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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 o f Wales, 1969  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of 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  presenting  this  an a d v a n c e d  degree  the  shall  I  Library  f u r t h e r agree  for  scholarly  thesis at  purposes  his„representatives.  of  this  written  for  it  freely  permission  by  thesis  partial  the U n i v e r s i t y  make  that  in  may  financial  is  British for  for extensive by  gain  Department  Columbia  shall  the  that  not  requirements  Columbia,  I  agree  r e f e r e n c e and copying  t h e Head o f  understood  permission.  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  of  available  be g r a n t e d  It  fulfilment of  of  or  that  study.  this  thesis  my D e p a r t m e n t  copying  for  or  publication  be a l l o w e d w i t h o u t  my  ii  ABSTRACT Methods o f search i n marine f i s h e r i e s a r e reviewed, and aspects of v a r i a b l e s necessary t o evaluate e x p l o i t a t i o n s t r a t e g i e s a r e 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 a r e made.  P o s s i b l e school patterns 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 o f the data variance.  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.  Tests 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 a r e 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 tion.  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 -  lated. D e c i s i o n methods so as t o i n c r e a s e c a t c h a r e c o n s i d e r e d f o r three situations: (1) Where the p o s i t i o n of a school i s known with 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 o r remain s e a r c h i n g on the present grounds. (2) When the Bayes estimate of catch 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 minimizing 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  considered of a present o r absent nature, and i t i s d e s i r e d to minimize the  iii 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 o t h e r workers i s a l s o d i s c u s s e d .  TABLE OF CONTENTS Page TITLE PAGE  1  ABSTRACT  11  TABLE OF CONTENTS  iv  LIST OF FIGURES  v  LIST OF GRAPHS  vi  LIST OF TABLES  v1i  ACKNOWLEDGMENTS.  vi  11  1.0  INTRODUCTION  1  2.0  SEARCH STRATEGIES: A GENERAL REVIEW  2  3.0  SYSTEMS VARIABLES  4.0  3.1 S i z e o f Search Area 3.2 F i s h Density SCHOOL PATTERN AND SIZE 4.1 4.2 4.3  16  ]^ " 24  24  P a t t e r n o f School Occurrence 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 School S i z e and U n c e r t a i n t y .  ^ ^5 36  5.0  DECAY OF INFORMATION  37  6.0  DECISION MAKING  58  7.0  6.1 6.2  Royce and Otsu (1955) Tuna Data C r a i g and Graham (1965) Tuna Data  6.3  Vancouver Trawl F i s h e r y  59 j£ Bi  DISCUSSION  7.1 System V a r i a b l e s 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 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 7.7 Decisionmaking LITERATURE CITED.  88  88  ^  jjjj  ^ ^ 'J?' 106  1  V  LIST OF FIGURES FIGURE  Page  1  Search p a t t e r n s given by Yodovich and Boral (1968)  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  5  "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 I II  III IV  V  Page Range l i m i t of school occurrence at specified probability  52  Probability of direction as a function of angular difference to modal direction  55  Area of possible school occurrence as a function of time  62  Strategy as whether to change grounds or remain on present grounds  65  Relative loss when changing grounds and when remaining  88  vii  LIST OF TABLES TABLE I  Page Means, variances, goodness of f i t and entropy for specified distributions  II  Ranked order of f i t of distributions for different sample lengths  III  29  31  Ratios of variance to mean, and variance to distribution  IV V VI  VII  variance, for specified distributions and sample l e n g t h s . . . .  3 3  Summary of vector diagrams of Royce and Otsu (1955),.  4 3  Data for test of K^O  4 5  Data for analysis of Royce and Otsu (1955) vector observations of tuna school movement  ^  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  \  viii  ACKNOWLEDGMENTS I would l i k e 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 i k e to acknowledge Dr Wilimovsky for suggestion of the topic. Special thanks go to Mr Neil G i l b e r t , Institute of Animal Resource Ecology, for discussions of s t a t i s t i c a 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 , or 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 expectations based on past experience.  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 catch 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 catch 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 catch 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 intend 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 evaluated 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  -2abundance, 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 o f search o r 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 resources i n h a b i t i n g a d e f i n e d area o f the ocean". my purposes, I i n c l u d e sonar search as f i s h i n g gear).  (For  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 t o some extent 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 operators 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 pot e n t i a l catch aspects o f t h e 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 o f 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 lacks any s y s t e m a t i c planning and as such may g i v e r i s e t o 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 o f search i n f i s h e r i e s has v a r i a b l e  interpretation. 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 marine c o n d i t i o n s such as s a l i n i t y , temperature o r plankton a r e conducive t o t h e presence o f 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 o r previous experience i n d i c a t e s t h a t they may be expected t o 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 t h e f i r s t type o f prognosis b u t 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 nonuniform, 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 t o e x p l o i t concent r a t i o n s o f f i s h may f i r s t have t o expend e f f o r t i n l o c a t i n g f i s h concentrations.  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 species  which form d i s c r e t e s c h o o l s , the f i s h a r e 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 extent. 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 years a t d i f f e r e n t l o c a t i o n s , together 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) o r densely concentrated herring fisheries.  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 t h e same p o r t or owned by the same company may i n d i cate t o 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 a r e at (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 reconnaissance 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 unoperational i n inclement weather conditions. 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 ies 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 .  fisher-  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 experience.  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 schools  i s e s s e n t i a l to the success o f a haul (Minne, 1970; Kodera,  S t e i n b e r g , 1970;  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 vessels.  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 hinder 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 with t h a t possible  by conventional 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 miles dependi 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 given 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), hatched areas represent fish schools.  Cross  (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_ given.  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 commerc 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 t o 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 o f 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° t o 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 g i v e n 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 experience. 