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Properties of age structure models for harvested populations with applications to yellowfin tuna Thunnus… Allen, Robin Leslie 1972

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PROPERTIES OF AGE STRUCTURE MODELS FOR HARVESTED POPULATIONS WITH APPLICATIONS TO YELLOWFIN TUNA Thunnus a l b a c a r e s (BGNATERRE) AND HARP SEALS Paggphilus 3£oenlandicus (ERXLEBEN) by Robin L e s l i e A l l e n B.Sc. Hons., V i c t o r i a U n i v e r s i t y of Wellington, 1966 A T h e s i s Submitted In P a r t i a l F u l f i l m e n t Of The Requirements For The Degree Of Doctor Of Philosophy i n the Department of Zoology We Accept T h i s T h e s i s As Conforming To The Required Standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1972 In present ing th i s thes is in p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the Un ive rs i t y of B r i t i s h Columbia, I agree that the L i b r a r y sha l l make i t f r e e l y a v a i l a b l e for reference and study. I f u r t h e r agree that permission for extensive copying o f t h i s thes 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 representat i ves . It i s understood that copying or p u b l i c a t i o n o f t h i s thes i s f o r f i n a n c i a l gain sha l l not be allowed without my wr i t ten permiss ion . Department of The Un ive rs i t y of B r i t i s h Columbia Vancouver 8, Canada Date M+f &jLfc& t?7Z A b s t r a c t Dynamic p o p u l a t i o n models that i n c o r p o r a t e the age s t r u c t u r e of the p o p u l a t i o n can be used as a powerful t o o l f o r f i s h e r i e s management. In t h i s t h e s i s some of the p r o p e r t i e s and p o t e n t i a l uses of a c l a s s of these models i s examined. T h i s c l a s s can be de s c r i b e d as a l i f e t a b l e o p e r a t i n g on a p o p u l a t i o n with a s t o c k - r e c r u i t r e l a t i o n t h a t i s formed by m u l t i p l y i n g the egg p r o d u c t i o n by a f u n c t i o n of a l i n e a r combination of the numbers of f i s h of d i f f e r e n t ages. The study of the p r o p e r t i e s i n c l u d e d an a n a l y s i s of the s t a b i l i t y and the n o n - e q u i l i b r i u m behaviour of the model. T h i s behaviour was r e l a t e d to some types of f l u c t u a t i o n s observed i n f i s h p o p u l a t i o n s , s p e c i a l a t t e n t i o n was given to the l i m i t c y c l e s that occur i n p o p u l a t i o n s with an uns t a b l e e q u i l i b r i u m and t h i s i s considered as a p o s s i b l e a mechanism causing the r e g u l a r f l u c t u a t i o n s known as c y c l i c dominance i n sockeye salmon Oncorhyncus nerka (Walbaum). The f i s h e r i e s f o r y e l l o w f i n tuna i n the e a s t e r n P a c i f i c Ocean, and harp s e a l s i n the western North A t l a n t i c were used as examples to show how these models might be used by a management body. In the case of the y e l l o w f i n tuna the model agreed with r e s u l t s obtained using s u r p l u s p r o d u c t i o n models and i n a d d i t i o n suggested i t i s p o s s i b l e t o i n c r e a s e both catch and catch per u n i t e f f o r t by changing the open season from the beginning of the year t o l a t e r i n the year. The y i e l d could a l s o be i n c r e a s e d by using a f i s h i n g method that i n c r e a s e d the age of f i r s t c apture. However, t h i s would s u f f e r from the disadvantage of producing a much grea t e r v a r i a b i l i t y i n the y i e l d because of the conseguent r e d u c t i o n i n the number of year c l a s s e s making a s u b s t a n t i a l c o n t r i b u t i o n to the y i e l d . The a n a l y s i s o f the harp s e a l p o p u l a t i o n showed that t h i s p o p u l a t i o n i s being over-harvested, and that i f the s i z e of the pup harvest i s not reduced the e n t i r e pup pr o d u c t i o n would be harvested by 1980. I f the harvest r a t e s a re s u b s t a n t i a l l y reduced a s u s t a i n e d h a r v e s t c o u l d be taken, the best estimate of a s u s t a i n e d k i l l of pups only was 172,000. Because the esti m a t e s of the parameters are not very p r e c i s e an a n a l y s i s of the s e n s i t i v i t y of the r e s u l t s t o changes i n the parameters was made, and t h i s showed i f the estimates are not very accurate' c o n s i d e r a b l e e r r o r s c o u l d be made i n the p r e d i c t i o n s of the model. T h i s a n a l y s i s was used to compare s e v e r a l p o s s i b l e o b s e r v a t i o n s t h a t c o u l d be used to check the e f f e c t of a reduced harvest on the p o p u l a t i o n , and of these the c a t c h curves of the younger age groups o f f e r e d the best compromise between s e n s i t i v i t y to changes i n p o p u l a t i o n s i z e and ease of measurement. i v T a b l e of Contents Page Ab s t r a c t i i L i s t of Tables v L i s t of F i g u r e s v i Acknowledgement v i i i I n t r o d u c t i o n 1 The Model 7 Non-Equilibrium behaviour of the Model 15 The Model a p p l i e d to Y e l l o w f i n Tuna 29 Harvest S t r a t e g i e s f o r Y e l l o w f i n tuna 57 Harp s e a l s 71 D i s c u s s i o n 96 L i t e r a t u r e c i t e d 100 Appendix 1 107 Appendix 2 108 Appendix 3 109 V L i s t of t a b l e s Table Page 1. Estimated weight and e x p o n e n t i a l growth r a t e of y e l l o w f i n tuna. 34 2. Estimates of t o t a l m o r t a l i t y and f i s h i n g e f f o r t f o r y e l l o w f i n tuna. 34 3. Average weight and estimated f e c u n d i t y of female y e l l o w f i n tuna i n the t h i r d guarter of the year. 38 4. Catch of y e l l o w f i n tuna and f i s h i n g e f f o r t i n the Commission's y e l l o w f i n r e g u l a t o r y area; 1934-1968. 43 5. Numbers of age one year and a d u l t s e a l s i n samples at St Anthony and the Front. 77 6. Estimates of pup p r o d u c t i o n using Sergeant's method. 77 7. Harvest of harp s e a l s f o r the years 1938-1971. 79 8. Estimated harp s e a l p o p u l a t i o n h i s t o r y , 1938-1971. 80 9. P r o j e c t e d harp s e a l p o p u l a t i o n s f o r the years 1972-1980. 89 L i s t of F i g u r e s F i g u r e 1. Small o s c i l l a t i o n s about the e q u i l i b r i u m of the p o p u l a t i o n s . 2. Large f l u c t u a t i o n s about unstable e q u i l i b r i u m p o p u l a t i o n s . 3. F l u c t u a t i o n s about e q u i l i b r i u m p o p u l a t i o n s modified by a random v a r i a b l e . 4. S p e c t r a of s e r i e s of r e c r u i t s generated by p o p u l a t i o n s modified by random v a r i a b l e s . 5. The e a s t e r n t r o p i c a l P a c i f i c Ocean showing the CYBA and the s u b d i v i s i o n s A1 and A2. 6. Catch curves f o r the 1954-1964 y e l l o w f i n tuna year c l a s s e s . 7. Y i e l d per r e c r u i t i n the y e l l o w f i n tuna f i s h e r y . 8. Y i e l d and estimated y i e l d per r e c r u i t i n the y e l l o w f i n tuna f i s h e r y , 1932-1968. 9. S t o c k - r e c r u i t r e l a t i o n s generated by f i t t i n g the model t o data f o r 1932-1960 and f o r 1932-1968. 10. The s u r f a c e of the r e s i d u a l sum of sguares f o r the model with the data f o r area A1. 11. Y i e l d of y e l l o w f i n tuna from the CYRA. 12. Comparison of e q u i l i b r i u m y i e l d s f o r the l o g i s t i c and age s t r u c t u r e models. 13. Catch per u n i t of e f f o r t and e f f o r t f o r area A2, 1962-1970. 14. The spectrum f o r the y e l l o w f i n catch data from area A l . 15. The s u s t a i n e d y i e l d s f o r q u a r t e r l y gears. 16. The y i e l d f o r p o l i c i e s of maximizing the t o t a l y i e l d over p e r i o d s of 5 and 10 years. 17. The average y i e l d f o r p e r i o d i c h a r v e s t i n g with one, two, and three years between h a r v e s t i n g years. v i i 18. Harvest sequences from s i m u l a t i o n s of the y e l l o w f i n tuna p o p u l a t i o n with random v a r i a t i o n i n r e c r u i t m e n t . 68 19. Expected number of pups to be born to female s e a l s of d i f f e r e n t ages. 83 20. P r o j e c t i o n s of numbers i n the 1967 1969, 1970, and 1971 harp s e a l year c l a s s e s r e l a t i v e t o the 1968 year c l a s s i n the year 1972. 85 21. lumbers of harp s e a l s i n the year c l a s s e s of 1968-1971 i n the year 1972. 86 22. P r o j e c t e d pup pr o d u c t i o n using the three parameter s e t s . 87 23. P r o j e c t e d c a t c h curves f o r harp s e a l s i n 1976 and 1980. 90 24. S t o c k - r e c r u i t curves f o r harp s e a l s . 94 v i i i Acknowledgement I t i s a plea s u r e t o thank Dr P. A. L a r k i n f o r h i s guidance, f i n a n c i a l support, and enthusiasm f o r t h i s study. To Dr N. J . Wilimovsky are due thanks f o r h i s e f f o r t s i n d i r e c t i n g ray reading as w e l l as h i s advic e on my r e s e a r c h . Also thanks to the other members of my re s e a r c h committee Dr J. Kane, and Dr B. N. Moyls f o r t h e i r h e l p f u l c r i t i c i s m . Advice and i n f o r m a t i o n was f r e e l y given t o me by Dr J . J . Joseph and the s t a f f of the Inter-American T r o p i c a l Tuna Commission, and by Dr H. D. F i s h e r . Mr S. Borden, Mrs a. Kardynal, Mrs D. L a u r i e n t e , Br W. Webb of the Bi o l o g y Data Centre provided me with much a p p r e c i a t e d computing support. F i n a l l y I would l i k e to thank my wife J a n i c e f o r her encouragement and constant support over the time of t h i s study. 1 I n t r o d u c t i o n In t h i s t h e s i s I s h a l l explore the uses and p r o p e r t i e s of dynamic models which r e p r e s e n t a p o p u l a t i o n as a s e r i e s of year c l a s s e s , the members of which are reduced each year by n a t u r a l and f i s h i n g m o r t a l i t y . By dynamic I mean non - e q u i l i b r i u m models, t h a t i s models i n which the p o p u l a t i o n i n any year depends on the previous h i s t o r y of the p o p u l a t i o n . T h i s type of model i s of most value when the r e l a t i v e year c l a s s si2e i s more or l e s s f i x e d , apart from f i s h i n g m o r t a l i t y , soon a f t e r i t i s produced. E s s e n t i a l l y the model i s a l i f e t a b l e o p e r a t i n g on a p o p u l a t i o n with a s t o c k - r e c r u i t r e l a t i o n . T h i s approach combines the best f e a t u r e s of the y i e l d per r e c r u i t and l o g i s t i c models which are the most commonly used models at the present. These f e a t u r e s are the i d e n t i f i c a t i o n of elementary processes which can be independently estimated i n the y i e l d per r e c r u i t models, and the dynamic nature of the l o g i s t i c models. There are s e v e r a l advantages that can be gained from using dynamic models t h a t i n c o r p o r a t e age s t r u c t u r e , one of the most important of these i s t h a t they can be used to examine the e f f e c t s of a wide range of h a r v e s t p o l i c i e s . T r a d i t i o n a l l y harvest p o l i c i e s have been p r i m a r i l y d i c t a t e d by the nature of the f i s h e r y , with r e s t r i c t i o n s on types o f gear, t o t a l e f f o r t or c a t c h , being i n t r o d u c e d a f t e r i t became obvious that s t o c k s were being s e r i o u s l y d e p l e t e d . S i z e l i m i t s were i n t r o d u c e d i n i t i a l l y to p r o t e c t the spawning stock, and more r e c e n t l y to attempt to allow year c l a s s e s to reach t h e i r maximum biomass before being 2 harvested. P a r t l y because of t h i s e a r l i e r models as a r u l e have only allowed f o r management by changing the age of f i r s t capture and o v e r a l l f i s h i n g e f f o r t . Dynamic age s t r u c t u r e models can be used to study the e f f e c t s of these p o l i c i e s as w e l l as a wide range of a l t e r n a t i v e s such as h a r v e s t i n g only a few of the year c l a s s e s present or p o l i c i e s t h a t f l u c t u a t e i n time. A second use of the i n f o r m a t i o n on age s t r u c t u r e i s t h a t harvest p o l i c i e s may be designed to ensure that the age s t r u c t u r e of the stock i s kept near some d e s i r a b l e s t r u c t u r e . Often an i n t e n s i v e f i s h e r y on a s h o r t l i v e d f i s h r e s u l t s i n the r e d u c t i o n of the spawning stock to only one or two year c l a s s e s of s u b s t a n t i a l abundance. T h i s coupled with poor spawning c o n d i t i o n s over one or two years c o u l d cause a d r a s t i c d e c l i n e i n the s t o c k . I t i s l i k e l y t h a t t h i s was a c o n t r i b u t i n g f a c t o r i n the d e c l i n e of the C a l i f o r n i a s a r d i n e Sardinop_s c a e r u l a (Girard) f i s h e r y (Clark 1952) , and c o u l d e a s i l y happen i n other major f i s h e r i e s where the spawning stock c o n s i s t s of very few year c l a s s e s such as the anchoveta E n g r a u l i s r i n gens (Jenyns) f i s h e r y of Peru (Schaefer 1967a), or the menhaden B r e v p o r t i a tyrannus (Latrobe) f i s h e r y on the A t l a n t i c c o a s t of U.S.A. (Schaaf and Huntsman 1972). Another advantage of keeping as many year c l a s s e s as p o s s i b l e i n the stock i s t h a t i t reduces the e f f e c t of v a r i a t i o n i n i n d i v i d u a l year c l a s s s i z e s on the p o p u l a t i o n s i z e and the y i e l d . In many f i s h e r i e s , estimates of year c l a s s s t r e n g t h are made r o u t i n e l y and these estimates can be used e i t h e r as data f o r the model or as a check on how w e l l the model i s d e s c r i b i n g 3 the behaviour of the p o p u l a t i o n . when year c l a s s s i z e i s measured before or s h o r t l y a f t e r a year c l a s s i s f i r s t e x p l o i t e d the estimates can be used with the model to provide s h o r t term p r e d i c t i o n s of y i e l d . With t h i s i n f o r m a t i o n a management body should be able to take f u l l advantage of a l a r g e year c l a s s or to ensure that a weak incoming year c l a s s i s not over e x p l o i t e d . The main disadvantage of these models i s t h e i r complexity. They are not u s u a l l y amenable to a n a l y t i c treatment and as a r u l e r e g u i r e a l a r g e amount of computer time both f o r f i t t i n g the model and f o r examining the p r o p e r t i e s of the model. An a s s o c i a t e d problem i s t h e i r need f o r more parameters than are r e q u i r e d f o r e i t h e r the l o g i s t i c or y i e l d per r e c r u i t models. Heview of e a r l i e r work A simple l o g i c a l f o u n d a t i o n f o r y i e l d models was provided by R u s s e l l (1931) i n h i s statement that the s u r p l u s y i e l d i s the sum of growth and recruitment minus the n a t u r a l m o r t a l i t y . On the b a s i s of t h i s , y i e l d models can be c l a s s i f i e d as belonging to one of two types. These are the y i e l d per r e c r u i t models i n which parameters f o r the three processes are estimated s e p a r a t e l y and t h e i r e f f e c t s combined i n the model; and the l o g i s t i c types i n which parameters that d e s c r i b e the summed e f f e c t s of a l l processes i n the f i s h p o p u l a t i o n are estimated from catch and e f f o r t data. Although S u s s e l l ^ s statement provided the l o g i c a l b a s i s f o r y i e l d models, i t was preceded by Baranov (1918) who produced the f i r s t y i e l d per r e c r u i t y i e l d model. Baranov i n t r o d u c e d the i d e a 4 of instantaneous m o r t a l i t y and used a c u b i c f u n c t i o n of time to d e s c r i b e growth of a f i s h . His approach r e c e i v e d l i t t l e a t t e n t i o n i n the Western world u n t i l the l a t e 1950s when the method became popular f o l l o w i n g the work of Beverton and H o l t (Hulme et a l . 1947, Graham 1952, Beverton 1953). Although Baranov's mathematical f o r m u l a t i o n was i g n o r e d , the y i e l d per r e c r u i t model was employed i n a t a b u l a r form n o t a b l y by Thompson and B e l l (1934) and R i c k e r (1945). S i n c e the p u b l i c a t i o n of Beverton and H o l t ' s book (Beverton and H o l t 1957), both methods of a p p l y i n g the model have been used as a guide f o r management (Hennemuth 1961b, IPHC 1960, Ketchen and F o r r e s t e r 1966). The l o g i s t i c model was d e r i v e d by V e r h u l s t (1838) and independently by P e a r l and Reed (1920). Many important developments of t h i s simple model were made p r i n c i p a l l y by Lotka (1924) and V o l t e r r a (Scudo 1971) , one of the most important of these being the m u l t i s p e c i e s form of the model which has f o r a long time been the major model used i n s t u d i e s of m u l t i s p e c i e s f i s h e r i e s (Larkin 1963). The simple model was used as a y i e l d model by H j o r t et a l . (1933) and by Graham (1935). Schaefer (1954, 1957) provided a method of e s t i m a t i n g the parameters f o r t h i s model and h i s methods have been the b a s i s f o r the management of y e l l o w f i n tuna i n the eastern P a c i f i c by the Inter-American T r o p i c a l Tuna Commission. M o d i f i c a t i o n s of t h i s model have been suggested by P e l l a and Toralinson (1969) who i n t r o d u c e d a t h i r d parameter which allows the s u r p l u s production curve to be skewed, and by Fox (1970) who used an e x p o n e n t i a l r e l a t i o n between c a t c h per u n i t e f f o r t and e f f o r t . 5 In the P a c i f i c salmon f i s h e r y a d i f f e r e n t management approach was d i c t a t e d by the l i f e h i s t o r y of salmon and the nature of the c o a s t a l f i s h e r y . For the c o a s t a l f i s h e r y , salmon are only s u s c e p t i b l e to capture during a b r i e f time span before they spawn and d i e . In t h i s case the aim i s to capture as many f i s h as p o s s i b l e s u b j e c t to the c o n s t r a i n t of a l l o w i n g s u f f i c i e n t spawners to provide f o r subsequent g e n e r a t i o n s . The problem was then to f i n d a r e l a t i o n between the number of spawners i n a given spawning area and the number of o f f s p r i n g that would r e t u r n to spawn i n l a t e r years. B i c k e r (1954) suggested such a r e l a t i o n s h i p and s e v e r a l v a r i a n t s of t h i s were d e s c r i b e d l a t e r ( L a r k i n et a l . 1964). As w e l l as f o c u s i n g a t t e n t i o n on the r e l a t i o n between stock and recruitment these s t u d i e s have s t i m u l a t e d work on the problem of what management p o l i c i e s are best i n the face of u n c e r t a i n t y about the f i s h s t o c k s . (Ricker 1958, Tautz et a l . 