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Search and decision in fishing systems Shotton, Ross 1973

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SEARCH AND DECISION IN FISHING SYSTEMS by Ross Shotton B . S c , U n i v e r s i t y of Wales, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department o f Zoology We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA December, 1973 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb ia , I a g ree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s „ r e p r e s e n t a t i v e s . It i s u n d e r s t o o d tha t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i thou t my w r i t t e n p e r m i s s i o n . The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada Department i i ABSTRACT Methods o f search i n marine f i s h e r i e s are reviewed, and aspects o f v a r i a b l e s necessary to ev a l u a t e e x p l o i t a t i o n s t r a t e g i e s are c o n s i d e r e d . Estimates o f f i s h school d e n s i t i e s based on gas d i f f u s i o n p r i n c i p l e s are made. P o s s i b l e school p a t t e r n s are examined using the P o i s s o n , Poisson with added z e r o e s , Neyman.type A and Negative Binomial d i s t r i b u t i o n s . The Negative Binomial d i s t r i b u t i o n was found t o g i v e best f i t to the data used although the Neyman type.A d i s t r i b u t i o n gave a b e t t e r estimate of the data v a r i a n c e . School s i z e was reasonably w e l l d e s c r i b e d by a l o g d i s t r i b u t i o n , Mean square d i s p e r s i o n r a t e s , modal v e l o c i t i e s and d i r e c t i o n of movement f o r s e t s of o b s e r v a t i o n s on tuna schools are found. T e s t s on the s i g n i f i c a n c e o f d i r e c t i o n o f movement by groups of tuna school are done and those groups showing s i g n i f i c a n c e were t e s t e d f o r homogeneity of d i r e c -t i o n . Confidence i n t e r v a l s on the movement o f d i r e c t i o n are a l s o c a l c u -l a t e d . D e c i s i o n methods so as to i n c r e a s e c a t c h are c o n s i d e r e d f o r three s i t u a t i o n s : (1) Where the p o s i t i o n of a school i s known wit h some e r r o r , and i t must be decided whether to attempt t o l o c a t e i t or remain s e a r c h i n g on the present grounds. (2) When the Bayes estimate of cat c h r a t e on the p r e s e n t grounds i s l e s s than t h a t expected on an a l t e r n a t e ground, and the d e c i s i o n o f changing depends on mini m i z i n g an expected l o s s f u n c t i o n . (3) For the Vancouver trawl f i s h e r y where f i s h occurrence may be cons i d e r e d of a pres e n t o r absent n a t u r e , and i t i s d e s i r e d to minimize the i i i time spent steaming between grounds. The use o f entropy as a c r i t e r i o n o f search e f f e c t i v e n e s s as used by other workers i s a l s o d i s c u s s e d . TABLE OF CONTENTS Page TITLE PAGE 1 ABSTRACT 1 1 TABLE OF CONTENTS i v LIST OF FIGURES v LIST OF GRAPHS v i LIST OF TABLES v 1 i ACKNOWLEDGMENTS. v i 1 1 1.0 INTRODUCTION 1 2.0 SEARCH STRATEGIES: A GENERAL REVIEW 2 3.0 SYSTEMS VARIABLES 1 6 3.1 S i z e o f Search Area ] ^ 3.2 F i s h Density " 4.0 SCHOOL PATTERN AND SIZE 2 4 24 4.1 P a t t e r n o f School Occurrence ^ 4.2 U n c e r t a i n t y as a F u n c t i o n o f D i s t r i b u t i o n Type ^5 4.3 School S i z e and U n c e r t a i n t y . 3 6 5.0 DECAY OF INFORMATION 3 7 6.0 DECISION MAKING 5 8 6.1 Royce and Otsu (1955) Tuna Data 59 6.2 C r a i g and Graham (1965) Tuna Data j£ 6.3 Vancouver Trawl F i s h e r y B i 7.0 DISCUSSION 8 8 7.1 System V a r i a b l e s 8 8 7.2 P a t t e r n of School D i s t r i b u t i o n ^ 7.3 Choice o f Sample S i z e jjjj 7.4 Variance Estimates ^ 7.5 Entropy and f i s h d i s t r i b u t i o n ^ 7.6 Decay o f i n f o r m a t i o n 'J?' 7.7 D e c i s i o n m a k i n g 1 LITERATURE CITED. 106 V LIST OF FIGURES FIGURE Page 1 Search p a t t e r n s given by Yodovich and Boral (1968) 5 2 Optimal t r a c k f o r square area of uniform p r o b a b i l i t y o f f i s h d e t e c t i o n with one d i s c o n t i n u i t y "1? 3 Nature o f optimal t r a c k s as curve l e n g t h L i s i n c r e a s e d . . . ^ 4 Slope of maximum entropy removing curve where p r o b a b i l i t y o f occurrence i s some f u n c t i o n of d i s t a n c e from a p o i n t . . . ^ 5 Nature o f area i n which school would be expected i f d i r e c t i o n a l v e l o c i t y and modal d i r e c t i o n with c o n f i d e n c e l i m i t s known ^ vi LIST OF GRAPHS GRAPH Page I Range l imi t of school occurrence at specified probability 52 II Probability of direction as a function of angular difference to modal direction 55 III Area of possible school occurrence as a function of time 62 IV Strategy as whether to change grounds or remain on present grounds 65 V Relative loss when changing grounds and when remaining 88 v i i LIST OF TABLES TABLE Page I Means, variances, goodness of f i t and entropy for specified distributions 29 II Ranked order of f i t of distributions for different sample lengths 31 III Ratios of variance to mean, and variance to distr ibution variance, for specified distributions and sample lengths . . . . 3 3 IV Summary of vector diagrams of Royce and Otsu (1955),. 4 3 V Data for test of K^ O 4 5 VI Data for analysis of Royce and Otsu (1955) vector observations of tuna school movement ^ VII Expected density values for sample data ^ VIII Steaming time from Vancouver Port to and between grounds,.. . IX Possible fishing strategies for Vancouver trawl f i s h e r y . . . . . 8 6 \ v i i i ACKNOWLEDGMENTS I would l ike to thank my supervisor, Dr Walters for reviewing my thesis and offering suggestions. Also Dr Larkin and Dr Wilimovsky for comments and Mr Julian Reynolds for reviewing the text. I would also l ike to acknowledge Dr Wilimovsky for suggestion of the topic. Special thanks go to Mr Neil Gi lbert , Institute of Animal Resource Ecology, for discussions of s ta t i s t i ca l points and Dr Zidek, Department of Mathematics for discussions regarding Bayes methods. Also thanks to Miss Doreen Housteen for invaluable assistance in typing the thesis. 1.0 INTRODUCTION E f f e c t i v e d e c i s i o n making i n a system such as a f i s h e r y r e q u i r e s the a b i l i t y to determine the s t a t e of the system f o r the p a r t i c u l a r area of i n t e r e s t , o r to s p e c i f y the expected s t a t e of the system given e i t h e r some sampling d a t a , or p r i o r e x p e c t a t i o n s based on past e x p e r i e n c e . Using such knowledge or e x p e c t a t i o n s , a f i s h i n g o p e r a t o r , u s u a l l y the s k i p p e r , w i l l have to make d e c i s i o n s i n s i t u a t i o n s where ca t c h r e s u l t s are a t v a r i a n c e w i t h those expected; such as e i t h e r to change grounds i n the e x p e c t a t i o n of encountering b e t t e r c a t c h r a t e s , or to continue f i s h i n g on the same grounds i n the a n t i c i -p a t i o n of i n c r e a s e d catch r a t e s . A b i l i t y to make p r i o r p r e d i c t i o n s o f ex-pected c a t c h r a t e s i s d i r e c t l y r e l a t e d to one of the c r u c i a l aspects of f i s h -e r i e s management, t h a t of s t o c k assessment. T r a d i t i o n a l l y estimates of s t o c k s i z e have been made by " i n d i r e c t methods, such as from c a t c h per u n i t e f f o r t d a t a , which o f f e r o n l y l i m i t e d i n f o r m a t i o n as to d i s t r i b u t i o n . R e s u l t s from such data may only become a v a i l a b l e s e v e r a l y e a r s a f t e r the time they supposedly r e f l e c t . However, more d i r e c t methods, such as sonar and echo sounding, can p r o v i d e r e a l time data on a r e a l f i s h d i s t r i b u t i o n as w e l l as abundance. In t h i s t h e s i s I inte n d to c o n s i d e r methods of s p e c i f y i n g f i s h i n g systems i n terms of the d i s t r i b u t i o n and abundance o f the f i s h s t o c k s . Q u a n t i t a t i v e methods w i l l be c o n s i d e r e d f o r making d e c i s i o n s about expected c a t c h r a t e s , g i v e n some p r i o r e x p e c t a t i o n s which may be augmented by c a t c h data, As a r e s u l t of such d e c i s i o n methods, and hence e x p e c t a t i o n s o f f u t u r e c a t c h r a t e s ( i . e . f i s h abundance) methods w i l l be e v a l u a t e d so a s k i p p e r c o u l d make more r a t i o n a l d e c i s i o n s as to the a d v i s a b i l i t y of s h i f t i n g grounds, or a f i s h e r i e s a d m i n i s t r a t o r could make r e a l time judgments about c u r r e n t stock -2-abundance, and hence have some j u s t i f i a b l e b a s i s f o r immediate d e c i s i o n s as regards p o s s i b l e e f f o r t l i m i t a t i o n s o r displacement. 2.0 Search S t r a t e g i e s : A General Review To s p e c i f y the s t a t u s ( d i s t r i b u t i o n and abundance) o f f i s h stocks r e q u i r e s some form of search or sampling s t r a t e g y . The purpose o f " e x p l o r -a t o r y f i s h i n g " may d i f f e r depending on the aims o f the agency i n v o l v e d . A l v e r s o n (1971) d e f i n e s e x p l o r a t o r y f i s h i n g as "a planned study by f i s h i n g gear o f the f i s h r esources i n h a b i t i n g a d e f i n e d area o f the ocean". (For my purposes, I i n c l u d e sonar search as f i s h i n g g e a r ) . In reviewing such i n v e s t i g a t i o n s A l v e r s o n notes t h a t to some ex t e n t s c i e n t i s t - a d m i n i s t r a t o r s may be c h i e f l y i n t e r e s t e d i n o b t a i n i n g a general d e s c r i p t i o n o f a resource and i t s p o t e n t i a l on a long term b a s i s , whereas i n d u s t r i a l o p e r a t o r s w i l l be more concerned with immediate i n f o r m a t i o n on the d i s t r i b u t i o n a l and po-t e n t i a l catch aspects o f the r e s o u r c e . I t i s the l a t t e r type o f i n f o r m a t i o n which forms an i n d i s p e n s a b l e p a r t o f preinvestment s t u d i e s f o r f i s h e r i e s development, e i t h e r alone or combined with other methods of resource e v a l u -a t i o n ( S a e t e r s d a l , 1973). T h i s author a l s o notes t h a t e x p l o r a t o r y f i s h i n g o f t e n l a c k s any s y s t e m a t i c p l a n n i n g and as such may g i v e r i s e to incomplete and m i s l e a d i n g c o n c l u s i o n s . The concept of search i n f i s h e r i e s has v a r i a b l e i n t e r p r e t a t i o n . In a wider sense 'search' may i n c l u d e l o c a t i o n o f areas where mar-in e c o n d i t i o n s such as s a l i n i t y , temperature o r plankton are conducive t o the presence of one or more e x p l o i t a b l e s p e c i e s . In a narrower sense 'search' may be c o n s i d e r e d as the l o c a t i o n o f s c h o o l s o r d e n s i t i e s o f f i s h above some commercial t h r e s h o l d , i n an area where oceanographic c o n d i t i o n s or previous experience i n d i c a t e s t h a t they may be expected to occur. Here o n l y the -3-second, more r e s t r i c t i v e sense w i l l be c o n s i d e r e d . Japanese agencies have a system performing the f i r s t type o f prognosis but the scheme appears t o be o f l i t t l e s h o r t - t e r m use (Burbank and Douglas, 1969). Because f i s h d e n s i t y throughout a f i s h i n g area i s g e n e r a l l y non-uniform, both s p a t i a l l y and t e m p o r a l l y , v e s s e l s wishing to e x p l o i t concen-t r a t i o n s o f f i s h may f i r s t have to expend e f f o r t i n l o c a t i n g f i s h concen-t r a t i o n s . In demersal f i s h e r i e s a given s p e c i e s i s o f t e n p r e s e n t through-out i t s range a t any one time, although the geographical l o c a t i o n o f g r e a t -e s t f i s h d e n s i t y g e n e r a l l y shows a seasonal p a t t e r n . For p e l a g i c s p e c i e s which form d i s c r e t e s c h o o l s , the f i s h are o f t e n present i n only one p a r t o f t h e i r range a t any one time, and t h e i r presence i n a p a r t i c u l a r area may be c a p r i c i o u s t o some ex t e n t . In most c u r r e n t f i s h i n g o p e r a t i o n s , the s t r a t e g y i n v o l v e d i n l o -c a t i n g the general areas o f f i s h d e n s i t y has t r a d i t i o n a l l y been based on the accumulated experience o f the v e s s e l ' s s k i p p e r , i . e . h i s knowledge o f catches i n previous y e a r s a t d i f f e r e n t l o c a t i o n s , t o g e t h e r with c u r r e n t i n f o r m a t i o n on other v e s s e l s ' catches and l o c a t i o n s . C o - o r d i n a t i o n i n s e a r c h i n g by v e s s e l s pursuing the same f i s h v a r i e s w i t h f i s h e r i e s and f l e e t s . Operations may be c o m p e t i t i v e between a l l v e s s e l s , as i n the C h i l e a n anchoveta f i s h e r y (Sams, 1970) or densely concen-t r a t e d h e r r i n g f i s h e r i e s . There may e x i s t l i m i t e d c o - o p e r a t i o n between some v e s s e l s , e.g. those from the same p o r t or owned by the same company may i n d i -c a t e to each o t h e r the s t a t e o f f i s h i n g c o n d i t i o n s o r the area t h a t they are a t (Anon, 1972a). In some p e l a g i c f i s h e r i e s a e r i a l r econnaissance i s used t o l o c a t e c o n c e n t r a t i o n s o f f i s h (Green, P e r r i n and P e l r i c h , 1970; Sams, 1970; -4-Yodovich and B a r a l , 1968) and p r e s e n t work i n d i c a t e s t h a t such methods have much p o t e n t i a l f o r c e r t a i n f i s h e r i e s ( B u i l i s and Bendigo, 1970). However, most such a e r i a l techniques become un o p e r a t i o n a l i n inclement weather con-d i t i o n s . Search plans and c o - o p e r a t i o n between v e s s e l s i n demersal f i s h e r -i e s e x i s t i n Japanese and S o v i e t 'mothership' o p e r a t i o n s . F i s h i n g con-d i t i o n s may be evaluated by t a k i n g an e x p l o r a t o r y tow, although the general search area w i l l depend on past e x p e r i e n c e . However, the development o f echo sounders and sonar has allowed more r a p i d f o r e c a s t s , though i t i s not uncommon f o r echo t r a c e s to show no r e l a t i o n to subsequent catches (Dreaver, 1970). Hodder (unpublished) notes t h a t f o r the West A t l a n t i c haddock (Melanogrammus a e g l e f i n u s ) f i s h e r y the gear i s only shot i f echo sounders i n d i c a t e f i s h presence. In mid-water t r a w l i n g , l o c a t i o n o f f i s h s c h o o l s i s e s s e n t i a l to the success o f a haul (Minne, 1970; S t e i n b e r g , 1970; Kodera, 1970). The o p e r a t i n g c h a r a c t e r i s t i c s and economics o f the f i s h d e t e c t i n g devices t h a t o f f e r the most p o t e n t i a l preclude t h e i r use i n a l l commercial v e s s e l s . Such a p p a r a t u s - v a r i a b l e depth s e c t o r scanning sonar - has best performance when towed, and would n e c e s s a r i l y h i n d e r f i s h i n g o p e r a t i o n s . This sonar has been shown to be o p e r a t i o n a l a t speeds up to 20 knots which would enable l a r g e areas to be searched i n u n i t time compared wi t h t h a t p o s s i b l e by c o n v e n t i o n a l f i s h i n g v e s s e l s (Johnson and P r o c t o r , 1970). In d i s c u s s i n g the S o v i e t e x p e r i e n c e , Yudovich and Baral (1968) note t h a t t r a c k widths may be spaced from 25-30 miles to 80-100 mi l e s depend-i n g on c o n d i t i o n s . F i g u r e 1 shows the schemes o f s e a r c h t r a c k s g i v e n by Yudovich and B a r a l . No j u s t i f i c a t i o n f o r the design o f these t r a c k s i s -5-Figure 1 Search patterns given by Yodovich and Baral (1968), Cross hatched areas represent f ish schools. (a) Tracks for seeking lost shoal or one where co-ordinates have been communicated by the exploratory team. (b) Arrow gives direction of seasonal migration course. (c) Choice for (c) not explained. _6_ g i v e n . In d e t a i l e d surveys the S o v i e t method i s to choose an i n t e r v a l , a, between t r a c k s 'with a view to uncovering e i t h e r most or a l l o f the commer-c i a l accumulations o c c u r r i n g i n the r e g i o n o f i n v e s t i g a t i o n ' , such t h a t : a = 2D + r where D = width of the area uncovered r = minimum s i z e o f a commercial c o n c e n t r a t i o n . The maximum i n t e r v a l between the t r a c k s corresponds to the maximum s i z e o f an accumulation; the minimum i n t e r v a l a p p l i e s to a l l l o c a l s e a r c h , d u r i n g which a boat equipped w i t h an echo sounder seeks out a commercial s c h o o l ; f o r which r = 20 - 30 meters. For d e t a i l e d search conducted on p e l a g i c f i s h the t r a c k s are g e n e r a l l y s e t orthogonal to the d i r e c t i o n o f the isotherms and t r a v e r s e the c u r r e n t stream. S i n c e accumulations of demersal f i s h o f t e n extend along i s o b a t h s , search t r a c k s s h o u l d c r o s s them o b l i q u e l y . When t r a w l i n g i s c a r r i e d out on s l o p e s where the bottom may i n c l i n e a t 10-15°, the t r a w l i n g course i s s e t a t an angle o f 30-35° to the i s o b a t h s . No j u s t i f i c a t i o n i s given by Yudovich and Baral (1968) f o r these values which appear to be based on o p e r a t i o n a l exper-i e n c e . Burbank and Douglas (1969) i n an e x t e n s i v e r e p o r t on the Japanese F i s h e r i e s F o r e c a s t i n g System note t h a t t h e i r system i s i n essence an e x t r a -p o l a t i o n o f very r e c e n t c a t c h s t a t i s t i c s , i n f l u e n c e d by r e p o r t e d oceano-gr a p h i c measurements. Environmental i n f o r m a t i o n c o n s i d e r e d by the Japanese to be o f primary importance are c u r r e n t boundary l o c a t i o n s and movements, u p w e l l i n g s , s u r f a c e temperature and thermal s t r u c t u r e , and presence of zoo-plankton and micronekton. World-wide l o n g l i n e f o r e c a s t s are i s s u e d s e v e r a l -7-months i n advance on a wide g r i d s c a l e (10 J and a r e o f most use to opera-t o r s i n d e c i d i n g the general area to d e p l o y t h e i r s hips and c o n c e n t r a t e t h e i r e f f o r t s , and i s of l i t t l e h elp i n the f i n a l search and capture o p e r a t i o n s . C o a s t a l / o f f s h o r e f o r e c a s t s are i s s u e d e v e r y 8 days and g i v e environmental d a t a , r e c e n t catch data and f o r e c a s t e d high c a t c h p r o b a b i l i t y a r e a s . C o r r e -l a t i o n s o f catch with s p e c i f i c phenomena (such as gyre b r e a k - o f f i n the Kuroshio-Oyashio f r o n t r e g i o n ) have been developed but a g e n e r a l i z e d s t a t i s -t i c a l approach g i v i n g c a t c h p r o b a b i l i t y as a f u n c t i o n o f environmental para-meters f o r any given area has a p p a r e n t l y not been developed (Burbank and Douglas, 1968). North American workers have r e c o g n i z e d the n e c e s s i t y to plan search s t r a t e g i e s u sing s p e c i a l i z e d s e a r c h c r a f t (Johnson and P r o c t o r , 1970; N i c k e r s o n , 1970) and such methods a r e now o p e r a t i o n a l (Johanesson and Losse, 1973; Nickerson and Dowd, 1973). F i s h e r i e s s c i e n t i s t s have a p p r e c i a t e d t h a t a 'systems a n a l y s i s ' approach would pr o v i d e a u s e f u l method o f e v a l u a t i n g methods o f s p e c i f y i n g the s t a t e o f a f i s h i n g system ( S a i l a and Flowers, 1967). S a i l a (1969) s t r e s s e s t h a t a h o l i s t i c approach to f i s h e r i e s i s necessary and i d e n t i f i e s the system components he c o n s i d e r s important, though no a n a l y s i s i s attemp-t e d . T e p l i t s k i and S h e i n i s (1970) d e s c r i b e a S o v i e t a n a l y s i s which con-s i d e r s aspects i d e n t i f i e d by S a i l a , i . e . economic as w e l l as y i e l d f a c t o r s . These authors developed a model which e v a l u a t e s t h r e s h o l d catch values a t which redeployment of the f l e e t s s h o u l d o c c u r , given c e r t a i n i n f o r m a t i o n about c a t c h r a t e s and economic f a c t o r s , but do not s p e c i f y how the