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 oceanographic 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 o f zooplankton 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 operat o r s i n d e c i d i n g the general area to deploy t h e i r ships and concentrate t h e i r e f f o r t s , and i s of l i t t l e help 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 every 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 catch p r o b a b i l i t y areas.  Corre-  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 region) 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 catch 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 using 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; Nickerson, 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 provide 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 attempted.  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 should 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 catch 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 crux o f t h e i r problem i s t h a t there 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 operations can be considered 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 operations research 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 contacts and the p r o b a b i l i t i e s o f t h e i r reaching 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 , along 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 search. Following 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 in many d i r e c t i o n s .  A search a l g o r i t h m using operations 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 Scorpion (Richardson and Stone, 1971).  They use a p r i o r i s c e n a r i o s to  evaluate 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 areas. 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 proc 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 handling 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  recurrence 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 contacts 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-  distribution.  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 boats, 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 a r e 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.  H i s model uses renewal theory t o d e s c r i b e d e l a y before  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 , t h e 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 provides 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 represented 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 H(x)  n = - 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 . np. = 1—  -10Th 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 ing a s e 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 land 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 t o 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 t o 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 ) = - ^p(x J  n  1  ... x ) l o g n  p(x '...x ) 1  n  dL  The v a r i a b l e s x-j...x are the search r e s u l t s , t h a t i s the number o f f i s h n  s c h o o l s , along the curve, 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 obtained.  The amount o f 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  is  required. 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 .  Sherstnikov gives  -11p r o b a b i l i t y o f d e t e c t i o n , P^, as P = 1 - e ^ ' ^ d  where  <(.(x,y) = search d e n s i t y a t (x,y)  Koopman uses P = l - e ^ d  where  y  =  instantaneous p r o b a b i l i t y o f 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 ' ) dx dy 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 o f d e t e c t i o n ( S h e r s t n i k o v , 1968). - < f ) ( x  y )  3  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 L  (2.1)  Sherstnikov s t a t e s t h a t f o r curves y , such t h a t y  then  = f(x)  a o 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 t o 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 equation; 9F 9y  =  _d_ 3F " 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.  Sherstnikov deduces the i n t u i t i v e l y obvious  r e s u l t t h a t t o minimize equation 2.3, y = f ( x ) must be orthogonal t o 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 o f 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 specified.  It would seem reasonable to make the  assumption that <j>(x,y) is constant;  then i f p(x,y) could be specified, a l l  that need be done is to maximize the line 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 detection of f i s h P/S, Sherstnikov shows that a zig-zag pattern w i l l minimize H(L,X,Y) where:  Ha.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 confidence interval on the mean f i s h 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 s h occurrence. Sherstnikov (1968) also considers the situation where the probab i l i t y density of occurrence i s dependent on distance from some point. The probability of occurrence, at range r, direction <f>, is P(r,f)  = 2is  and the curvilinear entropy to be maximized is H (L,r,$) = / r  2  + r'  <t>°  subject to / r + r' d«f, = L c  <J>°  £  2  d* = L  -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 of 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)  Figure 3. Nature of optimal t r a c k s as curve length 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 2 , r  ;  r  c-|dr , , r  ^ W  r  72  x  -  x)  2~  C  2  " •  " l c  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 C2 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 . nature o f these search paths.  F i g u r e 4 i n d i c a t e s the  As r i n c r e a s e s the curves become  compressed.  Sherstnikov 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 area.  The s t a t e o f the grounds may be known, but there may  be no f i s h present.  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". t h e s i s i s aimed.  It i s towards examining some o f i t s components t h a t t h i s  (b)  (a)  Figure 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 < r ( S h e r s t n i k o v , 1968), g  b  where r = range o f d e t e c t i o n a t 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 a r e 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 area.  (2)  F i s h d e n s i t y o r number of commercially f e a s i b l e schools or centres 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 , considered i n chapter 4.  3.1  S i z e o f 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 a r e  g e n e r a l l y not f i x e d and may depend on season and/or oceanographic  conditions,  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 widely 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 o f 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 t o 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 t o 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.  This 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 o r the  -17d i s t a n c e to the grounds may be so great 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 occurs 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 fishery;  or minimal p r o b a b i l i t y of encountering 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  in the Black sea, the l i m i t s of t h e i r search were determined by the absence of observed e c h o - r e f l e c t i o n s of f i s h at 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.  If  f i s h were detected 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  Fish Density 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 confidence 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 devices 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 surveys, 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 with 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 experienced 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 with  sonar search have, i n one case, shown t h a t the number o f s u r f a c e schools observed was no i n d i c a t i o n of the t o t a l number o f schools 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 subs u r f a c e schools were 2.5 times more p l e n t i f u l than s u r f a c e schools 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 schools i n a f i s h e r y have been made from catch data, 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 to 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 schools i n a f i s h e r y . was based on two  The method  premises:  (1) That d i s t r i b u t i o n of schools i s random ( i . e . catches f o l l o w a poisson r e l a t i o n s h i p ) ; (2) That boats do not overlap i n t h e i r a c t i v i t i e s and no communic a t i o n occurs between them. From these premises Neyman evaluated 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  at time t , n schools 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 schools 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 of 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 of 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 of 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 schools l o c a t e d .  