e l a t i o n s h i p between egg d e p o s i t i o n and r e c r u i t m e n t . M c N e i l ' s h a b i t a t model of spawning s u c c e s s assumes a f i n i t e h a b i t a t of u n i f o r m d e s i r a b i l i t y i n which randomly spawning females may d i g 5 up redds as t h e i r d e n s i t y i n c r e a s e s . A c c o r d i n g t o t h i s model redd super i m p o s i t i o n i s the mechanism t h a t s e t s the n u m e r i c a l upper l i m i t t o emerging f r y by i n c r e a s i n g ' compensatory m o r t a l i t y , and t h u s maximum c a r r y i n g c a p a c i t y i s reached a s y m p t o t i c a l l y (see F i g u r e 1.1). The maximum c a r r y i n g c a p a c i t y of t h e h a b i t a t d e t e r m i n e s th e number of s u c c e s s f u l " e q u i v a l e n t " f e m a l e s , i . e . , t h o s e whose eggs a r e d e p o s i t e d but whose f r y do not s u s t a i n m o r t a l i t i e s due t o redd s u p e r i m p o s i t i o n a t a g i v e n d e n s i t y . I use the term " e f f e c t i v e f e m a l e s " t o convey the same conc e p t of the d e n s i t y dependence of f r y p r o d u c t i o n on the spawning p o p u l a t i o n . Problems of r e s o u r c e use c o n f l i c t , such as between f o r e s t r y and f i s h e r i e s (Toews and Moore 1982; T s c h a p l i n s k i and Hartman 1983), and the need f o r a d e c i s i o n - m a k i n g r a t i o n a l e have a l s o c r e a t e d a need f o r h a b i t a t based models of t h e t ype d e s c r i b e d . One c a t e g o r y of such models c a l c u l a t e s h a b i t a t c a p a c i t y based on t h e development and a p p l i c a t i o n of s p e c i e s s p e c i f i c b i o s t a n d a r d s which a r e r e g i o n a l or l o c a l s u r v i v a l r a t e s between s u c c e s s i v e l i f e s t a g e s (Salmonid Enhancement Program. D e s i g n c r i t e r i a f o r average p e r c e n t s u r v i v a l . I n t e r n a l p u b l i c a t i o n , March 1982). These s u r v i v a l r a t e s v a r y w i t h l o c a l c o n d i t i o n s of streams (Neave 1953) as w e l l as w i t h the p a r t i c u l a r run (Skud 1958, 1973; L i s t e r and Walker 1966; L a r k i n and McDonald 1968). The o t h e r c a t e g o r y of h a b i t a t models a t t e m p t s t o a s c e r t a i n c a r r y i n g c a p a c i t y u s i n g " p r o b a b i l i t y of use c r i t e r i a " (Bovee 1978). T h i s a p proach t o h a b i t a t uses a h y d r o l o g i c a l model, c o n c e n t r a t e s on a few i m p o r t a n t v a r i a b l e s (water d e p t h , v e l o c i t y , temperature and 6 F i g u r e 1.1 A l t e r n a t i v e f u n c t i o n a l r e l a t i o n s h i p s between egg d e p o s i t i o n and f r y p r o d u c t i o n i n P a c i f i c salmon (adapted from M c N e i l l 1964). EGGS DEPOSITED B. s u b s t r a t e c o m p o s i t i o n ) , and, f o r each l i f e s t a g e , a w e i g h t e d u s a b l e a r e a (WUA) i s c a l c u l a t e d . The sum of t h e s e WUA's i s an o v e r a l l measure of stream c a r r y i n g c a p a c i t y . The p o p u l a t i o n dynamics approach of M c N e i l , the b i o s t a n d a r d s approach of SEP and t h e WUA approach of Bovee can a l l be g e n e r a l i z e d i n t o a s i m p l e l i f e h i s t o r y model t h a t i n c o r p o r a t e s d e n s i t y dependent s t a g e s l i n k e d by d e n s i t y independent s u r v i v a l r a t e s . In c h a p t e r one I c o n s i d e r one such model t h a t a l l o w s the - i n c o r p o r a t i o n of i n f o r m a t i o n about l i f e h i s t o r y parameters such as a n t i c i p a t e d changes i n s u r v i v a l r a t e s of salmon a t v a r i o u s s t a g e s i n t h e i r l i f e c y c l e ; or i n f o r m a t i o n about the impact of h a b i t a t m o d i f i c a t i o n , i n t o management d e c i s i o n s . Two management t o o l s a r e c o n s i d e r e d i n t h i s a n a l y s i s , escapement and h a r v e s t r a t e , and I show t h a t the c h o i c e between them depends on t h e i n f o r m a t i o n a v a i l a b l e . I n s e c t i o n 1.1 the g e n e r a l i z e d s a l m o n i d model i s d e s c r i b e d f o r t h r e e c a s e s of h a b i t a t c o n s t r a i n t , and i n s e c t i o n 1.2, c o r r e s p o n d i n g e x p r e s s i o n s f o r o p t i m a l escapement and h a r v e s t r a t e s a r e d e r i v e d . The q u e s t i o n of how o p t i m a l escapements and h a r v e s t r a t e s change w i t h v a r i o u s parameters of the model i s then g r a p h i c a l l y p r e s e n t e d and c o n t r a s t e d i n s e c t i o n 1.3. The model i s extended i n c h a p t e r two so t h a t i t r e p r e s e n t s any h a r v e s t a b l e p o p u l a t i o n c h a r a c t e r i z e d by n s u c c e s s i v e d e n s i t y dependent s t a g e s . T h i s model i s d e r i v e d i n s e c t i o n 2.2, and i n s e c t i o n 2.3 the dependence of o p t i m a l s t o c k s i z e and h a r v e s t r a t e on parameters of any i n t e r m e d i a t e s t a g e i s a n a l y z e d . A g e n e r a l d i s c u s s i o n of r e s u l t s i s p r e s e n t e d i n t h e c h a p t e r t h r e e . 9 CHAPTER ONE L I F E HISTORY MODEL AND ANALYSIS 1.1 The g e n e r a l i z e d l i f e h i s t o r y model The model d e s c r i b e d i s based on d e n s i t y dependent r e l a t i o n s h i p s a t two c r i t i c a l l i f e h i s t o r y s t a g e s , spawning and r e a r i n g , and d e n s i t y independent s u r v i v a l r a t e s i n between. T h i s model i s i m p l i c i t i n M c N e i l ' s (1964) and the SEP (1982) c a p a c i t y models. Three s u r v i v a l f a c t o r s a r e c o n s i d e r e d : prespawning s u r v i v a l (s ), eg g - t o - s m o l t s u r v i v a l ( s ), and s m o l t -1 2 t o - a d u l t s u r v i v a l (s ). 3 E q u a t i o n s (1.1) t h r o u g h (1.6) a r e a s t e p w i s e d e s c r i p t i o n of such a h a b i t a t - b a s e d l i f e h i s t o r y model. S t a r t i n g from escapement of a g i v e n brood year t h r o u g h a d u l t s r e c r u i t e d from t h i s p a r t i c u l a r brood y e a r , (1.1) F = 0.5E s where F = number of female spawners, assuming a sex r a t i o of s 1:1; and E = t o t a l escapement; 10 (1.2) F = F s g s 1 where F = females a r r i v i n g t o the spawning grounds; 9 s = prespawning, d e n s i t y independent s u r v i v a l ; and, 1 a x F = (1.3) e b + x where F = e f f e c t i v e f e m a l e s ; e x = d e n s i t y of female spawners on the grounds a = maximum number of females t h a t c o u l d be accomodated i n spawning a r e a a v a i l a b l e A ; s and b = spawner h a l f s a t u r a t i o n c o n s t a n t r e f l e c t i n g e l a s t i c i t y of t e r r i t o r i a l b e h a v i o u r , or how q u i c k l y the s a t u r a t i o n c a p a c i t y a i s r e a c h e d . V a r i a b l e s a and x can be more e x l p l i c i t l y d e f i n e d as A s a = , where t = spawner t e r r i t o r y s i z e i n m V f e m a l e ; t and 11 F g X = . A s The s a t u r a t i o n c u r v e of e q u a t i o n (1.3) can r e p r e s e n t a number of b i o l o g i c a l h y p o t h e s e s . M c N e i l ' s (1964) model of random redd l o c a t i o n i n a u n i f o r m h a b i t a t w i l l produce such a c u r v e . T h i s c u r v e may be f o r m a l l y d e r i v e d from the P o i s s o n d i s t r i b u t i o n (Dixon and Massey 1969) or from a "clumped" n e g a t i v e b i n o m i a l d i s t r i b u t i o n w i t h parameter k=1.0 (Southwood 1966). I f t h e r e i s a g r a d a t i o n i n g r a v e l q u a l i t y , e i t h e r due t o f l o w or p r o b a b i l i t y of d r y