c a t c h i n g v e s s e l s should be redeployed. The c r u x o f t h e i r problem i s t h a t t h e r e i s no guarantee catch r a t e s w i l l be b e t t e r elsewhere. -8-The problem of search s t r a t e g i e s i n f i s h e r i e s o p e r a t i o n s can be c o n s i d e r e d as a subset of the general problem o f search theory. Koopman (1956a, 1956b, and 1957) p u b l i s h e d the f i r s t treatment o f search theory using o p e r a t i o n s r e s e a r c h methods. In the f i r s t paper (1956a) Koopman d i s c u s s e s the p o s i t i o n s , geometrical c o n f i g u r a t i o n s and motions o f the searchers and t a r g e t s with p a r t i c u l a r r e f e r e n c e to the s t a t i s t i c s o f t h e i r c o n t a c t s and the p r o b a b i l i t i e s o f t h e i r r e a c h i n g v a r i o u s s p e c i f i e d p o s i t i o n s . In the second paper (1956b) he c o n s i d e r s the p r o b a b i l i s t i c behaviour o f the instrument o f d e t e c t i o n (eye, r a d a r , sonar) when making a given passage r e l a t i v e to the t a r g e t . His t h i r d paper (1957) deals with the o v e r a l l r e s u l t , the p r o b a b i l i t y o f c o n t a c t under general s t a t e d c o n d i t i o n s , a l o n g with the p o s s i b i l i t y of o p t i m i z i n g the r e s u l t s by improving the methods o f d i r e c t i n g the s e a r c h . F o l l o w i n g Koopman's p i o n e e r i n g s t u d i e s , f u r t h e r work has developed i n many d i r e c t i o n s . A search a l g o r i t h m using o p e r a t i o n s a n a l y s i s methods was developed f o r unique s i t u a t i o n s i n the search f o r the n u c l e a r submarine S c o r p i o n (Richardson and Stone, 1971). They use a p r i o r i s c e n a r i o s to e v a l u a t e l i k e l i h o o d s o f occurrence f o r d i f f e r e n t search a r e a s . Using an approach more r e l a t e d to those d e s c r i b i n g e c o l o g i c a l pro-c e s s e s , Paloheimo, i n one model (1971a) s t u d i e s the success o f search f o r randomly or c o n t a g i o u s l y l o c a t e d prey. The model i s s p e c i f i e d by g i v i n g the d i s t r i b u t i o n of the s e a r c h i n g time r e q u i r e d to l o c a t e a prey or prey c l u s t e r s and the j o i n t d i s t r i b u t i o n o f h a n d l i n g time and numbers caught from each c l u s t e r s i g h t e d , a s i m i l a r approach to t h a t o f Neymans (1949). A r e c u r r e n c e d i s t r i b u t i o n f u n c t i o n f o r the d i s t a n c e t r a v e l l e d between s u c c e s s i v e c o n t a c t s i s deduced f o r s i t u a t i o n s with Poisson or c l u s t e r e d Poisson -9-d i s t r i b u t i o n . Paloheimo's second (1971b) paper c o n s i d e r s the e f f e c t of d e n s i t y and d i s t r i b u t i o n of prey on t o t a l accumulated c a t c h by the p r e d a t o r , where prey are randomly l o c a t e d on a plane with given d e n s i t y , or c l u s t e r e d with v a r i a b l e c l u s t e r s i z e . A s i m i l a r s e a r c h i n g time d i s t r i b u t i o n i s used together with f u n c t i o n s r e p r e s e n t i n g p r o b a b i l i t y o f d e t e c t i o n and/or c a p t u r e . These concepts may w e l l have a p p l i c a t i o n t o d e s c r i b i n g o p e r a t i o n s of i n d i -v i d u a l c a t c h e r b o a t s , once the d e n s i t y or expected c a t c h r a t e s were known, but h i s methods are not d i r e c t l y concerned with determining the s t a t e of a p r e d a t i o n system. His model uses renewal theory t o d e s c r i b e d e l a y b e f o r e the predator ( o r f i s h i n g boat) can continue s e a r c h . A f i s h i n g system can be p o s t u l a t e d i n terms o f a renewal system, i . e . i f a school has been caught i n some a r e a , the appearance of another school w i l l be s p e c i f i e d by some renewal f u n c t i o n . Pel l a (1969) a l s o uses renewal processes to con-s i d e r l e n g t h o f time between prey c a p t u r e s . His model a l s o i m p l i e s random d i s t r i b u t i o n o f prey. For s i t u a t i o n s which can be s p e c i f i e d by some p r o b a b i l i t y d i s t r i -b u t i o n , i n f o r m a t i o n theory p r o v i d e s a unique c r i t e r i o n f o r i n d i c a t i n g the amount o f u n c e r t a i n t y r e p r e s e n t e d by the p r o b a b i l i t y d i s t r i b u t i o n , r e f l e c t -ing the e x p e c t a t i o n t h a t a more f l a t t e n e d d i s t r i b u t i o n r e p r e s e n t s g r e a t e r u n c e r t a i n t y than a s h a r p l y peaked one ( K h i n t c h i n , 1957). The measure o f u n c e r t a i n t y , o r entropy, i s n H(x) = - I p ( x . ) l o q p(x.O j=l J J (Shannon, 1948) where p(x-) i s the p r o b a b i l i t y of some o b s e r v a t i o n X = x. J J K h i n t c h i n shows t h a t the maximum value o f u n c e r t a i n t y i s given when the p r o b a b i l i t y f u n c t i o n i s uniform, i . e . n - 1 p. = — -10-Th e measure o f entropy has been used i n d i v e r s e s i t u a t i o n s i n v o l v -i n g a se a r c h , e.g. by Danskin (1960) i n c o n s i d e r i n g the optimal d i s t r i b u t i o n o f a e r i a l reconnaissance e f f o r t a g a i n s t l a n d t a r g e t s . S h e r s t n i k o v (1968) uses entropy as a method o f d e v i s i n g search t r a c k s to l o c a t e f i s h , and i s concerned with those t r a c k s which remove most u n c e r t a i n t y from a search system. The assumptions t h a t he makes: (1) square search area (2) uniform p r o b a b i l i t y weaken the p o s s i b i l i t y o f a p p l i c a t i o n t o r e a l s i t u a t i o n s . No use i s made o f p r i o r i n f o r m a t i o n , such as p o s s i b l e p a t t e r n o f school d i s t r i b u t i o n , though the treatment i s extended to the s i t u a t i o n where the l o c a t i o n o f a f i s h c o n c e n t r a t i o n i s known but with some e r r o r . S h e r s t n i k o v d e f i n e s the un-c e r t a i n t y along some search curve L, as the ' c u r v i l i n e a r entropy'; H(L, X p ... x n ) = - J^p(x 1 ... x n ) l o g p(x 1'...x n) dL The v a r i a b l e s x-j...x n are the search r e s u l t s , t h a t i s the number o f f i s h s c h o o l s , along the c u r v e , and the values of x^ w i l l be determined by the curve. For d i f f e r e n t curves d i f f e r e n t amounts o f c u r v i l i n e a r entropy w i l l be o b t a i n e d . The amount of i n f o r m a t i o n o b t a i n e d , I, w i l l equal the uncer-t a i n t y removed; I = H, - H 2 where H-j = p r i o r entropy H2 = p o s t e r i o r entropy Of a l l p o s s i b l e curves o f given l e n g t h , t h a t one which maximizes i s r e q u i r e d . L i k e Koopman (1956b), S h e r s t n i k o v (1968) c o n s i d e r s t a r g e t d e t e c t i o n to be s t o c h a s t i c , and both use s i m i l a r f u n c t i o n s . S h e r s t n i k o v g i v e s -11-p r o b a b i l i t y of d e t e c t i o n , P^, as P d = 1 - e ^ ' ^ where <(.(x,y) = search d e n s i t y a t (x,y) Koopman uses P d = l - e ^ where y = instantaneous p r o b a b i l i t y of d e t e c t i o n . The p r o b a b i l i t y of f i n d i n g f i s h i n the r e g i o n S, w i l l be Pc = / p ( x , y ) ( l - e - < f ) ( x ' y ) ) dx dy 3 L where p(x,y) i s the p r o b a b i l i t y d e n s i t y of d e t e c t i o n ( S h e r s t n i k o v , 1968). The c u r v i l i n e a r entropy i s then d e f i n e d as H(L,x,y) = - /p(x,y) l o g p(x,y) dL (2.1) L S h e r s t n i k o v s t a t e s t h a t f o r curves y , such t h a t y = f ( x ) a o then I 1 + (y') dx = L (2.2) b The best s t r a t e g y w i l l be t h a t curve which minimizes 2.1 s u b j e c t to 2,2 which he s o l v e s using v a r i a t i o n a l c a l c u l u s such t h a t F = [X - p(x,y) l o g p ( x , y ) ] - 1 + ( y 1 ) 2 (2.3) where X i s the Lagrangian m u l t i p l i e r and F s a t i s f i e s E u l e r ' s e q u a t i o n ; 9F = _d_ 3F 9y " dx ay' u This r e s u l t s i n a complicated n o n l i n e a r d i f f e r e n t i a l equation f o r which s o l u t i o n i s very complex. S h e r s t n i k o v deduces the i n t u i t i v e l y obvious r e s u l t t h a t to minimize equation 2.3, y = f ( x ) must be orthogonal to p ( x , y ) , where p(x,y) = c; i . e . the curve should f o l l o w the d i r e c t i o n of s t e e p e s t ascent. The major weakness in equation 2 .3 , apart from i t s i n t r a c t i b i l i t y , is that p(x,y) must be specif ied. It would seem reasonable to make the assumption that <j>(x,y) is constant; then i f p(x,y) could be specif ied, a l l that need be done is to maximize the l ine integral across the f i e l d . How-ever, i f many local maxima occur the best result may only be obtained by iterative procedures. For a square search area, S, with constant probability of detect-ion of f ish P/S, Sherstnikov shows that a zig-zag pattern w i l l minimize H(L,X,Y) where: H a . x . » ) - / , ^ l o g ( ^ ) l t ( y . , z < I L 0 This agrees with the results of Yudovich and Baral (1968) and also those of Nickerson and Dowd (1973) who note that such a pattern minimizes the con-fidence interval on the mean f ish density estimate for a given length of survey. Figures 2 and 3 show optimal search curves for square areas of uniform probability of f i sh occurrence. Sherstnikov (1968) also considers the situation where the proba-b i l i t y density of occurrence is dependent on distance from some point. The probability of occurrence, at range r, direction <f>, is P(r,f) = 2is and the curvi l inear entropy to be maximized is H (L,r ,$) = / r 2 + r ' 2 d* = L <t>° subject to / rc + r'£ d«f, = L <J>° -13-Figure 2. Optimal t r a c k f o r square area of uniform p r o b a b i l i t y o f f i s h d e t e c t i o n with one d i s c o n t i n u i t y ( S h e r s t n i k o v , 1968) F i g u r e 3. Nature of optimal t r a c k s as curve l e n g t h L i s i n c r e a s e d ( S h e r s t n i k o v , 1968). then the optimal search curve i s given by c-|dr ; 2 , r , , r x 72 2~ C2 " • r r ^ W - x ) " c l where c-| the f i n a l p o s i t i o n of the search v e s s e l and x , the Lagrangian m u l t i -p l i e r can be determined from simultaneous equations given by S h e r s t n i k o v , and C 2 i s the i n i t i a l p o s i t i o n of the search v e s s e l . F i g u r e 4 i n d i c a t e s the nature o f these search paths. As r i n c r e a s e s the curves become compressed. S h e r s t n i k o v extends the treatment to where the p r o b a b i l i t y of d e t e c t i o n i s given by a normal d e n s i t y f u n c t i o n . The r e s u l t s are e s s e n t i a l l y s i m i l a r . Although the u n c e r t a i n t y o f d i f f e r e n t search s i t u a t i o n s may be e f f i c i e n t l y minimized using S h e r s t n i k o v ' s (1968) methods, a system ( i n our case, a f i s h i n g ground) which has the minimum entropy i s not n e c e s s a r i l y the b e s t f i s h i n g a r e a . The s t a t e o f the grounds may be known, but there may be no f i s h p r e s e n t . A more u s e f u l manner o f s p e c i f y i n g the p o t e n t i a l o f some f i s h i n g grounds may be obtained by using D e c i s i o n Theory such as those based on Bayesian methods. These a l l o w s u b j e c t i v e f e e l i n g s about the p o s s i b l e c o n d i t i o n o f a f i s h e r y to be c o n s i d e r e d , ( L i n d l e y , 1965a) and a l s o the p o s s i b l e l o s s e s t h a t may occur given a wrong c h o i c e (Mood and G r a y b i l l , 1965). Such methods a l s o appear to have much p o t e n t i a l as aides to manage-ment, by both s c i e n t i s t - a d m i n i s t r a t o r s and f i s h i n g i n d u s t r y o p e r a t o r s . S a i l a (1969) notes t h a t Bayes's methods f o r d e c i s i o n making may p r o v i d e a way f o r s k i l l e d fishermen to combine t h e i r own judgment with the r e s u l t s o f systems a n a l y s i s , and o u t l i n e s a suggested " F i s h e r y E x p l o i t a t i o n System Symbolic Model". I t i s towards examining some o f i t s components t h a t t h i s t h e s i s i s aimed. (a) (b) F i g u r e 4 Slope o f maximum entropy removing curve where p r o b a b i l i t y o f occurrence i s some f u n c t i o n o f d i s t a n c e from a p o i n t r g < r b ( S h e r s t n i k o v , 1968), where r = range o f d e t e c t i o n at some given p r o b a b i l i t y . 3.0 System V a r i a b l e s For a given f i s h e r y , three v a r i a b l e s are of p a r t i c u l a r importance to decision-making using search or sampling i n f o r m a t i o n : (1) S i z e of search a r e a . (2) F i s h d e n s i t y or number of commercially f e a s i b l e schools or c e n t r e s of f i s h c o n c e n t r a t i o n s . (3) P a t t e r n o f the d i s t r i b u t i o n of the s c h o o l s , c o n s i d e r e d i n chapter 4. 3.1 S i z e of Search Area The l i m i t s o f f i s h i n g areas when not determined by bottom type are g e n e r a l l y not f i x e d and may depend on season and/or oceanographic c o n d i t i o n s , P e l a g i c f i s h e r i e s may e x p l o i t stocks which during d i f f e r e n t seasons occur over w i d e l y separated a r e a s , e.g., the P a c i f i c tuna f i s h e r i e s , w h i l e f o r demersal f i s h e r i e s seasonal movements may be r e l a t i v e l y l o c a l , c h a r a c t e r -i z e d mainly by changes of depth. F i s h d i s t r i b u t i o n , and hence area o f a f i s h e r y , i s known to be d i r e c t l y a f f e c t e d by oceanographic c o n d i t i o n s , e s p e c i a l l y temperature. Lee (1952), f o r example, shows t h a t the d i s t r i b u t i o n o f cod (Gadus morhua) i n the Barents Sea i s c l o s e l y r e l a t e d to bottom water temperature, and McKenzie (1964) shows t h a t the h o r i z o n t a l d i s t r i b u t i o n l i m i t s of f o u r s p e c i e s o f tuna are temperature dependent. T h i s r e l a t i o n s h i p may not be s i m p l e , and may r e l y upon temperature g r a d i e n t ( S e c h e l , 1963). S a l i n i t y does not appear to be important i n d e l i m i t i n g of o f f s h o r e f i s h e r i e s . F i s h e r i e s o f t e n have range r e s t r a i n t s as a consequence of the type of gear used, i . e . p e l a g i c s p e c i e s may move out of range of v e s s e l s or the -17-d i s t a n c e to the grounds may be so g r e a t as to be uneconomic; i n demersal f i s h e r i e s , the depth may become too g r e a t f o r the type o f gear a v a i l a b l e . The r e g i o n i n which a f i s h e r y o c c u r s u s u a l l y changes i n some c y c l i c a l seasonal manner as the f i s h e s undergo f e e d i n g and breeding m i g r a t i o n s In an a c t u a l f i s h e r y reconnaissance the l i m i t s o f some f i s h e r y area would be d e f i n e d i n terms o f some minimal f i s h d e n s i t y , i n the case of a demersal f i s h e r y ; or minimal p r o b a b i l i t y of e n c o u n t e r i n g a school i n a p e l a g i c f i s h -ery s i t u a t i o n . Johanneson and Losse (1973) r e p o r t t h a t i n a sonar survey i n the Black s e a , the l i m i t s of t h e i r search were determined by the absence o f observed e c h o - r e f l e c t i o n s of f i s h a t e i t h e r the outer or i n n e r end o f a t r a c k , coupled with a sonar check over an a d d i t i o n a l range o f 1500 M. I f f i s h were d e t e c t e d the range was extended, i f not, the v e s s e l proceeded on the next t r a c k . For t h i s t h e s i s , the range o f search w i l l be c o n s i d e r e d small compared with the area o f the f i s h e r y and edge e f f e c t s w i l l be i g n o r e d . 3.2 F i s h D e n s i t y D i r e c t methods, based on sampling, o f e s t i m a t i n g the t o t a l number of schools w i t h i n a given f i s h e r y w i l l s u f f e r problems o f sampling e r r o r and c a l c u l a t i o n of c o n f i d e n c e l i m i t s compounded by the d i f f i c u l t y o f sampling the marine environment. However, f o r many s i t u a t i o n s complete coverage o f a given f i s h i n g ground with a c o u s t i c d e v i c e s may g i v e an a b s o l u t e i n d i c a t i o n of f i s h d e n s i t y , i f the a c o u s t i c r e s u l t s can be r e l a t e d to a c t u a l c a t c h e s . In some such s u r v e y s , agreement between a c o u s t i c estimates and estimates based on t r a d i t i o n a l methods have been i n reasonable accord (Anon, 1973), although i n others the c a l i b r a t i o n of c a t c h r e s u l t s w i t h echo r e t u r n s has been the g r e a t e s t source o f v a r i a b i l i t y i n p o p u l a t i o n e s t i m a t e s . (Thorne, Reeves and M i l l i k a n , 1971). However, i n demersal f i s h e r i e s , echo i n t e g r a t o r s f o r counting f i s h are becoming f u r t h e r r e f i n e d and more r e l i a b l e (Lahore and L y t t l e , 1970; Anon, 1973). For some s u r f a c e s c h o o l s , a e r i a l assessment of school numbers and biomass during both n i g h t and day o p e r a t i o n s has been found p o s s i b l e and ex-p e r i e n c e d s p o t t e r s are able to provide c o n s i s t e n t estimates of r e l a t i v e abundance ( S q u i r e , 1972). However, s u r f a c e search by v e s s e l s coupled w i t h sonar search have, i n one case, shown t h a t the number o f s u r f a c e s c h o o l s observed was no i n d i c a t i o n of the t o t a l number o f s c h o o l s i n the a r e a . Wolfe (1971) i n i n v e s t i g a t i o n s around Tasmania found t h a t l a r g e dense sub-s u r f a c e s c h o o l s were 2.5 times more p l e n t i f u l than s u r f a c e s c h o o l s i n summer and 4 times more so i n w i n t e r . In a d d i t i o n , he found t h a t schools of the s p e c i e s concerned may s u r f a c e on calm days both w i n t e r and summer, but not on a l l calm days. Methods of e s t i m a t i n g the number of s c h o o l s i n a f i s h e r y have been made from catch d a t a , but such models r e q u i r e assumptions about the mode o f o p e r a t i o n o f the v e s s e l s which are i n r e a l i t y u n l i k e l y , and may a l s o f a i l t o d i s t i n g u i s h between a l t e r n a t e p o s t u l a t e s . Neyman (1949) i n a study prompted by the d e c l i n e of the C a l i f o r n i a n s a r d i n e (Sardinops sagax) d e v e l -oped a model f o r e s t i m a t i n g the number of s c h o o l s i n a f i s h e r y . The method was based on two premises: (1) That d i s t r i b u t i o n of s c h o o l s i s random ( i . e . catches f o l l o w a p o i s s o n r e l a t i o n s h i p ) ; (2) That boats do not o v e r l a p i n t h e i r a c t i v i t i e s and no communi-c a t i o n occurs between them. From these premises Neyman eva l u a t e d f u n c t i o n s to g i v e the p r o b a b i l i t y t h a t a t time t , n s c h o o l s have been caught and a (n+1)th school i s being e x p l o i t e d , or t h a t n s c h o o l s have been caught and the v e s s e l i s s e a r c h i n g . Estimates of t o t a l number o f schools and/or d e n s i t y are d e r i v e d from these r e l a t i o n s h i p s . However, Neyman shows t h a t h i s method i s unable to d i s t i n g u i s h between the s i t u a t i o n s o f i n e f f i c i e n t , b o a t s and abundant f i s h , and t h a t o f e f f i c i e n t boats but a s c a r c i t y of f i s h . C e r t a i n o p e r a t i o n a l data are a l s o necessary; the time taken to l o c a t e the f i r s t s c h o o l , and the number o f s c h o o l s l o c a t e d . These should not be major problems, but the independence o f f i s h i n g boats,and non-overlap o f t h e i r a c t i o n s appear u n r e a l i s t i c assumptions. Estimates o f f i s h abundance may be d e r i v e d from t r a n s e c t surveys o f f i s h i n g grounds. A n a l y s i s o f the r e s u l t s w i l l depend on the method o f f i s h d e t e c t i o n . A c o u s t i c equipment does not r e q u i r e d i r e c t c o n t a c t between f i s h and sampling gear so d e n s i t y and v a r i a n c e estimates can be made from c a l i b r a t e d a c o u s t i c r e c o r d s . When the survey method depends on a d i r e c t i n t e r a c t i o n between f i s h and the sampling gear, as when t r o l l i n g , then d e n s i t y estimates w i l l depend on the nature o f the i n t e r a c t i o n . Skellam (1958) shows t h a t when sampling i n v o l v e s such i n t e r a c t i o n , an unbiased estimate o f d e n s i t y i s given by where X = d e n s i t y of f i s h s c h o o l s Z = no. o f encounters per u n i t time R = range w i t h i n which a school must approach the observer to e f f e c t an encounter. V = r e s u l t a n t v e l o c i t y -20-and r~I 7 V = V i T + (3.2) where u = average velocity of schools w = average velocity of the observer. Skellam notes that though (3.2) is not s t r i c t l y true, i t can be considered adequate for most practical applications. The two variables which must be estimated are R and V. Koopman (1956a) has obtained a similar result , using s l ight ly different assumptions to those of Skellam. If the distr ibution of the target track angles (direction with respect to some observer) is uniform, such that the average number with track angle between $ and d<j> is N d<j>/27r, where N targets occur per unit area, and the speed of the observer is w, with a perception range R, then, from Koopman, the number of encounters, N Q would be o 2* 0 so N = 0 N 2-n 2R/27rwd<f, 0 In Skellam's (1958) notation, n = JL 2u L» on 2 R * fz"m Z 2 * [V*]2lF= z 2R L " l t ' J 0 2RV In estimating R a number of assumptions are necessary, which may be demonstrated by use of Craig and GraJiam's (1965) data, l i s t i n g schools fished by t r o l l i n g , the school size and location of capture. -21 -I t i s assumed t h a t whenever a t r o l l l u r e i s w i t h i n range o f an a l b a c o r e (Thunnus a l a l u n g a ) , i t i s taken, although as Nakamura (1967) notes, success i n t r o l l i n g may depend to some degree on how hungry the f i s h a re. The range at which a tuna can see a t r o l l l u r e w i l l depend on i t s v i s u a l a c u i t y , the ambient l i g h t c o n d i t i o n s and the s i z e and c o l o u r o f the l u r e among ot h e r f a c t o r s , e.g. a r e f l e c t i v e l u r e would probably be d e t e c t e d at a g r e a t e r range than a n o n - r e f l e c t i v e l u r e . Nakamura (1968) under experimental c o n d i t i o n s i n which s k i p j a c k (Euthynnus pelamis) and kawakawa (Euthynnus a f f i n i s ) were t r a i n e d to d i s c r i m -i n a t e between v e r t i c a l l y and h o r i z o n t a l l y s t r i p e d images measured t h e i r v i s u a l a c u i t y as 0.180 and 0.135 r e s p e c t i v e l y . L i g h t i n g c o n d i t i o n s were e q u i v a l e n t t o those o c c u r r i n g when the sun i s unobscured and a t an a l t i t u d e of 65°, and the tuna at a depth of 36 m. In the absence of d e t a i l e d data about such c o n d i t i o n s d u r i n g the c o l l e c t i o n of C r a i g and Graham's d a t a , c o n d i t i o n s were assumed t o be s i m i l a r , and the v i s u a l a c u i t y f o r Ibacore taken as the mean of t h a t f o r awakawa and k i p j a c k , i . e . 