These should  not be major problems, but the independence of 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 of 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. fish detection.  A n a l y s i s of the r e s u l t s w i l l depend on the method o f  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 calibrated acoustic records.  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 of 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 of 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 schools Z = no. of 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 = resultant velocity  -20-  7  r~I  and V  =ViT +  (3.2)  where u = average velocity of schools w = average velocity of the observer. Skellam notes that though (3.2) i s 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 r e s u l t , using s l i g h t l y different assumptions to those of Skellam.  If the distribution 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 i s w, with a perception range R, then, from Koopman, the number of encounters, N would be o so  2*  N =  0  N 2-n 0  2R/ wd<f, 27r  0 In Skellam's (1958) notation, n L»  = JL  2u  f "m  2 R o n*  z  * [V*] = 2R " ' 2RV Z 2  2lF  L  l t  J  z  0  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.  Q  -21-  It i s assumed that whenever a t r o l l l u r e i s w i t h i n range of an albacore (Thunnus a l a l u n g a ) , i t i s taken, although as Nakamura (1967) notes, in t r o l l i n g may depend to some degree on how hungry the f i s h are.  success 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 other 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 detected 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  to those o c c u r r i n g when the sun i s unobscured and at 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 during the c o l l e c t i o n of C r a i g and Graham's data, c o n d i t i o n s were assumed to 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 length of 30 cm (approximately 12 i n . ) , the d i s t ance at which the l u r e would be perceived 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 of 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 fraught with  -22assumptions.  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 purely 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 a r e types o f t e s t o b j e c t s .  Presence o f feeding  markings on the s i d e o f tuna would a l s o extend the range o f 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 schools during 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 , information must n e c e s s a r i l y be drawn from several sources.  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 ) f o r y e l l o w f i n (Thunnus a l b a c o r e s ) , albacore and b l u e f i n (Thunnus - 1  thynnus) tuna".  P r e v i o u s l y i t had been thought t h a t t h e 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 , but t h a t y e l l o w f i n o f the same s i z e - 1  averaged only 50 cm s e c , and i n a l a t e r unpublished - 1  paper 46 cm s e c  - 1  (or  1.31 lengths s e c ) f o r y e l l o w f i n . - 1  Many workers have reported a r e l a t i o n between f o r k length and swimming speed.  Magnuson (1970) found t h a t t h i s r e l a t i o n was well 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 length o f the f i s h . b = constant 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 that have been  -23used 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 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 , - 1  then from equation 3.4, h D  _ l o g V - l o g 10 ' log 1 = 0.441  C r a i g and Graham (1965) r e p o r t t h a t o f 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  This value i s below the range given by Nishimura (1963).  Magnuson's un-  published value o f 1.31 length s e c 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 - 1  give a speed o f 86 cm s e c " , which i s higher than t h a t obtained with h i s 1  1970 r e l a t i o n . Now where:  V =  J u  2  +  w  2  w = speed o f f i s h = 63.2 cm s e c "  1  = 1.23 knots u = speed o f observer = 6 . 5 0 knots V =  J  1.2 + 6.5' fc  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 schools a r e 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 s c h o o l s be randomly and uniformly 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 obs 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 o f i n t e r e s t t h a t Royce and  -24Otsu (1955) i n observations on schools of s k i p j a c k , found t h a t the schools 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, during 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 =  71 292 x .088 x 6.62 x 2  x  = 0.201  schools ( n a u t i c a l mile)  .  T h i s estimate o f mean school d e n s i t y w i l l provide a b a s i s f o r d e c i s i o n making in 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 provide i n f o r m a t i o n on  heterogeneity 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.  This 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 process, 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 schools 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 of four 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 set 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  -25the 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 -  butions;  choice 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 of others.  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 could 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 conditions. 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.  Clustered distributions reflect  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 encounter 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 thems 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 . 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  In  -26number 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) distribution or a Neyman type A respectively (Patil and Joshi, 1968).  Both the Poisson and Poisson with added zeroes are  limiting forms of the Neyman Type A d i s t r i b u t i o n .  (Martin and K a t t i , 1965)  If the distribution of cluster size tends towards a log d i s t r i b u t i o n , then the overall distribution tends to that of a negative binomial rather than a Poisson-Poisson distribution (Pielou, 1969).  For a Poisson distribution  the probability of encountering r schools is given by P(r) =  where x = mean school density. When the distribution of school numbers follows a Poisson d i s t r i bution with added zeroes, the probabilities 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~ + c  ,  x  e  =  eA -  For a Poisson-Poisson d i s t r i b u t i o n ,  - 1  -27-  „k -m  r  P(r) = I  /.,\r-k  -kx  (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 i s t r i b u t i o n , the probabilities are given by  '> - ' $ J ^ i m r ) ' V  Pr  (Southward, 1965)  where x = mean school density k = distribution parameter dependent on degree of contagiousness For the Poisson distribution the variance is equal to the mean, X For a Poisson with added zeroes the mean is ex and the variance i s ex(l + x) - e x 2  2  The mean for both the Poisson-Poisson and Neyman Type  A distributions is mX with variance Binomial distribution is  mx(l +X).  Variance of the negative  X - X /k„ 2  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 until the f i r s t catch of the day was assumed equal to that time from last catch to f i n i s h of sampling for the  -28day.  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 (length) of the sampling u n i t could be chosen as d e s i r e d , and the sampling day was considered 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  miles were used and the number of schools i n each sample found from the d a i l y accumulated d i s t a n c e s .  