1969, A l l e n 1972). L e s l i e (1945, 1948) developed the use of the l i f e t a b l e as an animal p o p u l a t i o n model. Most of L e s l i e ' s work was concerned with l i n e a r equations and was a p p l i e d to the p r o p e r t i e s of age s t r u c t u r e when p o p u l a t i o n s e x h i b i t e x p o n e n t i a l growth. However L e s l i e showed t h a t c e r t a i n types of d e n s i t y dependent growth that produced s t a b l e or bounded p o p u l a t i o n s c o u l d be handled using h i s method. The important d i f f e r e n c e between L e s l i e ' s models and the t a b u l a r forms of the y i e l d per r e c r u i t models i s that L e s l i e ' s model has a s t o c k - r e c r u i t r e l a t i o n , a l b e i t a very simple one. L e s l i e ' s model has been used mainly i n the a n a l y s i s of p o p u l a t i o n numbers ( L e s l i e 1966, Mertz 1971), but can e q u a l l y w e l l be used as a y i e l d model ( L e f k o v i t c h 1970). The e f f e c t of 6 v a r i a t i o n of the r a t e s of m o r t a l i t y and r e p r o d u c t i o n i n a L e s l i e model have been examined by Demetrius (1969) and Goodman (197 1). The next l o g i c a l s t e p i n the development of t h i s group of models was to r e p l a c e the simple s t o c k - r e c r u i t r e l a t i o n with more complex ones, such as those used by Southward (1968) and Walters (1969). The p a r t s of these models d e a l i n g with the numbers i n the p o p u l a t i o n are very s i m i l a r to the p o p u l a t i o n models used f o r the study of problems of stock and r e c r u i t m e n t by R i c k e r (195U) and Ward and L a r k i n (1964). 7 The Model For a s i n g l e s p e c i e s the b a s i c p o p u l a t i o n model I s h a l l use i s N(1,t) = 2*[h (j)«N ( j , t - 1 ) ].S (1,t) (1a) I N(j,t) = S( j) •» (j-1,t-1) j = 2,3 .k. (1b) where N{j,t) i s the number of f i s h that have j u s t completed t h e i r j t h . year of l i f e at the end of year t . Throughout year t I s h a l l r e f e r to these f i s h as being of age j-1. The symbol s(j)» il - 2,....k ) r e p r e s e n t s the p r o p o r t i o n of f i s h of age j -1 t h a t s u r v i v e to age j , while S(1,t) i s the p r o p o r t i o n of eggs that s u r v i v e to become f i s h of age 1, and h<j) i s the age s p e c i f i c f e c u n d i t y . In g e n e r a l a l l the c o e f f i c i e n t s S (j) and h(j) could be f u n c t i o n s of the p o p u l a t i o n s t r u c t u r e but I s h a l l take a l l but S(1,t) to be c o n s t a n t s . There are many p o s s i b i l i t i e s f o r the form of S ( 1 , t ) , but I s h a l l c o n s i d e r only those of the form S(1,t) = f { V ( t ) } where V (t) i s a l i n e a r combination of the numbers of f i s h i n each age group at that time, t h a t i s V (t) = <5Zl (j) »N ( j , t) . V(t) can be thought of as a measure of p o p u l a t i o n d e n s i t y t h a t i s a s s o c i a t e d with egg or l a r v a l m o r t a l i t y . Of the p o s s i b l e s e t of combinations a f r e q u e n t l y a r i s i n g subset w i l l be S(1,t) = f { U ( t ) } , where U (t) = 2 T h (j) •« ( j , t ) which i s the number of eggs produced by the p o p u l a t i o n . T h i s s i t u a t i o n i s l i k e l y to a r i s e when d e n s i t y dependent m o r t a l i t y i s the r e s u l t of 8 competition between eggs or between the young of the year. T h i s type of r e g u l a t i o n i s the b a s i s of the s t o c k - r e c r u i t models of B i c k e r (1954) and Beverton and Holt (1956), and has been used i n s i m u l a t i o n s t u d i e s by Southward (1968) and Walters (1969). at the o p p o s i t e extreme t h e r e i s the p o s s i b i l i t y t h a t some of the age groups which c o n t r i b u t e to V (t) do not c o n t r i b u t e to egg production f o r that year. For example, t h i s i s a c h a r a c t e r i s t i c i n the b i o l o g y of pink salmon Oncorhyncus ^orbuscha (Walbaum) popula t i o n s (Bicker 1962). Other examples of the g e n e r a l case S(1,t) = f {V (t) }, could be the r e s u l t of p r e d a t i o n by a d u l t s of the same s p e c i e s , or by other predators whose numbers are determined by the number of j u v e n i l e s i n previous y e a r s . These mechanisms have been suggested by Regier et a l . (1969) and L a r k i n (1972). F i n a l l y an important s p e c i a l case i s that where f f V ( t ) } = c where c i s a constant. T h i s i s simply the case of L e s l i e ' s model. The magnitude of unexplained or u n p r e d i c t a b l e v a r i a t i o n i n recruitment to some f i s h s t o c k s (Hjort 1926, Murphy 1966, Hylen and Dragesund 1972) i s s u f f i c i e n t l y l a r g e t o make i t important to look at the e f f e c t of random f l u c t u a t i o n s i n r e c r u i t m e n t , and consequently i n some cases S(1,t) i s m u l t i p l i e d by a random v a r i a b l e . The model i s the r e s u l t of combining a s e t of assumptions d e s c r i b i n g the growth of a p o p u l a t i o n . Of n e c e s s i t y these assumptions s i m p l i f y the processes i n v o l v e d . For example, the f e c u n d i t y i s assumed to be a f u n c t i o n of age alone. T h i s should not be i n t e r p r e t e d as assuming t h a t the e f f e c t s of d e n s i t y , or 9 other f a c t o r s not even i n c l u d e d i n the model such as environmental c o n d i t i o n s , are not important but r a t h e r as s a y i n g that the e f f e c t of v a r i a t i o n from these sources i s of l e s s importance than the v a r i a t i o n i n s u r v i v a l of eggs to r e c r u i t s . Thus i t i s assumed that the e f f e c t s of v a r i a t i o n i n any of the processes that are summarised by the con s t a n t s S { j ) , h ( j ) , and l ( j ) w i l l have e f f e c t s that a r e of secondary importance i n comparison with the e f f e c t s of the changes i n S(1,t) that are due to changes i n d e n s i t y , or e g u i v a l e n t l y as f a r as the model i s concerned with the e f f e c t s of changes i n f e c u n d i t y . Many workers (for example Ni k o l s k y 1969) have s t r e s s e d the importance of c o n c e i v i n g of p o p u l a t i o n s i z e and changes i n s i z e as being a response to f a c t o r s such as f e c u n d i t y , v i a b i l i t y of eggs and l a r v a e , m o r t a l i t y , and growth, which i n turn vary i n a complicated manner i n response to environmental changes. From t h i s p o i n t c f view my assumptions w i l l appear to be an over s i m p l i f i c a t i o n of the t r u e s i t u a t i o n . However, before the e f f e c t s of changes i n a l l these processes can be a p p r e c i a t e d i t i s necessary t o look at the s i t u a t i o n when they are assumed to be c o n s t a n t . The standard formula f o r c a l c u l a t i n g annual y i e l d from the j t h age group i n year t i s where w (x) i s the weight a t age x, and F and M are the instantaneous r a t e s of f i s h i n g and n a t u r a l m o r t a l i t y f o r t h a t age group. I f growth i s a l s o approximated by an instantaneous Y t) Aw (j + x) »exp[- (F + M)»x]»dx 10 r a t e i . e . w(j*x) i s r e p l a c e d by w (j) *exp (G»x) then Y = F«H ( j , t ) «w (j) { 1-exp[G-F-M ]}/(G-F-M) The e q u i l i b r i u m p o p u l a t i o n Suppose there i s an e q u i l i b r i u m p o p u l a t i o n with U(1,t) = n. Then time may be dropped from equations (lb) g i v i n g , say H (2) = S (2) »n = P (2) »n H{3) = S(3)«S(2)«n = P(3)«n and n (j) = P {j) »n That i s P(j) i s the f r a c t i o n of those f i s h that s u r v i v e from age 1 to age j . S e t t i n g P(1) = 1. and s u b s t i t u t i n g the ex p r e s s i o n s above i n eguation (1a) g i v e s n = 2 T [ h ( j ) * P ( j ) - n ] . f { ^ [ l ( j ) . P ( j ) •n]} Or by w r i t i n g E = 5Ith(j)»P(j) ], and D = ^ [ l ( j ) * P ( j ) ] n = E*n»f{D»n], That i s n = f - l{1/E}»/D p r o v i d i n g f - 1 e x i s t s . For t h e case that f i s a non-negative m o n o t c n i c a l l y decreasing f u n c t i o n , f _ 1 e x i s t s only f o r non-negative arguments l e s s than f f O } , and thus the p o p u l a t i o n becomes e x t i n c t i f E the 11 average number of eggs produced by a r e c r u i t f a l l s to ( f { 0 } ) _ 1 . The s t o c k - r e c r u i t r e l a t i o n The most convenient measure of stock s i z e i s the egg production^ and i n t h i s and the next s e c t i o n the two terms are used synonymously. For a given stock s i z e U(t) = <E h (j) «N {j, t) there i s a range of values of d e n s i t y V(t) = ^ 1 (j) . N (j , t) . T h i s i m p l i e s t h a t the s t o c k - r e c r u i t r e l a t i o n N(1,t + 1) = U (t) »f {V (t) } i s d e s c r i b e d by a r e g i o n r a t h e r than by a curve as i s the case f o r the R i c k e r or Beverton and Holt s t o c k - r e c r u i t r e l a t i o n s . T h i s r e g i o n i s bounded above provided there i s no value of j such that l ( j ) = 0 while h (j) > 0. a s t o c k - r e c r u i t r e l a t i o n of t h i s type would be very d i f f i c u l t to d e t e c t from an examination of s p a w n e r - r e c r u i t data, and the e s t i m a t i o n of such a r e l a t i o n would be i m p r a c t i c a l without p r i o r estimates of "weighting c o e f f i c i e n t s " 1 ( j ) . In view of t h i s i t i s not s u r p r i s i n g t h at models of the r e l a t i o n between stock and r e c r u i t s have with few exceptions been l i m i t e d to the type N(1,t + 1) = U ft) »f {U (t) }. In p r a c t i c e , the v a r i a t i o n that i s t y p i c a l l y shown by s t o c k - r e c r u i t data makes i t d i f f i c u l t to decide which f u n c t i o n a l form best d e s c r i b e s the data. For examples of curves f i t t e d to s t o c k - r e c r u i t data, see F i g u r e s 18-24 of R i c k e r (1954), and F i g u r e s 15.14-15.18 of Beverton and Holt (1957). Because of the d i f f i c u l t y of choosing a p a r t i c u l a r form of s t o c k - r e c r u i t r e l a t i o n from a wide p o s s i b l e range of r e l a t i o n s i t seems to be best to make the c h o i c e from a r e s t r i c t e d set that c o n t a i n s the 12 major f e a t u r e s t h a t may be r e g u i r e d i n a p a r t i c u l a r r e l a t i o n . Such a s e t i s generated by the f o l l o w i n g three s u r v i v a l func t i o n s (a) f (U) = c .^ - L e s l i e (b) f (U) = a«(o* (1+b«U)-i Beverton and H o l t (c) f (0) = a»exp{-b»U} B i c k e r when the d e c i s i o n as to the type of s t o c k - r e c r u i t f u n c t i o n to be used i s made s o l e l y on the b a s i s of the spawner-recruit data the f o l l o w i n g r u l e s w i l l help i n choosing an a p p r o p r i a t e r e l a t i o n . I f r e c r u i t m e n t appears to i n c r e a s e p r o p o r t i o n a l l y with stock s i z e over the whole range of the data the L e s l i e type should be used. I f f o r l a r g e s t o c k s i z e s r e c r u i t m e n t i n c r e a s e s at a slower r a t e than stock s i z e or appears to be independent of stock s i z e the Beverton and Holt r e l a t i o n should be used, while the B i c k e r r e l a t i o n should be used i f r e c r u i t m e n t d e c l i n e s a f t e r the stock surpasses some c r i t i c a l s i z e . I t i s of course probable t h a t the c h o i c e of s t o c k - r e c r u i t r e l a t i o n w i l l depend on the range of the data. For example, data from a p o p u l a t i o n with a B i c k e r r e l a t i o n would be taken as an i n d i c a t i o n of the L e s l i e r e l a t i o n i f the data only showed the ascending l e f t hand limb, while data t h a t extended to the dome of the curve c o u l d be taken as coming from a Beverton and Holt r e l a t i o n . T h i s problem cannot be avoided without g e t t i n g a d d i t i o n a l i n f o r m a t i o n on the form of the r e l a t i o n , and shows the p o s s i b l e dangers i n e x t r a p o l a t i n g o u t s i d e the range of the 13 data. Within the range of the data i t i s u n l i k e l y that the r e l a t i o n chosen w i l l make much d i f f e r e n c e to the r e s u l t s obtained from the model i f i t i s im p o s s i b l e to d i s c r i m i n a t e between the three types. The e g u i l i b r i u m s t o c k - r e c r u i t r e l a t i o n The arguments about the s t o c k - r e c r u i t r e g i o n and i t s bounds extend to the e g u i l i b r i u m r e l a t i o n which i s i n g e n e r a l d e s c r i b e d by a r e g i o n , though f o r any given m o r t a l i t y schedule there i s a f i x e d e q u i l i b r i u m p o i n t or s e t of e q u i l i b r i u m p o i n t s . Suppose the n a t u r a l s u r v i v a l i s P ( j ) , and f i s h i n g m o r t a l i t y a l t e r s the s u r v i v a l to Q ( j ) . The new value of E«n = 5f[ h (j) »Q {j) ]»n i s su b j e c t to the c o n s t r a i n t Q(j + 1)/Q(j) * P ( j + 1)/P(j) as w e l l as t o 0 < Q(j+1) < Q(j) thus the r e g i o n of p o s s i b l e e g u i l i b r i u m p o s i t i o n s i s contained w i t h i n the r e g i o n d e s c r i b i n g the s t o c k - r e c r u i t r e l a t i o n . The s t a b i l i t y of the e q u i l i b r i u m Suppose a t the e q u i l i b r i u m the stock s i z e i s U, the de n s i t y V, and the po p u l a t i o n i s N. For small displacements from the e g u i l i b r i u m equations (1) may be approximated by a l i n e a r set of 14 equations c o n s i s t i n g of the f i r s t terms of the T a y l o r s e r i e s expansion of the terms i n equations (1). Let Z{t) = N(t)-M be the displacement of the p o p u l a t i o n from i t s e q u i l i b r i u m , then Z (t) = A»Z (t-1) {3) where A i s the matrix A ( i , j ) which c o n s i s t s of the p a r t i a l d e r i v a t i v e s of N { i , t + 1) with r e s p e c t to N ( j , t ) . That i s M U J ) = h ( j ) . f (V)+l(j)»0.f'(V) A <j+1, j) = S ( j ) . The s t a b i l i t y of the e q u i l i b r i u m of equations (1) i s the same as the s t a b i l i t y of the e q u i l i b r i u m of equation (3), p r o v i d i n g A i s not a c r i t i c a l matrix (Hahn 1967). Thus the e q u i l i b r i u m of (1) i s s t a b l e i f a l l the eigenvalues of A have modulus l e s s than one, and i s u n s t a b l e i f there i s an eigenvalue with modulus greater than one. I f the modulus of the dominant eigenvalue i s e x a c t l y one, A i s c r i t i c a l , and the s t a b i l i t y of the equations must be i n v e s t i g a t e d by other methods. An o u t l i n e of the computational procedures used to e v a l u a t e the behaviour of the model i s given i n Appendix 1. 15 Non-Equilibrium Behaviour Of The Model In t h i s s e c t i o n I w i l l examine the behaviour of the model a f t e r the p o p u l a t i o n has been perturbed from i t s e q u i l i b r i u m , and I s h a l l r e l a t e t h i s to some types of f l u c t u a t i o n s shown by f i s h p o p u l a t i o n s . Large s c a l e f l u c t u a t i o n s due to v a r i a t i o n i n year c l a s s s i z e i n important f i s h e r i e s have a t t r a c t e d a l o t of i n t e r e s t , and there have been many sugg e s t i o n s as to t h e i r causes (Hjort 1926, N i k o l s k y 1961, Ward and L a r k i n 1964, Hylen and Dragesund 1972). Undoubtedly there i s a v a r i e t y of causes of these f l u c t u a t i o n s , and one p o s s i b i l i t y i s th a t i n some cases the type of d e n s i t y dependent p o p u l a t i o n processes d e s c r i b e d by the model play a major pa r t i n causing the f l u c t u a t i c n s . To check whether t h i s i s l i k e l y i n a s p e c i f i c i n s t a n c e i t i s necessary t o be f a m i l i a r with the p r o p e r t i e s of the model i n no n - e g u i l i b r i u m s i t u a t i o n s . B i c k e r (1954) made an e m p i r i c a l i n v e s t i g a t i o n of the case when S(1,t) = f { 0 ( t ) } and found that a p e r t u r b a t i o n from the e g u i l i b r i u m was f r e q u e n t l y f o l l o w e d by o s c i l l a t i o n s which may e i t h e r p e r s i s t or d i e away. His main r e s u l t s r e l a t i n g to the unst a b l e e g u i l i b r i u m can be summarised as: (1) P e r s i s t e n t o s c i l l a t i o n s only occur when the g r a d i e n t of the s t o c k - r e c r u i t r e l a t i o n a t the replacement p o i n t i s l e s s than -1. (2) The " p e r i o d " of the o s c i l l a t i o n s i s approximately twice the mean age of r e p r o d u c t i o n , and i s not much a l t e r e d by changes i n the s t o c k - r e c r u i t curve. 16 (3) The "amplitude" of the o s c i l l a t i o n s decreases with i n c r e a s i n g stock s i z e , and a l s o i s l a r g e r when the age of f i r s t r e p r o d u c t i o n i s delayed. I have used q u o t a t i o n marks about the words p e r i o d and amplitude as the o s c i l l a t i o n s shown i n R i c k e r ' s paper are not s t r i c t l y p e r i o d i c . N e v e r t h e l e s s , the p a t t e r n o f the f l u c t u a t i o n s i s s u f f i c i e n t l y r e g u l a r to make the terms u s e f u l i n d e s c r i b i n g the motion. The necessary c o n d i t i o n f o r the unstable e q u i l i b r i u m was d e r i v e d above but I do not know of an a n a l y t i c e x p r e s s i o n of the second and t h i r d p r operty. On the other hand these p r o p e r t i e s are i n t u i t i v e l y reasonable, f o r the age s t r u c t u r e i n the model provides a time l a g which produces o s c i l l a t i o n s which w i l l have a p e r i o d of the same order of magnitude as the l a g , and the more year c l a s s e s t h e r e are, the l e s s a f l u c t u a t i o n i n a s i n g l e year c l a s s w i l l a f f e c t the sum. Near the e q u i l i b r i u m p oint, the behaviour of the s o l u t i o n s of equations (1) can be approximated by the s o l u t i o n of equation (3). I t i s more convenient to work with the companion matrix of the c h a r a c t e r i s t i c e q u a t i o n of A. That i s the matrix B where B(1,j) = A{1,j).P<j) = b ( j ) say B ( j * 1 , j ) = 1, with c h a r a c t e r i s t i c equation 17 r - £{r *.b (j) } = 0. <4) t In most cases the e i g e n v a l u e s w i l l be d i s t i n c t ana can be denoted i n order of d e c r e a s i n g modulus as r (1),r<2),....r (k), and then the general s o l u t i o n of (4) i s f Z ( l . t ) = 5"{ K{j) . r (j) } where the c o n s t a n t s K(j) are determined by the i n i t i a l c o n d i t i o n s . In g e n e r a l the r (k) w i l l be complex numbers which can be w r i t t e n as r (j) = H (j) -exp{ i » e (j) }. The corresponding terms i n the s o l u t i o n , t h a t i s those c o n t a i n i n g r ( j ) and i t s complex conjugate w i l l o s c i l l a t e with a period T ( j ) = 27T /e(j). These o s c i l l a t i o n s w i l l be d e c r e a s i n g or i n c r e a s i n g i n amplitude depending on whether R(1) i s l e s s than or g r e a t e r than one. S i m u l a t i o n s showed that the dominant ro o t or p a i r of r o o t s g e n e r a l l y mask the o t h e r s i n the general s o l u t i o n , and consequently the p e r i o d of s m a l l o s c i l l a t i o n s i s as a r u l e be given by 8{1), the argument of the dominant e i g e n v a l u e . The p a r t of the s o l u t i o n i n v o l v i n g the dominant e i g e n v a l u e i s p r o p o r t i o n a l t o R (1) »cos{€ (1) * t } . If R(1) i s j u s t s l i g h t l y l e s s than one, o s c i l l a t i o n s w i l l p e r s i s t f o r a long time a f t e r an i n i t i a l d i s t u r b a n c e , while i f B(1) i s s m a l l any o s c i l l a t i o n s w i l l disappear s w i f t l y . 