0.158, V i s u a l a c u i t y i s d e f i n e d as the r e c i p r o c a l o f the v i s u a l angle measured i n minutes, and i s the angle subtended a t the eye by the s i z e o f the viewed o b j e c t . Hence v i s u a l angle = Q ] 5 G = 6.348 Assuming a l u r e l e n g t h o f 30 cm (approximately 12 i n . ) , the d i s t -ance at which the l u r e would be p e r c e i v e d under the c o n d i t i o n s d e s c r i b e d by Nakamura (1968) i s R = ta¥[OW2j" (3*3) = 162.5 M The use o f the v i s u a l a c u i t y or v i s u a l angle i n equation ,3 i s f r a u g h t with -22-assumptions. Pirenne (1962) notes t h a t the d e f i n i t i o n o f v i s u a l a c u i t y , as used here, i s p u r e l y an o p e r a t i o n a l d e f i n i t i o n , there being as many v i s u a l a c u i t y measurements as there are types of t e s t o b j e c t s . Presence o f f e e d i n g markings on the s i d e of tuna would a l s o extend the range of p e r c e p t i o n , but would not a l t e r the minimum d i s t a n c e a t which t r o l l s were l o c a t e d . C r a i g and Graham encountered 71 scho o l s d u r i n g 292 hours t r o l l i n g , , To estimate u, the average v e l o c i t y o f the s c h o o l s , i n f o r m a t i o n must n e c e s s a r i l y be drawn from s e v e r a l s o u r c e s . Nishimura (1963) s t a t e s t h a t i t was found t h a t "the maximum speed was i n the range 2-3 knots (103-154 cm s e c - 1 ) f o r y e l l o w f i n (Thunnus a l b a c o r e s ) , a l b a c o r e and b l u e f i n (Thunnus  thynnus) tuna". P r e v i o u s l y i t had been thought t h a t the maximum speed o f the tuna exceeded 10 knots, but t h a t t h e i r speed was much slower under normal c o n d i t i o n s . Magnuson (1966) found t h a t kawakawa o f 36 cm f o r k l e n g t h , when i n c a p t i v i t y averaged 80 cm s e c - 1 , but t h a t y e l l o w f i n o f the same s i z e averaged only 50 cm s e c - 1 , and i n a l a t e r unpublished paper 46 cm s e c - 1 ( o r 1.31 le n g t h s s e c - 1 ) f o r y e l l o w f i n . Many workers have r e p o r t e d a r e l a t i o n between f o r k le n g t h and swimming speed. Magnuson (1970) found t h a t t h i s r e l a t i o n was we l l d e s c r i b e d by an equation o f the form, V = 10 l b cm s e c " 1 (3.4) where V; = speed o f f i s h with p e c t o r a l s c o n t i n u o u s l y extended; 1 = f o r k l e n g t h o f the f i s h . b = cons t a n t The shape o f the curve d e s c r i b i n g the minimum speed o f kawakawa i n r e l a t i o n to f o r k length was c l o s e l y s i m i l a r t o the shapes o f curves t h a t have been -23-used to r e l a t e endurance speeds o f other f i s h to t h e i r l e n g t h . For a speed o f 48 cm s e c - 1 f o r y e l l o w f i n o f 35 cm f o r k l e n g t h , then from equation 3.4, h _ l o g V - l o g 10  D ' l o g 1 = 0.441 C r a i g and Graham (1965) r e p o r t t h a t of the albacore l e n g t h f r e q u e n c i e s , 91% were between 63-69 cm, with a mean length o f 65.36 cm. Assuming b has the same value f o r a l b a c o r e , then V = 10 x 6 5 . 3 6 0 ' 4 4 1 V = 63.2 cm s e c - 1 . T h i s value i s below the range given by Nishimura (1963). Magnuson's un-pu b l i s h e d value o f 1.31 length s e c - 1 f o r y e l l o w f i n , i f used f o r a l b a c o r e would give a speed of 86 cm s e c " 1 , which i s higher than t h a t obtained with h i s 1970 r e l a t i o n . Now V = J u 2 + w 2 where: w = speed o f f i s h = 63.2 cm s e c " 1 = 1.23 knots u = speed o f observer =6.50 knots V = J 1.2fc + 6.5' 6.62 knots. Skellam notes t h a t h i s method i m p l i e s t h a t the paths o f the scho o l s are r e c t i f i a b l e , which seems a reasonable assumption. The r e l a t i o n f o r the r e l a t i v e v e l o c i t y , equation 3.2, r e q u i r e s t h a t the d i r e c t i o n s o f the sc h o o l s be randomly and un i f o r m l y d i s t r i b u t e d with r e s p e c t to the motion o f the ob-s e r v e r , and even i f t h i s i s not s t r i c t l y t r u e , equation 3.2 s t i l l remains a good e s t i m a t o r o f the r e l a t i v e v e l o c i t y . I t i s of i n t e r e s t t h a t Royce and -24-Otsu (1955) i n o b s e r v a t i o n s on s c h o o l s of s k i p j a c k , found t h a t the s c h o o l s were randomly d i s t r i b u t e d over the scanning area, and t h a t t h e i r movement, dur i n g the p e r i o d o f o b s e r v a t i o n , appeared to approximate random motion. Hence from equation 3.1 x = 71 x 292 x .088 x 6.62 x 2 = 0.201 schools ( n a u t i c a l mile) . T h i s e s t i m a t e o f mean school d e n s i t y w i l l p r o v i d e a b a s i s f o r d e c i s i o n making i n s e c t i o n 6.2 4.0 School P a t t e r n and S i z e Sample r e s u l t s from t r a n s e c t surveys w i l l p r o v i d e i n f o r m a t i o n on h e t e r o g e n e i t y o f f i s h school s i z e and d i s t r i b u t i o n which w i l l enable d e t e r -mination o f l o c a l abundance on f i s h i n g grounds. T h i s may be used i n d e t e r -mining f i s h i n g t a c t i c s , e.g., i f the school d i s t r i b u t i o n o f a s p e c i e s f o l l o w s a known c l u s t e r i n g p r o c e s s , and i f the mean area such a c l u s t e r may occupy i s a l s o known, then i f a school i s l o c a t e d at the edge o f a search bond, the p o s s i b i l i t y o f other s c h o o l s i n the v i c i n i t y may be s p e c i f i e d . 4.1 P a t t e r n of School Occurrence There appear to be no p u b l i s h e d s t u d i e s on the a c t u a l p a t t e r n o f f i s h school d i s t r i b u t i o n . However, t r a n s e c t data f o r many f i s h i n g areas e x i s t which may be examined to c o n s i d e r the nature o f school p a t t e r n . Here t r a n s e c t data o f C r a i g and Graham (1965) w i l l be used to examine goodness o f f i t o f f o u r d i s t r i b u t i o n s . In choosing a d i s t r i b u t i o n to d e s c r i b e a s e t o f d a t a , i t i s d e s i r a b l e t h a t i t should have some b i o l o g i c a l meaning as w e l l as p r o v i d i n g a good f i t to -25-the data. However, the same s e t o f data may f i t s e v e r a l d i f f e r e n t d i s t r i -b u t i o n s ; c h o i c e of a d i s t r i b u t i o n may be d i f f i c u l t because of p o s s i b l e ambiguous r e l a t i o n s h i p s between d i s t r i b u t i o n s and d a t a , and because o f the i n t e r r e l a t i o n s h i p o f the d i s t r i b u t i o n s . The f o l l o w i n g d i s t r i b u t i o n s are c o n s i d e r e d here: (1) Poisson (2) Poisson with added zeroes. (3) Neyman Type A (or Poisson-Poisson) (4) Negative Binomial A Poisson d i s t r i b u t i o n would be expected when the number o f s c h o o l s per area i s low and the appearance o f a school i s independent o f the presence o f o t h e r s . T h i s s i t u a t i o n would be t h a t with l e a s t assumptions about the nature of d i s t r i b u t i o n , i . e . the most degenerate s i t u a t i o n , and provides a u s e f u l base f o r comparison with other d i s t r i b u t i o n s . A Poisson d i s t r i b u t i o n with added zeroes c o u l d be expected when some o f the sample u n i t s are not i n h a b i t a b l e by the sample i n d i v i d u a l s , i . e . f i s h s c h o o l s . Such a s i t u a t i o n may a r i s e i f d i s t r i b u t i o n of schools i s dependent on prey items which are r e s t r i c t e d to c e r t a i n p a r t s o f the sample a r e a , p o s s i b l y due to oceanographic c o n d i t i o n s . C l u s t e r e d d i s t r i b u t i o n s have c e r t a i n appeal i n t h a t b i o l o g i c a l meaning may be a t t r i b u t e d to the parameters. C l u s t e r e d d i s t r i b u t i o n s r e f l e c t the e x p e c t a t i o n t h a t i f one school i s encountered, then one expects to en-counter another with g r e a t e r than average p r o b a b i l i t y (Neyman, 1949). This could a r i s e from the h e t e r o g e n e i t y of the environment or i f prey items them-s e l v e s are d i s t r i b u t e d i n r e l a t i o n to environmental h e t e r o g e n e i t i e s . In two-parameter c l u s t e r d i s t r i b u t i o n s , one parameter d e f i n e s the expected -26-number of clusters per sample unit , and the other, the number of schools per cluster. For clustered distributions of a Poisson-Poisson type, depending on whether the size of the clusters is large or small , the best description may be given by Thomas's (1949) distr ibution or a Neyman type A respectively (Pati l and Joshi , 1968). Both the Poisson and Poisson with added zeroes are l imit ing forms of the Neyman Type A distr ibut ion. (Martin and K a t t i , 1965) If the distr ibution of cluster size tends towards a log d ist r ibut ion, then the overall distr ibution tends to that of a negative binomial rather than a Poisson-Poisson distr ibution (Pielou, 1969). For a Poisson distr ibution the probability of encountering r schools is given by P(r) = where x = mean school density. When the distr ibution of school numbers follows a Poisson d i s t r i -bution with added zeroes, the probabil i t ies are given by p<°> - t^Jr-(Martin and K a t t i , 1965), where c is a function of the proportion of the samples which are uninhabitable. If e is the proportion of habitats which are uninhabitable then, e~x + c , e = e -A - 1 For a Poisson-Poisson d ist r ibut ion, -27-r „k -m /. ,\r -k -kx P(r) = I (Thomas 1949) k=l where X = mean number of schools per cluster m = mean number of clusters per sample and for situations of small X, when a Neyman type A is expected to give a better description P(r) = (0 r + -^4- + -Ajf - + + > (Shenton, 1949) where x' - me~x For a negative Binomial d ist r ibut ion, the probabil it ies are given by P'r> - ' $ J^imr ) ' V (Southward, 1965) where x = mean school density k = distr ibution parameter dependent on degree of contagiousness For the Poisson distr ibution the variance is equal to the mean, X For a Poisson with added zeroes the mean is ex and the variance is ex(l + x) - e 2 x 2 The mean for both the Poisson-Poisson and Neyman Type A distributions is mX with variance mx(l +X). Variance of the negative Binomial distr ibution is X - X2 /k„ From Craig and Graham's 1965 data the distances between successive tuna strikes were calculated from co-ordinates of latitude and longitude, for each day. It is assumed that sampling was carried out for 12 hours per day which gives reasonable agreement for distance travelled by the search vessel (19 days at 6.5 knots per 12 hours per day). As the time at which catches were made was not recorded, the time period unti l the f i r s t catch of the day was assumed equal to that time from last catch to f in ish of sampling for the -28-day. For purposes of sampling the accumulated d i s t a n c e from the beginning of the day was c a l c u l a t e d f o r each c a t c h r e c o r d . The s i z e ( l e n g t h ) of the sampling u n i t c o u l d be chosen as d e s i r e d , and the sampling day was c o n s i d e r e d to c o n s i s t of an i n t e g r a l number o f sampling u n i t s , so that the d i s t a n c e sampled per day was a t l e a s t 78 n a u t i c a l m i l e s , i . e . 12 hours a t 6.5 knots. Sample lengths of 5, 10, 15, 20 and 25 m i l e s were used and the number of s c h o o l s i n each sample found from the d a i l y accumulated d i s t a n c e s . O v e r e s t i m a t i o n of sample d i s t a n c e per day was g r e a t e s t f o r the 25 m i l e l e n g t h (22 m i l e s ) . The d i s t r i b u t i o n s were f i t t e d to the sample data by the f o l l o w i n g maximum l i k e l i h o o d methods: Poisson,the sample mean was used as the d i s t r i -b u t i on mean; Poisson with added ze r o e s ; method of M a r t i n and K a t t i (1965); P o i s s o n - P o i s s o n , method of Thomas (1949); Neyman Type A, method o f Shenton (1949); and Negative B i n o m i a l , method of Southward (1965) with v a r i a n c e estimates o f k and P from Haldane (1941). Table I l i s t s f o r each d i s t r i b u t i o n and sample l e n g t h s i z e the value f o r goodness of f i t o f sample data to t h a t estimated from the d i s t r i b u t i o n . For the Poisson d i s t r i b u t i o n , best f i t was obtained with a sample length o f 5 m i l e s , and next best f i t with a l e n g t h of 20 m i l e s ; f o r the Poisson with added z e r o e s , best f i t was obtained with a sample l e n g t h of 25 m i l e s , and goodness o f f i t i n c r e a s e d with sample s i z e . The P o i s s o n -Poisson and Neyman Type A d i s t r i b u t i o n s gave b e s t f i t s f o r the 5 m i l e sample l e n g t h , then f o r the 15 m i l e , while the Negative Binomial gave the r e v e r s e . A l l d i s t r i b u t i o n s but the Poisson with added zeroes showed a s i m i l a r trend between sample s i z e and goodness of f i t . TABLE Means, v a r i a n c e s , goodness o f f i t and Poisson D i s t r i b u t i o n Sample Length Mean D i s t r i b u t i o n Variance 5 0.234 0.234 10 0.467 0.467 15 0.623 0.623 20 0.934 0.934 25 0.750 0.750 Poisson with added zeroes Sample Length Added Zeroes Parameter Mean Dis 5 0.299 1.145 10 0.409 1.479 15 0.482 1.614 20 0.571 1.919 25 0.538 1.781 I entropy f o r given d i s t r i b u t i o n s Data Variance 2 X d.f. Entropy 0.239 1.605 2 0.591 0.621 12.812 4 0.894 0.963 27.400 5 1.041 1.662 9.713 6 1.266 1.150 18.893 4 1.141 t r i b u t i o n Variance Data Variance X d . f . Entropy 0.617 0.239 17.155 2 0.685 1.134 0.621 8.453 4 1.019 1.428 0.963 8.549 5 1.183 1.997 1.662 7.670 6 1.410 1.747 1.150 5.635 4 1.294 Table I continued Poisson-Poisson Sample Length Mean Distribution Mean Sample Variance Distr ibuti on Variance m Var(m) X Var(X) 2 X d.f . Entropy 5 0.234 0.237 0.239 0.228 0.228 0.001 0.041 0.003 0.137 2 0.599 10 0.467 0.496 0.621 0.379 0.379 0.003 0.308 0.015 1.952 4 0.940 15 0.623 0.602 0.963 0.488 0.488 0.006 0.235 0.016 1.976 5 1.051 20 0.934 0.877 1.662 0.667 0.667 0.012 0.314 0.029 5.142 6 1.281 25 0.750 0.820 1.150 0.593 0.593 0.011 0.382 0.033 4.977 4 1.244 Neyman's Type A Sample Length Mean Sample Variance Distribution Variance m X 2 X d.f . Entropy 5 0.234 0.239 0.239 10.009 0.023 0.702 2 0.593 10 0.467 0.621 0.621 1.414 0.330 3.174 4 0.916 15 0.623 0.963 0.963 1.141 0.546 1.162 5 1.078 20 0.934 1.662 1.662 1.199 0.779 3.149 6 1.332 25 0.750 1.150 1.169 1.343 0.559 5.889 4 1.212 Table I continued Negative Binomial Sample Length Mean Sample Var. Distribution Var. 5 0.234 0.239 0.228 10 0.467 0.621 0.275 15 0.623 0.963 0.261 20 0.934 1.662 0.140 25 0.750 1.150 0.218 P Var. (p) k Var. (k) x 2 d.f . Entropy 0.023 0.000 10.009 0.616 0.710 2 0.593 0.412 0.001 1.414 0.000 3.987 4 0.915 0.581 0.002 1.141 0.000 0.839 5 1.077 0.850 0.002 1.099 0.000 2.877 6 1.335 0.709 0.003 1.058 0.000 7.046 4 1.191 Table II Ranked order of f i t of distributions for different sample length Sample Length Best F i t 2nd 3rd 4th Worst F i t 5 Poisson-Poisson Neyman A Negative Binomial Poisson Poisson + zeroes 10 Poisson-Poisson Neyman A Negative Binomial Poisson + zeroes Poisson 15 Negative Binomial Neyman A Poisson-Poisson Poisson + zeroes Poisson 20 Negative Binomial Neyman A Poisson-Poisson Poisson + zeroes Poisson 25 Poisson-Poisson Neyman A Negative Binomial Poisson + zeroes Poisson -32-Best o v e r a l l f i t s were: D i s t r i b u t i o n Sample l e n g t h ( L i n e a r i n t e r p o l a t i o n ) ( n a u t i c a l m i l e s ) a Negative Binomial 15 0.97 Poisson-Poisson 5 0.94 Negative Binomial 5 0.73 Neyman's Type A 5 0.73 Table II l i s t s the d i s t r i b u t i o n s i n order o f goodness of f i t f o r each sample l e n g t h . As co n f i d e n c e l i m i t s are a necessary a d j u n c t to management d e c i s i o n s , c h o i c e of a d i s t r i b u t i o n whose v a r i a n c e r e f l e c t s the a c t u a l data v a r i a n c e i s an important c o n s i d e r a t i o n , as well as t h a t of being able to d e s c r i b e the a c t u a l d i s t r i b u t i o n of sample r e s u l t s . For a Poisson d i s t r i b u t i o n the v a r i a n c e i s equal to the mean. From Table I, i t can be seen t h a t t h i s i s n e a r l y so f o r the 5 m i l e sample length s i t u a t i o n . However, as the sample length i n c r e a s e s , the v a r i a n c e i n c r e a s e s a t a g r e a t e r r a t e than the mean. Table III shows trends i n r a t i o s o f mean and v a r i a n c e as sample length i n c r e a s e s . As would be expected, the best f i t to a Poisson d i s t r i b u t i o n occurs f o r a sample length of 5 m i l e s . The modal value o f the v a r i a n c e i s f o r the 20 m i l e sample, but t h i s may be due to round-o f f e r r o r i n the 25 m i l e sample r e s u l t s . For the Poisson with added z e r o e s , the d i s t r i b u t i o n v a r i a n c e con-s i s t e n t l y overestimated the a c t u a l data v a r i a n c e . However, as sample l e n g t h i n c r e a s e d , the degree of o v e r e s t i m a t i o n decreased, c l o s e s t f i t being f o r a sample length of 20 m i l e s . -33-TABLE I I I Rat i o s o f v a r i a n c e to mean and v a r i a n c e to d i s t r i b u t i o n v a r i a n c e f o r s p e c i f i e d d i s t r i b u t i o n s , and sample l e n g t h s . P o i s s o n Sample length ( m i l e s ) 5 10 15 20 25 Poisson with added zeroes Sample length ( m i l e s ) 5 10 15 20 25 Negative Binomial Sample length ( m i l e s ) 5 10 15 20 25 V a r i a n c e mean 1.021 I. 330 1.546 1.779 1.533 V a r i a n c e D i s t r i b u t i o n v a r i a n c e 0.387 0.547 0.674 0.866 0.646 V a r i a n c e D i s t r i b u t i o n v a r i a n c e 1.048 2.258 3.689 I I . 