Overestimation 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 length (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 -  bution mean; Poisson with added zeroes;  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 distribution.  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 length of 20 m i l e s ;  for  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 Poisson-  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 reverse.  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 I  Means, v a r i a n c e s , goodness o f f i t and entropy f o r given d i s t r i b u t i o n s 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  Data Variance  2  d.f.  Entropy  2  0.591  5  0.234  0.234  0.239  X 1.605  10  0.467  0.467  0.621  12.812  4  0.894  15  0.623  0.623  0.963  27.400  5  1.041  20  0.934  0.934  1.662  9.713  6  1.266  25  0.750  0.750  1.150  18.893  4  1.141  Poisson with added zeroes Sample Length  Added Zeroes Parameter  Mean  Dis t r i b u t i o n Variance  5  0.299  1.145  0.617  0.239  17.155 2  0.685  10  0.409  1.479  1.134  0.621  8.453 4  1.019  15  0.482  1.614  1.428  0.963  8.549 5  1.183  20  0.571  1.919  1.997  1.662  7.670 6  1.410  25  0.538  1.781  1.747  1.150  5.635 4  1.294  Data Variance  X  d.f.  Entropy  Table I continued Poisson-Poisson Mean  Distribution Mean  Sample Variance  Distributi on Variance  m  Var(m)  X  5  0.234  0.237  0.239  0.228  0.228  0.001  0.041  0.003  2 X 0.137  10  0.467  0.496  0.621  0.379  0.379  0.003  0.308 0.015  15  0.623  0.602  0.963  0.488  0.488  0.006  0.235  20  0.934  0.877  1.662  0.667  0.667  0.012  25  0.750  0.820  1.150  0.593  0.593  Sample Length  Var(X)  d.f.  Entropy  2  0.599  1.952  4  0.940  0.016  1.976  5  1.051  0.314  0.029  5.142  6  1.281  0.011  0.382  0.033  4.977  4  1.244  m  X  Neyman's Type A  5  0.234  0.239  0.239  10.009  0.023  2 X 0.702  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  Sample Length  Mean  Sample Variance  Distribution Variance  d.f.  Entropy  2  0.593  Table I continued Negative Binomial Sample Length  Mean Sample Var.  Distribution Var.  P  Var.  (p)  k  Var.  (k)  x  2  d.f.  Entropy  5  0.234  0.239  0.228  0.023  0.000  10.009  0.616  0.710  2  0.593  10  0.467  0.621  0.275  0.412  0.001  1.414  0.000  3.987  4  0.915  15  0.623  0.963  0.261  0.581  0.002  1.141  0.000  0.839  5  1.077  20  0.934  1.662  0.140  0.850  0.002  1.099  0.000  2.877  6  1.335  25  0.750  1.150  0.218  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  -32Best o v e r a l l f i t s were: Distribution Negative Binomial  Sample length (nautical miles)  (Linear interpolation) a  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 o f f i t f o r each sample l e n g t h . As confidence l i m i t s are a necessary adjunct to management d e c i s i o n s , choice 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 o f being able to d e s c r i b e the actual 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 situation.  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  at a g r e a t e r r a t e than the mean.  Table I I I 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 roundo 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 zeroes, the d i s t r i b u t i o n v a r i a n c e cons 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 Ratios o f v a r i a n c e to mean and variance 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 .  Poisson Sample length (miles)  Variance mean  5  1.021  10  I. 330  15  1.546  20  1.779  25  1.533  Poisson with added zeroes Sample length (miles)  Variance Distribution variance  5  0.387  10  0.547  15  0.674  20  0.866  25  0.646  Negative Binomial Sample length (miles)  Variance D i s t r i b u t i o n variance  5  1.048  10  2.258  15  3.689  20  II. 871  25  5.275  -34-  TABLE I I I (cont'd)  Poisson-Poisson Sample length (miles)  Variance Distribution variance  5  1.048  10  1.639  15  1.973  20  2.492  25  1.939  Neyman's Type A Sample length (miles)  Variance Distribution Variance  5  1.000  10  1.000  15  1.000  20  1.000  25  0.984  -35-  The variance o f 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 of f i t ;  f o r example, p r o b a b i l i t y o f type I e r r o r f o r the 20 m i l e sample  length was 0.364 with a variance 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 o f type I e r r o r f o r the 10 mile sample length 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 .  This discrepancy  showed the same trend as the x f o r the goodness o f f i t .  However, the  2  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 ation. 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 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 a r e 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 , o r 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 . This c o u l d be due t o underestimation 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, Poisson-  -36Poisson, 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 schools or c o n c e n t r a t i o n s v a r i e s , and hence schools 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 occur, and t h e i r s i z e . I f A represents 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  k  l o g p + £p. k  = H(A) + z p H k  H  k  k  k  k  q.  z  n  l o g q. }  (B)  ( K h i n t c h i n e , 1957)  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 encountering school o f s i z e n  (B) can be regarded as the c o n d i t i o n a l entropy o f the scheme B, given  scheme A, the number of schools 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 log 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 using the method o f F i s h e r , Corbett and W i l l i a m s (1943). E(n) = £  x  For the log s e r i e s ,  n  where E(n) = expected number of schools with x f i s h , and  •37-  x  where  FTT  =  p  = parameter of the negative binomial distribution  a  =  (k-1)!  k = power.factor of the negative binomial d i s t r i b u t i o n .  From catch sizes given by Craig and Graham (1965), then, E(n)  =  (o.9157)  n  A goodness of f i t test gives a probability of Type I error equal to .19, (linear 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 distribution of numbers of schools per sample, 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 i z e s , and the remainder due to the uncertainty 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 i s h concentrations. Due to movement of the f i s h after the i n i t i a l location, their position w i l l  -38-  be known w i t h d e c r e a s i n g c e r t a i n t y as time passes; an i n c r e a s i n g f u n c t i o n o f time.  the entropy w i l l be  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 o f 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 o f 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 t o the manner o f movement o f f i s h s c h o o l s . c r i b e t h i s movement.  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 t o desC e r t a i n movements w i l l be c o n s i d e r e d t o be model  independent, such as c y c l i c a l movement due t o 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 o f two analogies: 1) The K i n e t i c theory o f gas movement, o r 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 constant v e l o c i t y over a l a r g e body. For t h e second model, S a i l a and Flowers (1969) make an analogy between conduction o f heat from a source moved a t 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 t o develop a q u a s i - s t a t i o n a r y s t a t e , and movement o f f i s h in 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 - ti .  - J L -  4Tik  e x p  e  x  21*  p  _ ^  e  X  p  - £  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 o f 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 r = x2 + y 2 + z 2 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 its possible realism. 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 large. 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 ) = exp { - r / n a } 2  2  2  2r/na  2  0 <. r < -  -40where: 2 a = mean square dispersion per generation (or unit move) analogous with the mean square velocity in Maxwell's d i s t r i b u t i o n , n  = number of generations (or unit moves)  After n generations (or unit moves) the proportion lying outside some c i r c l e of radius R is P = / exp [ - r / n a ] 2rdr/na R 2  2  2  = exp {-R /na } 2  2  For our s i t u a t i o n , we are concerned with the probability of a school being within some c i r c l e of radius R, rather than the proportion of some population, but the method is s i m i l a r . Skellam (1951) also notes that the maximum likelihood estimate of a i s given by 2  a = III 2  nv  where n = number of generations v = number of observed values of r. Jones (1959) gives an alternate form for this estimate a  where  2  =  1 [  zV  .  (rrcose)  2  ]  (  5  J  )  t = time interval n = number of schools Beverton and Holt (1957) appear to be the f i r s t to consider d i s -  persion of f i s h 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 fish such as spawning migrations and their primary interest, that of local interchange in  -41which 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 i s made.  t h a t a t times a random change i n d i r e c t i o n  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_  9t it  1_ v£  A4 n * n  .  ,j^C (  ~T 9 ax  +  j^C\  ()  V Ol  52  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 considered 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 negligible.  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  =  =  nd  2  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 o t h e r 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 .  -42Jones re-expresses the p r o b a b i l i t y of occurrence as a f u n c t i o n of time and d i s t a n c e : R P(R,t) = 1 - exp (-5-)  2  (5.3)  Equation 5.3 enables the p r o b a b i l i t y of a school'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 fied. Measurements of the displacement of f i s h schools over s h o r t time periods appear r a r e i n the l i t e r a t u r e . recorded observations on s k i p j a c k .  However, Royce and Otsu (1955) have  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 detected 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 could be de-  t e c t e d and movement of schools while being watched appeared to random motion.  approximate,  They then recorded d i r e c t o b s e r v a t i o n s on schools 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 while being observed.  Movements of  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 of 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) give v e c t o r diagrams showing movement of schools f o r 7 s e t s of o b s e r v a t i o n s .  The number of 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 schools which were observed f o r a f u l l 10. minutes.  However, the length of 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 10 mins  April 1 6 - 1 7  June  July  5 mins  No. of schools showing no change of position  14  3  4  7  20-21  22  4  4  14  25-26  19  3  2  14  27 - 28  11  3  1  7  13 - 14  30  5  5  20  29-30  27  3  5  19  28  3  4  21  9 - 10  -44-  Royce and Otsu's (1955) data can be used t o examine how t h e r e s u l t a n t v e l o c i t y o f a number o f schools may be determined with confidence estimates on the d i r e c t i o n o f movement, and how such estimates can be used in 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 o f Royce and Otsu's (1955) data t o 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 a r e necessary: (1)  That t h e 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 t h e 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 t o t e s t the hypothesis k = 0, a g a i n s t the alternative  k > 0 ; i . e . t o determine i f t h e 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 processes.  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 t h e i n d i v i d ual d i r e c t i o n s , where; 2 z cos e + and  D  1  where  -  z  2 sin o  2  T  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 o f o b s e r v a t i o n s  The r e s u l t s a r e 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 V I . 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 t o 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 could 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) Series  No. o f schools  No. o f z C r i t i c a l value z for moving schools f o r a l l schools o f z, a=.05 moving schools  C r i t i c a l value 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*  13 - 14  30  10  0.4347  2.9957  +  1.3040  2.9187  29 - 30  27  8  0.0894  2.9957  +  0.3016  2.9014  9. - 10  28  7  0.7485  2.9957  +  2.9936  2.8819  June July +  For number o f schools g r e a t e r than 24, z f o r n = » used i n t a b l e s o f 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 Direction cose e  sine  Distance r  Time t  rcose  0.2924  0.9563  0.505  10  0.148  ' /t 1.530  - .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  0.9976  0.0698  0.245  5  0.244  0.720  .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  .9483  - .9799  .277  5  .263  3.324  73 96.5  2  A p r i l 20 - 21 4 88.5  281.5  TABLE VI (cont'd)  cose  sine  Distance r  0.9925  0.1219  0.236  5  0.234  0.668  .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  Di recti on 6  Time t  rcose  April 25 - 26 7 9.5  April 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  e  Distance  Time t  cose  sine  r  54  0.5878  0.8090  0.904  87  .0523  .9986  .307  90  .0000  1.0000  92  - .0349  103  j  rcose  r \  0.531  4.903  5  .162  1.131  1.398  10  .000  8.388  .9994  .920  10  - .032  5.078  - .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  ine 13 -14  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  TABLE VI (cont'd) Direction e  cose  sine  r  0.8039  0.5948  68  .3746  106  2  t  rcose  r /  1.012  10  0.814  6.145  .9304  .289  5  .108  1.002  .2756  .9613  1.000  10  - .276  6.000  211  .8572  .5150  .388  10  - .333  .903  231  .6293  .7771  .269  5  - .169  .868  307  .6018  .7986  .277  5  .167  .921  333.5  .5519  .4462  .280  5  .154  .941  336.5  .9171  .3987  .289  5  .265  1.002  31.5  .8526  0.5225  0.260  5  0.222  0.811  51  .6293  .7771  1.429  10  .899  12.252  70.5  . 3338  o9426  .723  10  .241  3.136  92  .0349  .9994  .350  5  - .012  1.470  105  .2588  .9659  .350  5  - .091  1.470  309  .6293  .7771  .302  5  .190  1.094  354  .9945  .1045  1.010  10  1.000  6.120  t  June 29 -30 36.5  July 9 - 1 0  -50July 9-10 (see Table IV), at an a = 0.05. Watson polar vectors.  and Williams (1956) give a test for the homogeneity of two 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 N(N-1)(N-2)[(R +R ) 1  (N -R ) 2  where:  2  [N R  2  2  2  - R ]  2  -  2  [R  -(R Ro) ] r  2  (5.4)  (R^-Ro ) ] 2  2  N = Number of vectors = 11 for April 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)  where RIx  zr cose  R.  zr sine  and similarly 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  = 3.841.  2  1..05  Hence R-, and R can be considered as belonging to the same popu?  lation. 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 , a r e c a l c u l a t e d f o r a l l data combined, as well 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 a  o  = 1.172 m i l e s  2  -  a  p  = 3.610 miles  hour  1  hour 2  -  1  S i m i l a r l y f o r the data o f A p r i l 2 a 2 = 0.896 miles hour ?  f o r a l l schools, f o r moving schools o n l y  27-28 1  and J u l y  9-10  combined, then  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 schools w i l l be „  _ £ r cos  27-28  if( 5  m i l e s hour"  1  V = 1.731 miles hour"  1  =  and f o r A p r i l  6  Ft  V  and J u l y  0.493 9-10  A s k i p p e r , i n c o n s i d e r i n g whether t o 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, R  (5.6)  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 schools.  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 o f the school movement i s known with c e r -  t a i n t y , and i s o f 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  GRRPH I RRNGE LIMIT OF SCHOOL OCCURRENCE RT SPECIFIED PROBABILITY  o ui.  MEAN SQURRE DISPERSION =3.610  a C3 .  CO  21  a  + ,+  CD  CC cr:  ^ ^vx****  +  4-vXXX x*  X><  1 0.0  10.0  1  1  1  1  1  20.0  30.0  40.0  50.0  60  TIMECHOURS)  +  CONFIDENCE L I M I T S = 0.50  X  CONFIDENCE L I M I T S = 0.05  -53p 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 center o f t h e 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 V t i n e, where  the r e s u l t a n t d i r e c t i o n ,  e = a r c tan'*; "° z r 'cose r  ( S a i l a and Flowers, 1968).  si  (5.7) '  v  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 lack o f observations and/or because the d i r e c t i o n o f f i s h movement probably involves some s t o c h a s t i c element.  