i n g up, and f i s h choose the b e s t h a b i t a t f i r s t , a s i m i l a r s a t u r a t i o n c u r v e w i l l ,be produced. I m p e r f e c t t e r r i t o r i a l i t y of spawners would a l s o produce such a c u r v e , as would a l a r g e number of v a r i a t i o n s on random m a t i n g , t e r r i t o r i a l i t y and v a r i a t i o n i n g r a v e l q u a l i t y . The common t h r e a d i n the p r e c e e d i n g hypotheses i s f a l l i n g m a r g i n a l f r y p r o d u c t i o n as spawners' d e n s i t y i n c r e a s e s . T h i s i s s i m i l a r t o the concept of f a l l i n g m a r g i n a l p r o d u c t i v i t y as used i n economic p r o d u c t i o n f u n c t i o n s . The e f f e c t i v e females a r e c a r r i e d t o the next s t e p of egg p r o d u c t i o n , (1.4) 12 where e = t o t a l number of eggs l a i d by a brood y e a r ; f = f e c u n d i t y , as eg g s / f e m a l e , and smolt p r o d u c t i o n ; and es ( s u r v i v i n g s m o l t s ) 2 S = min \ (1.5) A c (max smolt c a p a c i t y ) r where S = number of s u r v i v i n g s m o l t s ; s = e g g - t o - s m o l t s u r v i v a l ; 2 A = r e a r i n g a r e a a v a i l a b l e i n square m e t e r s ; r c = maximum number of s m o l t s t h a t can r e a r s u c c e s s f u l l y per u n i t a r e a ; and f i n a l l y , (1.6) R = Ss 3 where R = a d u l t s r e c r u i t e d t o a brood year s = s m o l t - t o - a d u l t s u r v i v a l . 3 E q u a t i o n (1.4) g e n e r a t e s eggs from e q u i v a l e n t f e m a l e s ; i n e q u a t i o n ( 1 . 5 ) , where smolt r e a r i n g c o n s t r a i n t s may be met, the s m a l l e r of the two a l t e r n a t i v e numbers i s c a r r i e d i n the 13 subsequent s t e p (1.6) of a d u l t p r o d u c t i o n . 1.2 D e r i v a t i o n of o p t i m a l escapement and h a r v e s t r a t e The p r e v i o u s s e c t i o n summarized the l i f e c y c l e of a salmon p o p u l a t i o n i n s i x s t e p s . From the s e s t e p s we now d e r i v e an e x p r e s s i o n f o r a d u l t salmon p r o d u c t i o n as a f u n c t i o n of escapement. Symbols have the same meaning as i n the p r e v i o u s s e c t i o n . U s i n g e q u a t i o n s ( 1 . 4 ) , ( 1 . 5 ) and (1.6) r e c r u i t m e n t may be e x p r e s s e d as R = min • F f s s e 2 3 A c s r 3 (1.7) S u b s t i t u t i n g f o r e f f e c t i v e females from e q u a t i o n s ( 1 . 1 ) , (1.2) and ( 1 . 3 ) , R - min < 0.5afs s s E 1 2 3 bA + 0.5s E s 1 (1.8a) ^ A c s = R , r 3 2 (1.8b) 14 g i v i n g two e x p r e s s i o n s f o r a d u l t p r o d u c t i o n . E q u a t i o n (1.8a) c a l c u l a t e s the maximum number of a d u l t s produced where r e a r i n g h a b i t a t p r e s e n t s no c o n s t r a i n t s , and where the o n l y c o n s t r a i n t e n c o u n t e r e d i s spawning h a b i t a t . T h i s regime would be v a l i d f o r s a l m o n i d s t h a t do not undergo l o n g p e r i o d s of r e a r i n g i n f r e s h w a t e r h a b i t a t , as i n t h e c a s e of p i n k ( 0. gorbuscha ) and chum ( 0. k e t a ) salmon. The l i m i t i n g f a c t o r i n t h i s case i s the spawning a r e a and the form and parameters of the f u n c t i o n a l r e l a t i o n s h i p t h a t d e s c r i b e s i t s s a t u r a t i o n . E q u a t i o n (1.8b) e x p r e s s e s a d u l t s r e c r u i t e d under a r e a r i n g a r e a c o n s t r a i n t ; sockeye ( 0. nerka ) and coho ( 0. k i s u t c h ) a r e examples of t h i s form. I n b o t h of t h e s e c a s e s i t i s d e s i r a b l e and f e a s i b l e t o d e t e r m i n e o p t i m a l escapement, d e f i n e d as t h a t l e v e l of escapement f o r which s u s t a i n a b l e s u r p l u s p r o d u c t i o n i s maximized. Thus d e f i n e d , o p t i m a l escapement e n s u r e s t h a t the h a b i t a t p r o d u c t i o n c a p a c i t y , as f o r m u l a t e d , i s a t i t s s a t u r a t i o n l e v e l . The term " o p t i m a l " i s r a t h e r s p e c i a l i s e d i n t h i s c o n t e x t , as i t r e f e r s t o t h e h a b i t a t -f i s h component of the f i s h e r y ; t h e h a r v e s t i n g s e c t o r and r e l a t e d economic a s p e c t s a r e e x c l u d e d from t h i s d e f i n i t i o n . The d e r i v e d f u n c t i o n R=f(E) i n F i g u r e 1.2 has t h e f a m i l i a r B e v e r t o n and H o l t form and i s e x p r e s s e d i n e q u a t i o n (1.8a) where R i s the number 1 of a d u l t s produced s u b j e c t t o no r e a r i n g c o n s t r a i n t . To c a l c u l a t e o p t i m a l escapement we d i f f e r e n t i a t e R w i t h r e s p e c t t o 1 escapement 15 F i g u r e 1.2 B e v e r t o n and H o l t type r e c r u i t m e n t f u n c t i o n r e p r e s e n t i n g a f u n c t i o n a l r e l a t i o n s h i p between a d u l t s * r e c r u i t e d and escapement. E i s t h e o p t i m a l 1 * escapement t h a t maximizes a d u l t r e c u i t m e n t (R ) under 1 * no r e a r i n g a r e a c o n s t r a i n t ( e q u a t i o n 1.112). E i s 2 the o p t i m a l escapement under e f f e c t i v e r e a r i n g a r e a * c o n s t r a i n t ( e q u a t i o n 1.14, and E i s the o p t i m a l 3 escapement f o r the r e c t i l i n e a r model ( e q u a t i o n 1.16). ESCAPEMENT 17 dR 0.5abfs s s A 1 1 2 3 s = > o (1.9) 2 dE (bA + 0.5s E) s 1 the second d e r i v a t i v e 2 2 d R 0.5ab£s s s A 1 1 2 3 s = _ < 0 (1.10) 2 3 dE (bA + 0.5s E) s 1 c o n f i r m s t h a t a d u l t s i n c r e a s e w i t h escapement, and they do so a t a d e c r e a s i n g r a t e . The h a r v e s t a b l e s u r p l u s i n t h i s c u r v i l i n e a r model i s maximized ( R i c k e r 1973) by s e t t i n g e q u a t i o n (1.9) e q u a l t o one and s o l v i n g f o r E, t h u s dR 0.5abfs s s A 1 1 2 3 s = = 1. (1.11) 2 dE (bA + 0.5s E) S 1 * The escapement d e r i v e d from t h i s e x p r e s s i o n (E ) i s the 1 o p t i m a l escapement f o r the case where r e a r i n g a r e a does not 18 * p r e s e n t a c o n s t r a i n t , because E maximizes s u r p l u s p r o d u c t i o n 1 (see F i g u r e 1.2) • 1/2 0.5abfs s s A b A 1 2 3 s s 0. 5s 1 I now d e r i v e an e x p r e s s i o n f o r o p t i m a l escapement when r e a r i n g a r e a c o n s t r a i n s t h e maximum number of a d u l t s . I t can be noted i n F i g u r e 1.2 t h a t the f u n c t i o n R=f(E) shows i n c r e a s i n g a d u l t r e t u r n s w i t h i n c r e a s i n g escapement s u b j e c t t o the c o n s t r a i n t of r e a r i n g a r e a t h a t s e t s t h e maximum r e t u r n i n g a d u l t s a t R . T h i s means t h a t beyond some c e i l i n g escapement 2 * E , a d u l t r e t u r n s R a r e c o n s t a n t . T h i s c e i l i n g may be s e t by 2 p h y s i c a l space and food a v a i l a b i l i t y , r e s u l t i n g i n reduced s u r v i v a l and/or premature downstream e m i g r a t i o n of s m o l t s (Chapman 1962, 1966; Mason and Chapman 1965). Another h y p o t h e s i s t h a t d r i v e s the r e a r i n g a r e a c o n s t r a i n t i s t o assume a s a t u r a b l e f u n c t i o n a l r e l a t i o n s h i p s i m i l a r t o t h a t of the spawning a r e a . A two-stage d e n s i t y dependent model would r e s u l t and t h e number of s u r v i v i n g s m o l t s from e q u a t i o n (1.5) c o u l d be r e p l a c e d by 19 a' (es ) 2 S = (1.5a) b' + (es ) 2 where a' «= maximum number of s m o l t s t h a t c o u l d be accomodated i n the r e a r i n g a r e a a' = A /c , and. r b' = smolt h a l f s a t u r a t i o n parameter b' = 1 smolt/m 2. I f the f i r s t proposed d e n s i t y dependent mechanism of a n u m e r i c a l c e i l i n g h o l d s , then r e a r r a n g i n g e q u a t i o n ( 1 . 