871 5.275 -34-TABLE I I I (cont'd) P o i s s o n - P o i s s o n Sample length ( m i l e s ) 5 10 15 20 25 Variance Neyman's Type A Sample length ( m i l e s ) 5 10 15 20 25 D i s t r i b u t i o n v a r i a n c e 1.048 1.639 1.973 2.492 1.939 Variance D i s t r i b u t i o n V a r i a n c e 1.000 1.000 1.000 1.000 0.984 -35-The v a r i a n c e of the Negative Binomial d i s t r i b u t i o n a l s o c o n s i d e r -ably underestimated the a c t u a l sample v a r i a n c e as can be seen from Table I. G r e a t e s t d i s c r e p a n c y occurred f o r the 20 m i l e sample l e n g t h . The d i f f e r -ence between sample and d i s t r i b u t i o n v a r i a n c e appeared r e l a t e d to goodness o f f i t ; f o r example, p r o b a b i l i t y of type I e r r o r f o r the 20 m i l e sample length was 0.364 with a v a r i a n c e to d i s t r i b u t i o n v a r i a n c e r a t i o o f 11.871, whereas p r o b a b i l i t y of type I e r r o r f o r the 10 mi l e sample l e n g t h was 0. 443, with a v a r i a n c e to d i s t r i b u t i o n v a r i a n c e r a t i o o f only 2.258. The Poisson-Poisson d i s t r i b u t i o n underestimated the a c t u a l sample v a r i a n c e with i n c r e a s i n g e r r o r as sample s i z e i n c r e a s e d . T h i s d i s c r e p a n c y showed the same trend as the x 2 f o r the goodness of f i t . However, the v a r i a n c e f o r the Neyman type A d i s t r i b u t i o n showed e x c e l l e n t p r e d i c t i o n o f the sample v a r i a n c e f o r a l l sample s i z e s . Hence, i f e s t i m a t i o n o f con-f i d e n c e i n t e r v a l s i s an important c r i t e r i o n i n c h o i c e o f a d i s t r i b u t i o n , then accuracy o f v a r i a n c e e s t i m a t i o n should a l s o be an important c o n s i d e r -a t i o n . 4.2 U n c e r t a i n t y as a f u n c t i o n of d i s t r i b u t i o n type The u n c e r t a i n t y o f a s t o c h a s t i c s i t u a t i o n d e s c r i b e d by a probab-i l i t y d e n s i t y f u n c t i o n w i l l depend on the d e n s i t y f u n c t i o n . Entropy f o r the d i f f e r e n t sample lengths and p r o b a b i l i t y d i s t r i b u t i o n s are l i s t e d i n Table 1. Entropy i s an i n c r e a s i n g f u n c t i o n of the sample l e n g t h , or mean d e n s i t y , f o r a l l d i s t r i b u t i o n s except f o r the sample length o f t w e n t y - f i v e m i l e s . T h i s c o u l d be due t o und e r e s t i m a t i o n o f the d e n s i t y by t h i s sample l e n g t h . Lowest values o f entropy were given by the Poisson d i s t r i b u t i o n , then, i n ascending o r d e r , by the Negative B i n o m i a l , Neyman's Type A, Po i s s o n --36-P o i s s o n , and Poisson with added zeroes. Values f o r the Negative Binomial and Neyman's Type A, were almost i d e n t i c a l (Graph I ) . For a l l d i s t r i b u t i o n s , entropy was an i n c r e a s i n g f u n c t i o n of mean school d e n s i t y . 4.3 School S i z e and U n c e r t a i n t y F u r t h e r u n c e r t a i n t y w i l l be i n h e r e n t i n a f i s h i n g system when the s i z e o f f i s h s c h o o l s or c o n c e n t r a t i o n s v a r i e s , and hence s c h o o l s are of d i f f e r -ent value to the fisherman. In such a s i t u a t i o n , the u n c e r t a i n t y i s a f u n c t i o n o f the number of schools t h a t may o c c u r , and t h e i r s i z e . I f A r e p r e s e n t s the p r o b a b i l i t y d i s t r i b u t i o n of the number o f schools o f f i s h and B the p r o b a b i l i t y d i s t r i b u t i o n of p a r t i c u l a r s i z e d s c h o o l s , then, H(AB) = - { z p k l o g p k + £p. z q. l o g q. } k k n = H(A) + z p k H k (B) ( K h i n t c h i n e , 1957) k where p k = p r o b a b i l i t y o f encountering k schools q n = p r o b a b i l i t y o f e n c o u n t e r i n g school o f s i z e n H k (B) can be regarded as the c o n d i t i o n a l entropy o f the scheme B, g i v e n scheme A, the number of s c h o o l s i n the sample. The catch s i z e s given by C r a i g and Graham (1965) appear to f o l l o w a l o g s e r i e s d i s t r i b u t i o n , so such a d i s t r i b u t i o n was f i t t e d u s i n g the method o f F i s h e r , C o r b e t t and W i l l i a m s (1943). For the l o g s e r i e s , E(n) = £ x n where E(n) = expected number of schools with x f i s h , and •37-x = F T T p = parameter of the negative binomial distr ibution a = (k-1)! where k = power.factor of the negative binomial d ist r ibut ion. From catch sizes given by Craig and Graham (1965), then, E(n) = ( o . 9 1 5 7 ) n A goodness of f i t test gives a probability of Type I error equal to .19, (l inear interpolation) which though not an especially good f i t , provides an adequate f i t for present purposes. The probability of encountering a school of size n w i l l be p ( n ) = " nlog(l-x) Then assuming a Poisson distr ibution of numbers of schools per sam-ple, the system entropy per sample w i l l be given by „,«, . 4f l o g (4£) • I 4f X - log (- i ^ j y O Using the value of A for the 5 mile sample length, H(AB) = 1.059 per sample unit . Hence 44.2% of the conditional entropy is due to the probability distribution describing the school s izes , and the remainder due to the un-certainty of the sample results . 5.0 Decay of Information A search system w i l l be considered here such that schools are located by some search vessel and the information communicated to catcher vessels which may then steam to the location of the f ish concentrations. Due to movement of the f ish after the i n i t i a l location, their position w i l l - 3 8 -be known wi t h d e c r e a s i n g c e r t a i n t y as time passes; the entropy w i l l be an i n c r e a s i n g f u n c t i o n of time. With time, the p r o b a b i l i t y o f r e l o c a t i n g some school of a p a r t i c u l a r s i z e or l a r g e r w i t h i n a s p e c i f i e d r e g i o n w i l l approach t h a t p r o b a b i l i t y given no p r i o r i n f o r m a t i o n . To s p e c i f y the p r o b a b i l i t y d i s t r i b u t i o n o f the p o s i t i o n of a school with time, once l o c a t e d , c o n s i d e r a t i o n must be given to the manner o f move-ment of f i s h s c h o o l s . One p a r t i c u l a r model w i l l be c o n s i d e r e d here to des-c r i b e t h i s movement. C e r t a i n movements w i l l be c o n s i d e r e d to be model independent, such as c y c l i c a l movement due to e i t h e r t i d a l or d i u r n a l rhythms, and w i l l be ignored here. Models d e s c r i b i n g f i s h movement have been based on one of two ana-l o g i e s : 1) The K i n e t i c theory o f gas movement, or 2) A heat d i f f u s i o n r e l a t i o n , where a heat source i s moved a t a const a n t v e l o c i t y over a l a r g e body. For the second model, S a i l a and Flowers (1969) make an analogy between conduction of heat from a source moved at c o n s t a n t v e l o c i t y over a l a r g e body, so as to develop a q u a s i - s t a t i o n a r y s t a t e , and movement of f i s h i n random d i r e c t i o n s as i n d i v i d u a l s , but whose c e n t e r o f mass i s moving i n a s p e c i f i e d d i r e c t i o n i n a three dimensional system. They show t h a t : e = t - t i . - J L - e x p _ ^ e X p - £ 4Tik e x p 21* r where e = temperature excess a t some p o i n t , equal to d i f f e r e n c e i n temperature, t . q = r a t e of heat flow from source -39-k = thermal c o n d u c t i v i t y v = speed of p o i n t source along the x a x i s e = x - VT x = time a = thermal d i f f u s i v i t y 2 2 2 r = x + y + z The f o l l o w i n g t r a n s f o r m a t i o n s were suggested by S a i l a and Flowers (1969): q/k = r a t e of flow o f f i s h from c e n t e r o f mass v = c o e f f i c i e n t of d i r e c t e d movement. (V o f Jones, 1959) a = c o e f f i c i e n t of d i s p e r s i o n (a o f Jones, 1959) e = c o n c e n t r a t i o n a t some p o i n t e = moving c e n t e r of f i s h mass along x a x i s S a i l a and Flowers make no f i t to r e a l data w i t h t h i s model, and o n l y suggest i t s p o s s i b l e r e a l i s m . Skellam (1951) examines the p r o b a b i l i t y d i s t r i b u t i o n o f the p o s i t i o n o f a p a r t i c u l a r descendant o f an i n d i v i d u a l a t some time, i f the descendants move i n a random manner. As n, the number of generations (or u n i t moves f o r our case) becomes l a r g e , the p r o b a b i l i t y d i s t r i b u t i o n approaches t h a t o f a normal. Skellam a l s o notes t h a t a s l i g h t s y s t e m a t i c d r i f t , no matter how s m a l l , i s u l t i m a t e l y the most important cause o f displacement when n i s l a r g e . From Skellam (1951) the p r o b a b i l i t y t h a t some i n d i v i d u a l , (or school) i s r ± dr from the c e n t e r or p o i n t o f o r i g i n a l l o c a t i o n , i s f ( r | n ) , a 2 ) = exp { - r 2 / n a 2 } 2 r / n a 2 0 <. r < --40 -where: 2 a = mean square dispersion per generation (or unit move) analogous with the mean square velocity in Maxwell's d ist r ibut ion, n = number of generations (or unit moves) After n generations (or unit moves) the proportion lying outside some c i rc le of radius R is P = / exp [ - r 2 /na 2 ] 2rdr/na2 R = exp {-R2/na2} For our s i tuat ion , we are concerned with the probability of a school being within some c i r c le of radius R, rather than the proportion of some population, but the method is s imi lar . Skellam (1951) also notes that the maximum likelihood estimate of a 2 i s given by a2 = III nv where n = number of generations v = number of observed values of r. Jones (1959) gives an alternate form for this estimate a 2 = 1 [ z V . (rrcose) 2 ] ( 5 J ) where t = time interval n = number of schools Beverton and Holt (1957) appear to be the f i r s t to consider d is -persion of f ish using the concepts of gas diffusion in d e t a i l . They d i s t -inguish between clearly defined directional movement of f ish such as spawn-ing migrations and their primary interest, that of local interchange in -41-which they c o n s i d e r the e f f e c t s o f heterogenous f i s h i n g m o r t a l i t y over some area on f i s h s t o c k s . They s t a t e t h a t the k i n e t i c theory o f gas d i f f u s i o n , as a p p l i e d to l o c a l d i s p e r s i o n of f i s h , appears q u i t e r o b u s t , and does not r e q u i r e r i g o r o u s analogy, merely t h a t a t times a random change i n d i r e c t i o n i s made. In t h e i r development from Skellam (1951), they suggest t h a t the 'mean f r e e path' be equated to 'movement from one food patch to another', The r a t e o f change o f c o n c e n t r a t i o n of f i s h they g i v e as 3C_ 1_ v£ ,j^C . j ^ C \ i t A n * 9 Ol 9t 4 n ( ~ T + V (52) ax ay where: V 2 — i s analogous to mean square v e l o c i t y , and V = e f f e c t i v e v e l o c i t y , analogous to r o f Skellam (1951) n = number o f movements i n random d i r e c t i o n s made i n u n i t time Beverton and H o l t ' s d i f f e r e n t i a l i s p r o p o r t i o n a l to -^ as movement i s only c o n s i d e r e d i n t o the 4 a d j a c e n t squares, c o r n e r e f f e c t s being c o n s i d e r e d n e g l i g i b l e . They d e f i n e a d i s p e r s i o n c o e f f i c i e n t , D as V 2 n = = nd2 i . e . , the r a t e of d i s p e r s i o n i s both a f u n c t i o n of the v e l o c i t y of the f i s h and the l e n g t h of the "mean square path". Two other authors have subse-q u e n t l y a p p l i e d Skellam's method t o a c t u a l f i s h e r i e s s i t u a t i o n s . Jones (1959) extended Skellam's methods to the examination of movement of tagged haddock i n the North Sea, and S a i l a and Flowers (1968) a p p l i e d the methods to movement of the American l o b s t e r (Homarus americanus) near Rhode I s l a n d . -42-Jones re-expresses the p r o b a b i l i t y o f occurrence as a f u n c t i o n of time and d i s t a n c e : R 2 P(R,t) = 1 - exp (-5-) (5.3) Equation 5.3 enables the p r o b a b i l i t y of a s c h o o l ' s occurrence w i t h -i n some range a t a c e r t a i n time a f t e r being o r i g i n a l l y l o c a t e d , to be s p e c i -f i e d . Measurements of the displacement of f i s h schools over s h o r t time p e r i o d s appear r a r e i n the l i t e r a t u r e . However, Royce and Otsu (1955) have recorded o b s e r v a t i o n s on s k i p j a c k . They reasoned t h a t , s i n c e s k i p j a c k are suspected o f c o n s i d e r a b l e m i g r a t i o n s , then such movement might be d e t e c t e d as a Doppler e f f e c t d u r i n g search. However, no such e f f e c t c o u l d be de-t e c t e d and movement of schools w h i l e being watched appeared to approximate, random motion. They then recorded d i r e c t o b s e r v a t i o n s on s c h o o l s which f r e q u e n t l y moved c o n s i d e r a b l e d i s t a n c e s w h i l e being observed. Movements o f schools f o r the f i r s t 10 minutes of o b s e r v a t i o n were p l o t t e d and i t was found t h a t l e s s than h a l f showed d e f i n i t e movement (presumably, change o f p o s i t i o n ) . U s u a l l y the schools were m i l l i n g about i n one p l a c e , though a t times movement was d i r e c t i o n a l . Royce and Otsu (1955) g i v e v e c t o r diagrams showing movement o f schools f o r 7 s e t s of o b s e r v a t i o n s . The number o f schools f o r which no change of p o s i t i o n was observed i s given and a l s o the number o f s c h o o l s which were observed f o r a f u l l 10. minutes. However, the l e n g t h o f times were not s p e c i f i e d f o r schools t h a t d i d move but not for.10 minutes, so these movements were assumed to be 5 minutes, t h a t i s , t = 0 2 ^° = 5 mins, was taken as the best estimate This i n f o r m a t i o n i s summarized i n Table IV. -43-TABLE IV Summary of vector diagrams of Royce and Otsu (1955) Series Number of schools Time No. of schools showing no 10 mins 5 mins change of position Apri l 1 6 - 1 7 14 3 4 7 2 0 - 2 1 22 4 4 14 2 5 - 2 6 19 3 2 14 27 - 28 11 3 1 7 June 13 - 14 30 5 5 20 2 9 - 3 0 27 3 5 19 July 9 - 10 28 3 4 21 -44-Royce and Otsu's (1955) data can be used to examine how the r e s u l t a n t v e l o c i t y o f a number of scho o l s may be determined w i t h c o n f i d e n c e estimates on the d i r e c t i o n o f movement, and how such estimates can be used i n making d e c i s i o n s i n a h a r v e s t i n g s i t u a t i o n . Before use of Royce and Otsu's (1955) data to determine v e l o c i t y o f tuna s c h o o l s , t e s t s o f the f o l l o w i n g hypotheses are necessary: (1) That the r e s u l t a n t d i r e c t i o n i s s i g n i f i c a n t l y d i f f e r e n t from zero a t some given p r o b a b i l i t y o f type I e r r o r . (2) That the r e s u l t a n t v e c t o r s f o r the seven s e t s o f o b s e r v a t i o n s are homogeneous. In (1) i t i s wished to t e s t the hypothesis k = 0, a g a i n s t the a l t e r n a t i v e k > 0 ; i . e . t o determine i f the r e s u l t a n t v e c t o r c o u l d a r i s e from random proc e s s e s . The parameter k i s a v a r i a b l e i n d i c a t i n g the degree o f c o n c e n t r a t i o n o f d i r e c t i o n o f the s c h o o l s . Greenwood and Durand (1955) t a b u l a t e values o f Z f o r t e s t s o f s i g n i f i c a n c e on k from the i n d i v i d -ual d i r e c t i o n s , where; 2 2 z cos e + z s i n o and D2 1 - T where R = r e s u l t a n t d i r e c t i o n o f u n i t v e c t o r s o f angle 6 n = number of o b s e r v a t i o n s The r e s u l t s are given i n t a b l e V, and e, s i n e and cos e i n t a b l e VI. When n i n c l u d e d those schools which were ' s t a t i o n a r y ' , the n u l l hypothesis had to be accepted f o r a l l s i t u a t i o n s , i . e . , k = 0. When only those schools which were c o n s i d e r e d t o be changing p o s i t i o n were c o n s i d e r e d , then the a l t e r n a t e hypothesis c o u l d be accepted f o r data c f A p r i l 27-28 and TABLE V Data f o r t e s t o f k f 0 from Royce and Otsu (1955) S e r i e s No. o f No. o f z C r i t i c a l value z f o r C r i t i c a l v a l u e s c h o o l s moving sch o o l s f o r a l l schools o f z, a=.05 moving schools o f z, a=.05 A p r i l 16 - 17 14 7 0.1285 2.9413 0.2570 2.8819 20 - 21 22 8 0.8104 2.9613 2.2284 2.9014 25 - 26 19 5 0.0633 2.9558 0.2407 2.8260* 27 - 28 11 4 1.1618 2.9262 3.1950 2.7979* June 13 - 14 30 10 0.4347 2.9957 + 1.3040 2.9187 29 - 30 27 8 0.0894 2.9957 + 0.3016 2.9014 J u l y 9. - 10 28 7 0.7485 2.9957 + 2.9936 2.8819 + For number o f s c h o o l s g r e a t e r than 24, z f o r n = » used i n t a b l e s of Greenwood and Durand (1955) * By l i n e a r e x t r a p o l a t i o n TABLE VI Data f o r a n a l y s i s o f Royce and Otsu's (1955) tuna movement o b s e r v a t i o n s A p r i l 16 - 17 D i r e c t i o n e cose s i n e Distance r Time t rcose ' 2 / t 73 0.2924 0.9563 0.505 10 0.148 1.530 96.5 - .1132 .9936 .260 5 - .029 0.811 132 - .6691 .7431 .300 5 - .200 1.080 281 .1908 - .9816 .500 5 .057 1.080 285.5 .2672 - .9636 1.500 10 .401 13.500 288 .3090 - .9511 .250 5 .077 .750 307.5 .6088 - .8039 .350 10 .213 6.000 A p r i l 20 - 21 4 0.9976 0.0698 0.245 5 0.244 0.720 88.5 .0262 .9997 .260 5 .007 .811 115 - .4226 .9063 .500 10 - .211 1.500 120 - .5000 .8660 1.328 10 - .664 10.582 122 - .5299 .8480 1.519 10 - .805 6.000 126 - .5878 .8090 1.012 10 - .595 6.145 142 - .7880 .6157 .255 5 - .201 3.060 281.5 .9483 - .9799 .277 5 .263 3.324 TABLE VI (cont'd) Di recti on Distance Time 6 cose sine r t rcose Apri l 25 - 26 7 0.9925 0.1219 0.236 5 0.234 0.668 9.5 .9863 .1650 1.082 10 1.067 7.024 142 .7880 .6157 .250 5 .197 .750 169 - .9816 .1908 .521 10 - .511 1.629 215 - .8192 - .5736 .950 10 .778 5.415 Apri l 27 - 28 65 0.4226 0.9063 0.525 10 0.222 1.634 116 - .4384 .8988 .250 5 - .109 3.308 131 - .6561 .7547 .930 10 - .610 5.189 180 1.0000 ,0000 .500 10 .500 1.500 TABLE VI (cont'd) irection Distance Time j e cose sine r t rcose r \ ine 13 -14 54 0.5878 0.8090 0.904 0.531 4.903 87 .0523 .9986 .307 5 .162 1.131 90 .0000 1.0000 1.398 10 .000 8.388 92 - .0349 .9994 .920 10 - .032 5.078 103 - .2250 .9744 .316 5 - .071 1.198 115.5 - .4305 .9026 .935 10 - .402 5.245 136 - .7193 .6947 1.822 10 -1.311 19.918 270.5 .0087 - 1.0000 .410 5 .004 2.017 281.5 .1994 - .9799 .315 5 .063 1.191 308 .6157 - .7880 .310 5 .191 1.153 Direction e June 29 -30 36.5 68 106 211 231 307 333.5 336.5 July 9 - 1 0 31.5 51 70.5 92 105 309 354 cose 0.8039 .3746 .2756 .8572 .6293 .6018 .5519 .9171 .8526 .6293 . 3338 .0349 .2588 .6293 .9945 TABLE VI (cont'd) 2 sine r t rcose r / t 0.5948 1.012 10 0.814 6.145 .9304 .289 5 .108 1.002 .9613 1.000 10 - .276 6.000 .5150 .388 10 - .333 .903 .7771 .269 5 - .169 .868 .7986 .277 5 .167 .921 .4462 .280 5 .154 .941 .3987 .289 5 .265 1.002 0.5225 0.260 5 0.222 0.811 .7771 1.429 10 .899 12.252 o9426 .723 10 .241 3.136 .9994 .350 5 - .012 1.470 .9659 .350 5 - .091 1.470 .7771 .302 5 .190 1.094 .1045 1.010 10 1.000 6.120 -50-July 9-10 (see Table IV), at an a = 0.05. Watson and Williams (1956) give a test for the homogeneity of two polar vectors. When the polar vectors are not known then the maximum l i k e -lihood estimate of the polar vector given by the resultant vector must be used. To test for equality, they show that where: N = Number of vectors = 11 for Apri l 27-28 and July 9-10 data, R-j.Pw, = Resultants of the two groups of vectors R = Total resultant vector For R, (April 27-28) N ( N - 1 ) ( N - 2 ) [ ( R 1 + R 2 ) 2 - R 2 ] [R - ( R r R o ) 2 ] (5.4) ( N 2 - R 2 ) [ N 2 R 2 - ( R ^ - R o 2 ) 2 ] where R Ix zr cose R. zr sine and similar ly Ro for July 9-10 data. then R Using the data in Table V, R = 4.420 Then from 5.4 » x 2 = 0.201 From Rohlf & Sokal (1969) x 2 = 3.841. 1..05 Hence R-, and R? can be considered as belonging to the same popu-lat ion . So as to provide some comparison, a , the mean square dispersion c o e f f i c i e n t and V, the d i r e c t i o n a l v e l o c i t y , are c a l c u l a t e d f o r a l l data combined, as we l l as t h a t f o r only A p r i l 27-28 and J u l y 9-10. For a l l data combined, using the values f o r r , t and c o s e i n Table V, then from equation 5.1 o 2 - 1 a = 1.172 m i l e s hour f o r a l l s c h o o l s , p 2 - 1 a = 3.610 miles hour f o r moving sch o o l s o n l y S i m i l a r l y f o r the data o f A p r i l 27-28 and J u l y 9-10 combined, then 2 2 - 1 a = 0.896 m i l e s hour f o r a l l s c h o o l s , and ? 