18 The elements of the f i r s t row of B may be p o s i t i v e or negative as f * [ V ] i s negative. In p a r t i c u l a r when 1 (j) = 0 whenever h (j) = 0 these elements w i l l be p o s i t i v e f o r s m a l l p o p u l a t i o n s , f o r as the stock s i z e approaches zero b ( j ) w i l l approach h(j)»f[V]. In t h i s case the dominant eigenvalue w i l l be r e a l and p o s i t i v e , and w i l l be greater than or l e s s than one depending on whether the sum of the elements i n the f i r s t row of B i s g r e a t e r than or l e s s than one. Now 2"tb(j)}= 5(P<JMtMJ)*f(V) + D - l ( j ) *i ]} = E»f(V) + D«(]»f'(V) = 1+D*0*f a (V) Thus the e g u i l i b r i u m i s s t a b l e i n t h i s case when the s u r v i v a l curve has a n e g a t i v e g r a d i e n t a t the e g u i l i b r i u m . The other case, t h a t i s the s u r v i v a l curve i n c r e a s i n g at the e g u i l i b r i u m corresponds to depensatory m o r t a l i t y (Ricker 1954), the r e s u l t of a p e r t u r b a t i o n from such an unstable p o s i t i o n i s a move to the v i c i n i t y of a d i f f e r e n t e q u i l i b r i u m or e x t i n c t i o n . Even though the dominant eigenvalue i s r e a l t here s t i l l c o u l d be minor o s c i l l a t i o n s caused by the eigenvalues of s m a l l e r modulus. In the s p e c i a l case where the number of eggs produced i s the argument of the s u r v i v a l f u n c t i o n then b{j) = ( f (U) +0*f * (0) }.P ( j ) * h ( j ) (5) 19 and thus a l l the elements i n the f i r s t row of B have the same si g n which i s of course the s i g n of the g r a d i e n t of the s t o c k -r e c r u i t curve. I f the curve has a descending r i g h t hand limb then f o r an e g u i l i b r i u m i n t h i s r e g i o n a l l the elements i n the f i r s t row of B w i l l be be n e g a t i v e , and the dominant eigenvalue may be complex i n d i c a t i n g t h a t motion near the e g u i l i b r i u m w i l l be o s c i l l a t o r y . In f a c t t h i s was the case i n a l l the s i m u l a t i o n s I have examined. I f the s t o c k - r e c r u i t curve i s steep enough the modulus of the dominant eigenvalue c o u l d be g r e a t e r than 1. and i n t h i s case the e g u i l i b r i u m would be unstable. A necessary but not s u f f i c e n t c o n d i t i o n f o r t h i s to occur can be found by noting that a necessary c o n d i t i o n f o r a root of eguation (4) to have modulus g r e a t e r than one i s that ? l b (j) | > 1 As b(j) < 0. f o r a l l j t h i s i s e q u i v a l e n t to ZTfMJ) } < -1 and using (5) we get 2T(b<j) } = If (0)+TJ.f «{U) ] ^ ( P ( j ) * h ( j ) } = [ f (U) +U*f« (U) > E Thus a necessary c o n d i t i o n f o r an unstable e q u i l i b r i u m i s 20 that E - i < -I f (D) + U « f (0) ] which means that the g r a d i e n t of the s t o c k - r e c r u i t r e l a t i o n must be steeper than the replacement l i n e n = E _ 1 » S . T h i s c o n f i r m s the o b s e r v a t i o n s of R i c k e r (1954) and Basasibwaki (MS.) who found t h a t a necessary c o n d i t i o n f o r an unstable e q u i l i b r i u m when the replacement l i n e had a g r a d i e n t of 1.0 was that the g r a d i e n t of the s t o c k - r e c r u i t r e l a t i o n should be l e s s than -1.0 at the e q u i l i b r i u m , and provides a b a s i s f o r B i c k e r ' s (1954) a s s e r t i o n t h a t a p o p u l a t i o n that i s unstable before h a r v e s t i n g can be s t a b i l i z e d by t a k i n g a s u f f i c i e n t l y l a r g e h a r v e s t , provided no age groups c o n t r i b u t e to d e n s i t y dependent m o r t a l i t y without c o n t r i b u t i n g to egg p r o d u c t i o n . These p o i n t s are i l l u s t r a t e d by the examples shown i n Figure 1. The three s e r i e s shown i n F i g u r e 1{a)-1 (c) show the f l u c t u a t i o n s f o l l o w i n g a p e r t u r b a t i o n of a p o p u l a t i o n with e i g h t age c l a s s e s and a s t o c k - r e c r u i t r e l a t i o n of the form n = a»0»exp (-b»U). T h i s p o p u l a t i o n s h a l l be r e f e r r e d t o as p o p u l a t i o n A and i t s parameters are given i n Appendix 1. With no f i s h i n g m o r t a l i t y the e q u i l i b r i u m i s unstable and the o s c i l l a t i o n s f o l l o w i n g the p e r t u r b a t i o n grow s w i f t l y , while with f i s h i n g m o r t a l i t i e s of 0.5 and 1.0 a p p l i e d from age 3 onwards the e q u i l i b r i u m i s s t a b l e . With F = 0.5 the e f f e c t s of the p e r t u r b a t i o n p e r s i s t f o r a long time, as i s expected on the b a s i s of the dominant eigenvalue having a modulus (0.99) c l o s e to one. The p e r i o d i n each case agrees with t h a t p r e d i c t e d using the argument of the dominant e i g e n v a l u e , and i s approximately twice the mean age of r e p r o d u c t i o n . The word approximately i s used p a r t l y because i t i s d i f f i c u l t to measure the p e r i o d and p a r t l y because of u n c e r t a i n t y i n how the mean age of 21 a F i g u r e 1 Small o s c i l l a t i o n s about the e q u i l i b r i u m of the p o p u l a t i o n s . 21b 22 r e p r o d u c t i o n should be d e f i n e d ( L e s l i e 1966). Hhen the argument of the s u r v i v a l f u n c t i o n i s not the number of eggs produced the r e l a t i o n between the mean age of spawning and the p e r i o d may break down, an example of t h i s i s shown i n F i g u r e 1 (d) which shows the f l u c t u a t i o n s about an unstable e g u i l i b r i u m f o r a p o p u l a t i o n with f i v e year c l a s s e s a l l of which c o n t r i b u t e to d e n s i t y dependent m o r t a l i t y while only the l a s t two produce eggs. T h i s p o p u l a t i o n s h a l l be r e f e r r e d to as p o p u l a t i o n B and i t s parameters are given i n appendix 1. In t h i s case the p e r i o d i s only two y e a r s . Another d i f f e r e n c e between t h i s case and the previous one i s t h a t the e q u i l i b r i u m i s s t a b l e when there i s no f i s h i n g m o r t a l i t y and becomes unstable as f i s h i n g m o r t a l i t y i n c r e a s e s . A major d i f f e r e n c e between the s t a b l e and unstable s i t u a t i o n s i s t h a t i n the unstable case the the o s c i l l a t i o n s cannot continue to grow as an upper l i m i t t o recruitment i s set by the shape of the s t o c k - r e c r u i t curve. However i n the s i m u l a t i o n s I have c a r r i e d out the p e r i o d of the p e r s i s t e n t o s c i l l a t i o n s appeared to be r e l a t e d to the argument of the eigenvalues whose modulus was greater than one. F i g u r e 2 shows s e c t i o n s of the s e r i e s generated by the unstable cases shown i n Figure 1 a long time a f t e r the i n i t i a l p e r t u r b a t i o n . In the f i r s t panel the f l u c t u a t i o n s i n p o p u l a t i o n A appear to be s t a b l e , while the second panel shows the p a t t e r n of the f l u c t u a t i o n s i n p o p u l a t i o n B changing from the two year p e r i o d i c i t y of the s m a l l o s c i l l a t i o n s t o a p a t t e r n with a dominant year c l a s s o c c u r i n g every four years. In the f i r s t 23a F i g u r e 2 Large f l u c t u a t i o n s about unstable e q u i l i b r i u m p o p u l a t i o n s . 24 example the period c o u l d be r e l a t e d to the argument of the dominant eige n v a l u e which has a perio d of 12.3 years and i n the second case the p e r i o d can be r e l a t e d to the second eigenvalue with a p e r i o d of 3.8 years. The f i g u r e s suggest the f l u c t u a t i o n s are s e t t l i n g i n t o l i m i t c y c l e s , however i n the s i m u l a t i o n s c a r r i e d out none of the f l u c t u a t i n g p o p u l a t i o n s ever s e t t l e d i n t o a f i x e d p a t t e r n . In a n a t u r a l p o p u l a t i o n the e f f e c t of d e n s i t y dependent po p u l a t i o n processes would be modified by random or u n p r e d i c t a b l e v a r i a t i o n such as those caused by weather. In many cases these would d i s t o r t the p a t t e r n s t o such an extent as to make them un r e c o g n i s a b l e and t h i s prompted Rick e r (1954) to remark that the only f i s h e r y which showed f l u c t u a t i o n s t h a t resembled those that would be caused by a steep descending s t o c k - r e c r u i t r e l a t i o n were those shown by the Georges Bank haddock Melanoqramus a e q l e f i n u s (Linnaeus). R i c k e r was only concerned with a s p e c i a l case of the s t o c k - r e c r u i t r e l a t i o n I am c o n s i d e r i n g , and when the more g e n e r a l p r o p e r t i e s are considered i t can be seen t h a t other f i s h e r i e s do show f l u c t u a t i o n s t h a t could be a t t r i b u t e d to the e f f e c t of the s t o c k - r e c r u i t curve. A good example i s provided by s i m u l a t i o n of the dynamics of sockeye salmon of the Adams and Skeena r i v e r s ( L a r k i n and McDonald 1968, L a r k i n 1971) which reproduced the c y c l i c dominance with a p e r i o d of f o u r or f i v e years that i s c h a r a c t e r i s t i c of these s t o c k s . The model used i n these i n v e s t i g a t i o n s was more complicated than the forms I am c o n s i d e r i n g but L a r k i n (1971) made the point that a s i m i l a r 25 r e s u l t could be obtained using a model which i n my n o t a t i o n can be w r i t t e n as N(1,t + 1) = U (t) •£{ V (t) }. In f a c t F i g u r e 2(b) shows a p a t t e r n of c y c l i c dominance with a 4 year p e r i o d generated by such a model. I t i s i n t e r e s t i n g that the simple assumption of d e n s i t y dependent m o r t a l i t y that i s a f u n c t i o n of the numbers i n e a r l i e r year c l a s s e s than those spawning i s s u f f i c i e n t to account f o r the c y c l i c dominance i n salmon p o p u l a t i o n s . V a r i o u s mechanisms have been suggested as causes of t h i s type of m o r t a l i t y , the most l i k e l y of which seems to be n a t u r a l p r e d a t i o n (Ward and L a r k i n 1964). Another p o s s i b l e example i s pr o v i d e d by the blue pike s t i z o s t e d i c n vj.treuro _gj.aucum (Hubbs) po p u l a t i o n of Lake E r i e which showed a p e r s i s t e n t p a t t e r n of o s c i l l a t i o n s over the p e r i o d 1915-1960. Regier et a l . (1969) suggested that t h i s was due to the e f f e c t of p r e d a t i o n on the young of the year by o l d e r f i s h coupled with the e f f e c t s of an e x t e n s i v e f i s h e r y , which i s the same type of s i t u a t i o n as that which produced the c y c l i c dominance shown i n F i g u r e 2 (b). However with the e x c e p t i o n of these few cases which show a c l e a r p a t t e r n i n t h e i r f l u c t u a t i o n s i t i s l i k e l y to be d i f i c u l t t o d e t e c t the e f f e c t s of f l u c t u a t i o n s about an unstable e q u i l i b r i u m . F i g u r e 3 shows the f l u c t u a t i o n s f o r the p o p u l a t i o n s A and B when they are modified by a random variable,, orodifiod-by or random v a r i a b l e . In both the unstable cases the o r i g i n a l p a t t e r n i s v i s i b l e , however i n the s t a b l e p o p u l a t i o n (Figure 3b) i t i s not p o s s i b l e to detect a e i g h t year c y c l e by eye. T h i s p e r i o d i c i t y can be however detected by s p e c t r a l a n a l y s i s of the 26 a F i g u r e 3 F l u c t u a t i o n s about e q u i l i b r i u m p o p u l a t i o n s modified by a random v a r i a b l e . 26 b ZD or CJ UJ or a or Lul m Y E A R 27 data and an example of the r e s u l t s of t h i s i s shown i n F i g u r e 4. Each spectrum i s taken from a 100 year seguence of the data shown i n F i g u r e 3, and i n each case the dominant frequency i s c l e a r l y shown. However, i n p r a c t i c e i t i s unusual to get a s e r i e s of data as long as these and u s u a l l y one must be content with y i e l d data r a t h e r than year c l a s s s t r e n g t h . Furthermore i n any long s e r i e s of data the i n h e r e n t p a t t e r n s are l i k e l y to be obscured by trends i n the p a t t e r n of the development of a f i s h e r y or by environmental changes. Figure U S p e c t r a of s e r i e s of r e c r u i t s generated by p o p u l a t i o n s modified by random v a r i a b l e s . The s p e c t r a l d e n s i t y i s i n u n i t s of (Recruits/1000)*. 29 The Model Applied To Y e l l o w f i n Tuna The y e l l o w f i n f i s h e r y i n the e a s t e r n P a c i f i c Ocean was chosen as an example f o r s e v e r a l reasons; t h e r e are catch and e f f o r t data a v a i l a b l e from the year 1934 onwards, there has been a c o n s i d e r a b l e amount of work done on the b i o l o g y of t h i s f i s h , i n f o r m a t i o n i s a v a i l a b l e cn the r e l a t i v e f e c u n d i t y , m o r t a l i t y r a t e s , growth r a t e s , and age composition of the s t o c k s . A d d i t i o n a l l y the parameters f o r both the l o g i s t i c model (Schaefer 1957, Schaefer 1961, IATTC 1971) and the Beverton and Holt model (Hennemuth 1961b) have been estimated, and thus the r e s u l t s of u s i n g my model may be compared with the r e s u l t s from these models. There are of course some problems a s s o c i a t e d with the data, the most s e r i o u s of these being the question of whether the f i s h e r y can be t r e a t e d as one which operates on a s i n g l e stock. Since 1934 there has been an i n c r e a s e i n the area e x p l o i t e d by the f i s h i n g f l e e t , and i n the l a t e 1950's there was a major change i n the f i s h i n g method from b a i t f i s h i n g to purse s e i n i n g . As w e l l as c o n f u s i n g the measurement of f i s h i n g e f f o r t , t h i s change a c c e l e r a t e d the o f f s h o r e spread as the l a r g e r purse s e i n e r s have a g r e a t e r range than the b a i t b o a t s . An account of these changes i s given i n a s e r i e s of r e p o r t s , (Martin 1962, A l v e r s o n 1963, C a l k i n s and Chatwin 1967, 1971). The I n t e r -American T r o p i c a l Tuna Commission (IATTC) has used an i n f o r m a l d i v i s i o n of the e x p l o i t e d area (IATTC 1971) that i s shown i n Figure 5. Dp to 1962 the f i s h e r y operated e n t i r e l y w i t h i n area A1, and a f t e r 1962 the f l e e t s t a r t e d e x p l o i t i n g the remaining 30 a F i g u r e 5 The eastern t r o p i c a l P a c i f i c Ocean showing the CYRA and the s u b d i v i s i o n s A1 and A2. The shaded area i s the h i s t o r i c remainder of the CRY A i s area A1 and the area A2. 30 b 100" 80' EASTERN PACIFIC YELLOWFIN TUNA REGULATORY AREA 40" UNITED STATES 31 p o r t i o n (A2) of the Commission's y e l l o w f i n r e g u l a t o r y area (CYRA). Since 1966 the f i s h e r y i n s i d e the CYRA has been r e g u l a t e d by a c l o s e d season which i s imposed a f t e r a quota has been f i l l e d . In recent years the f i s h i n g season has been s h o r t e r and t h i s has encouraged boats to f i s h area A3 o u t s i d e the CYRA 1 I t i s not known whether the e n l a r g e d f i s h i n g area c o n t a i n s one homogeneous or s e v e r a l d i s c r e t e s t o c k s , though the evidence from t a g g i n g , morphometric, and g e n e t i c s t u d i e s (Joseph et a l , 1964, Fink and B a y l i f f 1970) suggests there i s some but not complete mixing of f i s h from d i f f e r e n t areas, More r e c e n t l y the m i g r a t i o n of tagged y e l l o w f i n from c o a s t a l r e g i o n s to o u t s i d e the CYRA has been observed (IATTC 1971). On the other hand Schaefer (1967b) argues that the c l o s e r e l a t i o n s h i p between change i n e f f o r t and c a t c h per u n i t of e f f o r t f o r the f i s h e r y i n the CYRA suggests the y e l l o w f i n i n the CYRA belong t c a u n i t stock independent of the f i s h west of the CYRA. F u r t h e r , the average weights of y e l l o w f i n from the three areas was 49.2, 53.7, and 58.6 l b s i n 1969, and 30.6, 45.6, and 67.9 i n 1970. (IATTC 1971). These d i f f e r e n c e s appear to be caused by d i f f e r i n g p r o p o r t i o n s of f i s h of d i f f e r e n t ages, suggesting t h a t t o t a l m o r t a l i t y decreased with d i s t a n c e from shore, and hence that the s t o c k s i n the d i f f e r e n t areas were independent. Y e l l o w f i n from the e a s t e r n P a c i f i c Ocean have not yet been s u c c e s s f u l l y aged using hard p a r t s (Suzuki 1971), and hence most estimates i n v o l v i n g ageing of the f i s h , ( f o r example growth *In 1971 the season was c l o s e d on 9 A p r i l . V e s s e l s t h a t are i n port a t the time of c l o s u r e are allowed to make one more unregulated t r i p a f t e r unloading. 32 r a t e s and age composition of the stocks) are based on the method of modal p r o g r e s s i o n , i . e . the r e c o g n i t i o n of year c l a s s e s i n leng t h freguency data (Hennemuth 1961a). T h i s of course w i l l r e s u l t i n some e r r o r s of c l a s s i f i c a t i o n . Spawning a c t i v i t y i s l a r g e l y c o n f i n e d t o the f i r s t t h ree g u a r t e r s of the year (Orange 1961, Kume and Schaefer 1966, Kume and Joseph 1969). Kume and Joseph suggest spawning i s most in t e n s e during the second g u a r t e r , and I s h a l l f o l l o w them and a s s i g n age zero to be on the f i r s t of A p r i l . Hennemuth (1961a) a s s i g n s age 0 i n August, and so my ages are approximately f o u r months g r e a t e r than Hennemuth's ages. Growth Hennemuth (1961a) using l e n g t h data c o l l e c t e d from b a i t b o a t s during the p e r i o d 1954-1958 estimated the parameters f o r the von B e r t a l a n f y growth formula, and a f t e r c o n v e r t i n g from l e n g t h to weight obtained w(t)=218«[ 1.