Gumbel, Greenwood and Durand (1953) show that 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 ) rt  '  f ( e )  2  *i (k) 0  <-> 6  8  where: k = parameter, analogous t o the information 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 = modal or most l i k e l y d i r e c t i o n Q  I ( k ) = Bessel f u n c t i o n o f the f i r s t kind o f pure imaginary argument Q  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 t h e s i g n o f x and y o f the v e c t o r mean, ( x , y ) , where: x = n —  EX.  l  y = —n Ey. i J  where from the p o l a r  J  transformation, x = r cose y = r sine  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 vector a n g l e , e , which i s independent o f the number o f s t a t i o n a r y Q  -54schools, is eo = 64° 39' 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 = 33°42' Q  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 -  fy 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: a For a l l d a t a ,  fj>  =| x + y  ~J~'  N  _ 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 ) as a f u n c t i o n o f (e - 9 ) Q  k.  Q  f o r these two values o f  As can be seen, the curve with k f o r a l l data i s l e s s concentrated  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 operator 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 schools so as to o b t a i n a measure o f the arc 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 .  This 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  K - 0 . 1 8 6 8 P L L ORTR  „ «o.22B5 APRIL 2 7 - 2 8 RNO JULY 9 - 1 0 DflTfi K  a a'  180.0  T  120.0  T  60.0  0.0  60.0  DEGREES FROM MODAL DIRECTION  T  120.0  1  180.0  -56school 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 Williams (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 can be found, such t h a t Q  P(R > R | X) = Q  where  a  N z cos e..  R = resultant vector a = specified level of probability They show t h a t N-R  I^-V  ( ^  P(R > R I X) 0  (5.9)  I f a = . 0 5 , then f o r a l l data combined; R  =  o  from equation 5 . 9  6.987  N  Now  R cos(X - R ) Q  = . ^ c o s 6.  Q  (5.10)  (Watson and W i l l i a m s , 1 9 5 6 ) For  = 0.05,let - o  X  Where  6  05  = n a  ^  -  R  6  .05  °f confidence i n t e r v a l on d i r e c t i o n ,  a r c  Then from equation 5 . 1 0 , 9  .05  =  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  43°3'  away from t h e 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 then 9  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 9  .05  =  6 7 0 5 1  = 0.05,  '  For a p r o b a b i l i t y o f 0.5 e .50 = 44°48' 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 operator 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 this direction.  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 schools combined.  These f i g u r e s could 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 sets 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  well 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 searching. 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 only 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 \  -58At both l e v e l s o f s i g n i f i c a n c e , the confidence 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 confidence 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  evaluated f o r moving schools o n l y , the confidence i n t e r v a l f o r a l l observat 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 reversed with the confidence 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 schools c o n s i d e r e d ) , In s e c t i o n 6.1 the confidence i n t e r v a l s f o r the A p r i l - J u l y observ 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 ing 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 real 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.  Within 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 . for  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.  -59The 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 rates. This 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 , weighted 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 fuel 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 present catch r a t e and t h a t which e x i s t elsewhere.  may  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 regard to  fishing tactics: (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 data. (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  of 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  -60i 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 catcher vessel 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 catcher vessel 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 present 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 concentration. The l o c a l i t y i n which the school would be expected to occur, 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 vessel 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, A = =  2e  {n(Vt + R ) - n(Vt - R) } 2n 2  2  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  sections R = range given by equation  5.6  Graph III 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 and 6  nc  = 67°51  5Q  = 44°48'  -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 occurrence 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 i s h 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 o r i g i n a l l y at <J> = direction of school relative to i n i t i a l position of catcher vessel For situation (b) l e t m- be the minimum size of a f i s h school worth exploiting, and assume that the value of the school is proportional to the school s i z e , n. n _> m.  Hence, catcher vessels are concerned with schools size n,  Assume that the probability of encountering a school size n, is as  for Craig and Graham's (1965) data, adequately described by the log s e r i e s ,  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  -64w = cross-sectional area or width of search by vessel X = expected school density The expected number of schools of uttltzable size encountered during this period would be . v Atwz  n=m  x  h  - lionHn g ( l - xv )\  Hence, to be worthwhile changing grounds then x  n  ,  VAtWl  n=m  V.  <  1  (6.1)  -iog(l-x)  Implicit in the inequality 6 . 1 , is that the density of f i s h schools and the distribution of t h e school sizes i s similar for the two different areas.  Then for a given v e s s e l , t h e 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 t h e 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 s u c h cases, only i f the school is very  distant does i t become a better policy to remain searching on the present grounds. 6.2  Craig and Graham's (1965) Tuna Data In this section Bayes Method w i l l be used in making predictions 1  about f i s h density in relation t o f i s h i n g t a c t i c s .  The potential and  a p p l i c a b i l i t y of such methods are considered further in the discussion. The basis of this method i s to make estimates of f i s h density at any particular time dependent both on t h e 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  cn cn  i  n 0 40 8J) 12.0 16.0 20.0 T I M E TO S T E A M TO A L T E R N A T E GROUNDS (HOURS)  24.0  -66prior expectation of r e s u l t s .  Obviously at time t=0, the expected value  of the f i s h density w i l l be expressed e n t i r e l y by the prior density. Explicitly, i f X = population density over the f i s h i n g grounds and x . = sample r e s u l t s , then  P(A|x ) 1  « p(x.JA)  Lindley, 1965b)  p(A)  where: P.(*I.Xf)  = posterior probability density of A given the sample data  P(X.|A)  =  likelihood of the sample data i f  A  i s the actual f i s h  density ?M  = prior probability of A being the f i s h density.  For t r a c t a b 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 i s t r i b u t i o n , although better f i t s were obtained with the negative Binomial and Neyman's Type A d i s t r i b u t i o n . For some sample x . , the Poisson probability is *1  -A  P(X< =  X) - t  *  6  X  j'  At time t = i , the likelihood of obtaining the i sample results w i l l be ,  v  1  .  e" > * x  e ^ A  £  X  Xj  J  Inherent in formulating the prior distribution is the use of subjective feelings about the possible state of some system - a fishing ground in the absence of current data.  The subjective feelings of a skipper may  -67be 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 during the same p e r i o d i n previous seasons.  To make use o f a skipper'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 conditional density. Assume that the s k i p p e r ' s expected catch r a t e i s 0.25 schools per 5 mile sample. 