2 - 1 a = 3.178 miles hour f o r o n l y those schools moving. The d i r e c t i o n a l component, f o r a l l moving sch o o l s w i l l be „ _ £ r cos 6 if-V F t ( 5 = 0.493 m i l e s hour" 1 and f o r A p r i l 27-28 and J u l y 9-10 V = 1.731 miles hour" 1 A s k i p p e r , i n c o n s i d e r i n g whether to attempt r e l o c a t i o n o f a school found e a r l i e r by a search v e s s e l , w i l l be concerned w i t h how the s i z e o f the area i n which the school may o c c u r , a t a given l e v e l o f p r o b a b i l i t y , w i l l change with time. From equation 5.1 i t can be seen t h a t the range a t time t w i l l be, Graph I shows how R v a r i e s with t f o r p r o b a b i l i t i e s o f 0.05 and 0.50, Equation 5.0 makes no assumptions about d r i f t i n d i r e c t i o n by s c h o o l s . Equation 5.5 gives the magnitude o f the d i r e c t i o n a l d r i f t , but not the d i r e c t i o n . I f the d i r e c t i o n of the school movement i s known with c e r -t a i n t y , and i s of magnitude V, then a t time t a f t e r i n i t i a l l o c a t i o n , the R (5.6) GRRPH I RRNGE LIMIT OF SCHOOL OCCURRENCE RT SPECIFIED PROBABILITY o ui. a C3 . CO 2 1 a CD CC cr: MEAN SQURRE DISPERSION =3.610 + + ^ ,+ ^vx**** 4-vXXXX>< x* 1 1 1 1 1 1 0.0 10.0 20.0 30.0 40.0 50.0 60 TIMECHOURS) + CONFIDENCE LIMITS = 0.50 X CONFIDENCE LIMITS = 0.05 -53-p r o b a b i l i t y d i s t r i b u t i o n w i l l s t i l l be a b i v a r i a t e normal, as given by Jones (1959), except t h a t the c e n t e r o f the d i s t r i b u t i o n w i l l be d i s p l a c e d Vt i n the r e s u l t a n t d i r e c t i o n , e, where e = a r c tan'*; r' s i"° (5.7) z r cose v ' ( S a i l a and Flowers, 1968). In an o p e r a t i o n a l s i t u a t i o n the d i r e c t i o n o f the d r i f t i s u n l i k e l y to be known f o r c e r t a i n , e i t h e r because o f l a c k o f obser-v a t i o n s and/or because the d i r e c t i o n o f f i s h movement probably i n v o l v e s some s t o c h a s t i c element. Gumbel, Greenwood and Durand (1953) show t h a t the c i r -c u l a r normal d i s t r i b u t i o n provides a good d e s c r i p t i o n o f the p r o b a b i l i t y o f a school moving i n a p a r t i c u l a r d i r e c t i o n , such t h a t k cos(e - e r t) f ( e ) ' 2 * i 0 ( k ) <6-8> where: k = parameter, analogous to the i n f o r m a t i o n measure, and i n d i c a t e s the degree o f c o n c e n t r a t i o n 6 Q = modal or most l i k e l y d i r e c t i o n I Q ( k ) = Bessel f u n c t i o n o f the f i r s t kind o f pure imaginary argument The d i r e c t i o n o f the r e s u l t a n t i s determined by the s i g n o f x and y o f the v e c t o r mean, (x, y ) , where: x = — E X . n l y = — Ey. J n Ji where from the p o l a r t r a n s f o r m a t i o n , x = r cose y = r s i n e Using equation 5.7 and the data from Table V, then f o r a l l s c h o o l s , the mean v e c t o r a n g l e , e Q , which i s independent o f the number o f s t a t i o n a r y -54-s c h o o l s , i s e = 64° 39' o For the A p r i l 27-28 and J u l y 9-10 o b s e r v a t i o n s o n l y , then from equation 5.7; e Q = 33°42' For both r e s u l t s x" and y are p o s i t i v e , hence the r e s u l t a n t d i r e c -t i o n i s i n the f i r s t quadrant. Gumbel e t a l (1953) give t a b l e s t h a t s p e c i -f y k, the measure of c o n c e n t r a t i o n of d i r e c t i o n , i n terms o f the mean v e c t o r s t r e n g t h , a", where: fj> ~J~' a = N| x + y For a l l d a t a , _ a = 0.093 From the t a b l e s o f Gumbel e t . a l . (1953) then, the maximum l i k e l i -hood estimate o f k i s : k = 0.1868 For the A p r i l 27-28 and J u l y 9-10 v e c t o r s combined, a = 0.114 and from Gumbel e t a l (1953), k = 0.2285 Graph II p l o t s f ( e - Q Q) as a f u n c t i o n o f (e - 9Q) f o r these two values o f k. As can be seen, the curve with k f o r a l l data i s l e s s c o n c e n t r a t e d about the modal d i r e c t i o n than t h a t with k f o r the combined data of A p r i l 27-28 and J u l y 9-10. In a f i s h i n g system, an o p e r a t o r would be i n t e r e s t e d i n e s t a b l i s h -ing a confidence i n t e r v a l on the modal d i r e c t i o n o f the f i s h s c h o o l s so as to o b t a i n a measure o f the a r c i n which the d i r e c t i o n of movement would be expected a t some p r o b a b i l i t y l e v e l . T h i s combined with the r e l a t i o n f o r range (equation 5.4) would enable the area o f the expected l o c a t i o n o f the GRAPH II PROBABILITY DF DIRECTION RS FUNCTION OF ANGULAR DIFFERENCE TO MODAL DIRECTION CM tn a a' 180.0 T T 120.0 60.0 0.0 K - 0 . 1 8 6 8 PLL ORTR „ K«o.22B5 APRIL 2 7 - 2 8 RNO JULY 9 - 1 0 DflTfi 60.0 T 1 120.0 180.0 DEGREES FROM MODAL DIRECTION -56-school to be c a l c u l a t e d a t a given p r o b a b i l i t y l e v e l . Watson and Will i a m s (1956) give an a l g o r i t h m so t h a t a r e s u l t a n t v e c t o r R Q can be found, such t h a t P(R > R Q | X) = a where N z cos e.. R = r e s u l t a n t v e c t o r a = s p e c i f i e d l e v e l o f p r o b a b i l i t y They show t h a t N-R I^-V P(R > R 0 I X) ( ^ ( 5 . 9 ) I f a = . 0 5 , then f o r a l l data combined; from equation 5 . 9 R = 6 . 9 8 7 o N Now R Q cos(X - R Q) = . ^ c o s 6. ( 5 . 1 0 ) (Watson and W i l l i a m s , 1 9 5 6 ) For = 0 . 0 5 , l e t X - Ro - 6.05 Where 6 05 = n a ^ a r c °f con f i d e n c e i n t e r v a l on d i r e c t i o n , Then from equation 5 . 1 0 , 9 . 0 5 = 4 3 0 3 8 Hence the confidence zone f o r a = . 0 5 i n c l u d e s those angles l e s s than 4 3 ° 3 ' away from the r e s u l t a n t d i r e c t i o n . For a p r o b a b i l i t y o f 0 . 5 9 then e < 5 0 = 40-41' -57-With the A p r i l 27-28 and J u l y 9-10 data f o r a l l s c h o o l s , then f o r = 0.05, 9.05 = 6 7 0 5 1' For a p r o b a b i l i t y o f 0.5 e = 44°48' .50 Hence f o r a f i s h e r y to which Royce and Otsu's (1955) data p e r t a i n s an o p e r a t o r would have a v a i l a b l e i n f o r m a t i o n on the r a t e o f movement o f the s c h o o l s , t h e i r modal d i r e c t i o n , i f s i g n i f i c a n t , and confidence i n t e r v a l s on t h i s d i r e c t i o n . T h i s i n f o r m a t i o n w i l l be used i n s e c t i o n 6.1 as a b a s i s f o r determining p o s s i b l e f i s h i n g t a c t i c s . For the combined data of A p r i l 27-28 and J u l y 9-10 the mean square d i s p e r s i o n r a t e was 76.5% of t h a t f o r a l l data combined f o r a l l s c h o o l s . However, the d i r e c t i o n a l v e l o c i t y of the A p r i l - J u l y data was 351.1% t h a t f o r a l l s chools combined. These f i g u r e s c o u l d i n d i c a t e t h a t t o t a l movement by the schools represented by the two s e t s o f data i s not as d i f f e r e n t as the values f o r the d i r e c t i o n a l v e l o c i t y i n d i c a t e , but might be e x p l a i n e d by the f a c t t h a t movement by the schools of the A p r i l 27-28 and J u l y 9-10 observa-t i o n s had a g r e a t e r component i n d i r e c t i o n a l r a t h e r than random movement. There may w e l l be a f u n c t i o n a l r e l a t i o n s h i p between these two components o f movement with a b a s i s i n terms o f the f i s h a c t i v i t y , e.g. f e e d i n g or s e a r c h i n g . The confidence i n t e r v a l f o r a l l schools combined at the 95% l e v e l o f s i g n i f i c a n c e was o n l y 4°40' wider than t h a t f o r the 50% l e v e l o f s i g n i f i -cance, 86°6' versus 81°26'. However, the confidence i n t e r v a l o f the A p r i l -J u l y combined data a t the 95% l e v e l was c o n s i d e r a b l y g r e a t e r than f o r the 50% i n t e r v a l , 125°42" versus 89°36', a d i f f e r e n c e o f 3 6 ° 6 \ -58-At both l e v e l s o f s i g n i f i c a n c e , the c o n f i d e n c e i n t e r v a l o f the A p r i l - J u l y data i s longer than t h a t f o r a l l data combined. The reasons f o r the c o n f i d e n c e i n t e r v a l s f o r the t o t a l data being s m a l l e r than those f o r the combined homogeneous data are not c l e a r . They do not appear to be e x p l i c i t -l y r e l a t e d to the r e l a t i v e number of moving s c h o o l s . I f equation 5.5 i s e v a l u a t e d f o r moving sch o o l s o n l y , the confidence i n t e r v a l f o r a l l observa-t i o n s combined i s l o n g e r than t h a t f o r the A p r i l 27-28 and J u l y 9-10 combined d a t a , 120°41' versus 114°2' r e s p e c t i v e l y (compared with 86°6' and 125°42 ! r e s p e c t i v e l y f o r a l l s c h o o l s ) at the 95% l e v e l of s i g n i f i c a n c e . At the 50% l e v e l of s i g n i f i c a n c e , f o r moving s c h o o l s on l y , t h i s s i t u a t i o n i s r e v e r s e d with the c o n f i d e n c e i n t e r v a l f o r the A p r i l 27-28 and J u l y 9-10 data being g r e a t e r than f o r a l l o b s e r v a t i o n s combined, 74°8' versus 73°16' r e s p e c t i v e l y (compared with 89°36' and 81°26' r e s p e c t i v e l y f o r a l l s c h o o l s c o n s i d e r e d ) , In s e c t i o n 6.1 the c o n f i d e n c e i n t e r v a l s f o r the A p r i l - J u l y obser-v a t i o n s u s i n g a l l schools are used to develop a b a s i s f o r determining f i s h -i n g t a c t i c s . 6.0 D e c i s i o n Making In f i s h i n g , d e c i s i o n making i s g e n e r a l l y a continuous p r o c e s s . In r e a l s i t u a t i o n s s k i p p e r s may have to choose betv/een many p o s s i b l e a c t i o n s given c o n s t r a i n t s which are o f t e n dynamic i n nature. W i t h i n the bounds o f these c o n s t r a i n t s d e c i s i o n s are u s u a l l y based on past e x p e r i e n c e , or i n t u i t i o n , although i t i s not unknown f o r c o n s t r a i n t s to be i g n o r e d , i . e . f o r v e s s e l s to run out of f u e l on the r e t u r n t r i p , with d i s a s t r o u s conse-quences. -59-The options open to a s k i p p e r i n a t y p i c a l s i t u a t i o n might be: (1) to s t a r t or continue f i s h i n g i n some area (2) to change ground i n a n t i c i p a t i o n o f encountering h i g h e r catch r a t e s . T h is may i n v o l v e a c h o i c e between a number of grounds (3) to stop f i s h i n g and r e t u r n to p o r t (4) to change gear, e.g. demersal to p e l a g i c trawl (5) to steam to avoid p o s s i b l e bad weather A l l o f these c o n s i d e r a t i o n s may have to be c o n s i d e r e d s i m u l t a n e o u s l y , weigh-ted by f u r t h e r c o n s i d e r a t i o n s such as crew f a t i g u e . In d e c i s i o n making c e r t a i n f a c t o r s w i l l be i n v a r i a n t , f o r example bad weather or f u e l l i m i t a t i o n s . However the dominant c o n s i d e r a t i o n s w i l l g e n e r a l l y be the catch r a t e , i . e . f i s h d e n s i t y p r e s e n t , and here s t r a t e g i e s w i l l be c o n s i d e r e d i n terms of the p r e s e n t catch r a t e and t h a t which may e x i s t elsewhere. Three s i t u a t i o n s w i l l be d i s c u s s e d here with r e g a r d to f i s h i n g t a c t i c s : (a) Where the mean square d i s p e r s i o n r a t e , d i r e c t i o n a l v e l o c i t y and modal d i r e c t i o n with confidence l i m i t s are known, as f o r Royce and Otsu's (1955) data. (b) Where catch r e s u l t s from t r a n s e c t data i s a v a i l a b l e as f o r C r a i g and Graham's (1965) tuna d a t a . (c) When options between d i f f e r e n t grounds are a v a i l a b l e , as i n the Vancouver trawl f i s h e r y given c e r t a i n assumptions. 6.1 Royce and Otsu's (1955) TUha Data The range w i t h i n which a school may be expected at a given l e v e l o f p r o b a b i l i t y , i s given by equation 5.6 and equation 5.9 enables a confidence -60-i n t e r v a l on the expected d i r e c t i o n of school movement to be found f o r a given s i g n i f i c a n c e l e v e l . Two s i t u a t i o n s can now be c o n s i d e r e d : (a) A c a t c h e r v e s s e l i s a c e r t a i n d i s t a n c e , t hours steaming, from a p o s i t i o n where a f i s h c o n c e n t r a t i o n has been l o c a t e d and i t i s wished to s p e c i f y the area ( s i z e and p o s i t i o n ) i n which the f i s h are l i k e l y to be, at some l e v e l o f p r o b a b i l i t y , by the time o f a r r i v a l o f the c a t c h e r v e s s e l . (b) A c a t c h e r v e s s e l i s s e a r c h i n g on another ground, t hours steaming away and must decide i f a b e t t e r catch w i l l be obtained by s t a y i n g on i t s p r e s e n t grounds r a t h e r than steaming to the l o c a t i o n o f the f i s h c o n c e n t r a t i o n . The l o c a l i t y i n which the school would be expected to o c c u r , at some l e v e l o f p r o b a b i l i t y at time t a f t e r i t s i n i t i a l l o c a t i o n , would be centered around the p o i n t Vt from the position-where o r i g i n a l l y l o c a t e d , where V i s the d i r e c t i o n a l v e l o c i t y o f the s c h o o l , and t i s the time taken f o r the c a t c h e r v e s s e l to a r r i v e . The shape o f t h i s area i s i n d i c a t e d i n f i g u r e 5. The s i z e of t h i s area w i l l be, 2e {n(Vt + R ) 2 - n(Vt - R) 2} A = 2n = 48 VtR where 6 = h a l f confidence i n t e r v a l given by equation 5.10 measured i n s e c t i o n s R = range given by equation 5.6 Graph I I I p l o t s the change i n the s i z e of t h i s area with time f o r p r o b a b i l i t i e s o f 0.50 and 0.95 f o r the A p r i l 27-28 and J u l y 9-10 data where 5 Q = 44°48' and 6 n c = 67°51 -61-Figure 5 0 0 = position where originally located Nature of area in which school would be expected i f directional velocity and modal direction with confidence limits known. GRAPH I I I HREfl OF POSSIBLE SCHOOL OCCURRENCE AS R FUNCTION OF TIME -63-The time for the catcher vessel to reach the area of probable school occur-rence w i l l depend on the speed of the vessel , the distance to where the school was i n i t i a l l y located, and the speed and direction of the f ish school The time for the vessel to close on the school w i l l be T . J U T V where V = speed of catcher vessel v - -tan (180- <fr)D * ~ tane - tan (180-$) y = tan x D = distance between school and vessel when located 6 = direction vessel must go to close on the school, relative to direction school or ig inal ly at <J> = direction of school relative to i n i t i a l position of catcher vessel For situation (b) let m- be the minimum size of a f ish school worth exploit ing, and assume that the value of the school is proportional to the school s i ze , n. Hence, catcher vessels are concerned with schools size n, n _> m. Assume that the probability of encountering a school size n, is as for Craig and Graham's (1965) data, adequately described by the log ser ies, The expected number of schools that would be encountered on the present ground of the catcher vessel during this time w i l l be given by,-xvtw where v = speed of catcher vessel t = time to change grounds - 64 -w = cross-sectional area or width of search by vessel X = expected school density The expected number of schools of utt l tzable size encountered during this period would be . x h v A t w z inHn v \ n=m - log ( l - x ) Hence, to be worthwhile changing grounds then x n V A t W l , V . < 1 (6.1) n=m - iog ( l - x ) Implicit in the inequality 6 . 1 , is that the density of f ish schools and the distr ibution of t h e school sizes is s imilar for the two different areas. Then for a given vessel , the decision to seek the previously located school w i l l depend on the time to change grounds, a function of the distance and vessel speed. Using the estimates ofx , V and w obtained in section 2.2 graph IV plots the two sides of the inequality 6.2. Graph IV indicates, as would be expected, that as the number of schools which are of exploitable size become fewer, then the steaming time for which i t is s t i l l beneficial to change grounds increases. In such cases, only i f the school is very distant does i t become a better pol icy to remain searching on the present grounds. 6.2 Craig and Graham's (1965) Tuna Data In this section Bayes1 Method w i l l be used in making predictions about f ish density in relat ion to f ishing tact ics . The potential and appl icabi l i ty of such methods are considered further in the discussion. The basis of this method i s to make estimates of f ish density at any particular time dependent both on the sample results obtained and on some GRAPH IV IF CURVE LESS TfflN fcl . BEST 3TRRTEGY IS TO CHANGE GROUMDB M - MN1HL* REUttlVE SI2E CF SCHCflL WORTHWHILE TO^PLOIT i c n c n i n 0 4 0 8J) 12.0 16.0 20.0 T IME TO STEAM TO A L T E R N A T E GROUNDS (HOURS) 24.0 -66-prior expectation of resul ts . Obviously at time t=0, the expected value of the f i sh density w i l l be expressed entirely by the prior density. E x p l i c i t l y , i f X = population density over the f ishing grounds and x . = sample results , then P(A|x 1 ) « p ( x .JA) p (A) Lindley, 1965b) where: P.(*I.Xf) = posterior probability density of A given the sample data P ( X . | A ) = l ikel ihood of the sample data i f A i s the actual f ish density ?M = prior probabil ity of A being the f ish density. For t rac tab i l i t y i t w i l l be assumed that the catch results of Craig and Graham (1965) follow a Poisson d ist r ibut ion, although better f i t s were obtained with the negative Binomial and Neyman's Type A distr ibut ion. For some sample x . , the Poisson probability is -A * 1 P(X< = Xt) - -6 * X j ' At time t= i , the l ikel ihood of obtaining the i sample results w i l l be , v 1 e"x>X j* . e ^ A £ X J Inherent in formulating the prior distr ibution is the use of sub-jective feelings about the possible state of some system - a f ishing ground in the absence of current data. The subjective feelings of a skipper may -67-be based on many f a c t o r s , though probably most important w i l l be the catch s i z e d u ring the same p e r i o d i n previous seasons. To make use o f a s k i p p e r ' s p r i o r b e l i e f s they must be expressed q u a n t i t a t i v e l y i n terms o f some c o n d i -t i o n a l d e n s i t y . Assume t h a t the s k i p p e r ' s expected catch r a t e i s 0.25 sc h o o l s per 5 mi l e sample. ( I t i s known from the data t h a t the ac t u a l o v e r a l l mean i s 0.239). To s p e c i f y the p r i o r d e n s i t y f u n c t i o n i t i s r e q u i r e d t h a t the s k i p p e r s t a t e h i s "degree o f b e l i e f " i n the range o f values t h a t the mean catch r a t e , x, may take. L i n d l e y (1965) d i s c u s s e s the concept o f degrees o f b e ! i e f i n d e t a i l . The p r i o r d e n s i t y f u n c t i o n should cover the range 0 to °°, and p r e f e r a b l y be unimodal. Here a r - f u n c t i o n i s used as b e i n g c o n v e n i e n t , amenable to ma n i p u l a t i o n and with t a b l e s a v a i l a b l e . However, i t i s con-c e i v a b l e t h a t i n some s i t u a t i o n s the p r i o r d e n s i t y may not be unimodal. Under t h i s assumption, p r o b a b i l i t y o f X i s gi v e n by P( X) = e (a) { 6 2 ) T (a)a where a , a are the parameters o f the r - f u n c t i o n . The mean o f the d i s t r i b u t i o n i s (a+1) and the v a r i a n c e i s (a+l)a , To s p e c i f y the d i s t r i b u t i o n parameters, i t i s necessary to equate equation 6.2 to some value. Assume t h a t the s k i p p e r i s 95% c e r t a i n t h a t the catch r a t e w i l l be above some value X i . e . oo / f ( X ) dX = .95 X.05 -68-0 r \05 / f ( x ) dx = .05 (6.3) 0 The parameters o f 6.3 can then be s p e c i f i e d from incomplete r - f u n c t i o n t a b l e s , We have: • ; 0 5 ^ ' ? ' ° ' 1 dx =0.05 , 0 r (a) a Then f o r (a+ l)a 2 = 0.25, from Pearson's (1946) t a b l e s a = 0.300 a =0.192 ' To e v a l u a t e the P o s t e r i o r p r o b a b i l i t y d e n s i t y , we have i x . . - x i . , z x j " a (-)a_1 pUlx,) = e . X ~ — n x . l r (a) a - -AT Z X j a (£) / e X e q dx n x.! J r(a)a i i i ' - x ( i + r ) EX. + o - l i £X.