8 a ) , escapement becomes R b A s . (1.13) 0.5afs s s - 0.5s R 1 2 3 1 Giv e n the upper l i m i t of r e c r u i t s , s e t a t R «= R by t h e 2 a v a i l a b l e r e a r i n g a r e a , I s u b s t i t u t e f o r R from e q u a t i o n ( 1 . 8 b ) , a r r i v i n g a t a new e x p r e s s i o n f o r o p t i m a l escapement A cb A * r s E = . (1.14) 2 0.5s ( a f s -cA ) 1 2 r t 20 I f t h e model i s assumed t o o p e r a t e t h r o u g h a second d e n s i t y dependent s t a g e , as r e p r e s e n t e d by e q u a t i o n ( 1 . 5 a ) , then o p t i m a l escapement has a form s i m i l a r t o e q u a t i o n ( 1 . 1 2 ) , but one t h a t i n c l u d e s the new parameters a' and b', and t h e r e f o r e w i l l not be r e p e a t e d . E q u a t i o n (1.14) i s the o p t i m a l escapement when r e a r i n g a r e a c o n s t r a i n s salmon p r o d u c t i o n . I have a l s o c o n s i d e r e d a t h i r d p r o d u c t i o n model where r e c r u i t m e n t i s a r e c t i l i n e a r f u n c t i o n of escapement (dashed l i n e i n F i g u r e 1.2), r e p r e s e n t e d by R = min 0.5 E f s s s = R 1 2 3 3 A c s = R r 3 2 (1.15) and where the upper l i m i t t o p r o d u c t i o n i s a g a i n s e t o n l y by r e a r i n g h a b i t a t c o n s t r a i n t s . T h i s model however, i m p l i e s l i n e a r l y i n c r e a s i n g r e c r u i t m e n t up t o the r e a r i n g c a p a c i t y ; t h e r e a f t e r r e c r u i t m e n t i s c o n s t a n t . The o p t i m a l escapement i n t h i s c a s e i s A c r 0.5 f s s 1 2 (1.16) 21 Another t o o l i n the management of s a l m o n i d s i s the h a r v e s t r a t e . I f o p t i m a l h a r v e s t r a t e i s d e f i n e d as t h a t r a t e which maximizes th e l o n g term y i e l d , then f o r each of t h e models (1.12),(1.14) and ( 1 . 1 6 ) , the o p t i m a l h a r v e s t r a t e may be e x p r e s s e d as HR 1 - (1.17) n n where n=1,2 or 3, c o r r e s p o n d i n g t o the c u r v i l i n e a r , c o n s t r a i n e d c u r v i l i n e a r , and r e c t i l i n e a r models r e s p e c t i v e l y . S u b s t i t u t i n g E and R i n e q u a t i o n ( 1 . 1 7 ) , HR b t 0.5fs s s 1 2 3 1/2 (1.18) b A * s HR = 1 -2 0.5s s ( a f s -cA ) 1 3 2 r (1 .19) and 22 * 1 HR = 1 - . (1.20) 3 0.5 f s s s 1 2 3 1.3 A n a l y s i s of the e f f e c t of some key parameters I n t h e p r e c e d i n g s e c t i o n , e q u a t i o n s (1.12),(1.14) and (1.16) were d e r i v e d f o r o p t i m a l escapement c o r r e s p o n d i n g t o t h r e e regimes of c o n d i t i o n s . Based on e q u a t i o n ( 1 . 1 7 ) , o p t i m a l h a r v e s t r a t e s c o r r e s p o n d i n g t o t h e s e regimes a r e a l s o s p e c i f i e d i n e q u a t i o n s (1.18 ),(1.19) and ( 1 . 2 0 ) . How do o p t i m a l escapements and h a r v e s t r a t e s change when p r o d u c t i o n parameters change? A l t e r n a t i v e l y , what s h o u l d the o p t i m a l escapement and/or h a r v e s t r a t e be, g i v e n a c h a n g i n g s e t of p r o d u c t i o n parameters? I have a n a l y z e d t h i s q u e s t i o n f o r the r e l e v a n t parameters i n each regime f o r o p t i m a l escapement ( e q u a t i o n s ( 1 . 1 2 ) , ( J . 1 4 ) and ( 1 . 1 6 ) ) , and f o r o p t i m a l h a r v e s t r a t e ( e q u a t i o n s (1.1 8 ),(1.19) and ( 1 . 2 0 ) ) . These parameters a r e s , s , s , a, 1 2 3 b, A and A . Parameters s , s and s r e s p e c t i v e l y c o r r e s p o n d s r 1 2 3 t o p r e s p a w n i n g , e g g - t o - s m o l t and s m o l t - t o - a d u l t s u r v i v a l . H a b i t a t c a p a c i t y a i s e x p r e s s e d as the maximum number of females t h a t c o u l d be accommodated i n t h e spawning a r e a a v a i l a b l e ( t h i s parameter i s i n v e r s e l y r e l a t e d t o t h e female t e r r i t o r y s i z e t ) . The h a l f s a t u r a t i o n c o n s t a n t b i s a measure of d e n s i t y dependence, or p r o d u c t i v i t y of the s t o c k . The spawning a r e a i s A and t h e r e a r i n g a r e a i s A . O p t i m a l escapement and h a r v e s t s r 23 r a t e e x p r e s s i o n s were d i f f e r e n t i a t e d w i t h r e s p e c t t o each of t h e s e p a r a m e t e r s . The r e s u l t s of t h i s a n a l y s i s f o r o p t i m a l escapement a r e shown g r a p h i c a l l y i n F i g u r e s 1.3a t h r o u g h 1.3f f o r e q u a t i o n ( 1 . 1 2 ) , i n F i g u r e s 1.4a t h r o u g h 1.4g f o r e q u a t i o n (1.14) and i n F i g u r e s 1.5a t h r o u g h 1.5d f o r e q u a t i o n ( 1 . 1 6 ) . I t i s i m p o r t a n t t o note t h a t t h i s a n a l y s i s assumes t h a t o n l y the parameter i n q u e s t i o n i s c h a n g i n g w h i l e the o t h e r s a r e kept c o n s t a n t a t a d e f a u l t v a l u e , and t h a t the a b s o l u t e n u m e r i c a l v a l u e s a r e n o t , i n g e n e r a l , as r e l e v a n t as t h e form of the c u r v e s . The n u m e r i c a l example used t o g e n e r a t e t h e s e r e s u l t s i s based on t h e f o l l o w i n g p h y s i c a l c h a r a c t e r i s t i c s and SEP b i o s t a n d a r d s f o r a s t o c k of coho salmon ( 0. k i s u t c h ) i n the Salmon R i v e r . T h i s i s a t r i b u t a r y of t h e F r a s e r R i v e r and a " t y p i c a l " lower m a i n l a n d r i v e r i n B r i t i s h C o l u m b i a , w i t h e q u a l spawning and r e a r i n g a r e a s A = A = 83000 m2, and a female s r t e r r i t o r y s i z e of t = 10 m2 per f e m a l e . T h e r e f o r e the h a b i t a t c a r r y i n g c a p a c i t y i s a - (A / t ) = 8300 f e m a l e s . The h a l f s s a t u r a t i o n c o n s t a n t b i s 0.1 female/m 2, smolt r e a r i n g c a p a c i t y c i s 1 smolt/m 2 and female f e c u n d i t y f i s 2500 eggs/female. The d e f a u l t v a l u e s of the the s u r v i v a l r a t e s a r e s = 1 ( i m p l y i n g 1 t h a t t h e r e i s no prespawning m o r t a l i t y ) , s = 0.012, and s = 2 3 0.15 (SEP 1982). Under a c u r v i l i n e a r regime w i t h no r e a r i n g c o n s t r a i n t , o p t i m a l escapement E i s an i n c r e a s i n g f u n c t i o n of prespawning 1 24 F i g u r e 1.3 R e l a t i o n s h i p between o p t i m a l escapement and par a m e t e r s of t h e u n c o n s t r a i n e d c u r v i l i n e a r f u n c t i o n c o r r e s p o n d i n g t o e q u a t i o n 1.12. MAXIMUM EFFECTIVE FEMALES (»> HALF SATURATION CONSTANT (b) (1000) 26 F i g u r e 1.4 R e l a t i o n s h i p between o p t i m a l escapement and parameters of r e a r i n g c o n s t r a i n e d c u r v i l i n e a r f u n c t i o n c o r r e s p o n d i n g t o e q u a t i o n 1.14. 0 .6 1 .01 .02 .03 PRESPAWNING SURVIVAL (»,) EQQ-TO-8MOLT SURVIVAL <»„> • 2 0 10 20 SO CO SO REARING AREA <A.) 10* m* J 5 ' SPAWNING AREA (A,) 10 m 2 6 MAXIMUM EFFECTIVE FEMALE6 (•) 1000 10 a. O 8M0LT REARIG DEN8ITY (c> 28 F i g u r e 1.5 R e l a t i o n s h i p between o p t i m a l escapement and parameters of a c o n s t r a i n e d r e c t i l i n e a r f u n c t i o n c o r r e s p o n d i n g t o e q u a t i o n 1.16. 