0.239).  ( I t i s known from the data that the actual o v e r a l l mean i s  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 that 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  of be!ief 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 t o °°, and p r e f e r a b l y be unimodal. amenable t o manipulation  Here a r - f u n c t i o n i s used as being and with t a b l e s a v a i l a b l e .  convenient,  However, i t i s con-  c e i v a b l e that 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 given by P( ) X  = e  (a) T  { 6 2 )  (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 variance 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 t o 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 rate w i l l be above some value X  i.e.  oo  / f ( X ) dX = .95 X  .05  -68-  \05  0 r  / f ( x ) d x = .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  0  5  ^'?'°' r (a)  Then f o r ( a + l ) a  dx  1  ,  a  = 0.25,  2  =0.05  from Pearson's  (1946)  tables  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  pUlx,) =  x . .  - x i . , zxj  .  e  X  n x.l  /  ~  " a  X  i  e  q  EX. o - l  i +  +  ° x  dx  r(a)a  i  ' - ( r)  e  —  a (£)  n x.! J x  a_1  r (a) a  -AT j e X Z  (-)  -d4  J  i  i £X.+a )  J  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  (6.4)  -69-  E ( x | ) = / Xp(x|x.)dX  (6.5)  Xi  Substituting 6.4 for p.(x|x^) in 6 . 5 ,  -X(l + -) oo  O  E(x|x.) - / Xe 1  Ex. +a - 1 , IX.-kx J i_» J X^ (1 + a' dx  (1 +  n  r  (a  a>  + ix.) J  I + EX.  a  i + f  (6.6)  As for a r - d i s t r i b u t i o n : mean  = (a+l)a 2  variance = (a+l)o „ tnen  o  _ variance = ———— 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 distribution becomes larger, the relative contribution that the prior distribution 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 miles was used f o r these sample  results. 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 confidence 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 that 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 confidence 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  /• °  e  5  -x(i  EX.+ct-l  + -) X  q  r  x,  (E  1 EX.+ct 1 o)  (1+  J  -  dx  0.50  +<x)  w  v + J  I  U j  =  L  a ) _ _  r(EX .  *.50 / =  /  c n  50  I  1  Ex.+ a - l  + o)  e  X  '  dX  (6.7)  t h e n 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 ZXj+a-l e  -t(-4)i + a \ EX.+a  0  (  0  -X(i  +a)  j  1 S e t t i n g t = x ( i +a),  A  1 T  }  J  t /-  50  cn  e  _ t  i + a . EX.+a-l t dt J  have  TABLE VII Expected d e n s i t y values f o r sample d a t a , x ; 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 confidence i n t e r v a l o f .95 (BP ) ; 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 (B 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. s  g  5  n  Day 1 Sample No. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16  x  s  0.000 .000 .000 .000 .200 .333 .428 .500 .444 .500 .545 .588 .538 .500 .466 .438  x  B  P  .95  B  .95  B  X  .50  X  s  B  0.250 0.250 .210 0.000 0.020 0.158 0.000 .210 .181 .000 .018 .137 .000 .181 .000 .159 .159 .000 .014 .119 .141 .000 .013 .106 .000 .141 .225 .000 .046 .193 .000 .127 .295 .000 .086 .266 .000 .116 .352 .022 .123 .327 .000 .107 .401 .089 .162 .378 .125 .174 .351 .162 .149 .111 .373 .079 .414 .132 .182 .392 .100 .151 .091 .142 .451 .179 .220 .429 .482 .221 .242 .462 .083 .134 .456 .204 .235 .440 .077 .126 .432 .189 .220 .416 .071 .120 .411 .177 .211 .396 .067 .114 .391 .165 .200 .374 .062 .108  P  B .95  B .95  i.OOO 0.20  .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000  .018 .014 .013 .012 .010 .010 .037 .033 .030 .030 .027 .027 .024 .024 .022  X  .50  s  0.158 0.000 .137 .500 .119 .333 .106 .250 .096 .200 .088 .166 .081 .142 .125 .149 .139 .111 .129 .100 .122 .181 .114 ,166 .108 .153 .103 .143 .098 .133 .093 .125  Day 3 x B  P  .95  B  .95  g  B  .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  TABLE VII Conti nued Day 6 as f o r Day 5  Sample No.  x  s  0  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16  0.000 .000 .000 .000 .200 .167 .143 .250 .333 .300 .364 .333 .308 .286 .267 .250  Day 4 x P B .95  Day 5 B  .95  B  .50  X  s  X  p  B  0.250 0.250 .210 0.000 0.020 0.158 0.000 .210 .181 .000 .018 .137 .000 .181 .000 .159 .000 .014 .119 .159 .141 .000 .013 .106 .000 .141 .225 .000 .046 .193 .000 .127 .175 .000 .116 .205 .000 .041 .162 .000 .107 .189 .000 .041 .000 .098 .250 .000 .037 .149 .000 .092 .303 .000 .070 .209 .000 .086 .261 .283 .015 .099 .327 .014 .096 .246 .000 .080 .123 .290 .000 .076 .308 .059 .291 .055 .118 .273 .000 .071 .276 .051 .110 .258 .000 .068 .263 .047 .106 .246 .000 .064 .250 .044 .100 .235 .000 .047  B  .95 .95  B  X  .50  s  Day 7 x B  P  .95  B  .95  B  .50  i.250 i.OOO 0.020  .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000  .018 .014 .013 .012 .010 .010 .009 .008 .008 .008 .007 .007 .006 .006 .006  0.158 0.000 .137 .000 .000 .119 .106 .000 .096 .000 .088 .000 .143 .081 .074 .125 .070 .222 .065 .300 .061 .273 .057 .250 .054 .231 .214 .051 .200 .049 .046 .188  .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 .162 .189 .000 .041 .174 .000 .137 .149 .232 .000 .070 .209 .261 .283 .015 .099 .265 .014 .096 .246 .250 .013 .087 .230 .236 .012 .086 .218 .224 .011 .078 .207 .213 .010 .077 .197 .070 .187 .203 .009  TABLE  VII Continued  Day 9 as for Day 5 Day 8 nple No.  X  s  0  x B  Day 11 P  .95  B  .95  B  .50  X  s  0.250  X  B  P  B .95 .95  B  X .50  s  0.250  1  0.000  .210  0.000  0.020  0.158  1.000  2  .500  .305  .000  .070  .276  .500  3  .333  .280  .000  .056  .242  4  .250  .250  .000  .051  5  .200  .225  .000  6  .333  7  .371  x B  P .95  B .95  B  0.000  0.020  0.158  .50  1.250 0.000  0.077  0.319  0.000  .319  .000  .070  .276  .000  .181  .000  .018  .137  .333  .280  .000  .056  .242  .000  .159  .000  .014  .119  .216  .250  .250  .000  .051  .216  .250  .250  .000  .051  .216  .046  .193  .200  .225  .000  .046  .193  .200  .225  .000  .046  .193  .295 ' .000  .041  .175  .167  .205  .000  .041  .175  .167  .205  .000  .041  .175  .286  .270  .000  .077  .242  .143  .189  .000  .041  .162  .286  .270  .000  .077  .242  8  .250  .250  .000  .071  .226  .125  .174  .000  .037  .149  .375  .326  .019  .118  .300  9  .333  .303  .000  .070  .209  .111  .162  .000  .070  .209  .333  .303  .017  .106  .281  10  .300  .283  .000  .063  .195  .100  .151  .000  .063  .195  .300  .283  .015  .099  .261  11  .273  .265  .000  .058  .184  .181  .204  .000  .058  .184  .273  .265  .014  .096  .246  12  .250  .250  .000  .056  .172  .167  .192  .000  .056  .172  .250  .250  .013  .087  .230  13  .231  .236  .000  .052  .164  .231  .236  .012  .086  .218  ,308  .291  .055  .118  .273  14  .214  .223  .000  .051  .155  .214  .224  .011  .078  .207  .286  .276  .051  .110  .258  15  .200  .213  .010  .077  .197  .200  .219  .010  .077  .197  .267  .262  .047  .106  .246  16  .188  .203  .009  .070  .187  .250  .250  .044  .100  .235  .250  .250  .044  .100  .235  .210  TABLE  VII Continued  Day 12 as for Day 2 Day 14  Day 13 flple No.  X  s  0  X  B  P  .95  B  .95  B  .50  X  s  X  B  Day 15 P  .95  B  .95  B  .50  X  s  B  P  .95  B  .95  B  .50  0.250  0.250  0.250  X  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 Sample No.  x s  0  x  B  P  Day 18 .95  B  B .95  x .50  x s  0.250  1  0.000  2  .000  3  .210  B  Day 19  P B B .95 .95 .50  s  0.250  x  B  x  P .95  B .95  B  0.000  0.020  0.158  .50  0.250  0.000  0.020  0.158  0.000  .210  0.000  0.020  0.158  0.000  .181  .000  .018  .137  .500  .319  .000  .070  .276  .000  .181  .000  .018  .137  .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  .210  -76S u b s t i t u t i n g In 6.7,  t-(xJ.Va) W h e r e  *0.50  t. 50  J  E  ertt  X  +  a  _ |  J  '°  dt=0 5  '  (6 8)  7~T  =  I T  —  a  Hence the confidence i n t e r v a l a t 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. VII.  The r e s u l t s are given i n Table  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 t o provide 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  calculated.  I f r schools are caught from each sample so that r = 0, 1, 2, ...» and n R =  ir, 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. ) Then  f(R)  e-  n X  (nx) ^  R  with E(R) = nx, v a r (R) =nx Let X be the mean o f the sample data o b t a i n e d , then X =n , with R variance = 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 U.B.C, Vancouver, B.C.  