+a e ° x J -d4 ) J (6.4) r (lx,+a) J The expected v a l u e o f x a f t e r sample i w i l l be -69-E (x| X i) = / Xp(x|x.)dX (6.5) Substituting 6.4 for p.(x|x^) in 6 .5 , E(x|x.) - / 1 n -X(l + -) Ex. +a - 1 , IX.-kx oo O J i_» J (1 + a> Xe X^  (1 + a' dx r ( a + i x . ) J I a + E X . i + f (6.6) As for a r -d ist r ibut ion: mean = (a+l)a 2 variance = (a+l)o „ _ variance tnen o = — — — — mean For a given mean, a w i l l be proportional to the variance. The more vague are the feelings about the prior density of x, the larger the variance w i l l be; a w i l l become larger and a smaller. It can be seen from equation 6.6 that as the variance of the prior distr ibution becomes larger, the relative contribution that the prior distr ibution w i l l make to the expected value becomes smaller. As the number of observations increases, the expected value of the posterior probability w i l l approach the mean of the sample results. -70-The expected value of the f i s h d e n s i t y u s i n g equation 6.6 and the average f i s h d e n s i t y as samples were obtained f o r each day have been c a l c u l a t e d from C r a i g and Graham's (1965) data, and are l i s t e d i n Table VII f o r each day. A sample length o f 5 m i l e s was used f o r these sample r e s u l t s . A s k i p p e r or search o p e r a t o r , would be concerned with e s t a b l i s h i n g whether the expected value f o r the f i s h d e n s i t y i s w i t h i n some con f i d e n c e i n t e r v a l , f o r i f not i t may be a d v i s a b l e t o s h i f t grounds or recommend th a t the r e s t o f the f l e e t move to areas where a g r e a t e r f i s h d e n s i t y may be encountered. I t i s the lower l e v e l o f the c o n f i d e n c e i n t e r v a l which i s of r e l e v a n c e , as l a r g e r values of the f i s h d e n s i t y w i l l i n v o l v e no l o s s . For 50% confidence i n t e r v a l , then A cn - x ( i + -) EX.+ct-l 1 EX.+ct-/ • 5 ° e q X J (1+ o) 1 dx r (E x, +<x) w 0.50 v + I 1 I J A c n - X ( i + o) Ex.+ a - l = U j L a ) _ _ 0/ 5 0 e X ' d X (6.7) r(EX j. +a) 1 S e t t i n g t = x ( i +a), then f o r the e x p r e s s i o n under the i n t e g r a l we have Z X j + a - l *.50 e -t(-4)-0 i + a i + a / 1 \ EX.+a t c n . EX.+a-l = ( T} J /- 5 0 e _ t t J d t TABLE VII Expected d e n s i t y values f o r sample d a t a , x s ; Bayes estimate X g , and lower value o f confidence i n t e r v a l f o r Poisson d i s t r i b u t i o n f o r a c o n f i d e n c e i n t e r v a l o f .95 (BP g 5 ) ; s i m i l a r l y f o r the Bayes confidence i n t e r v a l o f .95 n (B g j ) , and f o r a 50% c o n f i d e n c e l i m i t (B f o r s u c c e s s i v e days o f sampling. Day 1 Sample x x P No. s B .95 0 0.250 1 0.000 .210 0.000 2 .000 .181 .000 3 .000 .159 .000 4 .000 .141 .000 5 .200 .225 .000 6 .333 .295 .000 7 .428 .352 .022 8 .500 .401 .089 9 .444 .373 .079 10 .500 .414 .132 11 .545 .451 .179 12 .588 .482 .221 13 .538 .456 .204 14 .500 .432 .189 15 .466 .411 .177 16 .438 .391 .165 B B X X .95 .50 s B 0.250 0.020 0.158 0.000 .210 .018 .137 .000 .181 .014 .119 .000 .159 .013 .106 .000 .141 .046 .193 .000 .127 .086 .266 .000 .116 .123 .327 .000 .107 .162 .378 .125 .174 .149 .351 .111 .162 .182 .392 .100 .151 .220 .429 .091 .142 .242 .462 .083 .134 .235 .440 .077 .126 .220 .416 .071 .120 .211 .396 .067 .114 .200 .374 .062 .108 P .95 B .95 B .50 X s i.OOO 0.20 0.158 0.000 .000 .018 .137 .500 .000 .014 .119 .333 .000 .013 .106 .250 .000 .012 .096 .200 .000 .010 .088 .166 .000 .010 .081 .142 .000 .037 .149 .125 .000 .033 .139 .111 .000 .030 .129 .100 .000 .030 .122 .181 .000 .027 .114 ,166 .000 .027 .108 .153 .000 .024 .103 .143 .000 .024 .098 .133 .000 .022 .093 .125 Day 3 x P B B B .95 .95 .50 .250 .210 0.000 0.020 0.158 .319 .000 .070 .276 .280 .000 .056 .242 .250 .000 .051 .216 .225 .000 .046 .193 .205 .000 .041 .175 .189 .000 .041 .162 .174 .000 .037 .149 .162 .000 .033 .139 .151 .000 .030 .129 .198 .000 .058 .184 .192 .000 .056 .172 .181 .000 .052 .164 .172 .000 .051 .155 .163 .000 .047 .147 .156 .000 .046 .139 Day 6 as f o r Day 5 Day 4 Sample x x P No. s B .95 0 0.250 1 0.000 .210 0.000 2 .000 .181 .000 3 .000 .159 .000 4 .000 .141 .000 5 .200 .225 .000 6 .167 .205 .000 7 .143 .189 .000 8 .250 .250 .000 9 .333 .303 .000 10 .300 .283 .015 11 .364 .327 .014 12 .333 .308 .059 13 .308 .291 .055 14 .286 .276 .051 15 .267 .263 .047 16 .250 .250 .044 TABLE VII Day B B X X .95 .50 s B 0.250 0.020 0.158 0.000 .210 .018 .137 .000 .181 .014 .119 .000 .159 .013 .106 .000 .141 .046 .193 .000 .127 .041 .175 .000 .116 .041 .162 .000 .107 .037 .149 .000 .098 .070 .209 .000 .092 .099 .261 .000 .086 .096 .246 .000 .080 .123 .290 .000 .076 .118 .273 .000 .071 .110 .258 .000 .068 .106 .246 .000 .064 .100 .235 .000 .047 Conti nued 5 p .95 B .95 B .50 X s i.OOO 0.020 0.158 0.000 .000 .018 .137 .000 .000 .014 .119 .000 .000 .013 .106 .000 .000 .012 .096 .000 .000 .010 .088 .000 .000 .010 .081 .143 .000 .009 .074 .125 .000 .008 .070 .222 .000 .008 .065 .300 .000 .008 .061 .273 .000 .007 .057 .250 .000 .007 .054 .231 .000 .006 .051 .214 .000 .006 .049 .200 .000 .006 .046 .188 Day 7 x P B B B .95 .95 .50 i.250 .210 0.000 0.020 0.158 .181 .000 .018 .137 .159 .000 .014 .119 .141 .000 .013 .106 .127 .000 .012 .096 .116 .000 .010 .088 .189 .000 .041 .162 .174 .000 .137 .149 .232 .000 .070 .209 .283 .015 .099 .261 .265 .014 .096 .246 .250 .013 .087 .230 .236 .012 .086 .218 .224 .011 .078 .207 .213 .010 .077 .197 .203 .009 .070 .187 TABLE VII Continued Day 9 as for Day 5 Day 8 nple X x P No. s B .95 0 0.250 1 0.000 .210 0.000 2 .500 .305 .000 3 .333 .280 .000 4 .250 .250 .000 5 .200 .225 .000 6 .333 .295 ' .000 7 .286 .270 .000 8 .250 .250 .000 9 .333 .303 .000 10 .300 .283 .000 11 .273 .265 .000 12 .250 .250 .000 13 .231 .236 .000 14 .214 .223 .000 15 .200 .213 .010 16 .188 .203 .009 B B X X .95 .50 s B 0.250 0.020 0.158 1.000 .371 .070 .276 .500 .319 .056 .242 .333 .280 .051 .216 .250 .250 .046 .193 .200 .225 .041 .175 .167 .205 .077 .242 .143 .189 .071 .226 .125 .174 .070 .209 .111 .162 .063 .195 .100 .151 .058 .184 .181 .204 .056 .172 .167 .192 .052 .164 .231 .236 .051 .155 .214 .224 .077 .197 .200 .219 .070 .187 .250 .250 P .95 B .95 B .50 X s 0.000 0.077 0.319 0.000 .000 .070 .276 .000 .000 .056 .242 .000 .000 .051 .216 .250 .000 .046 .193 .200 .000 .041 .175 .167 .000 .041 .162 .286 .000 .037 .149 .375 .000 .070 .209 .333 .000 .063 .195 .300 .000 .058 .184 .273 .000 .056 .172 .250 .012 .086 .218 ,308 .011 .078 .207 .286 .010 .077 .197 .267 .044 .100 .235 .250 Day 11 x P B B B .95 .95 .50 1.250 .210 0.000 0.020 0.158 .181 .000 .018 .137 .159 .000 .014 .119 .250 .000 .051 .216 .225 .000 .046 .193 .205 .000 .041 .175 .270 .000 .077 .242 .326 .019 .118 .300 .303 .017 .106 .281 .283 .015 .099 .261 .265 .014 .096 .246 .250 .013 .087 .230 .291 .055 .118 .273 .276 .051 .110 .258 .262 .047 .106 .246 .250 .044 .100 .235 TABLE VII Continued Day 12 as for Day 2 Day 13 Day 14 Day 15 flple X X P B B X X P B B X X P B B No. s B .95 .95 .50 s B .95 .95 .50 s B .95 .95 .50 0 0.250 0.250 0.250 1 0.000 .210 0.000 0.020 0.158 0.000 .210 0.000 0.020 0.158 0.000 .210 0.000 0.020 0.158 2 .000 .181 .000 .018 .137 .500 .319 .000 .070 .276 .000 .181 .000 .018 .137 3 .000 .159 .000 .014 .119 .333 .280 .000 .056 .242 .000 .159 .000 .014 .119 4 .500 .359 .000 .106 .322 .500 .359 .000 .106 .322 .000 .141 .000 .013 .106 5 .600 .422 .030 .147 .390 .400 .324 .000 .096 .293 .000 .127 .000 .012 .096 6 .833 .563 .220 ,256 .532 .333 .295 .000 .086 .266 .000 .116 .000 .010 .088 7 .857 .598 .282 .290 .571 ,429 .352 .022 ,123 .327 .142 .189 .000 .041 .162 8 .875 .629 .331 .322 .603 .375 .326 .019 .118 .300 .250 .250 .000 .071 .226 9 1.000 .725 .452 .404 .701 .333 .303 .017 .106 .281 .333 .303 .017 .106 .281 10 1.111 .809 .554 .472 .787 .300 .283 .015 .099 .261 .500 .414 .132 .182 .392 11 1.091 .815 .504 .448 .737 .364 .327 .014 .096 .246 .455 .407 .120 .170 .367 12 1.083 .826 .525 .467 .752 .333 .308 .059 .123 .290 .417 .366 .110 .162 .347 13 1.077 .841 .544 .472 .765 .308 .291 .055 .118 .273 .385 .362 .102 .152 .327 14 1.000 .797 .505 .458 .730 .286 .276 .051 .110 .258 .357 .344 .094 .146 .309 15 .933 .757 .471 .429 .690 .333 .312 .088 .137 .294 .333 .312 .088 .137 .294 16 .875 .722 .442 .416 .657 .312 .297 .083 .131 .281 .312 .297 .083 .131 .281 TABLE VII Continued Day 16 as for Day 2 Day 17 Day 18 Day 19 Sample x x P B B x x P B B x x P B B No. s B .95 .95 .50 s B .95 .95 .50 s B .95 .95 .50 0 0.250 0.250 0.250 1 0.000 .210 0.000 0.020 0.158 0.000 .210 0.000 0.020 0.158 0.000 .210 0.000 0.020 0.158 2 .000 .181 .000 .018 .137 .500 .319 .000 .070 .276 .000 .181 .000 .018 .137 3 .000 .159 .000 .014 .119 1.000 .524 .050 .184 .484 .000 .159 .000 .014 .119 4 .000 .141 .000 .013 .106 .750 .467 .038 .164 .431 .000 .141 .000 .013 .106 5 .000 .127 .000 .012 .096 .600 .422 .030 .147 .390 .000 .127 .000 .012 .096 6 .167 .209 .000 .041 .175 .500 .384 .025 .134 .354 .000 .116 .000 .010 .088 7 .143 .188 .000 .041 .162 .429 .353 .022 .123 .327 .143 .189 .000 .041 .162 8 .250 .250 .019 .118 .300 .375 .326 .019 .118 .300 .250 .250 .000 .071 .226 9 .333 .303 .147 .200 .420 .333 .301 .017 .106 .281 .333 .303 .017 .106 .281 10 .500 .414 .132 .182 .392 .300 .283 .015 .099 .261 .400 .349 .071 .146 .327 11 .454 .407 .179 .220 .429 .364 .327 .065 .131 .306 .367 .327 .065 .131 .306 12 .500 .424 .164 .200 .404 .333 .308 .059 .123 .290 .333 .308 .059 .123 .290 13 .461 .401 .152 .189 .382 .462 .401 .102 .152 .327 .308 .291 .055 .118 .273 14 .429 .380 .141 .180 .363 .429 .380 .141 .180 .363 .286 .276 .051 .110 .258 15 .400 .361 .131 .170 .344 .467 .411 .177 .211 .396 .267 .262 .047 .106 .246 16 .375 .344 .123 .162 .330 .438 .392 .165 .200 .374 .250 .250 .044 .100 .235 -76-S u b s t i t u t i n g In 6.7, t. 50 E X + a _ | t-(x.Va) J e r t t J dt=0'5° (6'8) J W h e r e *0.50 = 7~T I T — a Hence the confidence i n t e r v a l at any time w i l l depend on the number o f samples and the sample r e s u l t s . The i n t e g r a l i n equation 6.8 has no c l o s e d form, so a numerical s o l u t i o n i s necessary. The r e s u l t s are given i n Table VII. S i m i l a r l y a lower confidence on X f o r a p r o b a b i l i t y o f ,05 was c a l -c u l a t e d and i s l i s t e d i n Table V I I . So as to pro v i d e a comparison to the confidence l i m i t on Bayes's e s t i m a t e , confidence l e v e l s on the sample mean values were a l s o c a l c u l a t e d . I f r schools are caught from each sample so t h a t r = 0, 1, 2, ...» and n R = i r , 1 then R w i l l be a poisson v a r i a b l e with mean nx (N. G i l b e r t , pers comm. ) e - n X ( n x ) R Then f(R) ^ with E(R) = nx, var (R) =nx Let X be the mean o f the sample data o b t a i n e d , then X = - , with n R v a r i a n c e = n The lower confidence l i m i t f o r 95% i n t e r v a l w i l l be 1 I n s t i t u t e o f Animal Resource Ecology, U.B.C, Vancouver, B.C. -77-R . TV where t = t - t a b l e value f o r 2 - t a i l e d t e s t at 90% c o n f i d e n c e i n t e r v a l . These values are l i s t e d i n Table V I I . The 50% c o n f i d e n c e l i m i t i s given by the sample mean, x From Table VI, as would be a n t i c i p a t e d , the expected value based on the Bayes estimate i s g r e a t e r f o r the i n i t i a l samples, i n which no s c h o o l s were encountered than t h a t based on the sample d a t a . A s i m i l a r t r e n d i s shown by the values f o r the lower confidence l i m i t on the mean value f o r p r o b a b i l i t i e s o f 95% and 50%. I f the number o f schools encountered per day are few, as f o r Days 2, 3, 5 or 7, then the Bayes estimate o v e r e s t i m a t e s the number o f s c h o o l s t h a t are encountered. When the number o f s c h o o l s caught i s g r e a t e r than the mean, then the Bayes estimate underestimates t h a t a c t u a l l y o c c u r r i n g . Comparisons between the 50% l i m i t f o r the Bayes e s t i m a t e and the 50% l i m i t f o r the Poisson e s t i m a t e , given by the sample mean, show a s i m i l a r r e l a t i o n s h i p to t h a t between the sample mean and the expected Bayes v a l u e , except t h a t the 50% l i m i t i s s l i g h t l y s m a l l e r than the expected Bayes value, The 95% lower c o n f i d e n c e l i m i t f o r the Poisson d i s t r i b u t i o n i s always l e s s than t h a t f o r the Bayes l i m i t at the same p r o b a b i l i t y , except on Day 13 when the g r e a t e s t number o f s c h o o l s were encountered. However, s i g n i f i c a n c e values a t c e r t a i n l e v e l s o f p r o b a b i l i t y do not provide a b a s i s f o r d e c i s i o n making per se. A s s o c i a t e d with any p a r t i -c u l a r value o f f i s h d e n s i t y w i l l be some d e c i s i o n , e.g. to change grounds or not, o r to d i r e c t o t h e r v e s s e l s to a l t e r n a t e grounds i n the chance o f encountering b e t t e r c a t c h r a t e s . A s s o c i a t e d w i t h each d e c i s i o n w i l l be a -78-p o s s i b l e l o s s , as the d e c i s i o n on x might not be the best s i n c e x i s a random v a r i a b l e . In making a d e c i s i o n on X, i t i s hoped to minimize any p o s s i b l e l o s s (or maximize any u t i l i t y ) . In a f i s h i n g system a l o s s c o u l d accrue i n three ways: (1) In r e t u r n i n g to p o r t when cat c h r a t e s d e c l i n e d , when they may have subsequently improved. Such a s i t u a t i o n might a r i s e when there was i n s u f f i c i e n t f u e l to change grounds. The l o s s would be p r o p o r t i o n a l to the d i f f e r e n c e i n catch r a t e s o b t a i n e d compared with those expected on the next t r i p . (2) In changing grounds u n n e c e s s a r i l y , as when f i s h d e n s i t y has been underestimated. (3) O v e r e s t i m a t i n g the catch r a t e and remaining on the prese n t grounds when a high e r catch r a t e c o u l d be expected on a l t e r n a t e grounds. In t h i s t h e s i s only s i t u a t i o n s 2 and 3 w i l l be c o n s i d e r e d , i . e . the s k i p p e r has a choice o f only two d e c i s i o n s , t o change grounds, d c , or to remain, d r . In changing grounds u n n e c e s s a r i l y , the l o s s w i l l then be propor-t i o n a l to f i s h i n g time l o s t i n changing grounds plus the product o f d i f f e r -ence i n catch r a t e s between grounds p l u s the time necessary to "make a t r i p " on the a l t e r n a t e ground, i . e . L c = c [ t x + t t ( x - * ) ] (6.9) where = l o s s i n changing grounds c = constant of p r o p o r t i o n a l i t y t = time to change to a l t e r n a t e grounds X = a c t u a l f i s h d e n s i t y on present grounds tj.= time to 'make a g r i p ' on a l t e r n a t e grounds $ = expected catch r a t e on the a l t e r n a t e ground. -79-I f the f i s h d e n s i t y on the a l t e r n a t e ground i s expressed as a f u n c t i o n o f the d e n s i t y on the present ground, i . e . , X = a * (6.10) then 6.9 can be expressed as l_ c = c * [ a t + t t (a - 1)] In a r e a l f i s h e r y , t ^ , the time to 'make a t r i p ' i s a complex v a r i a b l e dependent on such thin g s as crew f a t i g u e , market p r i c e s , r a t e at which the catch i s obtained as we l l as the a c t u a l f i s h d e n s i t y encountered. The s o l i d l i n e s i n Graph V shows how r e l a t i v e l o s s v a r i e s as a f u n c t i o n o f t f o r d i f f e r e n t values o f a, i f i t takes 10 days to make a t r i p a t a f i s h d e n s i t y o f i> . When the catch r a t e i s ov e r e s t i m a t e d , and a hi g h e r catch r a t e would be found elsewhere, then the l o s s i n c u r r e d w i l l be p r o p o r t i o n a l to the product o f the d i f f e r e n c e between catch r a t e s and the time to 'make a t r i p ' s t a y i n g on the present grounds, l e s s the amount o f catch t h a t would be caught w h i l e the v e s s e l would otherwise be changing grounds, i , e , L r = c [ t t ' U-A) - tA] (6.11) where t ' t = time to make a t r i p on the present ground. Then equation 7.12 can be expressed as L r = c [ t t ' (*-att'<J>) - a t * ] = c * [ t t ' ( l - a ) - a t ] The d i s c o n t i n u o u s l i n e s i n graph V shows how the l o s s when remaining v a r i e s with a, i n equation 6,10. For each d e c i s i o n t h e r e w i l l be an 'expected l o s s ' due to un-c e r t a i n t y about the a c t u a l f i s h d e n s i t y . The best d e c i s i o n w i l l be t h a t which minimizes the expected l o s s (Mood and G r a y b i l l , 1965) -80-in CO CO I — I D I— <X I UJ Q; in a . i a i GRAPH V - RELATIVE LOSS WHEN CHANGING - RELATJVE LOSS WHEN REMAINING P f f l J L - — - — • — ^ — — rv-\ 4 — — " — " i i 1 1 1 0.0 5.0 10.0 15.0 20.0 TIME TO CHANGE GROUNDS (HOURS] 25.0 -81-The expected loss for a decision d , Is F ( d ) = / L(d, x) p (x|x) dx (Lindley, 1965b) From 6.9 , the expected loss in changing grounds w i l l be L"c = / c [tx + t t ( x -$) ] p (x|x) dx where p(x|x) is given by equation 7.6 Let a = i + -Then and b = EX. + a ~-Xa,b-l b L c = fc [tx + t t (X-*)] dX , a b , b+1 , b+1 , b . " r%) { r<b> + Vl> *M ~ V<?> ^ (6.12) From 6.11 the expected loss due to remaining w i l l be - ^Xa .b^ l . b L R = / c [ f t (t - x ) - tx] [ e * ( a ) a ] dx . .b . b , b+1 , b+1 = ffiT) { vt^V r ( b * l ) - t « t • ( ! ) r(b)-t(I) r(b) } (6.13) If L c < L r , then d £ is a better strategy than d r and vice versa. For d to be the better strategy, then 6.12 < 6.13 Removing similar functions from both expressions then a a(t ttj; + t ' ^ ) 5 0 b K 2t + t , + t ' ' -82-substituting for b and a , then (oi+l).Ct t^.t.tt tip) . j x . < — — — — ~ * a J a(2t + t t + t't ) From the inequality 7.16, i t can be seen that for a given Ex . the greater the expected f ish density on the alternate grounds,'!' , and/or the shorter the steaming time to these grounds, t ; then the larger w i l l be the right hand side of the expression, and hence,, more l i ke l y that the inequality does hold. In this case the better strategy w i l l be for the skipper to change grounds. However, as the time to steam to the alternate grounds increases, and/or the smaller the expected f ish density on.ithe alternate grounds then for a given EX. the less l i k e l y is the inequality to hold and hence the w better strategy is to remain on the present grounds. Similarly for a given i>, and t, the smaller the combined catch after some time i , i . e . the lower the f ish density encountered on tihe present grounds, then the better is the strategy to change grounds. Conversely, the greater Zx. at some J time i , then the better strategy is to remain f ishing where at present. As mentioned ear l ie r , the time to make a t r i p , t t is d i f f i c u l t to quantify. If i t is considered solely in terms of the catch rate, then t t - * If the density on the present grounds i s considered, solely in terms of the catch obtained, Ex. , then J 1 t V x j H If the density is considered in terms.of the Bayes expected estimate, then t ' t = . t t . ( a V » ; i X j ? -83-However, in reality, the time to make a trip will involve factors such as crew morale, fuel or food supplies, and market prices. On an actual tr ip, a skipper is less likely to change grounds i f the vessel has been at sea for some time, though this will depend on the risk-taking propensity of the skipper (Cove, 1973). Even i f satisfactory catch rates are obtained on an alternate ground, catch independent factors such as fuel or food may limit the time spent there. In such cases, t^ . will be easily specified. The value of the time to change grounds will depend on the fish-ery. For the Vancouver trawl fishery maximum steaming time between any two grounds is 38 hours, though most changes occur on one ground and involve 0.5 to 1 hours steaming. In the north-west Atlantic, changing grounds may involve steaming from Newfoundland to Greenland, then back to Newfoundland grounds involving 70 hour steaming periods, while changing position on one ground will more likely involve periods no greater than 3-4 hours. 