30 s u r v i v a l . But as shown i n F i g u r e 1.3a, where o p t i m a l escapement i s p l o t t e d a g a i n s t prespawning s u r v i v a l s , t h e r e i s a t h r e s h o l d 1 below which i t i s o p t i m a l ( u s i n g t h e p r e s e n t d e f i n i t i o n ) t o f i s h t h e p a r t i c u l a r s t o c k t o e x t i n c t i o n . F o r v a l u e s of s below the 1 t h r e s h o l d , the r e c r u i t m e n t f u n c t i o n l i e s below the replacement l i n e and t h e r e f o r e the s t o c k i s not v i a b l e . Note t h a t t h i s d r a s t i c r e s u l t depends on the assumption t h a t the independent v a r i a b l e (s i n t h i s case) i s c o n s t a n t h e n c e f o r t h . In a m u l t i -1 p e r i o d c o n t e x t where s v a r i e s over t i m e , t h i s r e s u l t may not 1 h o l d . The o p t i m a l h a r v e s t r a t e f o r t h i s u n c o n s t r a i n e d c u r v i l i n e a r model (see e q u a t i o n (1.18) and F i g u r e 1.6a), has a c r i t i c a l t h r e s h o l d of prespawning s u r v i v a l below which the s t o c k c o u l d not p e r s i s t . Above t h i s t h r e s h o l d , o p t i m a l h a r v e s t r a t e i n c r e a s e s a s y m p t o t i c a l l y as prespawning s u r v i v a l i n c r e a s e s . In t h e graphs of F i g u r e s 1.3b and 1.3c o p t i m a l escapement i s a l s o an i n c r e a s i n g f u n c t i o n of s and s ; a g a i n w i t h a 2 3 c r i t i c a l t h r e s h o l d of s u r v i v a l . Under t h i s regime of no r e a r i n g h a b i t a t c o n s t r a i n t , o p t i m a l escapement i s p o s i t i v e l y l i n e a r l y r e l a t e d t o t h e spawning a r e a a v a i l a b l e , and hence t o the c a r r y i n g c a p a c i t y of the spawning h a b i t a t (parameter a ) , as shown i n F i g u r e s 1.3d and l . 3 e . O p t i m a l escapement under t h i s regime i s , however, n o n l i n e a r l y r e l a t e d t o t h e h a l f s a t u r a t i o n parameter b ( F i g u r e 1 . 3 f ) . T h i s parameter r e f l e c t s the degree of d e n s i t y dependence of e f f e c t i v e f e m a l e s . A s t o c k w i t h a 31 F i g u r e 1.6 R e l a t i o n s h i p between o p t i m a l h a r v e s t r a t e and p arameters of u n c o n s t r a i n e d c u r v i l i n e a r f u n c t i o n c o r r e s p o n d i n g t o e q u a t i o n 1.18. 32 33 s m a l l b v a l u e i n d i c a t e s a h i g h e r p r o d u c t i v i t y than a s t o c k w i t h a l a r g e b v a l u e , and t h u s the i n i t i a l s l o p e of the p r o d u c t i o n f u n c t i o n r i s e s more s t e e p l y and the the h a b i t a t s a t u r a t i o n c a p a c i t y i s reached sooner w i t h more p r o d u c t i v e s t o c k s than w i t h l e s s p r o d u c t i v e ones. O p t i m a l escapement i n c r e a s e s f o r b < 0.05, then d e c l i n e s a t a s l o w e r r a t e f o r b > 0.05 ( F i g u r e 1 . 3 f ) . Under the regime of r e a r i n g a r e a c o n s t r a i n t ( e q u a t i o n * 1.14), o p t i m a l escapement E i s i n v e r s e l y r e l a t e d t o 2 prespawning and e g g - t o - s m o l t s u r v i v a l , t o spawning a r e a and hence t o the parameter a, as shown i n F i g u r e s 1.4a, 1.4b, 1.4d, and 1.4e r e s p e c t i v e l y . O p t i m a l escapement here i s l i n e a r l y r e l a t e d t o the h a l f s a t u r a t i o n c o n s t a n t b ( F i g u r e 1.4g). In t h i s regime o p t i m a l escapement i s an i n c r e a s i n g f u n c t i o n of r e a r i n g a r e a A , and maximum r e a r i n g d e n s i t y c, r e f l e c t i n g the r f a c t t h a t f o r a g i v e n s t o c k , m a r g i n a l i n c r e a s e s i n p r o d u c t i o n r e q u i r e ever l a r g e r r e a r i n g h a b i t a t c a p a c i t y , ( F i g u r e s 1.4c and 1.4 f r e s p e c t i v e l y ) . O p t i m a l escapement i n the c o n s t r a i n e d r e c t i l i n e a r model i s i n v e r s e l y r e l a t e d t o prespawning and egg-t o - s m o l t s u r v i v a l ( F i g u r e s 1.5a and 1.5b), but independent of s m o l t - t o - a d u l t s u r v i v a l ( F i g u r e 1.5c). As r e a r i n g parameter c i n c r e a s e s , so does t h e p r o d u c t i v e c a p a c i t y of t h e h a b i t a t , hence l i n e a r l y i n c r e a s i n g o p t i m a l escapement ( F i g u r e 1.5d). F i g u r e s 1.6, 1.7 and 1.8 show the r e l a t i o n s h i p of o p t i m a l h a r v e s t r a t e t o parameters i n e q u a t i o n s ( 1 . 1 8 ) , (1.19) and (1.20) c o r r e s p o n d i n g t o t h e t h r e e models r e s p e c t i v e l y . The e f f e c t of e g g - t o - s m o l t s u r v i v a l (s ) on the o p t i m a l h a r v e s t r a t e 2 34 F i g u r e 1.7 R e l a t i o n s h i p between o p t i m a l h a r v e s t r a t e and p arameters of the c o n s t r a i n e d c u r v i l i n e a r f u n c t i o n c o r r e s p o n d i n g t o e q u a t i o n 1.19. g .06 -1 .16 •* HALF SATURATION CONSTANT (b) 36 F i g u r e 1.8 R e l a t i o n s h i p between o p t i m a l h a r v e s t r a t e and p a rameters of the r e c t i l i n e a r f u n c t i o n c o r r e s p o n d i n g t o e q u a t i o n 1.20. 38 has a s i m i l a r s a t u r a t i o n form f o r a l l t h r e e models ( F i g u r e 1.6b, 1.7b, and 1.8b), however the c e i l i n g i s reached sooner i n the r e c t i l i n e a r model. The o p t i m a l h a r v e s t r a t e i s i n v e r s e l y r e l a t e d t o 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 (b) i n b o t h c a s e s of the c u r v i l i n e a r model ( F i g u r e s 1.6f and 1.7g), but does not depend on b i n the r e c t i l i n e a r model. Spawning a r e a (A ) and e f f e c t i v e s females (a) have no i n f l u e n c e on t h e h a r v e s t r a t e i n the u n c o n s t r a i n e d c u r v i l i n e a r model ( F i g u r e s 1.6d and 1 . 6 f ) , and o n l y a s m a l l e f f e c t i n t h e c o n s t r a i n e d case ( F i g u r e s 1.7d and 1.7e). Changes i n the r e a r i n g a r e a (A ) have an impact on the r o p t i m a l h a r v e s t r a t e o n l y i n t h e c o n s t r a i n e d c u r v i l i n e a r model. The s e n s i t i v i t y of o p t i m a l escapement and o p t i m a l h a r v e s t r a t e t o v a r i o u s parameters of the t h r e e models a r e shqwn i n T a b l e I and T a b l e I I r e s p e c t i v e l y . S e n s i t i v i t y i s d e f i n e d as the p e r c e n t change i n o p t i m a l escapement or h a r v e s t r a t e i n response t o a 1 p e r c e n t i n c r e a s e i n the d e f a u l t v a l u e of the parameter c o n s i d e r e d . I n t h i s sense s e n s i t i v i t y i s e q u i v a l e n t t o t h e concept of e l a s t i c i t y as used i n economic t e r m i n o l o g y , and measures the r e l a t i v e importance of parameters a t d i f f e r e n t p o i n t s . T a b l e I shows t h a t o p t i m a l escapement i s h i g h l y s e n s i t i v e t o e g g - t o - s m o l t s u r v i v a l (s ) and s m o l t - t o - a d u l t 2 s u r v i v a l (s ) under th e u n c o n s t r a i n e d c u r v i l i n e a r model; 3 however, under th e c o n s t r a i n e d c u r v i l i n e a r model the s e same parameters have n e g l i g i b l e o r n u l l e f f e c t s . T a b l e I a l s o shows t h a t o t h e r parameters have o n l y s m a l l e f f e c t s i n a l l t h r e e e I S e n s i t i v i t y of o p t i m a l escapement t o parameters of the t h r e e models r e p r e s e n t i n g change i n o p t i m a l escapement due t o an i n c r e a s e i n parameter v a l u e s f o r c u r v i l i n e a r , c o n s t r a i n e d c u r v i l i n e a r , and r e c t i l i n e a r models. Numbers i n p a r e n t h e s e s a r e p e r c e n t changes i n o p t i m a l escapement f o r a 1 p e r c e n t i n c r e a s e i n d e f a u l t parameter v a l u e s . 