-exp{-0.6 (t-0.85) ] ] 3 l b s (5) where t i s measured i n years from August. T h i s r e l a t i o n d e s c r i b e s the data well f o r f i s h o l d e r than 1.5 years but underestimates the average weight of younger f i s h i n the catches (see Hennemuth 1961a F i g u r e 30). T h i s of course may be due t o the s e l e c t i o n by the f i s h i n g method of the f a s t e r growing young f i s h , but whatever the reason t h i s would t o some extent b i a s y i e l d per r e c r u i t c a l c u l a t i o n s based on eguation 33 (5). Thus f o r ages of 5, 6 and 7 q u a r t e r s I have used the mean of succeeding mid quarter average weights given i n Table 5 of Davidoff (1969). For younger f i s h I have assumed a uniform growth r a t e from 0.01 l b s at the end of the f i r s t q u a r t e r . Table 1 g i v e s the weights at the beginning of each quarter and the c o r r e s p o n d i n g e x p o n e n t i a l growth r a t e s during the q u a r t e r . T h i s growth r a t e agrees with the r a t e s estimated by Davidoff (1963) using data c o l l e c t e d from b a i t b o a t s and purse s e i n e r s during the p e r i o d 1951-1961, and Kume and Joseph (1969) using data c o l l e c t e d from the Japanese long l i n e f i s h e r y during the period 1964-1967. A c o n s i d e r a b l y lower growth r a t e f o r tagged f i s h was found by Schaefer e t a l . (196 1). T h i s c o u l d be due to the e f f e c t of tagging on growth, or because of the e f f e c t on the s i z e s e l e c t i v e f i s h e r y of the modal p r o g r e s s i o n e s t i m a t e s . M o r t a l i t y Estimates of m o r t a l i t y of y e l l o w f i n have been p u b l i s h e d by Hennemuth (1961b), Schaefer et a l . (1961), Fink (1965), Davidoff (1965), and B a y l i f f (1971). A summary of these e s t i m a t e s i s given i n Table 2. The estimates vary c o n s i d e r a b l y and, although some of the v a r i a t i o n can be e x p l a i n e d i n terms of events i n the f i s h e r y , they do not p r o v i d e good estimates of n a t u r a l and f i s h i n g m o r t a l i t y . The e s t i m a t e s give no assurances that n a t u r a l m o r t a l i t y has remained constant over the p e r i o d 1953-1963. Figure 6 shows the c a t c h curves f o r the year c l a s s e s 1954-1964, using data given by Davidoff (1969). In the absence of good estimates of m o r t a l i t y I s h a l l f o l l o w Schaefer (1967b) and i n v e s t i g a t e the conseguences of assuming n a t u r a l m o r t a l i t i e s of 34 Table 1 Estimated weight and e x p o n e n t i a l growth r a t e of y e l l o w f i n tuna. Qtr. Weight l b s Growth r a t e Qtr Weight l b s Growth 5 2.7 0.675 19 149.8 0.055 6 5.3 0.524 20 158.3 0.047 7 9.0 0.348 21 165.8 0.040 8 12.7 0.597 22 172.5 0.030 9 23. 0 0.433 23 178.5 0.029 10 35.5 0.329 23 183.6 0.025 11 49.4 0.257 25 188.2 0.021 12 63.8 0.205 26 192.2 0.018 13 78.3 0. 166 27 195.6 0.015 14 92.3 0. 135 28 198.7 0.013 15 105.7 0. 112 29 201.3 0.011 16 118.2 0.093 30 203.6 0.010 17 129.8 0.078 31 205.5 0.008 18 140.3 0.065 32 207.2 Table 2 Estimates of t o t a l m o r t a l i t y and f i s h i n g e f f o r t f o r y e l l o w f i n tuna. Year (a) (b) <<3) E f f o r t 1953 - - - 1.55 36,356 1954 1 .72 - - - 36,356 1955 1 .72 - - 2.26 17,198 1956 1 .72 - 1.93 1.43 27,204 1957 1 .72 1.20 1.32 1.54 26,768 1958 1 .72 1. 75 0.88 1.54 31,135 1959 - 1.28 0.77 1.43 28,271 1960 - 5. 17 6.59 1. 19 35,841 1961 - - - 1.64 4 1,646 (a) Hennemuth (1961b) average over a l l areas 1954-1958. (b) Fink (1965) Baja C a l i f o r n i a . (c) Fink (1965) Northern Peru. (d) Davidoff (1965) average over a l l areas. The f i s h i n g e f f o r t i s gi v e n i n standard b a i t b o a t days. Figure 6 Catch curves f o r the 1954-1964 y e l l o w f i n tuna year c l a s s e s . 36 0.55, 0.8, and 1.05 with the corresponding c a t c h a b i l i t y c o e f f i c i e n t s 0.48»10-*, 0.38»10-«, and 0.28*10-*, per standard b a i t b o a t day. For the years 1961-1968 the t o t a l e f f o r t i s a v a i l a b l e i n terms of standard b a i t b o a t days as w e l l as standard purse s e i n e days, and thus i t i s p o s s i b l e to estimate the corresponding c a t c h a b i l i t y c o e f f i c i e n t s f o r standard purse s e i n e days. These c o e f f i c i e n t s were obtained by m u l t i p l y i n g those f o r b a i t b o a t s by the r e g r e s s i o n c o e f f i c i e n t of b a i t b o a t e f f o r t on purse s e i n e e f f o r t f o r the years 1961-1968. T h i s gave c a t c h a b i l i t y c o e f f i c i e n t s of 0.80»10~*, 0,63*10-*, and 0.«6«10-* f o r purse s e i n e r s . Hennemuth (1961a) g i v e s the age of f i r s t capture as 20 months, which corresponds to my n i n t h q u a r t e r . However the data given by Davidoff (1969) i n h i s Table 2 shows that most year c l a s s e s become f u l l y vunerable i n t h e i r e ighth quarter (quarter f i v e i n h i s T a b l e ) . T h i s Table shows no c l e a r d i f f e r e n c e i n the age of f i r s t capture during the period when the f i s h e r y changed from b a i t b o a t s to purse s e i n e r s , and consequently I am assuming that the change d i d not e f f e c t the age of f i r s t c a p t u r e . In D a v i d o f f s Table there are a c o n s i d e r a b l e number of f i s h caught before the e i g h t h q u a r t e r , t h i s i s p a r t l y the r e s u l t of the extended spawning p e r i o d , and consequent e r r o r i n the age d e s i g n a t i o n of the f i s h , and p a r t l y because r e c r u i t m e n t to the f i s h e r y i s not k n i f e edged. Because the number of f i s h captured before t h e i r e i g h t h quarter i s s m a l l i t i s not worth e s t i m a t i n g a s e l e c t i o n curve and thus f o r both the b a i t b o a t and purse s e i n e f i s h e r i e s i t i s assumed t h a t c a t c h a b i l i t y i s zero u n t i l the e i g h t h q u a r t e r which corresponds to an age of f i r s t capture of 37 1.75 years. Fecundity Joseph (1963) estimated the r e l a t i o n between the weight of mature female f i s h and the number of maturing eggs as; Eggs ( m i l l i o n s ) = 0.106 • 0.046«Weight(lbs) Orange (1961) s t a t e s that at a l e n g t h of 120cm (age 3) 50?? of the f i s h are maturing, while f i s h l a r g e r than 140cm (age 4 and older) are a l l mature, and h i s F i g u r e 6 suggests about "lOIL of f i s h of l e n g t h 80cm (age 2) are mature. The average f e c u n d i t y f o r each age group can be c a l c u l a t e d using t h i s i n f o r m a t i o n as i s shown i n Table 3. The freguency of spawning i s not known, though i t i s l i k e l y that female y e l l o w f i n spawn at l e a s t twice a year (Schaefer et a l . 1963). However, p r o v i d i n g the freguency does not change with age i t w i l l be absorbed i n the c o n s t a n t s c and k i n equation 6(a) below. The model The y i e l d per r e c r u i t can be estimated from my model by d i v i d i n g the e q u i l i b r i u m y i e l d by the e q u i l i b r i u m number of r e c r u i t s . T h i s i s shown i n F i g u r e 7 together with the y i e l d per r e c r u i t f o r the Beverton and H o l t model using 0.8 as the n a t u r a l m o r t a l i t y . The two models g i v e very s i m i l a r r e s u l t s and I s h a l l use my estimates when I r e f e r to y i e l d per r e c r u i t below. Figure 8 shows the y i e l d , e f f o r t , and estimated y i e l d per r e c r u i t f o r 38 Table 3 Averaqe weight and f e c u n d i t y of female y e l l o w f i n tuna i n the t h i r d guarter. Age groups 2 3 4 5 6 7 8 Weight l b s 23-0 79.3 129.8 165.3 188. 2 201. 3 208.7 Eggs/mature female x10* 1. 17 3.71 6.08 7.73 8.76 9.37 9.71 % Mature 10% 50% 100?? 100% 100% 100% 100% R e l a t i v e Fecundity. 0.12 1.85 6.08 7.73 8.76 9.37 9.71 F i g u r e 7 Y i e l d per r e c r u i t i n the y e l l o w f i n tuna f i s h e r y . 40 a F i g u r e 8 Y i e l d and estimated y i e l d per r e c r u i t i n the y e l l o w f i n tuna f i s h e r y , 1932-1968. Y i e l d Y i e l d per r e c r u i t 40 b 41 the years 1934-1968. I t i s c l e a r t h at a c o n s i d e r a b l e amount of compensation has taken p l a c e during the p e r i o d , i . e . y i e l d has been i n c r e a s i n g while y i e l d per r e c r u i t has been constant or s l i g h t l y d e c r e a s i n g . T h i s suggests that r e c r u i t m e n t has been i n c r e a s i n g t o compensate f o r the f a l l i n y i e l d per r e c r u i t . Huch the same e f f e c t i s seen when the values 0.55 and 1.05 are used f o r n a t u r a l m o r t a l i t y , and thus i t i s a p p r o p r i a t e to use the Ricker s t o c k - r e c r u i t r e l a t i o n i n the model. The model i s then; N(1,t) = c» £ [ h ( i ) • N(i,t-1) ]»exp{-k»£[h(i) •*! ( i , t-1) ]} (6a) N(j, t ) = exp{-Z (j) }-N<j-1,t-1) j = 2,3, , t l . (6b) And the annual y i e l d i s given by C = C[F (i) »N ( i , t ) • (1.-exp{-Z (i) })/Z (i) ] (numbers) (7a) X = Z [F (i)»N(i,t) »w{i)» (1.-exp{B (i) })/B (i) ] (weight) (7b) where the f i s h i n g m o r t a l i t y F ( i ) i s the product of the c a t c h a b i l i t y g (i) and the e f f o r t X, Z (i) = H ( i ) + F ( i ) , and B(i) = G ( i ) - F ( i ) - M ( i ) , 42 Es t i m a t i o n of the parameters c and k The unknown parameters c and k i n eguation (2a) can be estimated by minimizing the f u n c t i o n Y ( i ) - [ & i ( c , k , X , N O ) ] } 2 . Where A i (c,k,X,|i 0) i s the value of the annual y i e l d given by the model i n year i , with p o p u l a t i o n J| 0 at the beginning of the f i r s t year, and X the vector g i v i n g the f i s h i n g e f f o r t f o r each year. N° i s unknown and co u l d be t r e a t e d as a s e t of parameters to be e s t i m a t e d . However as the e x p l o i t a t i o n r a t e i n the e a r l y years of the f i s h e r y was g u i t e low, i t probably d i d not a f f e c t the p o p u l a t i o n much and so I have assumed that N° i s the e q u i l i b r i u m p o p u l a t i o n with the f i s h i n g e f f o r t f o r 1934. The model was f i t t e d u s i n g c a t c h and e f f o r t data shown i n Table 4 f o r the two s e t s of years 1934-1960 and 1934-1968. Both s e t s were t r i e d to see i f there i s a d i f f e r e n c e which could be r e l a t e d to the change i n f i s h i n g method from b a i t f i s h i n g to purse s e i n i n g and the subsequent r a p i d i n c r e a s e of the area e x p l o i t e d i n r e c e n t y e a r s . In a d d i t i o n the f i t t i n g was t r i e d f o r each of the th r e e s e t s of m o r t a l i t y data. The best r e s u l t was obtained using M= 1.05, and i n t h i s case the estimates of c and k were 2.34«10-s and 1.23»10 - * 3 f o r the years 1934-1960, and 3.35-10- 5 and 1.69«10 ~ 1 3 f o r the years 1934-1968. F i g u r e 9 shows the corresponding s t o c k - r e c r u i t r e l a t i o n s , and the r e s i d u a l sum of sguares about the r e g r e s s i o n and mean f o r each case are shown below. 43 Table 4 Catch of y e l l o w f i n tuna and f i s h i n g e f f o r t i n the Commission's y e l l o w f i n r e g u l a t o r y area 1934-1968. Year Catch E f f o r t 1934 60,913 5,879 1935 72,294 6,295 1936 78,353 6,771 19 37 91,522 8,233 1938 78,288 6,830 1939 110,4 18 10,488 1940 114,590 10,801 1941 76,841 9,584 1942 41,965 5,961 1943 50,058 5,930 1944 64,869 6,475 1945 89, 194 9,377 1946 129,701 13,958 1947 160, 151 20,383 1948 206,993 24,781 1949 200,070 23,923 1950 224,810 31,856 1951 186,015 18,403 1952 195,277 34 ,834 1953 140,042 36,356 1954 140,033 26,228 1955 140,865 17, 198 1956 177,026 27,204 1957 163,020 26,768 1958 148,450 31, 135 1959 140,484 28,271 1960 244,331 35,841 1961 230,886 41,646 1962 174,063 42,248 1963 145,469 33,303 1964 203,882 42,090 1965 180,086 43,228 1966 182,294 40,393 1967 178,944 33,814 1968 225,000 39,199 The catch i s i n thousands of l b s and the e f f o r t i s i n standard b a i t b o a t days. F i g u r e 9 S t o c k - r e c r u i t r e l a t i o n s generated by f i t t i n g the model to data f o r 1932-1960 and f o r 1934-1968. E G G S * 10* 45 Source About r e g r e s s i o n About mean F i t t o the years 1934-1960 Sum of squares Mean square d.f. R a t i o 1.79*101* 7.18*101* 25 8.37*1016 32.2«10i* 26 0.22 Source About r e g r e s s i o n About mean F i t t o the years 1934-1968 Sum of sguares Mean square d.f. R a t i o 2.85*1016 8.63*101* 33 11.2*10i 6 32.9*101* 34 0.26 Contours f o r the r e s i d u a l sum of squares s u r f a c e f o r the model f i t t e d to the data f o r 1934-1960 are shown i n F i g u r e 10, the s u r f a c e i s a r i d g e with s t e e p l y s l o p i n g s i d e s i n d i c a t i n g t h a t the estimates of c and k are s t r o n g l y c o r r e l a t e d . T h i s means that the maximum height of the s t o c k - r e c r u i t curve, given by x = c/ke, i s more c r i t i c a l t o the goodness of the f i t than i s the l o c a t i o n of the maximum, given by (J = 1/k. F i g u r e 11 shows the annual y i e l d s with the best f i t of the model using each of the data s e t s . Both the mean sguare about the r e g r e s s i o n , and the r a t i o of mean sguare about the r e g r e s s i o n to mean square about the mean i n d i c a t e the f i t of the model to the r e s t r i c t e d data set i s b e t t e r than the f i t to the f u l l s e t of data. These d i f f e r e n c e s are not very l a r g e and because of the s t r u c t u r e of the model i t i s not p r a c t i c a l to t r y to apply a s t a t i s t i c a l t e s t t o them. 46 a Fi g u r e 10 The s u r f a c e of the r e s i d u a l sum of squares f o r the model with the data f o r area A1. The contours f o r (19 ,20 ,30,50, 100)x10 l b s 2 are shown. 46 b •xr x o _ u> O O t - k » - * H » l - » l - * ' U • • • • • » • • ( n c s o r u ^ c n o a o H 1 1 1 1 1 1 1 1 1 1 1 1 1 r F i g u r e 11 Y i e l d of y e l l o w f i n tuna from the CYRA. Expected values using the f i t t o the data f o r 1934-1960 Expected values using the f i t to the data f o r 1934-1968 Data V 48 However the marked drop i n the p r e d i c t i o n of y i e l d a f t e r 1960 by the model f i t t e d to the data f o r 1934-1960 i s an i n d i c a t i c n t h a t the stock being e x p l o i t e d i n c r e a s e d a f t e r 1960, suggesting t h a t f i s h i n the area that was e x p l o i t e d a f t e r 1960 are t o some extent p a r t of s t o c k s s e p a r a t e from those e x p l o i t e d before 1960. I s h a l l r e f e r to the model as the model f o r A1 or the model f o r the CYRA depending on whether the parameters used with i t are those obtained from the data f o r 1934-1960, or from the data f o r 1934-1968. If year c l a s s s i z e i s not much a l t e r e d by v a r i a t i o n s i n n a t u r a l m o r t a l i t y the r e s i d u a l s about the r e g r e s s i o n would be c o r r e l a t e d , i . e . the e f f e c t of a s m a l l or l a r g e year c l a s s should p e r s i s t f o r two or three years, and t h i s should cause runs of o b s e r v a t i o n s g r e a t e r than or l e s s than the expected values to be longer than i f the r e s i d u a l s were d i s t r i b u t e d independently. For the model f o r A1 there are 4 runs of o b s e r v a t i o n s above the expected values and 4 runs below, the l o n g e s t run being the 9 above f o r the years 1939-1947. The one sample runs t e s t ( S i e g e l 1956) shows t h a t t h i s i s a s i g n i f i c a n t departure at the 5% l e v e l from what would be expected with a n u l l hypothesis of independently d i s t r i b u t e d r e s i d u a l s . The mean le n g t h of the runs i s j u s t over 3 years and t h i s i s a l s o c o n s i s t e n t with the hypothesis t h a t the r e s i d u a l s are mainly due to the e f f e c t of v a r i a t i o n s i n the r e l a t i v e year c l a s s s t r e n g t h . Much the same e f f e c t s are shown by the model f o r the CYRA. These c o r r e l a t i o n s may of course be caused s o l e l y by the s t r u c t u r e of the model r a t h e r being present i n the data. However, the comparison with the g e n e r a l p r o d u c t i v i t y model which i s dicussed 49 below makes t h i s seem u n l i k e l y . F i g u r e 12 shows the e q u i l i b r i u m y i e l d p r e d i c t e d by the model f o r area A1 compared t o t h a t of the l o g i s t i c model (IATTC 1963) f i t t e d t o the data f o r the years 1934-1961. The two models agree w e l l f o r values of e f f o r t l e s s than 24,000 days which i s the r e g i o n most of the data i s from. The d i f f e r e n c e between the models f o r l a r g e r values of e f f o r t p r o v i d e s a reminder of the dangers of e x t r a p o l a t i n g o u t s i d e the range of the data. I t i s a l s o i n t e r e s t i n g t o compare the r e s u l t s from the model f o r A1 with those obtained by P e l l a and Tomlinson (1969). They f i t t e d t h e i r g e n e r a l p r o d u c t i v i t y model to the data f o r the years 1934-1967 and found they could e x p l a i n Q3% of the v a r i a t i o n about the mean of the data (R i n t h e i r n o t a t i o n and 1.- the value of the r a t i o given above). As they estimated f o u r parameters from the data i t not s u r p r i s i n g t h a t t h i s i s b e t t e r than the 74% I obtained f o r the years 1934-1968 with two estimated parameters. As w e l l as e x p l a i n i n g about the same amount of the v a r i a n c e the f i t of the two models i s s u r p r i s i n g l y s i m i l a r i n view of the d i f f e r e n c e s between them. The years t h a t each model overestimated the data are shown below. P e l l a and Tomlinson 1941-47 1952-54 1956-59 1962 A l l e n 1934 1940-45 1952-54 1957-59 1962-63 Thus a l l but 6 of the 33 r e s i d u a l s agree i n s i g n and t h i s i s a s i g n i f i c a n t departure from what would be expected i f the two s e t s of r e s i d u a l s were independent. In a d d i t i o n P e l l a and Tomlinson's r e s i d u a l s show the same s e r i a l c o r r e l a t i o n as those F i g u r e 12 Comparison of e g u i l i b r i u m y i e l d s f o r l o g i s t i c and age s t r u c t u r e models. O ITI E00 + T H O U S A N D S OF B A I T B O A T D A Y S 51 f o r the model f o r A1. These s i m i l a r i t i e s support the c o n t e n t i o n that the s t r u c t u r e shown by the r e s i d u a l s i s i n h e r e n t i n the data, and i s not the r e s u l t of f i t t i n g an age s t r u c t u r e model. The h y p o t h e s i s that the s t o c k s i n area A1 should be t r e a t e d s e p a r a t e l y from those i n the other areas can be t e s t e d using the p r e d i c t i o n s of y i e l d by the model f o r A1, u s i n g e f f o r t data f o r A1 from 1962-1970. The mean squared d i f f e r e n c e between the p r e d i c t e d and observed y i e l d s i n A1 i n t h a t time p e r i o d was 7 . 