6.3 Vancouver Trawl Fishery Once catch data becomes available on some ground, a best strategy may involve moving to one of a number of different grounds each with a particular ty., and t.. and t t l . . However in some fisheries fish are often either practically absent or present in acceptable quantities. The grounds exploited by Vancouver trawlers appear to f i t this description. (D. Guine*, pers. comm.). In such a situation the decision to change grounds will depend on the presence or absence of f ish . * Skipper, Bon Accord II Vancouver, B.C. -84-Certain assumptions w i l l be made here. In the actual f ishery, season is a factor in choice of grounds, e.g. Hecate Straits is favoured more in winter. Also the probabil it ies of encountering f ish and times required to search the grounds are different for- the different areas. Here season w i l l be ignored and i n i t i a l l y , the probability of encountering f ish and time to determine each ground w i l l be considered equal for a l l areas. Table VIII l i s t s the possible fishing grounds, their steaming time from each other, and the steaming time from Vancouver, in hours, at 9 knots. Table IX diagrammatical^ shows the possible options a skipper has on leaving Vancouver port. For each possible strategy there w i l l be a possible cost represented as the steaming time. The search w i l l end at any ground where f ish are encountered. The best strategy w i l l be that for which the sum of the possible costs (time) for covering the fishing grounds, together with times to return home i f f ish are encountered, is least. This possible cost is given in Table VIII as c; i t can be seen that the best strategy is to go Vancouver-Lower West Coast - Goose Island - Hecate Stra i t - Vancouver, the worst Vancouver - Hecate Stra i t - Lower West Coast - Goose Island - Vancouver. These results would not be unexpected from examination of Table VIII. However, i f the searching times and probabil it ies of f ish occur-rence at the different grounds are not uniform, then the results may not be so obvious from casual considerations. Let T L , Tg and T^  be average time required to determine presence or absence of f ish at the Lower West Coast, Goose Island and Hecate Straits grounds respectively and P L , PQ and P H be the probabil it ies of encountering f i s h . Let D.. be the steaming time from position i to position j , e.g. DV| = steaming time from Vancouver to Lower TABLE VIII Steaming time between grounds and from Vancouver Port Lower West Coast Vancouver Island Goose Island Hecate Straits Vancouver port Lower West Coast, Vancouver Island Goose Island Hecate Straits 17 24 38 36 24 62 38 19 19 Personal communication; Skipper D. Guine, Bon Accord I I , Vancouver TABLE IX P o s s i b l e s t r a t e g i e s . Distances between p o s i t i o n s are steaming times between p o s i t i o n s . Distances above p o s i t i o n s are the accumulated d i s t a n c e s , and those below, the accumulated d i s t a n c e plus steaming d i s t a n c e back to p o r t . Van = Vancouver P o r t G.I.= Goose I s l a n d L.W.C. = Lower West Coast, Vancouver I s l a n d H.S. = Hecate S t r a i t s EC 233 265 309 295 405 359 -87-West Coast, Vancouver I s l a n d . The p o s s i b l e r e l a t i v e l o s s f o r each s t r a t e g y can then be e v a l u a t e d i n terms o f the D.^ ., P. and T-. Then f o r a p a r t i c u l a r s t r a t e g y , s 1 J k - f <°U' pf T1>-By e v a l u a t i n g S... f o r a l l p o s s i b i l i t i e s i j k the best s t r a t e g y , t h a t f o r minimum S, can be determined. As an example, f o r the s t r a t e g y o f moving Vancouver - Lower West Coast - Goose I s l a n d - Hecate S t r a i t s , the p o s s i b l e r e l a t i v e l o s s w i l l be SLGH = D V L + T L + P LD L + (1-P L) { D L G + T Q + P ^ + (1-Pg) [ D G H + T^ + D H V ] } (7.17) I f the time i t takes t o determine the s t a t e o f f i s h i n g on the grounds i s known, then the r e l a t i v e p r o b a b i l i t i e s o f e n c o u n t e r i n g f i s h can be determined a t which one s t r a t e g y becomes p r e f e r a b l e to another. For example, c o n s i d e r the two s t r a t e g i e s p o s s i b l e i f f i s h were found to be absent on the Lower West Coast grounds. A s k i p p e r can proceed to e i t h e r the Goose I s l a n d grounds, and i f no f i s h are encountered, then t o the Hecate S t r a i t grounds, o r i n the r e v e r s e o r d e r , Assume t h a t the time to determine the grounds i s p r o p o r t i o n a l to t h e i r s i z e . P.A. L a r k i n * ( p e r s . comm.) gi v e s t h e i r s i z e as: Ground R e l a t i v e s i z e R e l a t i v e time to determine Hecate S t r a i t s 12 36 Goose I s l a n d 4 12 Lower West Coast 1 3 So as to compare the s t r a t e g i e s , S ^ and S^Q can be determined * Dept. of Zoology U n i v e r s i t y o f B r i t i s h Columbia -88-as i n equation 7,17. However, the component DVL + T L + P L P V L + ^ ~ P L ^ l s c o m n l o n to both s t r a t e g i e s and may be i g n o r e d . Then, from t a b l e V I I I , SLGH = DLG + V D G V P G + O ~ P G ^ D G H + TH + V = 153 - 81P Q SLHG = DLH + TH + DHV PH + ( 1 " PH><DHG + T G + DGV> = 141 - 5P H Hence f o r S ^ G to be a p r e f e r a b l e s t r a t e g y , SLHG < SLGH o r P R > 2.2 - 16.2P G and f o r the to be p r e f e r a b l e , P G > p H - 2.2 16.2 Even i f P^ = 1, i t i s s t i l l a b e t t e r s t r a t e g y to go to the Goose I s l a n d grounds f i r s t , u n l e s s , P G < 0.074 Th i s i s not unexpected i n t h a t from the steaming times given i n t a b l e VIII i t can be seen t h a t the Goose I s l a n d grounds are almost on route to the Hecate S t r a i t grounds from the Lower West Coast grounds. 7.0 D i s c u s s i o n 7.1 System V a r i a b l e s For e s t i m a t i n g school d e n s i t y over a survey area using Skellam's (1958) r e l a t i o n (equation 2.1) i n the s i t u a t i o n i n which C r a i g and Graham's -89-(1965) data was c o l l e c t e d , the estimate o f range, R, was the d i s t a n c e at which a l b a c o r e would make a s t r i k e at the l u r e . . T h i s would be dependent on t h e i r v i s u a l a c u i t y and hence ambient l i g h t c o n d i t i o n s , v i s u a l c h a r a c t e r i s t -i c s of the l u r e , and p h y s i o l o g i c a l aspects o f the a l b a c o r e , such as degree o f hunger, i . e . , w i l l i n g n e s s to s t r i k e . With a search method using sonar, d e t e c t i o n o f f i s h would be independent of p h y s i o l o g i c a l and b e h a v i o u r a l c h a r a c t e r s o f the s p e c i e s i n v o l v e d . With m e c h a n i c a l l y scanning sonar, s c h o o l s could be missed due to the slow r a t e o f scan (.005 r.p. second). However, with e l e c t r o n i c a l l y s e c t o r scanning sonar, r a t e s of scan of 500 r.p. second are p o s s i b l e and hence the search t r a c k i s completely covered (Johnson & P r o c t o r , 1970). Development i n sonar e n g i n e e r i n g has r e s u l t e d i n sonar t h a t can d e t e c t sc h o o l s o f zero d e c i b e l t a r g e t s t r e n g t h at 3000-5000 M. (Gerhardsen e t . a l . (1972)). In such s i t u a t i o n s v a r i a t i o n i n l a t e r a l range may be c o n s i d e r -a b l e , and of importance f o r purposes o f r e l o c a t i o n . The school may be anywhere i n the l a t e r a l range, r < 2z s i n (|) where r = range z = l a t e r a l range (or depth f o r v e r t i c a l d i r e c t i o n o f sonar beam) e = beam width (Tucker and Welsby (1964). By narrowing the beam width, the u n c e r t a i n t y o f p o s i t i o n w i l l be c o r r e s p o n d i n g l y reduced. Hence i n a sonar s e a r c h , as opposed to t r o l l i n g , the range o f d e t e c t i o n w i l l depend on the s p e c i f i c a -t i o n s o f the sonar equipment and on those sea-water c h a r a c t e r i s t i c s a f f e c t -i n g t r a n s m i s s i o n (mainly temperature) r a t h e r than on p h y s i o l o g i c a l f e a t u r e s o f the t a r g e t f i s h . For estimates using equation 2.1 the sonar range under the ambient c o n d i t i o n s could be o b t a i n e d by c a l i b r a t i o n , u s ing methods -90-such as those d e s c r i b e d by Johanneson and Losse (1973). In some s i t u a t i o n s u n c e r t a i n t y about the i d e n t i t y o f the s p e c i e s causing the echo may occur. However methods are being developed t h a t enable the s p e c i e s to be determined by a n a l y s i s of the p o l a r b a c k - s c a t t e r i n g p a t t e r n (Hearn, 1970) or from the nature o f the echo-graph ( B e s t , 1964). C r a i g and Graham's (1965) data was obtained at a speed o f 6.5 knots At t h i s speed the component due to the f i s h speed, w, of the r e s u l t a n t v e l o -c i t y o f both f i s h and search v e s s e l , V, was 6.71%. At a speed of 11 knots as p o s s i b l e with R.G. Dowd's system ( d i s c u s s e d i n s e c t i o n 7.3), the compon-ent of V due to w i s 2.51%, and at 20 knots, as i s p o s s i b l e with L.W. P r o c t o r ' s system, i t would be o n l y 0.78%. Hence any e r r o r i n determining the speed o f f i s h becomes i n c r e a s i n g l y l e s s important, as does the value o f t h e i r speed i t s e l f , as the speed o f search i n c r e a s e s . At 20 knots, the com-ponent w c o u l d be c o n s i d e r e d n e g l i g i b l e . As tuna are r e l a t i v e l y f a s t -swimming f i s h the r e l a t i v e importance of w f o r o t h e r , more s l o w l y swimming s p e c i e s would be even l e s s . One o f the c o n d i t i o n s o f equation 2.1 i s t h a t d i r e c t i o n o f move-ment i s random, although Skellam (1950) notes t h a t t h i s need not be s t r i c t l y t r u e . I f , however, the magnitude and d i r e c t i o n o f the s c h o o l s can be d e t e r -mined, as f o r the A p r i l 27-28 and J u l y 9-10 data of Royce and Otsu (1955), then the e f f e c t i v e speed o f the search v e s s e l can be c a l c u l a t e d as: 7 2 2 u + v - 2uv cos where u l = e f f e c t i v e speed o f search v e s s e l u = speed o f search v e s s e l v = speed o f s c h o o l s -91-<j> = angle between u and v measured from u to v cl o c k w i s e (Koopman, 1956; Skellam, 1958). As d i s c u s s e d e a r l i e r , as u becomes large,' the c o n t r i b u t i o n o f v to u' w i l l become s m a l l e r . For a given v, maximum c o n t r i b u t i o n to u' from v w i l l be when $ = i r , i . e . COSTT = -1. When a l l data are combined, an o v e r a l l mean may be o f l e s s value than means c a l c u l a t e d f o r l i m i t e d areas where f i s h d e n s i t i e s are uniform. In such a s i t u a t i o n i t may be more u s e f u l to have s t r a t i f i e d e s t imates o f f i s h d e n s i t y as they w i l l o f f e r b e t t e r estimates o f l o c a l d e n s i t i e s . S t r a t i f i c a t i o n o f d e n s i t y i n t o f o u r and three groups has been done i n s u r -veys by Johanesson and Losse (1973) and Thorne, Reeves, and M i l l i k a n (1971), The o b j e c t i v e o f these authors was, however, the r e d u c t i o n o f v a r i a n c e estimates o f t o t a l biomass r a t h e r than p r e d i c t i o n s o f d e n s i t i e s i n l o c a l areas and r e q u i r e d more i n t e n s i v e sampling than might be f e a s i b l e with e x p l o r a t o r y work. An a l t e r n a t e method of d e s c r i b i n g v a r i a b l e d e n s i t y with area i s to d e s c r i b e the v a r i a t i o n o f the mean value by some d i s t r i b u t i o n ( P i e l o u , 1969). The mean may be d e s c r i b e d by any standa r d curve which i s non-ne g a t i v e , unimodal and p o s s i b l y skewed. I f the sample r e s u l t s are random and a Pearson type III d e n s i t y d e s c r i b e s the d i s t r i b u t i o n o f the mean, then the r e s u l t a n t d i s t r i b u t i o n o f sample r e s u l t s i s t h a t o f a Negative Binomial ( P i e l o u , 1969). In c o n s i d e r i n g v a r i a n c e estimates o f the mean d e n s i t y , , from equation 2.1, Skellam (1958) notes t h a t the v a r i a n c e w i l l be p r o p o r t i o n a l to the v a r i a n c e o f the number of encounters during a t r a n s e c t , z. In C r a i g and Graham's (1965) s i t u a t i o n , i n essence only one t r a n s e c t i s made. However, Skellam notes t h a t f o r many t r a n s e c t s , or f o r a p a r t i t i o n e d t r a n s e c t -92-no methodology has been developed to evaluate the variance, and that the relationship for the variance appears complex. Whereas the expected number of encounters did not appear to depend on the shapes of the transect paths, the variance of the number of encounters did. Skellam conjectured that i f the targets swept across the observer's space without any special tendency to double back or execute osci l latory movements, then the number of encounters per unit time would be a Poisson variate. If the targets were aggregated in groups of a certain s i ze , g, then the variance would be gX. Any folding back by the targets would increase the variance in a manner similar to aggregation. Also, any heterogeneities would increase the size of the variance. Effects which were in part dependent on target speed and contributed to variance, would s imi lar ly be reduced in a search system where the search speed was much larger than that of the target, as discussed ear l ier in this section. 7.2 Pattern of School Distribution Choice of a model to describe the distr ibution of f ish may be entirely empirical, i . e . , that which gives best f i t to the data; or the distr ibution model may be chosen so as to describe some underlying environ-mental or ecological factors. Ideally the empirical and 'ecological ' models are the same, though to consider an 'ecological ' model measurements or postulates about the related factors are required. In a search and assessment study the main emphasis would be on description of catch data. The selection of the best distr ibution for a particular set of data is complicated by the interrelationships of many distributions and their occasional ambiguous relationship to the models used in formulating them. Also, different models can be used to derive the same distr ibution -93-(Gurland, 1958) and c o n v e r s e l y the same d i s t r i b u t i o n may even be d e r i v e d from c o n t r a d i c t o r y s e t s of p o s t u l a t e s ( C a s s i e , 1962). Mart i n and K a t t i (1965) f i t t e d the P o i s s o n , Poisson with added ze r o e s , Neyman type A, and Negative Binomial d i s t r i b u t i o n s to t h i r t y - f i v e s e t s of b i o l o g i c a l d a t a . They found t h a t the Neyman Type A and Negative Binomial d i s t r i b u t i o n s had wide a p p l i c a b i l i t y w h i l e the Poisson with added zeroes p r o v i d e d a good f i t to only a few se t s o f d a t a . For t h i s reason Martin and K a t t i c a u t i o n t h a t the u n d e r l y i n g model should be checked before using t h i s d i s t r i b u t i o n . They a l s o found t h a t no d i s t r i b u t i o n f i t t e d a l l data s e t s w e l l . From Table I I , i t can be seen t h a t f o r a l l s i z e s o f sample l e n g t h , the g e n e r a l i z e d d i s t r i b u t i o n s gave best f i t s , w hile the Poisson and Poisson with added zeroes p r o v i d e d the p o o r e s t . The b e s t f i t f o r the Poisson d i s t r i b u t i o n was given f o r the s m a l l -e s t sample l e n g t h , as would be expected, f o r the s m a l l e r the sample s i z e , the more random would be the sample r e s u l t s . The b e s t f i t f o r the Poisson w i t h added zeroes was f o r the 25 m i l e sample l e n g t h , r a t h e r unexpectedly, as t h i s l e n g t h r e s u l t e d i n the second fewest number o f zero c l a s s e s . For the Negative Binomial best f i t was f o r sample length o f 15 m i l e s ; f o r Neyman Type A, 15 m i l e s a l s o ; and f o r the P o i s s o n - P o i s s o n , 5 m i l e s . In f i t t i n g the Neyman type A l a r g e r values were obtained f o r the c l u s t e r s i z e s f o r a l l sample lengths except 5 miles than f o r the P o i s s o n - P o i s s o n . When a d d i t i o n a l i n f o r m a t i o n i s a v a i l a b l e , such as water temper-a t u r e , depth, c u r r e n t , bottom type , or plankton presence, which can be r e l a t e d to f i s h o c c u r r e n c e , methods of p a t t e r n r e c o g n i t i o n using c l u s t e r or p a t t e r n a n a l y s i s may be u s e f u l i n p r e d i c t i n g f i s h presence. Such -94-methods may provide a fundamental r a t h e r than e m p i r i c a l b a s i s f o r d e s c r i b -i n g the p a t t e r n o f f i s h d i s t r i b u t i o n . The p o t e n t i a l o f these methods would depend on the e x t e n t o f c a u s a t i o n or c o r r e l a t i o n o f p r e v i o u s l y c o l l e c t e d data to f i s h presence or absence. An a l t e r n a t i v e method f o r o b t a i n i n g the p r o b a b i l i t y of f i s h o c c u r r e n c e , given a number of f u n c t i o n s d e s c r i b i n g the p r o b a b i l i t y o f an event, f i s h o c c u r r e n c e , i n terms o f some v a r i a b l e , e.g. temperature, depth, e t c . , i s c o n s i d e r e d i n s e c t i o n 7.4. 7.3 Choice o f Sample S i z e One c r i t e r i o n f o r c h o i c e of sample s i z e i s t h a t s i z e which would i n d i c a t e any b i o l o g i c a l p a t t e r n such as some c l u s t e r i n g p r o c e s s , i f p r e s e n t . I f the sample u n i t i s l a r g e compared with the s c a l e o f h e t e r o g e n e i t y , sample values w i l l tend to the mean value f o r a l l samples with an expected decrease i n v a r i a n c e . I t i s o f i n t e r e s t t h a t the sample v a r i a n c e o f C r a i g and Graham's (1965) data i n c r e a s e d as a f u n c t i o n o f sample s i z e . I f the d i s t r i b u t i o n of s c h o o l s i s some c l u s t e r e d p a t t e r n , then a square sample area would be expected to minimize any edge e f f e c t s . However, i f the nature of the c l u s t e r e d d i s t r i b u t i o n i s known, as from previous sampling, then i f schools are l o c a t e d along the margin o f the sample a r e a , p r e d i c t i o n s may be made as to the occurrence o f o t h e r s c h o o l s o u t s i d e the sample area. An i n d i c a t i o n of the s c a l e of the school d i s t r i b u t i o n heterogene-i t y may be obtained by v a r y i n g the s i z e of the sampling u n i t and n o t i n g the s i z e a t which i n d i c a t i o n s o f non-randomness d i s a p p e a r o r decrease markedly. Greig-Smith (1964) d e s c r i b e s a technique based on s y s t e m a t i c sam-p l i n g . A g r i d of continuous quadrats i s used, each s i d e o f the g r i d -95-c o n s t s t i n g o f a number of g r i d u n i t s which i s a power o f 2, I f the number of i n d i v i d u a l s per g r i d u n i t i s determined, then f o r a random d i s t r i b u t i o n the mean square f o r a l l b l o c k s i z e s should be the same and equal to the d e n s i t y . I f the d i s t r i b u t i o n i s c o n t a g i o u s , the v a r i a n c e w i l l r i s e with i n c r e a s e of b l o c k s i z e u n t i l block s i z e i s e q u i v a l e n t to the areas o f the patches. I f the patches themselves are random or c o n t a g i o u s , the v a r i a n c e w i l l be maintained at t h i s l e v e l with i n c r e a s i n g block s i z e , I f the patch are r e g u l a r , the v a r i a n c e w i l l f a l l as b l o c k s i z e i s i n c r e a s e d f u r t h e r . I f more than one s c a l e of h e t e r o g e n e i t y i s p r e s e n t , t h i s behaviour w i l l be repeated as b l o c k s i z e i n c r e a s e s to reach the secondary h e t e r o g e n e i t y s c a l e Hence from examination of v a r i a n c e trend with changing b l o c k s i z e , i n f e r -ences about p a t t e r n s i z e can be made. Variance estimates w i l l depend on the type of d i s t r i b u t i o n used. I f some l i k e l y model i s proposed, the expected mean square and i t s standard e r r o r can be c a l c u l a t e d f o r each b l o c k s i z e . I f the expected mean squares do not d e v i a t e s i g n i f i c a n t l y , then the model can be regarded as a s a t i s f a c t o r y d e s c r i p t i o n o f the p a t t e r n . 