40 Parameter and v a l u e U n c o n s t r a i n e d C o n s t r a i n e d c u r v i l i n e a r c u r v i l i n e a r model model R e c t i l i n e a r model 1 1.0 s 2 0.012 s 3 0.15 A 1 s 83000 m2 A 1 r 83000 m2 t 10 m2 b 0.1 F /m2 9 c 1 smolt/m 2 i n c r e a s e s (0.5) i n c r e a s e s (14.6) i n c r e a s e s (14.6) i n c r e a s e s (1.0) no change (0) i n c r e a s e s (0.5) d e c r e a s e s (5.0) no change (0) d e c r e a s e s (1.0) d e c r e a s e s (1.5) no change (0) i n c r e a s e s (1.0) i n c r e a s e s (1.5) d e c r e a s e s (1.5) i n c r e a s e s (1.0) i n c r e a s e s (1.5) d e c r e a s e s (1.0) d e c r e a s e s (1.0) no change (0) no change (0) i n c r e a s e s (1.0) no change (0) no change (0) i n c r e a s e s (1.0) 1 Spawning and r e a r i n g a r e a s a r e p h y s i c a l parameters of the Salmon R i v e r ( B r i t i s h C o l u m b i a ) , not SEP b i o s t a n d a r d s . 41 T a b l e I I S e n s i t i v i t y of o p t i m a l h a r v e s t r a t e t o p a r a m e t e r s of the t h r e e models r e p r e s e n t i n g change i n o p t i m a l h a r v e s t r a t e due t o an i n c r e a s e i n parmeter v a l u e s under c u r v i l i n e a r , c o n s t r a i n e d c u r v i l i n e a r , and r e c t i l i n e a r models. Numbers i n p a r e n t h e s e s a r e p e r c e n t changes i n o p t i m a l h a r v e s t r a t e f o r a 1 p e r c e n t i n c r e a s e i n d e f a u l t parameter v a l u e s . 42 Parameter and v a l u e u n c o n s t r a i n e d c u r v i l i n e a r model c o n s t r a i n e d c u r v i l i n e a r model R e c t i l i n e a r model s 1 i n c r e a s e s i n c r e a s e s i n c r e a s e s 1.0 (1.0) (2.0) (0.3) s 2 i n c r e a s e s i n c r e a s e s i n c r e a s e s 0.012 (1.0) (3.0) (3.0) s 3 i n c r e a s e s i n c r e a s e s i n c r e a s e s 0.15 (1.0) (2.0) (0.3) A 1 s no change i n c r e a s e s no change 83000 m2 (0) (1.0) (0) A 1 r no change no change no change 83000 m2 (0) (0) (0) t d e c r e a s e s d e c r e a s e s no change 10 m2 (1.0) (3.0) (0) b d e c r e a s e s d e c r e a s e s no change 0.1 F /m2 9 (1.0) (2.0) (0) c no change d e c r e a s e s no change 1 smolt/m 2 (0) (1.0) (0) 1 Spawning and r e a r i n g a r e a s a r e p h y s i c a l parameters of the Salmon R i v e r ( B r i t i s h C o l u m b i a ) , not SEP b i o s t a n d a r d s . 43 models. In c o n t r a s t T a b l e I I shows t h a t o p t i m a l h a r v e s t r a t e i s most s e n s i t i v e t o changes i n the parameters of the c o n s t r a i n e d c u r v i l i n e a r model. '. These r e s u l t s show the importance of r e a r i n g c o n s t r a i n t s . I f t h e r e i s no r e a r i n g c o n s t r a i n t then t h e o p t i m a l escapement goes up w i t h h i g h e r prespawning s u r v i v a l or e g g - t o - s m o l t s u r v i v a l , but i f r e a r i n g i s l i m i t e d t h e o p t i m a l escapement goes down. F i n a l l y , the d i f f e r e n c e s between o p t i m a l escapement and o p t i m a l h a r v e s t r a t e a r e i l l u m i n a t i n g . H i g h e r p r e s m o l t s u r v i v a l r a t e s l e a d t o h i g h e r o p t i m a l escapements when r e a r i n g a r e a i s not l i m i t i n g , but t o lower o p t i m a l escapements when r e a r i n g a r e a i s l i m i t i n g . The o p t i m a l h a r v e s t r a t e i s h i g h e r f o r b o t h c a s e s . So i f we were u n c e r t a i n about r e a r i n g c o n s t r a i n t s , we might be b e t t e r o f f u s i n g a h a r v e s t r a t e p o l i c y . H a r v e s t r a t e s , f u r t h e r m o r e , do not depend upon spawning or r e a r i n g a r e a s , whereas optimum escapements do. H a r v e s t r a t e p o l i c i e s r e q u i r e s u b s t a n t i a l l y l e s s i n f o r m a t i o n than optimum escapement p o l i c i e s . 44 CHAPTER TWO EXTENSION OF THE MODEL 2.1 I n t r o d u c t i o n In c h a p t e r one a l i f e h i s t o r y model c o n s i s t i n g of two d e n s i t y dependent s t a g e s s e p a r a t e d by d e n s i t y independent s u r v i v a l r a t e s , and r e p r e s e n t i n g a g e n e r a l i z e d s a l m o n i d l i f e c y c l e was a n a l y z e d . In t h i s c h a p t e r t h e model i s extended t o a m u l t i s t a g e form such t h a t more i n t e r m e d i a t e l i f e h i s t o r y s t a g e s a r e f o r m a l l y i n c 6 r p o r a t e d . T h i s i s a c c o m p l i s h e d by r e g a r d i n g i n t e r m e d i a t e l i f e h i s t o r y s t a g e s as s u c c e s s i v e s t o c k and r e c r u i t m e n t r e l a t i o n s h i p s , such t h a t the output from one stage c o n s t i t u t e s the i n p u t t o the n e x t . U s i n g t h i s i d e a and employing compensatory and d e p e n s a t o r y e f f e c t s a t i n t e r m e d i a t e s t a g e s , L a r k i n e t aJL ( 1964) produced a number of t h e o r e t i c a l s t o c k and r e c r u i t m e n t c u r v e s based on a v a r i e t y of p r e m i s e s about l i f e h i s t o r y c h a r a c t e r i s t i c s . Ward and L a r k i n (1964) e x p l a i n e d t h e phenomenon of c y c l i c dominance i n t h e Adams R i v e r sockeye salmon ( 0. nerka ) i n terms of d e p e n s a t o r y m o r t a l i t y due t o p r e d a t i o n , d u r i n g the l a c u s t r i n e s t a g e . A l o g i c a l next s t e p i s t o d e r i v e an a n a l y t i c a l model t h a t i n c o r p o r a t e s a l l l i f e h i s t o r y s t a g e s . T h i s model w i l l then be used t o d e r i v e a f u n c t i o n a l form f o r t h e o p t i m a l s t o c k s i z e and h a r v e s t r a t e i n terms of the par a m e t e r s of any s t a g e i n t h e l i f e c y c l e . The o p t i m a l h a r v e s t r a t e and s t o c k s i z e a r e a g a i n d e f i n e d as those w h i c h maximize the l o n g term h a r v e s t a b l e s u r p l u s . The remainder of t h i s c h a p t e r i s d i v i d e d i n t o two s e c t i o n s : i n s e c t i o n 2.2 the g e n e r a l form of a m u l t i s t a g e s t o c k - r e c r u i t m e n t f u n c t i o n i s 45 d e r i v e d , w h i l e i n s e c t i o n 2.3 t h e r e l a t i o n s h i p between h a r v e s t r a t e , s t o c k s i z e and i n t e r m e d i a t e parameters i s p r e s e n t e d . The f u n c t i o n a l form and parameters a r e p r e s e n t e d i n a s l i g h t l y d i f f e r e n t f o r m u l a t i o n than i n the p r e v i o u s c h a p t e r . A p r o d u c t i v i t y parameter p r e p l a c e s 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 b; and a c a p a c i t y parameter c r e p l a c e s i t s c o u n t e r p a r t a . I n s t e a d of "escapement", I use t h e term " s t o c k s i z e " t o r e p r e s e n t numbers of spawning a d u l t s . 2.2 The m u l t i s t a g e s t o c k and r e c r u i t m e n t model C o n s i d e r B e v e r t o n and H o l t ' s s t o c k - r e c r u i t m e n t c u r v e shown i n F i g u r e 2.1 and w r i t t e n i n the form R = number of r e c r u i t s , S = number of spawners, p = p r o d u c t i v i t y parameter, and c = c a p a c i t y parameter. In a h y p o t h e t i c a l p o p u l a t i o n c o n s i s t i n g of two s t a g e s , j u v e n i l e s and a d u l t s , assume t h a t the above f u n c t i o n a l form h o l d s f o r b o t h s t a g e s , but t h a t the parameters a r e stage c where 46 F i g u r e 2.1 B e v e r t o n and H o l t type s t o c k and r e c r u i t m e n t c u r v e c o r r e s p o n d i n g t o e q u a t i o n 2 . 1 . RECRUITS (R) o CO TJ >. Z m CO CO 48 s p e c i f i c . Then 1 I + JL S c, where R = j u v e n i l e s produced 1 S = spawning a d u l t s , r e p r e s e n t s t h e f i r s t s t a g e ; and i + h_ R c2 1 where R = A d u l t s produced 2 R = j u v e n i l e s from s t a g e 1, and e q u a t i o n (2.3) r e p r e s e n t s 1 the second s t a g e . S u b s t i t u t i n g f o r R we get 1 ^ ( 1 ^ ) 5 T h i s may be r e p e a t e d f o r more s t a g e s and a g e n e r a l form f o r n 49 s t a g e s may be r e p r e s e n t e d by E q u a t i o n (2.5) may be r e w r i t t e n as where P„ 5 + £ 5 12 J.) P.