6 7 » 1 0 1 4 , which i s i n good agreement with the mean squared d i f f e r e n c e s obtained from the r e g r e s s i o n s above. The mean sguare of the r e s i d u a l s between the p r e d i c t i o n of y i e l d by the model fo r A1 using the data f o r a l l of the CYRA f o r the years 1962-1968 and the a c t u a l y i e l d s was 35.1•10 1*, i n d i c a t i n g t h a t the hypothesis g i v e s a b e t t e r e x p l a n a t i o n of the the d a t a than i s obtained under the assumption t h a t the f i s h i n areas A1 and A2 can be t r e a t e d as i f they belonged to a s i n g l e stock. The experimental f i s h i n g programme To t e s t the accuracy of the s u s t a i n e d y i e l d p r e d i c t e d by t h e i r model the Commission undertook an experimental f i s h i n g programme i n 1969. T h i s programme c o n s i s t e d of i n c r e a s i n g the guota to 240«10* l b s which was a l i t t l e more than the estimate of maximum s u s t a i n e d y i e l d (IATTC 1971). The guota was to remain i n e f f e c t u n t i l the catch per u n i t e f f o r t f e l l t o 3 s h o r t tons per standard purse seine day. The catch per u n i t e f f o r t expected on the b a s i s of the Commission's model (IATTC 1972) i s shown below, with the observed c a t c h per u n i t e f f o r t and y i e l d . 52 Year Catch per expected 1968 1969 5.1 1970 4.0 1971 3.0 Though the catch per u n i t of e f f o r t has d e c l i n e d over the period 1968-1972 the d e c l i n e has been l e s s than was expected on the b a s i s of the Commission 1s model. T h i s d i s c r e p a n c y becomes even l a r g e r when the e f f e c t of the high l e v e l s of mercury found i n some tuna on f i s h i n g s t r a t e g y i s considered. Because of higher mercury content of o l d e r f i s h , fishermen have been a v o i d i n g s c h o o l s of l a r g e f i s h and thus the c a t c h per u n i t of e f f o r t i n 1971 may underestimate the abundance compared t o the abundance est i m a t e s f o r e a r l i e r years (IATTC 1972). A d d i t i o n a l l y 1971 was an e x c e p t i o n a l l y good year f o r s k i p j a c k tuna Katsuwanus pelamis (Linnaeus). T h i s caused a s h i f t i n e f f o r t from y e l l o w f i n to s k i p j a c k which had the e f f e c t o f reducing the catch per u n i t of e f f o r t f o r y e l l o w f i n (IATTC 1972). The discrepancy may be accounted f o r using the assumption of separate s t o c k s . The maximum s u s t a i n a b l e y i e l d p r e d i c t e d by the model f o r A1 using gear P i s 167*10 6 l b s with a corresponding c a t c h per u n i t e f f o r t o f 5.8 tons per day. As the catc h f o r area A1 has not been g r e a t e r than the maximum su s t a i n e d y i e l d f o r the years 1968-1970, and as the catch per u n i t e f f o r t f o r A2 appears to be independent of e f f o r t and greater than 6 tons per day I would expect c a t c h per u n i t e f f o r t u n i t e f f o r t observed 6.1 6.0 6.0 4. 1 Y i e l d a l l CYEA 229 253 282 228 (10* lbs) A1 o n l y 167 114 160 120 53 to have remained at the 1968-1970 l e v e l . Thus, u n l e s s the e f f e c t s of the mercury l e v e l s and s k i p j a c k abundance on f i s h i n g s t r a t e g y can be shown to have caused the drop i n c a t c h per u n i t of e f f o r t observed i n 1971 and 1972, the r e s u l t s of the experimental f i s h i n g programme suggest that the t r u e s t o c k s t r u c t u r e i s between the extremes of no mixing and complete mixing, t h i s c o n t e n t i o n i s supported by the r e s u l t s of r e c e n t tagging experiments (IATTC 1972). The f i t t i n g of the model does not give any i n d i c a t i o n of how p a r t i a l mixing of s t o c k s may be o c c u r i n g . Two p o s s i b l e s i t u a t i o n s t h a t might be c o n s i d e r e d are that the two areas c o n t a i n s t o c k s that mix i n some years, or that there i s some mixing of f i s h from the two areas each year. Because the d i v i s i o n i n t o areas i s based on h i s t o r i c a l p a t t e r n s of the f i s h e r y caused mainly by the type of boat and f i s h i n g method used, r a t h e r than on the b i o l o g y of the f i s h i t i s t o be expected that t h i s d i v i s i o n i s not the most a p p r o p r i a t e f o r the f i s h e r y . Tagging experiments and o b s e r v a t i o n s o f g e n e t i c a l d i f f e r e n c e s should i n f u t u r e provide i n f o r m a t i o n that can be used as a r a t i o n a l b a s i s f o r the d i v i s i o n of the CYRA i n t o subareas c o n t a i n i n g separate or near separate s t o c k s . The outer areas F i g u r e 13 shows the c a t c h per u n i t e f f o r t a g a i n s t e f f o r t i n area A2 f o r the years 1962-1970. There i s no s i g n of a r e l a t i o n between the two q u a n t i t i e s and there i s no p o i n t i n attempting to f i t the model to t h i s data. The same c o n c l u s i o n a p p l i e s to F i g u r e 13 Catch per u n i t of e f f o r t and e f f o r t f o r area 1962-1970. 20 + IB IG 14 + 12 10 B3 G4 G7 + G5 GG + G9 + 70 B + G2 -+• 2 '3 4 5 5 7 B 9 T H O U S A N D S OF P U R S E S E I N E 3 D A Y S 10 55 area A3 f o r which there i s only three years data. The lack of a r e l a t i o n i n d i c a t e s that the impact of the f i s h e r y on the s t o c k s i n t h i s area i s not yet great enough to a f f e c t the c a t c h and e f f o r t s t a t i s t i c s . Thus i n c r e a s e d f i s h i n g e f f o r t i n these areas may be expected to p r o v i d e a p r o p o r t i o n a l l y i n c r e a s e d y i e l d . The dynamics of the model f o r A1 The model f o r Al i s s t a b l e f o r a l l v a l u e s of f i s h i n g m o r t a l i t y and thus the model does not p r e d i c t the e x i s t e n c e of l a r g e f l u c t u a t i o n s i n year c l a s s s t r e n g t h , and i n t h i s i t agrees with o b s e r v a t i o n s (Davidoff 1969). The p e r i o d c o r r e s p o n d i n g to the dominant eigenvalue was n e a r l y 10 years when there was no f i s h i n g and j u s t greater than U years f o r the h i g h e s t value of f i s h i n g m o r t a l i t y of about 1.2. Thus i f the s m a l l o s c i l l a t i o n s p r e d i c t e d by the model do occur they w i l l c o n t r i b u t e to the spectrum i n the range of 0.1 to 0.25 c y c l e s per year. In f a c t there i s a minor peak i n t h i s r e g ion of the spectrum which i s shown i n F i g u r e 14. However, t h i s peak i s s m a l l compared to the peak a s s o c i a t e d with f r e q u e n c i e s lower than 0.1 c y c l e s per year which r e f l e c t s the trends i n the development of the f i s h e r y . Thus the peak at medium f r e q u e n c i e s cannot be taken as s t r o n g evidence s u p p o r t i n q the d e s c r i p t i v e a b i l i t y of the model. 56 a F i g u r e 14 The s p e c t r a f o r the y e l l o w f i n catch data from area A1. The s p e c t r a l d e n s i t y i s i n u n i t s of ( y i e l d i n l b s / 1 0 8 ) * . 56b - l 1 1 1 — — — f 1 1 1 1 i 0'5 0-4 0'3 0-2 0.1 C Y C L E S P E R Y E A R 57 Harvest s t r a t e g i e s f o r y e l l o w f i n tana In t h i s s e c t i o n I w i l l explore the consequences of using d i f f e r e n t harvest s t r a t e g i e s with the model f o r the y e l l o w f i n f i s h e r y i n area A1. The e f f e c t of a f i s h e r y on a f i s h p o p u l a t i o n i s determined by two f a c t o r s . The f i r s t i s the gear s e l e c t i v i t y which determines how the gear w i l l a f f e c t f i s h of d i f f e r e n t s i z e s , and secondly the f i s h i n g e f f o r t which w i l l determine how much m o r t a l i t y i s a p p l i e d . I s h a l l use a v e c t o r ( c a l l e d the gear) to r e p r e s e n t the gear s e l e c t i v i t y of a p a r t i c u l a r f i s h i n g gear, and m u l t i p l y t h i s v e c t o r by a s c a l a r ( f i s h i n g e f f o r t ) to get the age s p e c i f i c q u a r t e r l y instantaneous m o r t a l i t y . The purse s e i n e f i s h e r y which has an age of f i r s t c apture of about 1.75 years (7 quarters) can be represented by a gear, say gear P, that has zeros i n the f i r s t seven p o s i t i o n s and u n i t s i n the remaining p o s i t i o n s . The s t r a t e g y that i s u s u a l l y examined f i r s t i s t h a t which g i v e s the maximum su s t a i n e d y i e l d . 1 However i t has been r e a l i s e d f o r some time that the maximum s u s t a i n e d y i e l d i s not n e c e s s a r i l y the best b i o l o g i c a l or economic p o l i c y (Dickie 1962) , and i t i s of i n t e r e s t to examine other p o l i c i e s t h a t may have advantages that compensate f o r t h e i r s m a l l e r y i e l d . The d e f i n i t i o n of what i t i s that makes a y i e l d optimum i s o u t s i d e the range of t h i s study and so I s h a l l c o n f i n e my a t t e n t i o n to maximizing y i e l d s s u b j e c t to v a r i o u s c o n s t r a i n t s . 1IATTC operates under the d i r e c t i o n of a i n t e r n a t i o n a l convention which r e q u i r e s t h a t the stocks be held at a l e v e l t h a t w i l l allow the maximum s u s t a i n e d y i e l d to be taken from them. 58 Sustained annual y i e l d s The maximum s u s t a i n e d y i e l d p o l i c y f o r the model was found by s e a r c h i n g the values of s u s t a i n e d y i e l d f o r d i f f e r e n t v a l u e s of f i s h i n g m o r t a l i t y f o r each guarter of the l i f e of the f i s h . The search was s u b j e c t to the c o n s t r a i n t that the instantaneous f i s h i n g m o r t a l i t y r a t e was not allowed to exceed 5.0 i n any guarter. The b e s t p o l i c y found was one with f i s h i n g m o r t a l i t y zero f o r the f i r s t eleven q u a r t e r s of the f i s h e s l i f e , 0. 1 f o r the t w e l f t h , and 5.0 f o r the remaining q u a r t e r s . T h i s p o l i c y g i v e s a y i e l d of 246«10 6 l b s , which i s c o n s i d e r a b l y b e t t e r than the maximum of about 165»10 6 l b s that can be achieved with the age of f i r s t capture f i x e d as i t i s now. Because a f i s h i n g m o r t a l i t y of 5.0 a p p l i e d f o r one q u a r t e r w i l l reduce a coh o r t to l e s s than one percent of i t s i n i t i a l s i z e t h i s p o l i c y i s not much d i f f e r e n t from one i n which a f i s h i n g m o r t a l i t y of 5.0 i s a p p l i e d i n the t h i r t e e n t h g uarter and on subsequent a n n i v e r s a r i e s , and i n f a c t the y i e l d f o r t h i s p o l i c y i s 245»10 6 l b s . The d i f f e r e n c e i n y i e l d between the p o l i c i e s i s i n s i g n i f i c a n t , and the second p o l i c y i s to be p r e f e r r e d because i t r e q u i r e s l e s s e f f o r t as f i s h i n g i s only c a r r i e d out i n one q u a r t e r of the year. I s h a l l c a l l the gear corresponding to the l a s t p o l i c y gear A, and t h i s has the value of 1. i n p o s i t i o n s 13,17,...,29 and zeros in the other p o s i t i o n s . C a l k i n s (1965) has o u t l i n e d the d i f f i c u l t i e s i n v o l v e d i n i n c r e a s i n g the age of f i r s t capture by i n c r e a s i n g the minimum s i z e l i m i t . He shows that because of the mixture of s i z e s and 59 sometimes s p e c i e s w i t h i n s c h o o l s , a minimum s i z e l a r g e r than the c u r r e n t minimum s i z e i s not f e a s i b l e with the present method of f i s h i n g . However i t i s p o s s i b l e to change the average s i z e of f i r s t c a p ture by opening the f i s h i n g season at some time other than the beginning of the year. F i g u r e 15 shows the y i e l d f o r gear A, together with the y i e l d s f o r a s e t of gears operated i n other q u a r t e r s . These gears form three l o g i c a l groups; (1) Those t h a t f i r s t capture f i s h i n t h e i r f o u r t h year, (A). (2) Those t h a t f i r s t capture f i s h i n t h e i r t h i r d year (B,C,D,and E) . (3) Those which takes f i s h which are s t i l l i n t h e i r second year, ( F ) . The only cause of d i f f e r e n c e s w i t h i n a group i s the d i f f e r e n c e i n y i e l d per r e c r u i t . T h i s i s because the e f f e c t of m o r t a l i t y on a year c l a s s at the end of a year i s independent of what time of the year the m o r t a l i t y occured. The age composition f o r the maximum e g u i l i b r i u m y i e l d f o r the gears A, C, and P i s shown below. I f the s t r e n g t h s of incoming year c l a s s e s were d i s t r i b u t e d as an independent random v a r i a b l e the c o e f f i c i e n t of v a r i a t i o n of annual y i e l d s wculd be somewhat l e s s than the c o e f f i c i e n t of v a r i a t i o n of year c l a s s s t r e n g t h . The square of the r a t i o of the l a t t e r c o e f f i c i e n t to the former i s egual to the sum of the squares of the p r o p o r t i o n s of each year c l a s s i n the catch (see Appendix 2) and the square r o o t of t h i s q u a n t i t y i s shown under the heading S. F i g u r e 15 The s u s t a i n e d y i e l d s f o r q u a r t e r l y gears. 61 Year o f l i f e 2 3 4 5 6 S Gear A 0.00 0.00 1. 00 0.00 0.00 1. 00 Gear C 0.00 0.67 0.25 0.06 0.01 0.72 Gear P 0. 12 0. 51 0.27 0.08 0.02 0.60 P r i o r t o 1966 the average o f the c a t c h per standard day's f i s h i n g f o r the f i r s t q u a r t e r was about 10-20% higher than the average over the whole year (Joseph 1970). T h i s suggests t h a t c a t c h a b i l i t y may be lower i n the l a t e r part of the year, and t h i s would be a good reason not to t r y h a r v e s t i n g only i n t h i s part of the year. However when the f i s h e r y i s spread over the year the model p r e d i c t s a drop of about the same s i z e i n c a t c h per standard days f i s h i n g as a r e s u l t of a d e c l i n e i n biomass of the stock and thus i t i s not necessary to p o s t u l a t e a drop i n c a t c h a b i l i t y . Another way of changing the age of f i r s t c apture i s to use a d i f f e r e n t f i s h i n g method. An example of t h i s i s the Japanese l o n g l i n e f i s h e r y i n the eastern P a c i f i c . The age of f i r s t c apture f o r t h i s f i s h e r y i s a l i t t l e more than a year g r e a t e r than that of the s u r f a c e f i s h e r y . The incoming year c l a s s e s are f i r s t e x p l o i t e d i n the second or t h i r d g u a r t e r of the year (Kume and Joseph 1969) and thus t h i s f i s h e r y very n e a r l y corresponds to the gear t h a t gives the maximum s u s t a i n e d y i e l d . 62 O p t i m i s a t i o n over a span of years In t h i s case we r e q u i r e t h a t the y i e l d be optimum over some period p o s s i b l y with some c o n s t r a i n t on the s t a t e of the s t o c k s a f t e r t h at p e r i o d . I f the aim i s maximization of y i e l d with no c o n s t r a i n t then i t i s c l e a r t h a t the s t o c k s would be destroyed at the end of the p e r i o d . F i g u r e 16 shows the best s t r a t e g i e s found f o r maximizing y i e l d s over periods of f i v e and ten years using gear P. As usual the s t r a t e g i e s were c o n s t r a i n e d so t h a t f i s h i n g m o r t a l i t y was l e s s than 5.0 per q u a r t e r . These p o l i c i e s gave y i e l d s of 121•10 7 and 220*10 7 l b s with annual averages of 242O0* and 220*106 l b s , both of which are b e t t e r than any s u s t a i n e d annual y i e l d t h a t can be achieved with gear P. T h i s type of h a r v e s t i n g i s s i m i l a r to the pulse h a r v e s t i n g that i s used by some t r a w l i n g f l e e t s which i n f l i c t a high m o r t a l i t y on f i s h i n an area i n a r e l a t i v e l y s h o r t time span (and then leave the a r e a ) . A major problem with t h i s type of h a r v e s t i n g i s t h a t i t i s p o s s i b l e that a stock would not recover a f t e r being s e v e r e l y depressed. T h i s could be the r e s u l t of c o m p e t i t i v e e x c l u s i o n which i s probably a t l e a s t a f a c t o r i n the l o s s of the s a r d i n e f i s h e r i e s of C a l i f o r n i a , Japan, and South A f r i c a , or simply through a f a i l u r e of the depressed stock to reproduce s u c c e s s f u l l y , as may be happening to the haddock s t o c k s of Georges Bank ( G r o s s l e i n and Hennemuth 1972). However i f these problems do not occur the model i n d i c a t e s that good year c l a s s e s would be formed two years a f t e r the c e s s a t i o n of h a r v e s t i n g , and t h a t the biomass would have recovered to i t s v i r g i n s t a t e f i v e years a f t e r the c e s s a t i o n of 63 a F i g u r e 16 The y i e l d f o r p o l i c i e s c f maximizing the t o t a l y i e l d over p e r i o d s of 5 and 10 years. The numbers below the histograms are g u a r t e r l y i n s t a n t a n e o u s f i s h i n g i n hundreths. the v a l u e s of m o r t a l i t y Xi cn 64 h a r v e s t i n g . In a d d i t i o n , by t h i s time four good year c l a s s e s would have formed and so a f t e r r e s t i n g f i v e years the pa t t e r n could be repeated. Thus the long term average annual y i e l d s f o r these p o l i c i e s are 121•10 6 l b s and 145*10* over p e r i o d s of ten and f i f t e e n years r e s p e c t i v e l y . P e r i o d i c h a v e s t i n g Walters (1969) has suggested that when there i s no way of of i n c r e a s i n g the age of f i r s t capture above some c r i t i c a l age a p e r i o d i c p o l i c y may provide the maximum long term y i e l d . To check i f t h i s i s the case when gear P i s being used I looked a t the conseguences of h a r v e s t i n g f o r one year and then l e a v i n g the stock untouched f o r per i o d s of one, two, and three y e a r s . Figure 17 shows the long term average y i e l d obtained with each of the three p a t t e r n s . The c a t c h composition a t the f i s h i n g m o r t a l i t y t h a t gave the maximum y i e l d f o r each p a t t e r n i s shown below. Year of l i f e 2 3 4 5 6 S One year c l o s u r e 0.10 0.52 0.27 0.08 0.02 0.60 Two year c l o s u r e 0.10 0.42 0.35 0.10 0.02 0.57 Three year c l o s u r e 0. 