7.4 Variance Estimates Greig-Smith (1964) notes t h a t when the sample p o i n t s are not randomly d i s t r i b u t e d , the v a r i a n c e i s n e i t h e r equal to the mean, nor pro-p o r t i o n a l to i t . For a reduced number of l a r g e r s i z e d quadrates, he found an i n c r e a s e i n the v a r i a n c e expressed as a p r o p o r t i o n o f the mean. A s i m i l a r r e s u l t o c c u r r e d w i t h C r a i g and Graham's (1965) data as shown below: -96-5ample length Variance/mean r a t i o 5 miles 1.021 10 1.330 15 1.546 20 1.780 25 1.533 Greig-Smith concluded t h a t the s a f e s t procedure i s to use the s m a l l e s t p r a c t i c a l quadrat s i z e . In C r a i g and Graham's (1965) d a t a , with the exce p t i o n o f the f i t f o r the Neyman type A d i s t r i b u t i o n , which gave e x c e l l -ent agreement f o r d i s t r i b u t i o n v a r i a n c e and data v a r i a n c e , the other d i s t r i -b utions f o r which f i t s were made had i n c r e a s i n g values o f the r a t i o o f sample v a r i a n c e to d i s t r i b u t i o n v a r i a n c e as sample s i z e i n c r e a s e d (Table I I I ) except f o r the 25 mi l e sample. The sample r e s u l t s obtained using C r a i g and Graham's (1965) data were not independent, but formed a s e r i a l arrangement o f samples. Hence i t co u l d be expected t h a t the sample v a r i a n c e would not c o r r e c t l y estimate the d i s t r i b u t i o n v a r i a n c e . Hogg and C r a i g (1968) show t h a t a b e t t e r estimate i s given by: S2 = S l J" 1 + 2 " r j where: S 2 = c o r r e c t e d estimate o f standard d e v i a t i o n S-| = standard d e v i a t i o n c a l c u l a t e d from sample r e s u l t s r . = s e r i a l c o r r e l a t i o n between sample values j u n i t s a p a r t . J Nickerson and Dowd (1973) found i n t h e i r sampling program t h a t u n d e r e s t v mated S^. A c o n s t r a i n t on equation 7.1 would be t h a t ; z r . >_ - 0.5 J """" The s i z e and s i g n of z r . w i l l depend on the nature o f the s e r i a l J c o r r e l a t i o n which w i l l i n turn depend on the s i z e o f the sample u n i t and -97-s c a l e o f the p o p u l a t i o n h e t e r o g e n e i t y . For r e l a t i v e l y small sample s i z e s , p o s i t i v e s e r i a l c o r r e l a t i o n would be expected; i f the sample s i z e was small r e l a t i v e to the s c a l e o f p o p u l a t i o n h e t e r o g e n e i t y , an empty sample would probably be f o l l o w e d by another. I f however the sample s i z e was s i m i l a r to t h a t o f the p o p u l a t i o n h e t e r o g e n e i t y s i z e , f o r example the area o f a c l u s t e r of s c h o o l s , and c o i n c i d e d with the c l u s t e r , then a sample 'quadrat' which covered a c l u s t e r would probably be f o l l o w e d by one t h a t c o n t a i n e d no scho o l s and a neg a t i v e s e r i a l c o r r e l a t i o n would be o b t a i n e d . To achieve a given c o n f i d e n c e i n t e r v a l on estimates o f f i s h den-s i t y , Nickerson and Dowd (1973) g i v e : A d d i t i o n a l survey d i s t a n c e _ D r CI-j r e q u i r e d r 1 £T~ - ^ -I 2 where: D r = d i s t a n c e covered i n r e f e r e n c e survey CI-j = confidence i n t e r v a l o b t a i n e d i n r e f e r e n c e survey CI2 = d e s i r e d confidence i n t e r v a l . In a s i m i l a r study t o t h a t o f Nickerson and Dowd (1973), o f acou-s t i c methods o f assessment o f demersal stock s i z e , Thorne, Reeves and Mi 11ikan (1971) adopt a d i f f e r e n t approach t o the problem o f v a r i a n c e e s t i -mation, t h a t o f s t r a t i f i e d sampling. Estimates o f the t o t a l f i s h abundance of a surveyed area i-s o b t a i n e d by p a r t i t i o n i n g the area i n t o sub-areas o r s t r a t a , with l i m i t s contoured a c c o r d i n g to the d e n s i t y o f the samples. The estimate o f t o t a l biomass i s given by the sum o f the estimates f o r each s t r a t a . The g r e a t e r the homogeneity o f s t r a t a d e n s i t y o b t a i n e d , then the s m a l l e r w i l l be the r e s u l t i n g e s timate o f v a r i a n c e , f o r each s t r a t a . The standard e r r o r o f the mean d e n s i t y estimate w i l l be -98-L N 2 Var ( y j Var (y ) = z ^ o — where: ^ s t = m e a n ^1S'1 d e n s i t y f ° r a ^ s t r a t a = number of u n i t s i n stratum N = t o t a l number of u n i t s y ^ = mean d e n s i t y f o r stratum h. (Cochran, 1963). The v a r i a n c e of Js^ depends only on the v a r i a n c e s of the estimates of the i n d i v i d u a l stratum means. I f i t were p o s s i b l e to d i v i d e a v a r i a b l e p o p u l a t i o n i n t o s t r a t a such t h a t the s t r a t a were o f c o n s t a n t den-s i t y , then the mean value could be estimated without e r r o r . A general con-sequence i s t h a t more sampling e f f o r t should be a l l o c a t e d to l a r g e r s t r a t a and those which are more v a r i a b l e . For optimum a l l o c a t i o n o f sampling e f f o r t , a p r i o r i i n f o r m a t i o n on stratum s i z e and v a r i a b i l i t y would be r e q u i -red. However, Thorne e t . a l . (1971) found t h a t f o r P a c i f i c Lake ( M e r l u c c i u s  p r o d u c t u s ) , the d i s t r i b u t i o n s of d e n s i t i e s as i n d i c a t e d by i n i t i a l sampling were not s t a b l e , and i t was very d i f f i c u l t to p r e d i c t stratum s i z e from i n i t i a l t r a c k s o f the survey. Hence attempts at optimum a l l o c a t i o n would be i n v a l i d a t e d . Cochran notes t h a t e r r o r i n stratum s i z e computation can i n t r o d u c e s e r i o u s b i a s to estimates of t o t a l p o p u l a t i o n . For t h i s reason Thorne e t . a l . s t r a t i f i e d the survey area based on p r o p o r t i o n a l a l l o c a t i o n . The gain i n p r e c i s i o n was 29% over t h a t f o r random sampling. Johanneson and Losse (1973) i n s t o c k assessment s t u d i e s i n the Black Sea used f o u r s t r a t a , a l s o p r o p o r t i o n a l l y a l l o c a t e d . For 95% con-f i d e n c e i n t e r v a l s , the v a r i a n c e estimates were 6.4%, 11.4%, 10.4% and 9,4% of the s t r a t a e s t i m a t e s . However, Thorne e t , a l . (1971) found the major -99-component o f v a r i a n c e i n the p o p u l a t i o n e s t i m a t e was due t o c a l i b r a t i n g a c o u s t i c records w i t h trawl c a t c h e s . For a 95% c o n f i d e n c e i n t e r v a l a + 50% range was o b t a i n e d on the r e g r e s s i o n c o e f f i c i e n t . 7.5 Entropy and F i s h D i s t r i b u t i o n The concept of entropy has been d i s c u s s e d i n the i n t r o d u c t i o n and the entropy f o r the d i f f e r e n t d i s t r i b u t i o n s i s d i s c u s s e d b r i e f l y i n s e c t i o n 3.2. As was seen, the entropy i s a f u n c t i o n of the d i s t r i b u t i o n type and mean school d e n s i t y , hence entropy provides an a l t e r n a t e e x p r e s s i o n f o r the expected frequency o f school numbers. S h e r s t n i k o v (1968) shows t h a t f o r a square search area o f uniform p r o b a b i l i t y o f t a r g e t d e t e c t i o n , then optimal search t r a c k s i n terms o f removal o f u n c e r t a i n t y can be determined. A l s o , when the p o s i t i o n o f a t a r g e t i s known with some e r r o r , then search curves can be d e v i s e d which minimize the system u n c e r t a i n t y . However, n e i t h e r the entropy o f some expected d i s t r i b u t i o n o f school occurrence or S h e r s t n i k o v ' s (1968) approach o f maximizing i n f o r m a t i o n gained along some search t r a c k o f f e r s any e x t r a guide to s t r a t e g i c d e c i s i o n s than does the v a l u e o f the expected school number. Entropy can be used however to p r o v i d e a c r i t e r i o n f o r c o n s t r u c t -i n g p r o b a b i l i t y d i s t r i b u t i o n s , which c o u l d be o f d i r e c t use i n d e c i s i o n making, on the b a s i s o f p a r t i a l knowledge. (Jaynes, 1957). That d i s t r i -b u t i o n which maximizes the entropy e s t i m a t e , s u b j e c t to c e r t a i n c o n s t r a i n t s , provides the l e a s t biased estimate p o s s i b l e on any given i n f o r m a t i o n . Using Lagrangian m u l t i p l i e r s , e, y, Jaynes (1957) shows t h a t p(x) = e " e f ( x > -100-i where e e = z exp[- y fCxJ] (7.2) j = l J Wilson (1970) shows that that function f(x) which has maximum en-tropy minimizes the l ikelihood function of a Bayes estimate, i . e . makes the weakest assumptions which are consistent with what is known. Using a -\ Bayes approach to determine the density function, a maximum likelihood \ estimate of the school density, x, is obtained and that function which gives the best f i t for X chosen. In maximizing entropy, that f(x) which gives the best f i t to p(x) is chosen. If f(x) is not in accord with rea l i t y , the form of f(x) can be changed, as long as i ts expected value is known. If the expected value is not known then s ta t i s t i ca l methods such as maximum likelihood must be used to estimate the parameter, v, of equation 8.3. Wilson (1970) believes that the procedure of maximizing entropy has three potential advantages: 1) Dealing with constraint equations rather than with the d i s t r i -bution function direct ly may allow a more consistent approach to be achieved in complex situations. 2) The constraint equations may have fundamental meaning which would f a c i l i t a t e understanding of the system, 3) It may be possible to proceed more directly to a dynamic model, using general principles of systems analysis. If the frequency of school occurrence is related to a number of variables, and can be expressed in terms of some function, f (x), for m variable m, then p(x^) may be expressed in terms of these functions; p(x.) = exp {-[e 0 + e 1 f 1 (x) + + e m f m (x)]} where e = Lagrangian mult ip l ier -101-(Jaynes, 1957), Such an e x p r e s s i o n would enable other v a r i a b l e s , such as temperature, s a l i n i t y , p l a n k t o n , to be used when the f u n c t i o n a l form had been e s t a b l i s h e d and the v a r i a b l e monitored. 7.6 Decay of Information. For any p a r t i c u l a r s i t u a t i o n , where a f i s h c o n c e n t r a t i o n has been l o c a t e d , r e l o c a t i o n by f i s h i n g v e s s e l s w i l l depend on the v e l o c i t y o f f i s h , time f o r f i s h i n g v e s s e l s t o a r r i v e , and the e f f e c t i v e n e s s o f f i s h f i n d i n g equipment. Such v e s s e l r e l a t e d f a c t o r s can be s p e c i f i e d . In some f u t u r e f i s h i n g system, i t may be t h a t the mean square r a t e o f d i s p e r s i o n , the d i r e c t i o n a l v e l o c i t y and the modal d i r e c t i o n o f a l l important s p e c i e s w i l l be known f o r the s e a s o n a l , t i d a l and d i e l p e r i o d s . Such movement parameters would a l s o be expected to be s p e c i f i c to d i s c r e t e stocks on the d i f f e r e n t grounds. D e t a i l e d knowledge o f these parameters may be o f much a s s i s t a n c e both i n d e v i s i n g sampling s t r a t e g y and i n t e r p r e t i n g i t . For a given den-s i t y o f f i s h , as the d i r e c t i o n a l v e l o c i t y and/or the mean square d i s p e r s i o n r a t e i n c r e a s e , then Koopman (1956a) shows t h a t at a given l e v e l o f search o r f i s h i n g e f f o r t , the r a t e o f encounters or catch r e s p e c t i v e l y w i l l i n c r e a s e . Where sampling i s by t r a w l i n g and the speed d i f f e r e n c e between f i s h and ve s s e l i s not g r e a t , then a r e l a t i v e l y g r e a t e r d i f f e r e n c e i n apparent d e n s i t y would r e s u l t . Koopman a l s o shows t h a t the number o f encounters w i l l depend on the r e l a t i v e t r a c k angle between v e s s e l and t a r g e t . Hence the modal d i r e c t i o n o f f i s h movement, as well as mean square d i s p e r s i o n r a t e and d i r e c t i o n a l v e l o c i t y , should be con s i d e r e d i n i n t e r p r e t a t i o n o f catch r e s u l t s . Beverton and H o l t (1957) show how t h e i r d i s p e r s i o n c o e f f i c i e n t , -102-(5.1), e q u i v a l e n t to the mean square d i s p e r s i o n c o e f f i c i e n t o f Skellam (1958) and Jones (1959), can be r e l a t e d to the p a t t e r n of food d i s t r i b u t i o n and type o f f e e d i n g behaviour. In the case o f demersal f i s h , i f the time spent f e e d i n g on some patch of food i s l a r g e r e l a t i v e to the time spent swimming between patches, then the r a t e o f d i s p e r s i o n i s i n v e r s e l y p r o p o r t i o n a l to a power o f the food abundance, b, where 1 < b <_ 2 depending on whether changes i n the d i s t r i b u t i o n p a t t e r n of the food i s a r e s u l t of the number o f food patches, b=l; or the s i z e o f the food patches, b=2. I f the f i s h feed w h i l e moving, as f o r p e l a g i c s p e c i e s , the e f f e c t on the r a t e of d i s p e r s i o n w i l l be l e s s pronounced. However, i f the food items are taken i n d i v i d u a l l y , t here w i l l be some decrease i n the r a t e of d i s p e r s i o n due to pauses when s e i z i n g each food item. Tests on the homogeneity of the d i r e c t i o n o f movement have two p o s s i b l e uses. Information obtained about one stock may be a p p l i c a b l e to another i f i t i s known they e x h i b i t s i m i l a r d i r e c t i o n o f movement. A l s o s i m i l a r i t i e s o r d i s s i m i l a r i t i e s i n b e h a v i o u r a l p a t t e r n s at d i f f e r e n t times f o r p a r t i c u l a r stocks may be d e t e c t e d by examination of d i r e c t i o n a l move-ment patterns at d i f f e r e n t p e r i o d s : s e a s o n a l , t i d a l o r d i e l . 7.7 D e c i s i o n making D e c i s i o n methods i n s e c t i o n 6 were d i s c u s s e d i n an o p e r a t i o n a l c o n t e x t , t h a t o f a s k i p p e r attempting to maximize his c a t c h , Three d i f f e r -ent but r e l a t e d circumstances were c o n s i d e r e d . In one s i t u a t i o n where a school had been l o c a t e d at some d i s t a n c e from a c a t c h e r v e s s e l , a r e l a t i o n was found from which d e c i s i o n s c o u l d be made on whether to continue s e a r c h i n g or to attempt to r e l o c a t e the school -103-found by the search v e s s e l . In development o f C r a i g and Graham's (1965) d a t a , a l o s s f u n c t i o n , a l b e i t r a t h e r simple, was minimized, and a parameter e s t i m a t i o n (the f i s h d e n s i t y , x) made i n c o n j u n c t i o n with p r i o r b e l i e f s and subsequent sample data. This s i t u a t i o n i s c h a r a c t e r i z e d by the continuous range which x may take. When x i s below some determined t h r e s h o l d l e v e l , then a bene-f i c i a l s t r a t e g y i s to change grounds. In the B.C. trawl f i s h e r y the s i t u a t i o n i s d i f f e r e n t i n t h a t r a t h e r than attempting to assess some parameter, given a p o s s i b l e l o s s f u n c t i o n , the s k i p p e r i s c o n f r o n t e d with a number of p o s s i b l e a c t i o n s , each with a p a r t i c u l a r chance o f success and a p a r t i c u l a r chance o f i n c u r r i n g some l o s s , and then i f not encountering f i s h , being faced with another s e t o f o p t i o n s . From estimates o f p o s s i b l e l o s s e s the b e s t branch o f the d e c i s i o n t r e e can be determined. The way i n which d e c i s i o n s on f i s h i n g t a c t i c s are made w i l l have important consequences, f o r f i s h e r i e s management. T r a d i t i o n a l l y , measure-ments o f catch per e f f o r t assume independence o f f i s h i n g e f f o r t . I f the manner i n which c o - o p e r a t i o n between v e s s e l s enhances t h e i r c a t c h , o r how a s k i p p e r makes d e c i s i o n s i n order t o l o c a t e above average f i s h c o n c e n t r a -t i o n s can be d e s c r i b e d , then those changes i n catch per e f f o r t which are independent of stock d e n s i t y may be b e t t e r i d e n t i f i e d . Recently s e v e r a l workers have noted the n e c e s s i t y f o r r a p i d management d e c i s i o n s i n s i t u a t i o n s o f v a r i a b l e degrees o f u n c e r t a i n t y and i n c o n j u n c t i o n with d i v e r s e o b j e c t -i v e s o f d i f f e r e n t i n t e r e s t e d groups. L a r k i n (1972) observes t h a t d e c i s i o n making i n f i s h e r i e s should i n v o l v e , e i t h e r d i r e c t l y or i n d i r e c t l y , c o n s u l t -a t i o n with a l l a f f e c t e d groups - i n d u s t r y , fishermen, e t h n i c groups, ' e x p e r t s ' , -104-s o c i a l groups and, i n a vague s o r t of way, the people at l a r g e . In a s i m i l a r v e i n , G u l l and (1971) says t h a t management must r e s o l v e a wide range of o f t e n c o n f l i c t i n g o b j e c t i v e s - p o l i t i c a l , s o c i a l , economic - but t h a t i n the past s c i e n t i f i c evidence has o f f e r e d f a r g r e a t e r o b j e c t i v i t y than s o c i a l or economic c o n s i d e r a t i o n s , and has thus been the only t o o l , r a t h e r than merely p r o v i d i n g one element, i n the d e c i s i o n making p r o c e s s . C r u t c h f i e l d (1973) b e l i e v e s t h a t no s i n g l e system of management w i l l be optimal f o r a l l members o f an i n t e r n a t i o n a l l y shared f i s h e r y , and acknowledges t h a t the problem o f r e a c h i n g a s a t i s f a c t o r y common agreement i s d i f f i c u l t . Even w i t h i n one country c o n f l i c t o f i n t e r e s t s may occur between segments o f the f l e e t t h a t use d i f f e r e n t types of gear, o r between d i f f e r e n t areas. F u r t h e r , economic problems may have to be c o n s i d e r e d , such as whether to exceed some l e v e l o f s u s t a i n a b l e y i e l d f o r the immediate economic r e t u r n s o r to decrease e f f o r t f o r delayed b e n e f i t s . However, i f d i f f e r e n t economic and s o c i a l goals are to be compared, then the p o s s i b l e outcomes of such choices must be expressed q u a n t i t a t i v e l y , f o r i n s t a n c e , as the "expected monetary v a l u e s " o f R a i f f a (1968). In t h i s manner d e c i s i o n t r e e s may be c o n s t r u c t e d which assess p o s s i b l e l o s s e s and gains f o r a s e r i e s o f r e l a t e d d e c i s i o n s i n a management s i t u a t i o n . For example i f i t i s wished to reduce e f f o r t i n a f i s h e r y , e x p l o i t e d by g types o f gear on S grounds f o r m months o f the y e a r , many d i f f e r e n t combinations o f g, S and m might give the r e q u i r e d r e d u c t i o n i n c a t c h , each, however, with a d i f f e r e n t l o s s depending on the s o c i a l and economic c o n s i d e r a t i o n s . G u l l and (1972a, 1972b) notes t h a t f i s h e r i e s b i o l o g y i s not an exact s c i e n c e and t h a t with the r a p i d development of f i s h e r i e s technology, manage-ment d e c i s i o n s are r e q u i r e d long b e f o r e those based on t r a d i t i o n a l methods o f catch and e f f o r t c o u l d be made. To dat e , up to f o u r years may occur -105-between a f i s h e r y reaching a c r i t i c a l stage and any management a c t i o n a f f e c t i n g the l e v e l of e f f o r t . In many f i s h e r i e s , the time between catches reaching a peak and d i s a p p e a r i n g i s c o n s i d e r a b l y l e s s than t h i s p e r i o d . Here s c i e n t i s t s cannot a f f o r d to wait f o r c o n c l u s i v e evidence, as by t h a t time such c o n c l u s i o n s may w e l l be post mortem i n nature. In such s i t u a t i o n s , use o f one's previous experience i s e s s e n t i a l , and Bayesian methods o f f e r the i d e a l and p o t e n t i a l l y very powerful approach o f i n c o r p o r a t i n g one's p r i o r e x p e c t a t i o n , expressed as degrees of b e l i e f , o f the e f f e c t o f some l e v e l o f e f f o r t on s t o c k s , augmented by subsequent data as i t becomes a v a i l a b l e . Using r e a l i s t i c l o s s f u n c t i o n s the consequences of d i f f e r e n t c h o i c e s f o r the f i s h stock s i z e can be explored and t h a t c h o i c e which minimizes p o s s i b l e l o s s be made. Here a g a i n , s o c i a l and economic c r i t e r i a may be i n c l u d e d i n the l o s s f u n c t i o n i f they can be expressed q u a n t i t a t i v e l y and i n c o r p o r a t e d with the p e r t i n e n t b i o l o g i c a l model. 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