- W f . i s a composite of t h e p r o d u c t i v i t y parameters of n s u c c e s s i v e s t a g e s , and From e q u a t i o n (2.8) we can s o l v e f o r t h e o v e r a l l c a p a c i t y 50 parameter C , r e s u l t i n g i n n c „ -- - 8 — ' (2-9) Thus, the c u m u l a t i v e c a p a c i t y parameter i s a f u n c t i o n of the composite p r o d u c t i v i t y parameter P and the s t a g e s p e c i f i c n c a p a c i t y c . i Next, I s o l v e f o r P and C r e c u r s i v e l y such t h a t the n n o v e r a l l p r o d u c t i v i t y and c a p a c i t y parameter up t o and i n c l u d i n g some st a g e n, may be d e f i n e d i n terms of t h e ( n - 1 ) t h s t a g e . Thus e q u a t i o n (2.8) may be e x p r e s s e d as (z.iz) 51 However, U 5 5 r * i - i ' v t h e r e f o r e e q u a t i o n (2.12) may be r e w r i t t e n as " ^ — * -D i v i d i n g e q u a t i o n (2.14) by P g i v e s n-1 p . J - + | , ( 2 , 5 ) and t h e o v e r a l l c a p a c i t y parameter may be e x p r e s s e d r e c u r s i v e l y as C n = — (Z . / 4 ) — — n - ' T h i s d e r i v a t i o n a l l o w s us t o w r i t e t h e s t o c k - r e c r u i t m e n t r e l a t i o n s h i p r e c u r s i v e l y such t h a t r e c r u i t s t o any stage n may 52 be e x p r e s s e d i n terms of the composite parameters up t o the ( n -1 ) t h s t a g e . Thus the r e c u r s i v e form of e q u a t i o n (2.6) becomes when s u b s t i t u t i o n s a r e made. 2.3 A n a l y s i s of i n t e r m e d i a t e parameters Having an e x p r e s s i o n f o r a m u l t i s t a g e s t o c k - r e c r u i t m e n t r e l a t i o n s h p , i t i s now f e a s i b l e t o t o examine the e f f e c t of changes i n i n t e r m e d i a t e p arameters on management d e c i s i o n s . As i n c h a p t e r one, I f i r s t d e r i v e an e x p r e s s i o n f o r optimum s t o c k s i z e and h a r v e s t r a t e from the g e n e r a l m u l t i s t a g e form of the r e c r u i t m e n t f u n c t i o n . Then I d i f f e r e n t i a t e e q u a t i o n (2.6) w i t h r e s p e c t t o S, s e t the d e r i v a t i v e e q u a l t o one t o maximize * s u r p l u s over r e p l a c e m e n t , and s o l v e f o r t h e o p t i m a l s t o c k S . Thus 53 and <- - (2i9) * U s i n g e q u a t i o n ( 2 . 8 ) , S may be r e w r i t t e n as Yz 1%. I f t h i s l a s t r e s u l t i s combined w i t h the g e n e r a l m u l t i s t a g e form r e p r e s e n t e d by e q u a t i o n ( 2 . 6 ) , the r e s u l t i n g o p t i m a l r e c r u i t m e n t i s f? ^ 5 / \ T h i s e x p r e s s i o n may be s i m p l i f i e d by s u b s t i t u t i n g f o r S from e q u a t i o n ( 2 . 1 9 ) , g i v i n g 54 To c a l c u l a t e t h e o p t i m a l h a r v e s t r a t e H from the above model I e x p r e s s the o p t i m a l s t o c k i n terms of H as (2-23) or 2.2+) S u b s t i t u t i n g f o r R from e q u a t i o n ( 2 . 2 2 ) , g i v e s r ]/2 (2.25. w h i c h c o u l d be made an e x p l i c i t f u n c t i o n of any component p , as J i n U - i -- y 2 fi "is I »• T h i s r e l a t i o n s h i p i s shown g r a p h i c a l l y i n F i g u r e 2.2 . The o p t i m a l s t o c k s i z e i n e q u a t i o n (2.20) may be c o n s i d e r e d 55 F i g u r e 2.2 R e l a t i o n s h i p between o p t i m a l h a r v e s t r a t e and p r o d u c t i v i t y parameter p f o r any i n t e r m e d i a t e s t a g e j J i n an n s t a g e s t o c k and r e c r u i t m e n t model c o r r e s p o n d i n g f t o - e q u a t i o n ( 2 . 2 6 ) . p i s the p r o d u c t i v i t y t h r e s h o l d j a t which h a r v e s t r a t e becomes p o s i t i v e . <5C 57 t o be f u n c t i o n of a s e r i e s of parameters p and c of any stage J J j , where 2 < j < n. Thus e q u a t i o n (2.20) may be r e w r i t t e n as an e x p l i c i t f u n c t i o n of any i n t e r m e d i a t e p r o d u c t i v i t y parameter p : j * ^ 5 - 1 (2.21) where and a r e c o n s t a n t s a s s o c i a t e d w i t h r e l a t i o n s h i p between S and p i s (7.11c) the j t h s t a g e . The f u n c t i o n a l t h e r e f o r e dependent on the 58 s i g n of t h e d e r i v a t i v e of e q u a t i o n (2.20) w i t h r e s p e c t t o p . and can be e i t h e r p o s i t i v e or n e g a t i v e depending on the a b s o l u t e v a l u e s of t h e component terms. E q u a t i o n (2.27) may be e x p r e s s e d g r a p h i c a l l y as shown i n F i g u r e 2.3 . L i k e w i s e , the o p t i m a l s t o c k s i z e e x p r e s s e d by e q u a t i o n (2.20) may be r e w r i t t e n i n terms of any i n t e r m e d i a t e c a p a c i t y parameter c such t h a t T h i s d e r i v a t i v e i s j where 59 F i g u r e 2.3 R e l a t i o n s h i p between o p t i m a l s t o c k s i z e and p r o d u c t i v i t y parameter p f o r any i n t e r m e d i a t e s t a g e j j i n an nstage s t o c k and r e c r u i t m e n t model c o r r e s p o n d i n g t t o e q u a t i o n ( 2 . 2 7 ) . p i s t h e t h r e s h o l d below which j i t i s o p t i m a l t o h a r v e s t t h e s t o c k t o e x t i n c t i o n . The i n f l e c t i o n p o i n t p i s where the d e r i v a t i v e i n e q u a t i o n j * (2.28) e q u a l s z e r o and S r e a c h e s a maximum. 61 E q u a t i o n (29) has the f a m i l i a r s a t u r a t i o n type form shown i n F i g u r e 2.4 w i t h an i n i t i a l s l o p e of 1 / l and an asymptote of 1 3 1 / 1 . 1 2 These r e s u l t s c o n f i r m the c o n c l u s i o n s reached i n c h a p t e r one and g e n e r a l i z e them t o any h a r v e s t a b l e p o p u l a t i o n w i t h i n t e r m e d i a t e l i f e h i s t o r y s t a g e s . 62 F i g u r e 2.4 R e l a t i o n s h i p between o p t i m a l s t o c k s i z e and c a p a c i t y parameter c f o r any i n t e r m e d i a t e s t a g e j i n an j n s t a g e s t o c k and r e c r u i t m e n t model c o r r e s p o n d i n g t o e q u a t i o n ( 2 . 2 9 ) . 63 CAPACITY (c. ) 64 CHAPTER THREE DISCUSSION Stock and r e c r u i t m e n t models a r e u s e f u l t o o l s f o r the management of e x p l o i t e d p o p u l a t i o n s , p a r t i c u l a r l y when mechanisms a c t i n g on i n t e r m e d i a t e l i f e h i s t o r y s t a g e s cannot be p r e d i c t e d or c o n t r o l l e d . The sum t o t a l of t h e s e mechanisms i s compressed i n two p a r a m e t e r s , namely p r o d u c t i v i t y of the s t o c k and h a b i t a t c a p a c i t y . However, th e s e parameters a r e assumed t o h o l d on the average; t h u s s t o c k and r e c r u i t m e n t models do not a l l o w the i n c o r p o r a t i o n of e x p e c t e d changes i n l i f e h i s t o r y e v e n t s . To the e x t e n t t h a t mechanisms a c t i n g d u r i n g l i f e h i s t o r y e v e n t s may be e l u c i d a t e d , i t i s p o s s i b l e t o d i s a g g r e g a t e the r e c r u i t m e n t f u n c t i o n i n t o c o n s t i t u e n t components. F a c t o r s g o v e r n i n g marine growth and s u r v i v a l ( L a s k e r 1979, 1981; Kruse and T y l e r 1983; K r e u t z e t a l . 