11 0.39 0.30 0. 16 0.03 0.53 Maximum average annual y i e l d s were 164*10*,151*10*, and 128*10* l b s f o r the p o l i c i e s of one, two, and three year c l o s u r e . Closure f o r one year g i v e s a y i e l d and c a t c h composition much l i k e those f o r gear P while the p o l i c i e s of c l o s u r e f o r two or three years have lower average annual y i e l d s . The lower y i e l d s are compensated f o r a l i t t l e by the f a c t t h a t 65 a F i g u r e 17 The average y i e l d f o r p e r i o d i c h a r v e s t i n g with one, two, and three years between h a r v e s t i n g years. IT) F I S H I N G M O R T A L I T Y I N H A R V E S T I N G Y E A R S 66 the y i e l d i s spread more evenly over the year c l a s s e s with a subsequent d e c l i n e i n v a r i a b i l i t y of y i e l d . T h i s kind of s t r a t e g y shares with the p r e v i o u s type the problem of what to do with the f l e e t i n the years when th e r e i s no f i s h i n g . One p o s s i b i l i t y i f there i s a group of more or l e s s independent s t o c k s i s to f i s h them i n t u r n . I f f o r example there are three independent sto c k s we could f i s h each f o r one year and then leave i t f o r two y e a r s , while f i s h i n g i n the other areas. In t h i s case we would get a s u s t a i n e d y i e l d t h r e e times as l a r g e as the average y i e l d of one of the areas. I f on the other hand there i s some t r a n s f e r of f i s h between s t o c k s , then a f i s h i n g e f f o r t a p p l i e d i n one area w i l l produce f i s h i n g m o r t a l i t y i n the other areas. Thus i f we had three stocks i d e n t i c a l to those i n A1 but whose members interchange so a f i s h i n g e f f o r t i n one area produces 60% of i t s f i s h i n g m o r t a l i t y i n that area and 20% i n each of the other areas, then the p a t t e r n of f i s h i n g m o r t a l i t y i n any area i s 0.6F, 0.2F, 0.2F i n s t e a d of F, 0, 0. The e f f e c t of t h i s i s of course to produce a p o l i c y whose e f f e c t i s midway between the s u s t a i n e d p o l i c y and the o n - o f f p o l i c i e s . The y i e l d f o r t h i s p o l i c y i s shown i n Figure 17. The r e l a t i o n s between s u s t a i n e d or average y i e l d and e f f o r t shown i n F i g u r e s 12, 14, and 16 are t y p i c a l of domed y i e l d curves. They have the property that each i n c r e a s e i n e f f c r t gets a smaller r e t u r n i n i n c r e a s e d y i e l d . Consequently, i f the c o s t of h a r v e s t i n g i s s u b s t a n t i a l compared to the value of the harvest the most economical s t r a t e g y i s to h a r v e s t a t some r a t e i e s s than that f o r the maximum s u s t a i n e d y i e l d . The r e d u c t i o n of 67 the marginal r e t u r n i s i l l u s t r a t e d by the f a c t t h a t the s t r a t e g i e s of using gear P, C, and P with a 2 year pause, with a h a r v e s t i n g r a t e of 67* of t h a t which gives the maximum y i e l d would give r e s p e c t i v e l y y i e l d s of 88%, 83%, and 90% of the maximum i n each case. Random v a r i a t i o n i n recruitment To get some idea of the e f f e c t of random v a r i a t i o n i n r e c r u i t m e n t , s i m u l a t i o n s of v a r i o u s p o l i c i e s were t r i e d with a v e r s i o n of the model which had the r e c r u i t s f o r each year m u l t i p l i e d by a random v a r i a b l e that was uniformly d i s t r i b u t e d between 0.5 and 1.5. The t a b l e below shows the r e s u l t s of 100 year s i m u l a t i o n s with the gears P,C, and A at the f i s h i n g m o r t a l i t y which gave the maximum y i e l d . A 25 year y i e l d seguence f o r each gear i s shown i n Figure 18. Y i e l d i n m i l l i o n s of l b s gear P gear C gear A Expected y i e l d 166 193 245 Mean y i e l d 162 182 229 Sample standard d e v i a t i o n 31 40 67 C o e f f i c i e n t of v a r i a t i o n 0. 19 0.22 0.29 The sample means are a l l s i g n i f i c a n t l y l e s s than the values expected on the b a s i s of the d e t e r m i n i s t i c model, and t h i s e f f e c t i n c r e a s e s as the maximum y i e l d i n c r e a s e s . The standard d e v i a t i o n of the y i e l d from gear P was l e s s than that from gear C, which i n t u r n was l e s s than the standard d e v i a t i o n of the F i g u r e 18 Harvest sequences from s i m u l a t i o n s of the y e l l o w f i n tuna p o p u l a t i o n with random v a r i a t i o n i n r e c r u i t m e n t . Y I E L D I N M I L L I O N S OF L B S M- ru ru © ui o tn © o o © M- ru ru u> o ui o ui o o o o o o ru LJ UJ ui o ui o o o oo c r 69 y i e l d from gear A. The expected values f o r the c o e f f i c i e n t of v a r i a t i o n assuming that the year c l a s s s t r e n g t h s are independently d i s t r i b u t e d were 0.17, 0.21, and 0.29 which are gu i t e c l o s e t o the r e s u l t s of the s i m u l a t i o n . Summary There are s e v e r a l f a c t o r s t h a t must be born in mind when d e c i d i n g what i s an optimum s t r a t e g y f o r f i s h i n g , and some of these f a c t o r s can be s t u d i e d with the a i d of the model. The f i r s t t h i n g t h a t i s u s u a l l y c o n s i d e r e d i s the t o t a l y i e l d and the c o s t of g e t t i n g i t . The nature of the purse seine f i s h e r y f o r y e l l o w f i n suggests that the c o s t of h a r v e s t i n g w i l l not r i s e as f a s t as the value of the y i e l d , and thus a t f i r s t s i g h t the maximum y i e l d would seem to be the i d e a l t a r g e t . However there are s e v e r a l problems a s s o c i a t e d with maximizing the y i e l d . F i r s t l y i t seems that i t may be necessary to change t o another f i s h i n g method which would almost c e r t a i n l y i n c r e a s e the c o s t of h a r v e s t i n g . Secondly there i s a general trend of i n c r e a s i n g v a r i a b i l i t y of annual y i e l d s as y i e l d i n c r e a s e s which would make the l a r g e r y i e l d s l e s s a t t r a c t i v e . F i n a l l y , s t r a t e g i e s with l a r g e t o t a l y i e l d s i n v o l v e the capture of o l d e r f i s h which may cause market problems because of the high l e v e l s of mercury i n these f i s h . The annual y i e l d from the present f i s h i n g method can be i n c r e a s e d s l i g h t y by changing the time f o r the unregulated f i s h e r y from the beginning of the year to the t h i r d or f o u r t h g u a r t e r . T h i s change would not i n c r e a s e the p r o p o r t i o n of o l d 70 f i s h i n the c a t c h , and would only s l i g h t y i n c r e a s e the v a r i a b i l i t y of the y i e l d . 71 Harp S e a l s The f i s h e r y f o r the harp s e a l on the east c o a s t of Canada provides an example of the use of the model with a l i n e a r stock r e c r u i t r e l a t i o n . T h i s simply i s L e s l i e ' s model or the method of l i f e t a b l e s , and i s i n c l u d e d i n t h i s t h e s i s as i t i l l u s t r a t e s the u s e f u l n e s s of age s t r u c t u r e data i n e s t i m a t i n g the dynamics of t h i s p o p u l a t i o n . The l i n e a r model i s used i n t h i s case because i t would not be expected t h a t d e n s i t y dependent e f f e c t s would be important while the p o p u l a t i o n remained c l o s e to i t s present low s i z e . Because harp s e a l s are long l i v e d and have a slow r a t e of populat i o n i n c r e a s e the p o p u l a t i o n w i l l not grow very r a p i d l y even i f the harvest i s stopped e n t i r e l y . A l s o i t i s u n l i k e l y t h a t the p o p u l a t i o n w i l l be much depressed by over h a r v e s t i n g i n f u t u r e y e a r s . L i f e h i s t o r y The l i f e h i s t o r y of the harp s e a l i s important i n r e l a t i o n to the harvest and i s o u t l i n e d i n the f o l l o w i n g summary which i s taken mainly from Sergeant (1963). Adult harp s e a l s s t a r t m i grating from A r c t i c waters i n November and move down the coast of Labrador r e a c h i n g the north of the Gulf of St Lawrence by January. They remain i n open water u n t i l the i c e begins t o form i n the l a t t e r p a r t of February, when the females form whelping groups on the i c e . There are two major whelping grounds, known as the Gu l f and Front, the f i r s t 72 being i n the Gulf of St Lawrence and the l a t t e r being i n the open sea to the east of Newfoundland. I n i t i a l l y the pups are covered with the white h a i r which makes t h e i r p e l t s so a t t r a c t i v e to the i n d u s t r y . T h i s begins t o loose n a f t e r a week and the pups moult at two weeks. At t h i s time they begin to enter the water and are deserted by t h e i r mothers. The a d u l t s breed i n l a t e March and s h o r t l y afterwards the s e a l s move onto the i c e to moult. Moulting patches begin t o form about the 10th of A p r i l and at t h i s time they are composed mainly of a d u l t males and bedlamers (immature s e a l s with a range of ages from 1 to about 5 years) which have j u s t a r r i v e d from the n o r t h . Adult females s t a r t j o i n i n g the groups at about the 20th of A p r i l and are f u l l y r e p r e s e n t e d i n about a week. F o l l o w i n g the moult the s e a l s f o l l o w the edge of the breaking i c e and r e t u r n to the A r c t i c . I t i s not c l e a r to what extent animals from the two herds i n t e r m i n g l e , though r e c e n t s t u d i e s i n d i c a t e t h a t while there may be some homing to the Gulf or Fr o n t t h e r e i s not a sharp d i v i s i o n between the two herds (Sergeant 1971). In t h i s study i t i s assumed t h a t the two herds form a s i n g l e breeding p o p u l a t i o n . One of the most important consequences of the harp s e a l behaviour i s t h a t because hunting i s c a r r i e d out a t a time when the s e a l s are congregated i n a few e a s i l y l o c a t e d dense groups of the cat c h per u n i t of e f f o r t does not g i v e a r e l i a b l e i n d i c a t i o n of po p u l a t i o n s i z e . Another important consequence i s the e f f e c t of the c l o s i n g date of the hunt on the sex r a t i o of the k i l l . Because a d u l t females do not j o i n the moulting groups 7 3 u n t i l l a t e A p r i l they can be p r o t e c t e d by c l o s i n g the season before they a r r i v e . Seal hunting Though some s e a l s are k i l l e d by landsmen on the c o a s t s of Canada and Greenland the major p a r t of the k i l l i s taken from whelping and moulting patches by s e a l e r s o p e r a t i n g from Canadian and Norwegian s h i p s . S t a t i s t i c s f o r the h a r v e s t are given i n the ICNAF S t a t i s t i c a l B u l l e t i n . From these f i g u r e s I have estimated the number of females k i l l e d every year (see Appendix 1). Over the l a s t ten years the s e a l e r s o p e r a t i n g from s h i p s have taken an average of 230,000 pups, and 50,000 o l d e r s e a l s . A s m a l l part of t h i s k i l l (approximately 8000 pups, 1000 bedlamers, and 1000 ad u l t s ) i s taken by Eskimos, from Canada and Greenland and I have assumed t h a t t h i s w i l l remain constant i n f u t u r e years. Thus t h i s k i l l w i l l appear i n a l l my p o p u l a t i o n p r o j e c t i o n s . The model and parameters. There i s no i n f o r m a t i o n as to whether harp s e a l s are monogamous (C.O.S.S. 1972), and so the model i s i n e f f e c t a model of the dynamics of the female p o p u l a t i o n , expressed i n terms of a 1:1 sex r a t i o . I f only a few males are r e q u i r e d f o r breeding a l l the estimates of k i l l s may be i n c r e a s e d by the number of s u r p l u s males that can be captured. 74 The model i s JO N(1,t*1) = ^ { H ( i ) . S ( i ) . K ( i , t ) ] N (i+1 , t + 1)=S (i) •» ( i , t ) i= 1,2, .... f 30. Where N ( i , t ) i s the number of s e a l s beginning t h e i r i t h year of l i f e i n year t , H(i) and S (i) are the age s p e c i f i c pregnancy and s u r v i v a l r a t e s . To use the model f o r p r e d i c t i v e purposes i t i s necessary to have estimates of b i r t h r a t e s , m o r t a l i t y and the p o p u l a t i o n s i z e and s t r u c t u r e i n at l e a s t one year. Pregnancy r a t e s can be taken from data given by Sergeant (1971) i n h i s Table 1. These r a t e s were rounded and d i v i d e d by 2 to g i v e the f o l l o w i n g r a t e s which are r e l a t i v e to a p o p u l a t i o n with a sex r a t i o of 1:1. Age 4 5 6 7 8 9 and over Pregnancy r a t e s .01 .11 .31 .42 .42 .45 T h i s data i s based on o b s e r v a t i o n s on s e a l s c o l l e c t e d i n the Gulf during the years 1966-1969, and g i v e s a s l i g h t y lower r e p r o d u c t i v e r a t e than had been observed i n e a r l i e r y e a r s . The m o r t a l i t y r a t e f o r a d u l t s e a l s was estimated by B i c k e r (MS.) who found a t o t a l annual m o r t a l i t y of 15%. T h i s r e f l e c t e d c o n d i t i o n s 10-20 years ago when the hunting m o r t a l i t y f o r a d u l t s was about 1%. D l l t a n g (MS.) using more r e c e n t samples found a 75 t o t a l m o r t a l i t y of M%. Hunting m o r t a l i t y had d e c l i n e d t o about 5% f o r the animals i n U l l t a n g ' s sample and so both e s t i m a t e s give a f i g u r e of about 8% f o r n a t u r a l m o r t a l i t y , which i s the f i g u r e used here. Estimates of the s i z e of the breeding p o p u l a t i o n are a v a i l a b l e from s e v e r a l sources of data. The most d i r e c t e stimates obtained are those from a e r i a l surveys of both whelping and moulting s e a l s . R e s u l t s of these surveys are d i s c u s s e d by Sergeant (1966, 1971) and a summary of the r e s u l t s i s Year A d u l t s Pups 1950 730,000 440,000 1959 370,000 240,000 1970 180,000 Because i t i s d i f f i c u l t to f i n d a l l the s e a l s i n photographs these e s t i m a t e s tend to be on the low s i d e but they do i n d i c a t e a l a r g e drop i n p o p u l a t i o n during the twenty year p e r i o d . I t i s p o s s i b l e to get a minimum estimate of pup p r o d u c t i o n i n any year simply by taking the sum of the members of t h a t year c l a s s that are harvested. T h i s procedure was used by O r i t s l a n d (1971) to get a minimum estimate of 292,000 pups f o r the 1967 year c l a s s . Another estimate can be made using a method based on the v a r i a t i o n c f the age composition of samples of s e a l s . T h i s method was suggested by Sergeant (1971) and modified by R i c k e r 76 (MS.). Assuming that the samples r e p r e s e n t the p o p u l a t i o n the r a t i o K (i) of age 1 to a d u l t s e a l s i n samples i s be given by K(i) = Q»{M(i)-C(i) }/M (i) where M (i) and C ( i ) are the pup production and k i l l i n year i and Q i s a constant. By f u r t h e r assuming t h a t the pup pro d u c t i o n i n years i and j was the same and egual to M, B i c k e r showed t h a t M = ( K ( i ) * C ( j ) - K ( j ) . C ( i ) }/{ K (i)-K (j) } In f a c t i f the pup pr o d u c t i o n i n the years i and j i s not same the estimate of M given above by the formula above would l i e between M (i) and M (j) s i n c e M (i) > M(j), i m p l i e s that M (i) > M > M(j). For M(i) = {[M(i)/M(j) ] . K ( i ) . C ( j ) - K ( j ) . C ( i ) )/{K ( i ) - K ( j ) } > M and thus M g i v e s an estimate between M(i) and M ( j ) . B i c k e r obtained a mean estimate of 273 thousand pups i n 1968 using a comparison of the 1968 year c l a s s with those of 1966, 1967, 1969 and 1970. The data used are shown i n Table 5, and i t can be seen that the pup k i l l i n 1968 was u n u s u a l l y low g i v i n g high v a l u e s of K f o r t h a t year c l a s s . B i c k e r suggested t h a t because t h i s year provided the g r e a t e s t c o n t r a s t with the other years, comparisons using i t would give the most r e l i a b l e r e s u l t s . However i t can a l s o be argued that because t h i s year i s u n l i k e the r e s t , the assumptions about the r a t i o s are l e s s l i k e l y to be 77 Table 5 Numbers of age one year and a d u l t s e a l s i n samples a t St Anthony and the F r o n t . St Anthony Front Year Pup k i l l age 1 a d u l t s r a t i o age 1 a d u l t s r a t i o c l a s s thousands 1966 257 18 261 0.069 176 377 0.467 1967 280 7 165 0.042 84 332 0.253 1968 158 87 68 1.279 62 33 1.879 1969 235 41 434 0.094 10 5 186 0.565 1970 226 39 285 0.137 Table 6 Estimates of pup p r o d u c t i o n using Sergeant's method, Number of pups i n thousands St Anthony Front 1970 1969 1968 1967 1969 1968 1967 1966 289 317 263 317 1966 362 290 307 1967 304 317 284 1967 317 299 1968 234 241 1968 268 1969 255 Each e s t i m a t e i s based on comparisons between the data f o r the years heading the row and column c o n t a i n i n g the esti m a t e . 