1982) p o i n t t o the p o t e n t i a l use of p h y s i c a l measurements l i k e wind v e l o c i t y ( L a s k e r 1980) or n e a r s h o r e temperature regimes (Kruse and Huyer 1983) t o e s t i m a t e c o n d i t i o n s of s u r v i v a l . To t h e e x t e n t t h a t marine s u r v i v a l of s a l m o n i d s i s c o u p l e d t o s p e c i f i c o c e a n o g r a p h i c v a r i a b l e s , a l b e i t w i t h a margin of u n c e r t a i n t y , then a h a b i t a t based model such as the one p r e s e n t e d here may be a u s e f u l t o o l . Whereas we a r e l e s s l i k e l y t o modify s m o l t - t o - a d u l t ocean s u r v i v a l r a t e s , we a r e b e g i n n i n g t o u n d e r s t a n d the f a c t o r s t h a t a f f e c t such r a t e s (ODFW 1982; C l a r k and M c C a r l 1983). I f we b e l i e v e t h a t e a r l y and c r i t i c a l ocean s u r v i v a l depends on the t e m p o r a l p a t t e r n of u p w e l l i n g , as i n t e r p r e t e d by n e a r - s h o r e t e m p e r a t u r e s t r u c t u r e , and i f we observe t h a t such a s t r u c t u r e 65 i s c h a n g i n g , we c o u l d i n c r e a s e or d e c r e a s e r e q u i s i t e escapement g o a l s i n r e s p o n s e . The i m p l i c a t i o n s of e x p e c t e d changes, such as an i n c r e a s e i n prespawning m o r t a l i t y , on management d e c i s i o n s a r e not r e a d i l y o b v i o u s ; i n d e e d they may appear t o be i n c o n f l i c t . For example, a f i s h e r m a n w i t h a s h o r t time h o r i z o n would advocate h i g h e r c a t c h e s b e f o r e an e x p e c t e d prespawning m o r t a l i t y t a k e s i t s t o l l . A r e g u l a t o r y agency, on the o t h e r hand, w i t h a l o n g term mandate f o r the r e s o u r c e would recommend h i g h e r escapements t o compensate f o r t h e h i g h e r e x p e c t e d prespawning m o r t a l i t y . U s i n g a s p e c i f i c d e c i s i o n r u l e , namely t o maximize the l o n g term h a r v e s t a b l e s u r p l u s , t h e r e s u l t s i n c h a p t e r one r e s o l v e t h i s apparent c o n f l i c t and i l l u s t r a t e , u s i n g t h r e e models, the e x p l i c i t i n c o r p o r a t i o n of e x p e c t e d changes i n l i f e h i s t o r y p a r a m e t e r s , such as s u r v i v a l r a t e s , i n management d e c i s i o n s . T h i s d e c i s i o n r u l e however, i m p l i c i t l y assumes t h a t b o t h manager and f i s h e r m a n a r e o p e r a t i n g under an i d e n t i c a l d i s c o u n t r a t e , namely z e r o . I n c o r p o r a t i n g a p o s i t i v e d i c o u n t r a t e w h i l e a p p l y i n g t h e d e c i s i o n r u l e may modify i t s outcome ( M o u s s a l l i and H i l b o r n 1983). Perhaps the most s u r p r i s i n g and c o u n t e r i n t u i t i v e r e s u l t of the a n a l y s i s i s the importance of r e a r i n g c o n s t r a i n t s . I f t h e r e i s no r e a r i n g c o n s t r a i n t then t h e o p t i m a l escapement goes up w i t h h i g h e r prespawning s u r v i v a l or e g g - t o - s m o l t s u r v i v a l , but i f r e a r i n g i s l i m i t e d t h e o p t i m a l escapement goes down. T h i s c o n c l u s i o n h o l d s f o r parameters of any i n t e r m e d i a t e s t a g e as shown i n F i g u r e 2.3. Most human i n t e r v e n t i o n w i l l a f f e c t t h e s e 66 s u r v i v a l r a t e s . H y d r o l o g i c a l m a n i p u l a t i o n may change prespawning s u r v i v a l due t o temperature changes, or egg-to-smolt s u r v i v a l due t o s i l t a t i o n or m o d i f i c a t i o n of f l o w . Stream e n r i c h m e n t , c a n a l i z a t i o n , bank p r o t e c t i o n e t c . s h o u l d a l l a f f e c t e g g - t o - s m o l t s u r v i v a l (Cederholm e t a l . 1981; S c r i v e n e r and Brownley 1981). Though th e s e e f f e c t s a r e documented ( L i s t e r and Walker 1981; Hartman et a l . 1982), t o my knowledge the c o n s i d e r a t i o n g i v e n t o such h y d r o l o g i c a l m a n i p u l a t i o n s , does not e x t e n d s u f f i c i e n t l y t o management i m p l i c a t i o n . In a l i f e h i s t o r y model such as t h e one p r e s e n t e d h e r e , d i s a g g r e g a t i n g the p r o d u c t i o n f u n c t i o n i n t o component s t e p s has f u r t h e r i m p l i c a t i o n s f o r s a l m o n i d enhancement a c t i v i t i e s . A l t e r n a t i v e p r o p o s a l s of enhancement, such as stream bed improvements, l a k e f e r t i l i z a t i o n , or bank p r o t e c t i o n t o name a few, may be e v a l u a t e d i n terms of t h e i r impact a t the s p e c i f i c s t a g e i n the l i f e h i s t o r y as w e l l as i n terms of o v e r a l l p r o d u c t i o n . To do t h i s , a v e r s i o n of t h i s model has been adapted t o a s p r e a d s h e e t c o n f i g u r a t i o n on a microcomputer ( M o u s s a l l i and H i l b o r n 1983). One outcome of such a s i m u l a t i o n i s t h a t i f a h a b i t a t b o t t l e n e c k e x i s t s such as smolt r e a r i n g space due t o t e r r i t o r i a l i t y , then p r o j e c t s t h a t i n c r e a s e f r y p r o d u c t i o n a r e m i s a l l o c a t e d . V a r i o u s enhancement p r o j e c t s a r e s i m u l a t e d over twenty y e a r s ' t i m e ; d i s c o u n t e d b e n e f i t s and c o s t s a r e e v a l u a t e d and compared. Thus used, t h i s model p r o v i d e s a c o m p u t a t i o n a l framework f o r e v a l u a t i n g a l t e r n a t i v e enhancement p r o p o s a l s . Inasmuch as l o c a l knowledge of stream reaches i s a v a i l a b l e , the outcome of such s i m u l a t i o n s i s i n c r e a s i n g l y 67 r e l i a b l e . D i s a g g r e g a t i n g a p r o d u c t i o n f u n c t i o n - i n the manner p r e s e n t e d a l l o w s the t h e e v a l u a t i o n of p o t e n t i a l l y p r e d i c t a b l e e f f e c t s of l o n g term a n t h r o p o g e n i c m o d i f i c a t i o n s t o anadromous f i s h h a b i t a t , such as t h o s e g e n e r a t e d by d e f o r e s t a t i o n ( H o l t b y and Hartman 1982; Hartman and H o l t b y 1982). In a t e n - y e a r study of a watershed on the west c o a s t of Vancouver I s l a n d , B r i t i s h C o lumbia, S c r i v e n e r and Andersen (1982) d i f f e r e n t i a t e d between two groups of l o g g i n g r e l a t e d e f f e c t s : I n t h e s h o r t term (1-15 y e a r s ) opening of the canopy i n c r e a s e s the r a t e of energy t r a n s f e r i n t o t h e b i o l o g i c a l community. P r i m a r y p r o d u c t i o n , d e n s i t i e s of b e n t h i c and d r i f t food organisms and p r e d a t o r s i n c l u d i n g s a l m o n i d s can a l l i n c r e a s e . These impacts d i s a p p e a r as the d r i v i n g f o r c e of s u n l i g h t on the stream and the i n c r e a s e d f r e e n u t r i e n t s a r e reduced by t h e second growth f o r e s t . The second group of h a b i t a t changes a r e u s u a l l y n e g a t i v e and l o n g term (35-50 y e a r s ) . They a r e most pronounced i n low g r a d i e n t streams where m e c h a n i c a l energy may not be adequate t o t r a n s p o r t i n t r o d u c e d f i n e s which then accumulate i n the g r a v e l d e s t r o y i n g the spawning h a b i t a t . I t i s p o s s i b l e t o use t h e p r e s e n t model i n a manner t h a t a l l o w s the l o n g term e f f e c t s t h e e x p l o i t a t i o n of one r e s o u r c e be e x p l i c i t l y i n c o r p o r a t e d i n management p l a n s of a n o t h e r one. As s p e c i e s i n t e r a c t i o n s become more t r a c t a b l e (Jones 1982; U r s i n 1982; Sheldon e t a l . 1982) a t i n t e r m e d i a t e s t a g e s i n l i f e h i s t o r y , d i s a g g r e g a t i n g a p r o d u c t i o n f u n c t i o n becomes more f e a s i b l e and t h u s may h e l p f o r m u l a t e m u l t i s p e c i e s f i s h e r i e s 68 management. T h i s s t u d y p o i n t s t o the importance of f a c t o r s a c t i n g a t i n t e r m e d i a t e l i f e h i s t o r y s t a g e s i n h a r v e s t a b l e p o p u l a t i o n s . As f u t u r e e m p i r i c a l s t u d i e s d e f i n e such f a c t o r s , they may be e x p l i c i t l y i n c o r p o r a t e d i n management d e c i s i o n s . 