78 true i n t h i s year. T h i s s u s p i c i o n i s strengthened by l o o k i n g a t the r a t i o s of the values of K f o r the F r o n t and St Anthony samples. T h i s r a t i o which should be constant has the values 6.8, 6.0, 1.5, 6.0 f o r the years 1966 to 1969. The estimates of M based on a l l the a v a i l a b l e comparisons are shown i n Table 6, which i l l u s t r a t e s that i f the 1968 comparisons are ignored the mean esti m a t e s would be c o n s i d e r a b l y l a r g e r . Using t h i s i n f o r m a t i o n an ICHAF assessment subcommittee (ICNAF 1971) estimated pup production as 400,000, 350,000, and 300,000 i n the years 1960, 1965, and 1970. These e s t i m a t e s were compared with p o p u l a t i o n s i m u l a t i o n s s t a r t i n g from a r b i t r a r y s t a b l e p o p u l a t i o n s i n 1938. The harvest f o r each year i s given i n Table 7 as the number of pups, bedlamers, and a d u l t s k i l l e d . For each year of the s i m u l a t i o n the k i l l of bedlamers and a d u l t s i s taken from the year c l a s s e s according to t h e i r r e l a t i v e s t r e n g t h . The r e s u l t s of the s i m u l a t i o n that was chosen as the best estimate of the p o p u l a t i o n h i s t o r y are shown i n Table 8. Harvest p o l i c i e s Even with the assumption of a l i n e a r stock r e c r u i t r e l a t i o n i n the model there i s a wide c h o i c e of p o s s i b l e harvest s t r a t e g i e s . For example i t i s p o s s i b l e to f i n d s u s t a i n e d harvests with d i f f e r e n t stock s t r u c t u r e s . L e s l i e (1945) showed that a s u s t a i n e d harvest c o n s i s t i n g of age c l a s s e s i n the same frequency as they appear i n the p o p u l a t i o n c o u l d be obtained using a harvest r a t e of 1-1/3 where a i s the dominant eigenvalue of the l i f e t a b l e matrix. In general an e q u i l i b r i u m h a r v e s t can 79 Table 7 Harvest of harp s e a l s f o r the years 1938-197 Numbers i n thousands Year Pups Bedlamers a d u l t s T o t a l 1938 221 34 7 262 1939 102 19 17 139 1940 132 32 13 177 1941 17 35 8 60 1942 2 20 5 26 1943 6 3 2 11 1944 6 10 8 24 1945 10 21 4 34 1946 73 18 9 100 1947 102 47 22 171 1948 137 45 39 221 1949 227 28 33 288 1950 226 22 28 276 1951 319 55 66 439 1952 198 33 60 292 1953 198 18 45 262 1954 175 27 65 267 1955 252 25 57 334 1956 34 1 15 35 391 1957 165 20 58 244 1958 14 1 48 100 289 1959 239 26 51 3 16 1960 170 30 75 275 1961 179 7 1 1 197 1962 214 35 63 313 1963 278 22 41 342 1964 273 22 45 340 1965 190 17 29 236 1966 257 23 41 321 1967 280 29 9 318 1968 158 21 6 185 1969 235 30 9 274 1970 226 22 14 262 1971 210 15 10 236 Table 8 Estimated harp s e a l p o p u l a t i o n h i s t o r y , 1938-1971, Numbers i n thousands Aqe group Year 0-1 1-2 2-3 3-4 4-5 5-6 a d u l t s t o t a l 1938 337 105 97 89 82 75 749 1534 1939 332 106 89 82 76 70 738 1493 1940 322 212 94 79 72 67 715 1559 1941 313 174 183 81 68 62 694 1575 1942 3 05 272 150 158 70 59 676 1690 1943 300 279 244 134 141 62 662 1822 1944 3 03 271 256 22 3 123 129 655 1961 1945 316 273 246 23 3 204 112 705 2089 1946 339 282 246 222 210 184 738 2221 1947 370 245 255 223 201 190 828 2313 1948 400 247 216 225 197 177 901 2363 1949 418 242 217 190 198 173 94 1 2380 1950 437 175 217 194 170 177 983 2354 1951 455 194 158 195 175 153 1030 2359 1952 451 126 168 136 168 151 1012 2210 1953 445 232 110 147 119 148 1001 2204 1954 444 228 208 99 132 107 1006 2224 1955 426 247 202 185 88 117 955 2221 1956 409 160 221 180 165 79 925 2139 1957 403 62 145 199 163 149 884 2006 1958 3 97 219 56 129 178 146 889 2013 1959 378 236 188 48 111 153 84 6 1960 1960 3 76 128 209 167 42 99 863 1885 1961 353 190 113 184 147 37 808 1830 1962 351 159 173 103 167 133 764 1850 1963 341 126 140 151 90 146 758 1752 1964 344 57 112 124 134 80 787 1638 1965 337 65 50 99 109 118 751 1529 1966 338 136 57 45 87 97 766 1525 1967 331 74 118 50 39 76 749 1436 1968 325 46 63 100 42 33 742 1352 1969 314 154 39 54 85 36 700 1383 1970 3 02 73 130 33 45 72 662 1318 1971 291 70 63 112 29 39 651 1255 81 be taken by using harvest r a t e s c ( i ) such t h a t the r e p r o d u c t i v e p o t e n t i a l i s egual to 1, i . e . 5:{H<i).Q<i) } = 1 Where Q (i) =S (1) •[ 1-c (1) ]»S (2) •[ 1-c <2) ]•. .. .S (i) *[ 1-c (i) ] Thus a s u s t a i n e d h a r v e s t of pups only can be taken by s e t t i n g c(1) = 1-1/E and c(i)=0 f o r i>1, where E i s the r e p r o d u c t i v e p o t e n t i a l with no hunting m o r t a l i t y . With the parameters I have used the dominant eigenvalue i s 2,03 which g i v e s a s u s t a i n e d harvest from a l l age groups of 51% of the p o p u l a t i o n . The value of E i s 2.95 which g i v e s a s u s t a i n e d harvest of pups on l y , of 66% of the pup p r o d u c t i o n . These s u s t a i n e d r a t e s of h a r v e s t only apply to s i t u a t i o n s where the s t a b l e age d i s t r i b u t i o n has a l r e a d y been reached, and of course t h i s i s not the case with t h i s stock. In f a c t t h i s p o p u l a t i o n i s now d e c l i n i n g , and because i t takes s i x years before pups j o i n the breeding p o p u l a t i o n r e d u c i n g the harvest of pups w i l l not a l t e r the pup p r o d u c t i o n u n t i l a f t e r 1977. The estimated pup production f o r the years 1972-1977 i s Pup p r o d u c t i o n i n thousands 1972 1973 1974 1975 1976 1977 278 266 261 254 244 242 The estimate of maximum su s t a i n e d y i e l d of pups only s t a r t i n g from the e s t i m a t e of the present s t o c k s t r u c t u r e was found by t r i a l and e r r o r to be 164 thousand pups (plus the 82 Eskimo k i l l of 8,000 pups and 2,000 o l d e r s e a l s ) . Another p o l i c y t h a t could be t r i e d i s t h a t of a l l o w i n g k i l l of 20 thousand a d u l t s and bedlamers i n a d d i t i o n to the Eskimo k i l l . In t h i s case the s u s t a i n e d y i e l d i s only 126 thousand pups and thus the d i f f e r e n c e of 38 thousand pups i s traded o f f a g a i n s t 20 thousand o l d e r s e a l s . A simple way of a s s e s s i n g the e f f e c t cn the p o p u l a t i o n of k i l l i n g female s e a l s of d i f f e r e n t ages i s to look at the expected number of pups that would be produced i n the remainder of the s e a l ' s l i f e , which i s shown i n F i g u r e 19. T h i s f i g u r e shows that from the p o i n t of view of maximizing r e p r o d u c t i o n the l e a s t d e s i r a b l e age groups to h a r v e s t are the 6 year o l d s , and t h a t the best age groups to harvest are the very o l d ones and the pups. To maximize p r o f i t the value of s e a l s of d i f f e r e n t ages would have to be c o n s i d e r e d , and i n a d d i t i o n the expected f u t u r e pup production would have to be d i s c o u n t e d . S e n s i t i v i t y of the model as a p p l i e d to harp s e a l s If a model of t h i s type i s to be used f o r management purposes i t i s important to know the s e n s i t i v i t y of i t s output to changes i n parameters. Furthermore, i t i s important to be able to make ob s e r v a t i o n s which can be compared with p r e d i c t i o n s of the model t o t e s t i t s performance. For these o b s e r v a t i o n s to be u s e f u l they must provide a c r i t i c a l t e s t of the model, i . e . the o b s e r v a t i o n s must be s e n s i t i v e t o changes i n the parameters. Two p o s s i b l e s e t s of o b s e r v a t i o n s are the s i z e s of age groups, and the r a t i o s of s i z e s of age groups. R a t i o s of the F i g u r e 19 Expected number of pups to be born to female s e a l s of d i f f e r e n t ages. E X P E C T E D NUMBER OF P U P S TO B E B O R N 84 numbers i n age groups have the advantage of being e a s i e r to measure than a c t u a l numbers, but they are not as s e n s i t i v e to changes i n parameters as the a c t u a l numbers a r e . I t e s t e d the s e n s i t i v i t y of the model using three s e t s of s u r v i v a l and pregnancy r a t e s as w e l l as a v a r i e t y of i n i t i a l p o p u l a t i o n s i 2 e s . The s e t s of parameters were; (1) The estimates given above. (2) A s e t with a m o r t a l i t y r a t e of 6% and pregnancy r a t e s i n c r e a s e d by 10%. (3) A s e t with a m o r t a l i t y r a t e of 10% and pregnancy r a t e s decreased by 10%. The s u s t a i n e d y i e l d s of pups only with these three s e t s of parameters were r e s p e c t i v e l y 66%, 78%, and 49% of pup p r o d u c t i o n . F i g u r e s 20 (a)-(d) show the model's p r e d i c t i o n s of s t r e n g t h s of the year c l a s s e s 1967, 1969, 1970, and 1971 r e l a t i v e to the 1968 year c l a s s , as a f u n c t i o n of the 1970 pup production f o r the three s e t s of parameters. The r a t i o s shown are those p r e d i c t e d by the model f o r the year 1972. The r a t i o f o r other years w i l l vary s l i g h t y because of v a r i a t i o n s i n the harvest of bedlamers taken from d i f f e r e n t age groups. F i g u r e 21 (a)-(d) shows the p r e d i c t i o n s of numbers i n 1972 f o r the d i f f e r e n t parameters as a f u n c t i o n of the 1970 pup p r o d u c t i o n . Neither the numbers nor r a t i o s of numbers i n 1972 are very s e n s i t i v e to the d i f f e r e n c e s between the parameters, but they are both f a i r l y s e n s i t i v e to changes i n p o p u l a t i o n s i z e as measured by the 1970 year c l a s s s t r e n g t h . Figure 22 shows the pup p r o d u c t i o n p r o j e c t i o n s f o r each of the parameter s e t s with a F i g u r e 20 P r o j e c t i o n s of numbers i n the 1967, 1969, 1970, and 1971 harp s e a l year c l a s s e s r e l a t i v e to the 1968 year c l a s s i n the year 1972. Parameter s e t 1 • Parameter s e t 2 - - - -Parameter s e t 3 — — — (A) Ratio of 1967 to 1968 year c l a s s . (B) Ratio of 1969 to 1968 year c l a s s . (C) Ratio of 1970 to 1968 year c l a s s . JD) Ratio of 1971 to 1968 year c l a s s . 86 a F i g u r e 21 Numbers of harp s e a l s i n the year c l a s s e s i n the year 1972. of 1968-1971 Parameter set 1 Parameter s e t 2 - - -Parameter s e t 3 (ft) 1968 year c l a s s . (B) 1969 year c l a s s . (C) 1970 year c l a s s . (D) 1971 year c l a s s . F i g u r e 22 P r o j e c t e d pup pro d u c t i o n u s i n g the three parameter s e t s . The k i l l s f o r the year 1972 onwards are 126,000 pups and 20,000 o l d e r s e a l s i n a d d i t i o n to the Eskimo k i l l of 10,000 s e a l s N U M B E R S I N THOUSANDS 88 1970 year c l a s s s t r e n g t h of about 300,000, and a h a r v e s t of 134,000 pups plus 22,000 o l d e r s e a l s a f t e r 1971. although these p r o j e c t i o n s are s u b s t a n t i a l l y d i f f e r e n t a f t e r about 10 or 15 years, i t would be d i f f i c u l t i f not impossible t o choose between them before 1976, by which time the pup production u n t i l 1982 i s determined. Thus an e r r o r i n a s s e s s i n g the s t a t e of the p o p u l a t i o n at t h i s stage w i l l be hard to d e t e c t , and could l e a d to severe damage to the s t o c k s . In view of the p o s s i b l e e r r o r i n the estimates of the s t a t e of the s t o c k s i t i s e s s e n t i a l that the b e s t p o s s i b l e o b s e r v a t i o n s be made over the next few years to ensure t h a t the p o p u l a t i o n i s not f u r t h e r depressed by o v e r - h a r v e s t i n g . Table 9 shows the p r o j e c t i o n s of numbers i n the youngest age groups f o r the years 1972-1980 f o r each of the three parameter s e t s . In each case the 1970 pup p r o d u c t i o n was approximately 300,000 and the annual h a r v e s t f o r the years 1972 onwards was 134,000 pups and 22,000 o l d e r s e a l s . There are l a r g e d i f f e r e n c e s between the three p r o j e c t i o n s , but as noted above, d i r e c t o b s e r v a t i o n s of the p o p u l a t i o n s i z e may not be accurate enough to d i s t i n g u i s h between these three p o s s i b i l i t i e s . I t i s however e a s i e r to d e t e c t r e l a t i v e d i f f e r e n c e s between numbers i n age groups which can be e x h i b i t e d as c a t c h curves. The catch curves f o r the years 1976 and 1980 f o r the three p r o j e c t i o n s are shown i n F i g u r e 23. There i s very l i t t l e d i f f e r e n c e between the p r e d i c t i o n s of c a t c h c a r v e s i n 1972 f o r the three s e t s of parameters, but F i g u r e 23 shows the d i f f e r e n c e s become more marked with time and that by 1980 there are l a r g e d i f f e r e n c e s between the t h r e e c u r v e s . Table 9 P r o j e c t e d Harp S e a l P o p u l a t i o n For The Years 1972-1980. Numbers In Thousands Parameter Set (1) Age group Year 0-1 1-2 2-3 3-4 4-5 5-6 a d u l t s t o t a l 1972 278 74 61 55 98 25 618 1209 1973 266 132 65 54 49 87 574 1 227 1974 261 121 118 59 49 44 588 1 240 1975 254 117 108 106 52 44 564 1 244 1976 244 110 105 97 95 47 54 1 1239 1977 241 101 99 94 87 85 524 1 231 1978 243 98 91 89 84 78 54 1 1225 1979 247 100 88 82 80 76 553 1226 1980 250 104 90 79 73 72 561 1229 Parameter Set (2) Age group Year 0-1 1-2 2-3 3-4 4-5 5-6 a d u l t s t o t a l 1972 278 73 59 52 96 18 556 1 130 1973 270 135 66 53 47 87 521 1 178 1974 271 127 123 60 48 43 552 1225 1975 270 129 117 113 55 44 540 1267 1976 265 128 118 107 104 51 532 1 303 1977 270 123 117 108 93 95 529 1342 1978 282 128 114 108 100 91 568 1390 1979 297 139 118 105 100 92 602 1452 1980 312 153 128 108 96 92 634 1525 Parameter Set (3) Age group Year 0-1 1-2 2-3 3-4 4-5 5-6 a d u l t s t o t a l 1972 269 68 59 55 96 30 672 1250 1973 252 121 59 51 48 84 614 1230 1974 241 106 106 52 45 42 609 1201 1975 228 96 93 92 45 39 568 1 162 1976 213 85 84 81 81 39 53 0 1113 1977 203 71 74 74 71 70 495 1058 1978 197 62 62 65 64 62 491 1003 1979 192 57 54 54 56 56 481 9 50 1980 186 52 49 47 47 49 466 895 90 a F i g u r e 23 P r o j e c t e d c a t c h curves f o r harp s e a l s i n 1976 and 1980. 90 b 0.3 .. 0*2 • O-i-. A G E I N Y E A R S 91 The three s e t s of parameters are of course only a smal l sample from the p o s s i b l e range o f the types of e r r o r s t h a t may have been made i n e s t i m a t i n g the stock c o n d i t i o n . However, r e g a r d l e s s of the type of e r r o r made the c a t c h curves w i l l provide an i n d i c a t i o n of whether recruitment to the p o p u l a t i o n i s d e c r e a s i n g , constant, or i n c r e a s i n g . T h i s i s because trends i n r ecruitment w i l l a l t e r the apparent m o r t a l i t y r a t e that i s shown by the c a t c h curve. I f recruitment i s i n c r e a s i n g the d i f f e r e n c e s between numbers i n s u c c e s s i v e age groups shown by the curve w i l l be grea t e r than that caused by m o r t a l i t y alone, and the apparent m o r t a l i t y w i l l be higher than i t would be i f there were no trend i n r e c r u i t m e n t . The opposite e f f e c t occurs when recru i t m e n t i s d e c l i n i n g . Dsing F i g u r e 23 as a guide i t can be seen that f o r the harvest schedule used above we co u l d say the p o p u l a t i o n was being over harvested i f the average s l o p e of the observed c a t c h curve was l e s s than 0.04 i n 1976 or l e s s than 0.11 i n 1980. Unless methods of o b t a i n i n g d i r e c t e stimates of p o p u l a t i o n s i z e are improved i n the next few years i t would be best to concentrate on o b t a i n i n g a c c u r a t e c a t c h curves to be used as i n d i c a t o r s of tr e n d s i n r e c r u i t m e n t . Density dependent e f f e c t s I t i s of i n t e r e s t to sp e c u l a t e about harvest p o l i c i e s t h a t might have been a p p l i e d a t the i n i t i a t i o n of the f i s h e r y t h a t c o u l d have ensured a l a r g e r s u s t a i n e d h a r v e s t . According to 92 Chafe (1923) between 500,000 and 700,000 pups were k i l l e d a n n u a l l y i n the 1840's. Beari n g i n mind the lower e f f i c i e n c y of hunting at t h a t time a f i g u r e of one m i l l i o n would seem to be a reasonable estimate f o r the pup production i n the unhatvested p o p u l a t i o n . Because of the l a r g e range of po p u l a t i o n s i z e i t i s now necessary to c o n s i d e r the e f f e c t of p o p u l a t i o n s i z e cn the parameters. A reasonable simple assumption i s that p o p u l a t i o n s i z e has a l i n e a r e f f e c t cn pregnancy r a t e s and no e f f e c t on s u r v i v a l . Then the r e p r o d u c t i v e p o t e n t i a l E w i l l be a l i n e a r f u n c t i o n of pup production and we can write E = a-b»n where E i s the r e p r o d u c t i v e p o t e n t i a l when the po p u l a t i o n has a pup pr o d u c t i o n of n. The corresponding s u s t a i n e d harvest of pups i s then C = n« (E-1)/E The constants a and b can be evaluated using the assumption t h a t E=1 when the pup production i s one m i l l i o n and that E = 2.95 when the production i s 300,000. T h i s g i v e s values of a and b of 3.78 and 2.78 and a s u s t a i n e d harvest of n» (1-n)/(1.36-n). The maximum s u s t a i n e d harvest i s thus 320,000 pups which i s taken from a p o p u l a t i o n with a pup production of 660,000, more than twice as l a r g e as the present p o p u l a t i o n . I f the stock s i z e 0 i s measured i n terms of the number of pups needed to maintain the stock then 93 n = E«U which can be combined with the r e l a t i o n between E and n g i v i n g n/U = a-b«n or n = a»0/ (1+b*U) Thus the assumption of a l i n e a r d e c l i n e i n f e c u n d i t y l e a d s to the Beverton and H c l t type of stock r e c r u i t r e l a t i o n which i s shown i n Fig u r e 24. The s u s t a i n e d y i e l d of pups at any stock s i z e i s the d i f f e r e n c e between the stock r e c r u i t curve and the replacement l i n e . Of course other assumptions could have been made about the nature of the e f f e c t s of pop u l a t i o n s i z e on the l i f e h i s t o r y parameters and these would lead to d i f f e r e n t stock r e c r u i t c u rves. However, as any curve i s c o n s t r a i n e d to pass through the three p o i n t s o, A, and B i t i s u n l i k e l y t h a t r a d i c a l l y d i f f e r e n t c o n c l u s i o n s would r e s u l t i f a d i f f e r e n t curve was used. For comparison the B i c k e r curve t h a t passes through the three p o i n t s i s a l s o shown on Fi g u r e 24. T h i s curve has a parameter of 1.2 and g i v e s a maximum s u s t a i n e d y i e l d of 420,000 pups at a pup production of 820,000 pups. 94 a F i g u r e 24 S t o c k - r e c r u i t curves f o r harp s e a l s . 0«2 0.4 O'S 0-8 1-0 1>2 1-4 1.6 U N I T S OF S T O C K 95 The dynamics of the model f o r harp s e a l s With a stock r e c r u i t r e l a t i o n of the type n = U»f (U) such as those shown i n F i g u r e 24 i t i s u n l i k e l y t h a t the model would be u n s t a b l e . With the two cases shown the model i s s t a b l e f o r a l l values of f i s h i n g m o r t a l i t y , as the Beverton and H o l t curve always g i v e s a s t a b l e p o p u l a t i o n and a B i c k e r curve must have a parameter g r e a t e r than 2 to make the model u n s t a b l e . A stock-r e c r u i t curve of t h i s type would have to have a much more pronounced dome and steeper r i g h t hand limb than the B i c k e r curve i n F i g u r e 24 i f i t were to produce an u n s t a b l e e q u i l i b r i u m i n t h i s p o p u l a t i o n . Even i f t h i s i s the case f o r t h i s p o p u l a t i o n such an i n s t a b i l i t y would only occur a t low f i s h i n g m o r t a l i t i e s and thus i s of l i t t l e importance i n the present s i t u a t i o n . On the other hand i f the s t o c k - r e c r u i t r e l a t i o n were of the more g e n e r a l type n = U«f (V) then i t i s p o s s i b l e t o get f l u c t u a t i o n s with a high harvest r a t e . For example i f d e n s i t y dependent m o r t a l i t y o c c u r s as a r e s u l t of i n t e r a c t i o n s betweeen a d u l t and immature s e a l s at the winter f e e d i n g grounds (Sergeant 1971), high hunting m o r t a l i t i e s c o u l d lead to f l u c t u a t i o n s i n year c l a s s s t r e n g t h . 