69 LITERATURE CITED Bakun, A. 1973. C o a s t a l u p w e l l i n g i n d i c e s , west c o a s t of No r t h A m e r i c a , 1946-1971. U.S. Dept. Commerce, NOAA Tech. Rept. NMFS SSRF 671. 103pp. Bakun, A. 1975. D a i l y and weekly u p w e l l i n g i n d i c e s , west c o a s t of N o r t h America,1967-1973. U.S. Dept. Commerce, NOAA Tech. Rept. NMFS SSFR 693. 114pp. B e v e r t o n , R.J.H. and S . J . H o l t . 1957. On t h e dynamics of e x p l o i t e d f i s h p o p u l a t i o n s . U.K. M i n . A g r . , F i s h . I n v e s t i g . , S e r . 2, 19:533 p. Bovee, K.D. 1978. P r o b a b i l i t y of use c r i t e r i a f o r the f a m i l y s a l m o n i d a e . I n s t r e a m Flow I n f o r m a t i o n Paper No. 3,4. U.S. F i s h and W i l d l i f e S e r v i c e , F o r t C o l l i n s , C o l o r a d o . Cederholm, D.J., L.M. R e i d , and E.O. S a l o . 1981. C u m u l a t i v e e f f e c t of l o g g i n g road sediment on s a l m o n i d p o p u l a t i o n s i n C l e a r w a t e r R i v e r , J e f f e r s o n County, Washington. Pages 38-74 I_n P r o c e e d i n g s from the c o n f e r e n c e Salmon Spawning g r a v e l : a renewable r e s o u r c e i n the P a c i f i c n o r t h w e s t ? S t a t e of Washington Water R e s e a r c h C e n t e r , Washington S t a t e U n i v e r s i t y , P u l l m a n , Washington. Report 39. Chapman, D.W. 1962. A g g r e s s i v e b e h a v i o u r i n j u v e n i l e coho salmon as a cause of e m i g r a t i o n . J . F i s h . Res. Board Can. 19:1047-1080. Chapman, D.W. 1966. Food and space as r e g u l a t o r s of s a l m o n i d p o p u l a t i o n s i n streams . Amer. Nat. 100:345-357. C l a r k , C.W. 1973. The economics of o v e r e x l p o i t a t i o n . S c i e n c e 181:630-634. C l a r k , C.W., and G. Munro. 1978. Renewable r e s o u r c e management and e x t i n c t i o n . J . Env. Econ. Manag. 5:198-205. 70 C l a r k , J . and B. M c C a r l . 1983. An i n v e s t i g a t i o n of the r e l a t i o n s h i p between Oregon coho salmon ( Oncorhynchus  k i s u t c h ) h a t c h e r y r e l e a s e s and a d u l t p r o d u c t i o n u t i l i z i n g law of the minimum r e g r e s s i o n . Can. J . F i s h . Aquat. S c i . 40:516-523. C u s h i n g , D.H. 1969. The f l u c t u a t i o n of y e a r - c l a s s e s and the r e g u l a t i o n of f i s h e r i e s . F i s k e r i d i r . S k r . S e r . Havunders. 15:368-379. C u s h i n g , D.H. 1973a Stock and r e c r u i t m e n t and t h e problem of d e n s i t y - d e p e n d e n c e . Rapp. P.-V. Reun. Cons. I n t . E x p l o r . Mer 164:142-155. C u s h i n g , D.H. 1973b. R e c r u i t m e n t and the p a r e n t s t o c k i n f i s h e s . U n i v . Wash. P r e s s , S e a t t l e , Wa. 197 p. D i x o n , W.J. and F . J . Massey J r . 1969. I n t r o d u c t i o n t o s t a t i s t i c a l a n a l y s i s . T h i r d e d i t i o n . M c G r a w - H i l l Book Company, New York, N.Y. 638 pp. G u l l a n d , J.A. 1965. S u r v i v a l of t h e youngest s t a g e s of f i s h , and i t s r e l a t i o n t o y e a r - c l a s s s t r e n g t h . Spec. P u b l . ICNAF 6:363-371. G u l l a n d , J.A. 1974. The management of marine f i s h e r i e s . U n i v . Wash. P r e s s , S e a t t l e , Wa. 198 p. Hartman, G.F. and L.B. H o l t b y . 1982. An o v e r v i e w of some b i o p h y s i c a l d e t e r m i n a n t s of f i s h p r o d u c t i o n and f i s h p o p u l a t i o n r esponses t o l o g g i n g i n C a r n a t i o n C r e e k , B r i t i s h Columbia. Pages 348-374 I_n G. Hartman [ed.] P r o c e e d i n g s of the C a r n a t i o n Creek Workshop, a 10 y e a r Review, 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, B.C. F e b r u a r y 24-26, 1982.. Hartman, G.F., B.C. Andersen, and J . S c r i v e n e r . 1982. Seaward movement of coho salmon (Oncorhynchus k i s u t c h ) f r y i n C a r n a t i o n Creek, an u n s t a b l e c o a s t a l stream i n B r i t i s h C o l umbia. Can. J . F i s h . Aquat. Sci.39:588-579. 71 Hempel, G. 1965. On the importance of l a r v a l s u r v i v a l f o r the p o p u l a t i o n dynamics of marine foo d f i s h . C a l i f . Coop. Oceanic F i s h . I n v e s t . , Rep. 10:13-23. H i l b o r n , R. and C.J. W a l t e r s . 1977. D i f f e r i n g g o a l s of salmon management on the Skeena R i v e r . J . F i s h . Res. Board Can. 34:64-72. H j o r t , J . 1914. F l u c t u a t i o n s i n t h e g r e a t f i s h e r i e s of N o r t h e r n Europe viewed i n the l i g h t of b i o l o g i c a l r e s e a r c h . Rapp. P.-v. Reun. Cons. I n t . E x p l o r . Mer. 20:1-228. H o l t b y , L.B. and G.F. Hartman. 1982. The p o p u l a t i o n dynamics of coho salmon ( Oncorhynchus k i s u t c h ) i n a West Coast r a i n f o r e s t stream s u b j e c t e d t o l o g g i n g . Pages 308-347 In G. Hartman [ed.] P r o c e e d i n g s of the C a r n a t i o n Creek Workshop, a 10 year Review, 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, B.C. F e b r u a r y 24-26, 1982. J o n e s , R. 1982. S p e c i e s i n t e r a c t i o n s i n the N o r t h Sea, Pages 48-63. In M.C. Mercer [ed.] M u l t i s p e c i e s approaches t o f i s h e r i e s management a d v i c e . Can. Spec. P u b l . F i s h . Aquat. S c i . 59. Keeney, R.L., and H. R a i f f a . 1976. D e c i s i o n s w i t h m u l t i p l e o b j e c t i v e s . 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Mer 173:212-230. L a s k e r , R. 1979. The e f f e c t of weather and o t h e r e n v i r o n m e n t a l v a r i a b l e s upon l a r v a l f i s h s u r v i v a l l e a d i n g t o r e c r u i t m e n t of t h e N o r t h e r n anchovy. Pages 127-129 I_n C l i m a t e and F i s h e r i e s , P r o c e e d i n g s of a Workshop, C e n t e r f o r Ocean Mgmt. S t u d i e s , U n i v . Rhode I s l a n d . L a s k e r , R. 1981. F a c t o r s c o n t r i b u t i n g t o v a r i a b l e r e c r u i t m e n t of the n o r t h e r n anchovy ( E n g r a u l i s mordax) i n the C a l i f o r n i a C u r r e n t : c o n t r a s t i n g y e a r s , 1975 th r o u g h 1978. ICES Symp. on the E a r l y L i f e H i s t o r y of F i s h . Woods H o l e , Mass., A p r i l 1979. Rapp. P.-v. Reun. Cons. I n t . E x p l o r . Mer, 178:375-387. L i s t e r , D.B., and C.E. Walker. 1966. 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S i g n i f i c a n c e of e a r l y emergence, e n v i r o n m e n t a l r e a r i n g c a p a c i t y , and b e h a v i o u r a l e c o l o g y of j u v e n i l e coho salmon i n stream c h a n n e l s . J . F i s h . Res. Board Can. 22:173-190. M c N e i l , W.J. 1964. Redd s u p e r i m p o s i t i o n and egg c a p a c i t y of p i n k salmon spawning beds. J . F i s h . Res. Board Can. 21(6):1385-1396. M o u s s a l l i , E. f and R. H i l b o r n . 1983. Salmonid h a b i t a t e v a l u t i o n model on v i s i c a l c : documented user manual and program. C o o p e r a t i v e F i s h e r i e s Research U n i t . Report No. 12. I n s t i t u t e of A nimal Resource E c o l o g y . U n i v . B r i t i s h C o l umbia, Vancouver. Neave, F. 1953. P r i n c i p l e s a f f e c t i n g the s i z e of p i n k and chum salmon p o p u l a t i o n s i n B r i t i s h C olumbia. J . F i s h . Res. Board Can. 9 ( 9 ) : 4 5 0 - 4 9 l . Neave, F. 1958. Stream e c o l o g y f i s h . 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