96 D i s c u s s i o n In t h i s study the p r o p e r t i e s and uses of a c l a s s of p o p u l a t i o n models that are u s e f u l i n f i s h e r i e s management has been examined. These models which are r e f e r r e d to as dynamic age s t r u c t u r e d models are e s s e n t i a l l y a l i f e t a b l e modified by a stock r e c r u i t r e l a t i o n . The recruitment i s made up of the product of the number of eggs produced by the p o p u l a t i o n and a s u r v i v a l term which i s a f u n c t i o n of the stock " d e n s i t y " , which was d e f i n e d to be a l i n e a r combination of the numbers of f i s h of v a r i o u s ages. Because of the assumption that the n a t u r a l m o r t a l i t y of o l d e r f i s h i s constant t h i s d e n s i t y dependent s u r v i v a l r e g u l a t e s the model p o p u l a t i o n . A wide range of s u r v i v a l f u n c t i o n s has been suggested i n the l i t e r a t u r e f o r t h i s r o l e , but u n l e s s there i s some knowledge of the mechanisms i n v o l v e d i t i s probably best to r e s t r i c t the c h o i c e of f u n c t i o n used to one which lead s to one of the L e s l i e , Beverton and H o l t , or R i c k e r s t o c k - r e c r u i t r e l a t i o n s . One of the most important b e n e f i t s t h a t can be got from using these models i s that i t i s p o s s i b l e to use them to i n v e s t i g a t e the e f f e c t s of a much wider range of h a r v e s t s t r a t e g i e s than can be i n v e s t i g a t e d with e i t h e r the y i e l d per r e c r u i t or s u r p l u s p r o d u c t i o n models. T h i s wider range i n c l u d e s the important p o s s i b i l i t i e s of h a r v e s t i n g cnly a p o r t i o n of the age c l a s s e s present and p o l i c i e s which f l u c t u a t e i n time. Another major b e n e f i t i s a s s o c i a t e d with the age s t r u c t u r e of the model, i . e . harvest p o l i c i e s may be designed to ensure as many age groups as p o s s i b l e make a s i g n i f i c a n t c o n t r i b u t i o n to 97 the p o p u l a t i o n which w i l l minimize the e f f e c t of f o r t u i t o u s f l u c t u a t i o n s i n year c l a s s s t r e n g t h on y i e l d or r e c r u i t m e n t . Furthermore with an age s t r u c t u r e d model f u l l use can be made of year c l a s s s t r e n g t h data that i s r o u t i n e l y c o l l e c t e d f o r many f i s h e r i e s . T h i s data can be used as data f o r the model or as a t e s t of the performance of the model. The use of t h i s type of model i s a p p r o p r i a t e when the r e l a t i v e year c l a s s s i z e i s more or l e s s f i x e d , apart from changes caused by f i s h i n g m o r t a l i t y , soon a f t e r i t i s produced. T h i s requirement i s s a t i s f i e d i f v a r i a t i o n i n the number of f i s h r e c r u i t e d i s l a r g e r than the v a r i a t i o n i n year c l a s s s i z e due to v a r i a t i o n s i n the n a t u r a l m o r t a l i t y , and i f t h e r e are no s t r o n g i n t e r a c t i o n s between year c l a s s e s once r e c r u i t e d . The approach i s of course a p p l i c a b l e to other animals than f i s h as i s suggested by the c h o i c e of a s e a l p o p u l a t i o n as an example. In f a c t i t c o u l d be used as a management model f o r any po p u l a t i o n that e x h i b i t s age s t r u c t u r e with the r e q u i r e d p r o p e r t i e s . On the other hand the approach i s of course of no use f o r those t r o p i c a l f i s h populations i n which which reproduce throughout the year, and thus do not have seperate age groups. The dynamic nature of the model makes i t s u i t a b l e f o r use i n c o n j u n c t i o n with an experimental approach t o h a r v e s t i n g , i n which harvest p o l i c i e s are thought of as experimental designs. The o b j e c t of the experiment i s to understand the dynamics of the p o p u l a t i o n as a necessary step i n o p t i m i z i n g the y i e l d . The most obvious way of experimenting i s to explore p o l i c i e s t h a t are l i k e the one which i s thought to be the best. T h i s i s r e a l l y 98 r e f i n i n g the estimates of the parameters over a s m a l l region which i s thought to c o n t a i n the optimum p o l i c y , and i s what the IATTC i s doing with t h e i r experimental f i s h i n g programme. An a l t e r n a t i v e approach which i s l i k e l y to be a more u s e f u l way of g e t t i n g i n f o r m a t i o n when not much i s known of the s t o c k s i s to v i o l e n t l y s t r e s s the p o p u l a t i o n with the i n t e n t i o n of l e a r n i n g i t s behaviour under a wide range of s i t u a t i o n s . T h i s would be a much f a s t e r method of f i n d i n g out about the dynamics of the p o p u l a t i o n and of t e s t i n g the model than i s the method of examining the e f f e c t of s m a l l changes i n f i s h i n g p o l i c i e s . The major shortcoming of t h i s type of model i s i t s complexity, and the consequent d i f f i c u l t y i n examining the behaviour of the model. A s s o c i a t e d with t h i s i s the need f o r more parameters than are r e q u i r e d f o r the l o g i s t i c or y i e l d per r e c r u i t models. One of the p r i n c i p a l f e a t u r e s of the model i s the idea of the p o p u l a t i o n r e g u l a t i o n by a s t o c k - r e c r u i t r e l a t i o n . Although t h i s concept i s w e l l entrenched i n f i s h e r i e s l i t e r a t u r e no r e a l l y c o n v i n c i n g mechanisms causing t h i s type of d e n s i t y dependence have been found. T h i s i s probably due to the major i n t e r e s t of f i s h e r i e s workers being i n management, as opposed to the i n t e r e s t of workers i n other f i e l d s such as s m a l l mammel populat i o n s i n how, or whether p o p u l a t i o n s are r e g u l a t e d . Because of t h i s i t i s necessary that r e s e a r c h be c a r r i e d out to t e s t whether the assumption of the e x i s t e n c e of a s t o c k - r e c r u i t r e l a t i o n i s h e l p f u l i n g i v i n g a reasonable d e s c r i p t i o n of the p o p u l a t i o n . 99 One of the most i n t e r e s t i n g p o s s i b i l i t i e s f o r f u t u r e development of age s t r u c t u r e d models i s t h e i r a p p l i c a t i o n to h a r v e s t i n g i n m u l t i p l e s p e c i e s f i s h e r i e s . M u l t i s p e c i e s forms of the model w i l l of course more complex than s i n g l e s p e c i e s forms, however the p r i n c i p a l d i f f i c u l t y i s l i k e l y to l i e i n e s t a b l i s h i n g the form of the i n t e r a c t i o n between p o p u l a t i o n s , which seems even more d i f f i c u l t than i t i s to measure the e f f e c t of d e n s i t y dependent m o r t a l i t y . Thus i t i s to be expected t h a t the m u l t i s p e c i e s forms of the model w i l l only be developed i n s i t u a t i o n s where i t i s reasonable at l e a s t to p o s t u l a t e a s t r o n g i n t e r a c t i o n between p o p u l a t i o n s . F i n a l l y i t has f o r a long time been accepted that the use of mathematical models i s e s s e n t i a l f o r the r a t i o n a l management of f i s h e r i e s , and progress i n t h i s f i e l d i s thus a s s o c i a t e d with the development of models which are r e a l i s t i c i n the sense t h a t t h e i r behaviour mimicks that of the p o p u l a t i o n as f a i t h f u l l y as p o s s i b l e . 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R i c k e r as, Concerning the causes of f l u c t u a t i o n s i n the abundance of f i s h e s . F i s h . Res. Bd. Canada T r a n s l a t i o n s e r i e s No. 389. 1969. Theory of f i s h p o p u l a t i o n dynamics as the b i o l o g i c a l background f o r r a t i o n a l e x p l o i t a t i o n and management of f i s h e r y r e s o u r c e s . O l i v e r and Boyd. Edinburgh, 323pp. Orange, C» J . 1961. Spawning o f y e l l o w f i n tuna and s k i p j a c k as i n f e r r e d from s t u d i e s of gonad development. B u l l . I n t e r -American T r o p i c a l Tuna Comm., 5 (6):459-526. O r i t s l a n d T. 1971. Progress r e p o r t on Norwegian s t u d i e s of harp s e a l s at Newfoundland. ICNAF S e r i a l No. 2509 28pp. P e a r l , R. and L. J. Reed. 1920. On the r a t e of growth of the po p u l a t i o n of the United S t a t e s s i n c e 1790 and i t s mathematical r e p r e s e n t a t i o n . Proc. N a t l . Acad, of Science. 6:275-288. P e l l a , J . J , , and P. K. Tomlinson. 1969. A g e n e r a l i z e d stock p r o d u c t i o n model. B u l l . Inter-American T r o p i c a l Tuna Comm., 13 (3) :421-496. Regier, H. A., V. C. Applegate, and R. A. Ryder. 1969. The ecology and management of the walleye i n western Lake E r i e . Great Lakes F i s h e r y Commission, Tech, Rep. 15:1-101. Ricker, W. E. 1945. A method of e s t i m a t i n g minimum s i z e l i m i t s f o r o b t a i n i n g maximum y i e l d . Copeia, 1945 (2) :84-94. 1954. Stock and r e c r u i t m e n t . J . F i s h . Res. Bd. Canada, 11 (5):559-623. 1958. Maximum s u s t a i n e d y i e l d s from f l u c t u a t i n g environments and mixed s t o c k s . J . F i s h . Res. Bd. Canada, 15 (5) : 991-1006. 1962. R e g u l a t i o n of abundance of pink salmon p o p u l a t i o n s . Symposium on Pink Salmon, U n i v e r s i t y Of B r i t i s h Columbia. 1960. E d i t e d By N. J . Wilimovsky. Pp 155-201. MS. Comments on the West A t l a n t i c harp s e a l herd and p r o p o s a l s f o r the 1972 harvest 28pp, R u s s e l l , E. S. 1931. Some 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 on the o v e r f i s h i n g problem. J , Cons. Int. E x p l o r . Mer., 6:3-27. Schaaf, W. E., and G. R. Huntsman, 1972. E f f e c t of f i s h i n g on the A t l a n t i c menhaden stock. Trans. Amer. F i s h . S o c , 101 (2):290-297. 105 Schaefer, M. B. 1954. Some aspects of the dynamics of pop u l a t i o n s important to the management of commercial marine f i s h e r i e s . B u l l . Inter-Amer. Trop. Tuna Comm., 1 (2):26-46. 1957. A study of the dynamics of y e l l o w f i n tuna i n the e a s t e r n t r o p i c a l P a c i f i c Ocean. B u l l . Inter-American T r o p i c a l Tuna Comm., 2 (6):245-285. 1967a. Dynamics of the f i s h e r y f o r the anchoveta g n g r a u l i s r i n q e n s , of Peru. I n s t i t u t e Del Mar Del Peru, 1 (5) : 192-304. 1967b. F i s h e r y dynamics and the present s t a t u s of the y e l l o w f i n tuna p o p u l a t i o n of the eastern P a c i f i c Ocean. B u l l . Inter-American T r o p i c a l Tuna Comm., 12 (3) : 89- 136. Schaefer, M. B., G. C. Broadhead, and C. J . Orange. 1963. Synopsis on the b i o l o g y of y e l l o w f i n tuna. Thunnus (Neothunnus) a l b a c a r e s (Bonnaterre) 1788 ( P a c i f i c Ocean). Proceedings of the World S c i e n t i f i c Meeting on the b i o l o g y of tunas and r e l a t e d s p e c i e s . 2-14 J u l y 1962, F.A.O. F i s h e r i e s Reports, 6 (2) 1963 :538-561. Schaefer, M. B., and E. M. Chatwin, and G. C. Broadhead. 1961. Tagging and recovery of t r o p i c a l tunas, 1955-1959. B u l l . Inter-American T r o p i c a l Tuna Comm., 5 (5): 341-454. Scudo, F. M. 1971. V i t o V o l t e r r a And T h e o r e t i c a l Ecology. T h e o r e t i c a l P o p u l a t i o n B i o l o g y , 2:1-23. Sergeant D. E. 1963. Harp s e a l s and the s e a l i n g i n d u s t r y . Canadian Audubon, 25 (2):29-35 1966. On the populat i o n dynamics o f western harp s t o c k s . ICNAF, S e r i a l No. 1749. 14pp. 1971. C a l c u l a t i o n of production of harp s e a l s i n the Western North A t l a n t i c . ICNAF, S e r i a l No. 2476 23pp. S i e g e l , S. 1956, Nonparametric s t a t i s t i c s f o r the b e h a v i o r a l s c i e n c e s . , McGraw-Hill Book Company, Inc. New York. 312pp. Southward, G. M. 1968. A s i m u l a t i o n of management s t r a t e g i e s i n the P a c i f i c h a l i b u t f i s h e r y . Rep. I n t . Pac. H a l i b . Comm., 47: 5-70. Suzuki, Z. 1971 comparison of growth parameter e s t i m a t e s f o r the y e l l o w f i n tuna i n the P a c i f i c Ocean. B u l l . Far Seas F i s h . Res. l a b . , 5 :89- 105. ( E n g l i s h synopsis.) Tautz, A., P. A. L a r k i n , and W. E. R i c k e r . 1969. Some e f f e c t s of simulated long term environmental f l u c t u a t i o n s on maximum su s t a i n e d y i e l d . J . F i s h . Bd. Canada, 26:2715-2 726. Thompson, W. F., and F. fl. B e l l . 1934. B i o l o g i c a l s t a t i s t i c s of 106 the P a c i f i c h a l i b u t f i s h e r y . (2) E f f e c t o f changes i n i n t e n s i t y upon t o t a l y i e l d and y i e l d per u n i t of gear. I n t e r n a t i o n a l F i s h e r i e s Commission Report, 8:1-49. U l l t a n g , 0. MS. Estimates of m o r t a l i t y and p r o d u c t i o n of harp s e a l s at Newfoundland. 13pp. V e r h u l s t , P. F. 1838. N o t i c e sur l a l o i que l a p o p u l a t i o n s u i t dans son accroissement. Correspondence mathematic e t physigue, 10:113-121. Walters, C. J . 1969. A g e n e r a l i z e d computer s i m u l a t i o n model f o r f i s h p o p u l a t i o n s t u d i e s . Trans. Amer. F i s h . S o c , 98 (3) :505-512. Ward, F. J . , and P. A. L a r k i n . 1964. C y c l i c dominance i n the Adams R i v e r sockeye salmon. Int. Pac. Salmon F i s h Comm. Progr. Rep. 11, 116pp. 107 Appendix 1 Although the eguations (1) given on page 8 d e f i n e the model some e x t r a i n f o r m a t i o n d e s c r i b i n g t h e i r implementation i s given here to make i t p o s s i b l e to d u p l i c a t e the computations. In a d d i t i o n the parameters used f o r the p o p u l a t i o n s A and B used i n the t h i r d chapter are g i v e n . The b a s i c scheme of the model i s shown below f o r the case of an annual spawner. r "* ~ ~ ~-% I I n i t i a l p o p u l a t i o n i s | | input or c a l c u l a t e d | t 1 I V I r T | C a l c u l a t e number of | Yes | r e c r u i t s with o p t i o n | < -, | of m u l t i p l y i n g by a | | | random d e v i a t e | J L j , i I I r T I C a l c u l a t e y i e l d f o r the| No / i s i t 1 time p e r i o d and update | < ^ t h e end | the p o p u l a t i o n | a yeaty u 1 1 i L ^ J These computations were c a r r i e d out i n double p r e c i s i o n a r i t h m e t i c on the IBM /360 computer operated by the UBC computing c e n t r e . The psuedo-random d e v i a t e s were generated by a uniform random number generator that i s d e s c r i b e d i n the document " DBC RANDOM December 1970", which i s d i s t r i b u t e d by the computing c e n t r e . A copy of the program i s a v a i l a b l e from the author. 108 The p o p u l a t i o n s A and E These popu l a t i o n s were c o n s t r u c t e d t o have an e q u i l i b r i u m p o p u l a t i o n with 1 0 0 0 r e c r u i t s with no f i s h i n g m o r t a l i t y and a instantaneous annual n a t u r a l m o r t a l i t y of 0 . 2 . The s t o c k r e c r u i t r e l a t i o n f o r p o p u l a t i o n A i s n = c « 0»exp (-k «0 ) with c = 4 . 0 2 2 , and k = 0 . 0 0 0 2 9 5 . The f e r t i l i t y schedule was 1 , 2 , 4 , 8 , 1 2 , 1 4 , f o r ages 3 upwards. The stoc k r e c r u i t r e l a t i o n f o r p o p u l a t i o n B was S n = c»U«exp£-k» SLH (i) ] with c = 3 . 0 1 7 , and k = 0 . 0 0 1 0 0 4 . The f e r t i l i t y schedule was 1 0 and 1 2 f o r ages 4 and 5 r e s p e c t i v e l y . The random v a r i a t i o n i n the p o p u l a t i o n was simulated by m u l t i p l y i n g by a psuedo-random number uniformly d i s t r i b u t e d between 0 . 1 and 1 . 9 . Appendix 2 The annual y i e l d from the model can be w r i t t e n as Y(t+r) = k* F p ( j ) • I ( 1 , t + j ) where Y<t) and N ( 1,t) are the y i e l d and r e c r u i t m e n t i n year t, p{j) i s the p r o p o r t i o n of the y i e l d made up of f i s h of age j , and k i s a constant chosen so ^,p(j} = 1 . 0 109 I f N(1,t) i s a sequence of independent random v a r i a b l e s with v a r i a n c e s 2 then the v a r i a n c e of ¥ (t) i s S 2 = k 2 . 2 P ( J ) 2 « S 2 and i f N i s the expected value of N(1,t) the square of the c o e f f i c i e n t of v a r i a t i o n of Y(t) i s s 2 / k 2 « N 2 = s 2» £ p (j) 2/N 2 < s 2 / N 2 / ft N thus i f the recruitment i s a sequence of independent random v a r i a b l e s the c o e f f i c i e n t of v a r i a t i o n of y i e l d i s l e s s than the c o e f f i c i e n t of v a r i a t i o n of the recruitment. When a non t r i v i a l stock r e c r u i t r e l a t i o n i s used i n the model the r e c r u i t m e n t i n s u c c e s s i v e years i s no l o n g e r independent and so the e x p r e s s i o n above i s only approximate. Appendix 3 Treatment of the harp s e a l harvest s t a t i s t i c s . The h a r v e s t s t a t i s t i c s f o r harp s e a l s f o r the years 1938-1970 were taken from the ICNAF s t a t i s t i c a l b u l l e t i n s f o r 1969 and 1970, and estimates of the 1971 harvest were given to me by Dr, H. D. F i s h e r (personal communication). The s t a t i s t i c s are 110 d i v i d e d i n t o s e v e r a l c a t e g o r i e s which can be combined t o produce the f o l l o w i n g s t a t i s t i c s . (a) Pup k i l l . (b) S e a l s one year and o l d e r . Denmark, Norway and the O.S.S.H. (c) Bedlamers Canada. (d) Adult s e a l s Canada. (e) T o t a l k i l l . I have assumed that the group (fc) was made up of 20% bedlamers and 80% a d u l t s i n the years 1938 to 1966, and 50% bedlamers and 50% a d u l t s i n the years 1967 to 1971. T h i s change i n the age composition i s the r e s u l t of changes i n hunting p r a c t i c e (Dr. W. E. R i c k e r personal communication). D i f f e r e n c e s between the t o t a l k i l l and the sum of the i n d i v i d u a l f i g u r e s f o r pups, bedlamers, and a d u l t s were made up by a d j u s t i n g the f i g u r e s f o r bedlamers and a d u l t s a c c o r d i n g to the formulae below 0» = 0» (T-P)/<B+0) B» = B* (T-P)/(B+0) where T i s the t o t a l k i l l , B i s the k i l l of bedlamers, P i s the pup k i l l , and primes denote adjusted f i g u r e s . Estimates of females i n each category were then made using the f o l l o w i n g p r o p o r t i o n s of females. Pups 50% 111 Bedlamers 50% a d u l t s 40% 1938-1966 15% 1967-1969 34% 1970-1971 The reason f o r the changing sex r a t i o s i n the h a r v e s t i s v a r i a t i o n i n the c l o s i n g date of the hunt, which was i n t r o d u c e d i n 1967. as mentioned above the e f f e c t of an e a r l y c l o s i n g date i s to reduce the number of females i n the a d u l t h a r v e s t . The estimates of the p r o p o r t i o n of females i n the k i l l i s taken from O r i t s l a n d (1971). 

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