Ecology,  -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% confidence 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% confidence 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 schools were encountered than t h a t based on the sample d a t a .  A similar  trend 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 overestimates 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 estimate 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 confidence 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 catch r a t e s .  A s s o c i a t e d with 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 ( o r maximize any u t i l i t y ) .  In a f i s h i n g system a l o s s could accrue  i n three ways: (1) In r e t u r n i n g t o port when catch rates 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 rates obtained 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) Overestimating  the catch r a t e and remaining on the present  grounds when a higher catch r a t e could 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 considered, 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 , o r t o remain, d . c  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 proport 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 rates between grounds plus the time necessary t o "make a t r i p " on the a l t e r n a t e ground, i . e . L where  c  = c [tx+ t ( x - * ) ] t  = l o s s i n changing grounds c = constant o f p r o p o r t i o n a l i t y t = time t o change t o a l t e r n a t e grounds X = actual f i s h d e n s i t y on present grounds tj.= time t o '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.  (6.9)  -79I 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 * [ a t + t (a - 1 ) ] c  t  In a r e a l f i s h e r y , t ^ , the time t o 'make a t r i p ' i s a complex v a r i a b l e dependent on such things as crew f a t i g u e , market p r i c e s , r a t e a t which the catch i s obtained as well as the actual 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 t o 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 rate i s overestimated,  and a higher 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 t o the product o f the d i f f e r e n c e between catch rates 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 while the vessel would otherwise be changing grounds, i , e , L = c [ t ' U-A) - t A ] r  (6.11)  t  where t ' = time to make a t r i p on the present ground. t  Then equation 7.12 can be expressed as L  r  = c [ t ' (*-at '<J>) - a t * ] t  t  = c*[t '(l-a) - at] t  The discontinuous 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 there w i l l be an 'expected l o s s ' due t o unc e r t a i n t y about the actual 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 that  which minimizes the expected l o s s (Mood and G r a y b i l l , 1965)  -80-  GRAPH V - RELATIVE LOSS WHEN CHANGING  - RELATJVE LOSS WHEN REMAINING  PfflJL-—-—•—  in  CO CO ^  —  —  I—ID  I—  <X  I UJ  Q;  in  rv-\  4——"—"  ai .  a i 0.0  i 5.0  i 10.0  1  15.0  1  20.0  TIME TO CHANGE GROUNDS (HOURS]  1  25.0  -81The 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 [tx + t ( x - $ ) ] p (x|x) dx c  t  where p(x|x) is given by equation 7.6 Let  a = i + -  and  b = EX. + a  Then L  = fc [tx + t  c  , b  (X-*)]  ~-Xa,b-l b  , b+1  a  . " r%)  t  , b+1  <> Vl>  {  dX  r b  +  ,b  *M ~ V<?>  ^ (6.12)  From 6.11 the expected loss due to remaining w i l l be L =/ c [f R  ( t - x ) - tx] [  t  ..b  ffiT) t^V  =  {  v  ^Xa.b^l . b * ] dx e  ( a )  a  . b , b+1 , b+1 r(b*l)-t« •(!) r(b)-t(I) r(b) } t  (6.13)  If L < L , then d is a better strategy than d and vice versa. c  r  £  r  For d to be the better strategy, then 6.12 < 6.13 Removing similar functions from both expressions then a a(t tj; t  5 0  b  K  +  t'^)  2t + t , + t ' '  -82substituting for b and a , then jx.  <  J  (oi+l).Ct ^.t.tt ip) t  —  —  a(2t + t  t  t  + t'  —  —  )  t  ~  .  * a  From the inequality 7.16, i t can be seen that for a given E x . the greater the expected fish 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 k e 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 fish 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 i s 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 fish density encountered on tihe present grounds, then the better i s the strategy to change grounds.  Conversely, the greater Zx. at some J  time i , then the better strategy is to remain fishing where at present. As mentioned e a r l i e r , the time to make a t r i p , t quantify.  t  is d i f f i c u l t to  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 1  J  t  V  x  j  H  If the density is considered in terms.of the Bayes expected estimate, then t'  t  =.t .(aV»;iXj? t  -83However, 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 t r i p , 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 f i s h 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 . . tl  However in some fisheries fish are often  either practically absent or present in acceptable quantities. exploited by Vancouver trawlers appear to f i t this description. pers. comm.).  (D. Guine*,  In such a situation the decision to change grounds will  depend on the presence or absence of f i s h .  * Skipper, Bon Accord II Vancouver, B.C.  The grounds  -84Certain assumptions w i l l be made here.  In the actual fishery,  season i s a factor in choice of grounds, e . g . Hecate Straits i s favoured more in winter.  Also the probabilities of encountering f i s h 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 fish 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 fish 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 fish are encountered, i s least. Table VIII as c ;  This possible cost i s given in  i t can be seen that the best strategy is to go Vancouver-  Lower West Coast - Goose Island - Hecate S t r a i t - Vancouver, the worst Vancouver - Hecate S t r a 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 probabilities of fish occur-  rence at the different grounds are not uniform, then the results may not be so obvious from casual considerations.  Let T , Tg and T^ be average time L  required to determine presence or absence of f i s h at the Lower West Coast, Goose Island and Hecate Straits grounds respectively and P , PQ and P be the L  probabilities of encountering f i s h . position i to position j , e . g . D  V|  H  Let D.. be the steaming time from = steaming time from Vancouver to Lower  TABLE  VIII  Steaming time between grounds and from Vancouver Port Lower West Coast Vancouver Island Vancouver port  Goose Island  17  Lower West Coast, Vancouver Island Goose Island  24  Hecate Straits  38  Personal communication; Skipper D. Guine, Bon Accord II,  Hecate Straits  36  62  24  38 19  19  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 distance plus steaming d i s t a n c e back to p o r t . Van = Vancouver Port L.W.C. = Lower West Coast, Vancouver Island G.I.= Goose I s l a n d H.S. = Hecate S t r a i t s EC 233  265  309  295  405  359  -87West 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 evaluated 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  1Jk-  <°U' f 1>p  f  T  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 Island - 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 S  LGH = D  VL  + T + P D + (1-P ) { D L  L  L  L  LG  + T + P ^ Q  + (1-Pg) [ D  G H  + T^ +  D ]} HV  (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, consider 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 reverse order,  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 * (pers.  comm.) gives t h e i r s i z e as: Ground  Relative size  Hecate S t r a i t s  R e l a t i v e time to determine  12  36  Goose Island  4  12  Lower West Coast  1  3  So as to compare the s t r a t e g i e s , S ^ * Dept. of Zoology U n i v e r s i t y o f B r i t i s h Columbia  and S^Q  can be determined  -88-  as i n equation 7,17. D  VL  +  L  T  +  However, the component P  L VL P  ^ ~ L^  +  P  ls  c o m n l o n  to both s t r a t e g i e s and may be ignored. S  LGH = LG D  V GV G  +  D  P  = 153 - 81P S  LHG  =  D  LH  +  T  H  +  D  = 141 - 5P Hence f o r S ^  G  O ~ G^ GH  +  P  D  H  +  T  +  T  Then, from t a b l e V I I I , V  +  Q  HV H P  +  " H>< HG  (1  P  D  G  +  D  GV>  H  to be a p r e f e r a b l e s t r a t e g y , S  LHG  or and f o r the  <  S  LGH  P > 2.2 - 16.2P R  G  to be p r e f e r a b l e , P > H-  2.2  p  G  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 < 0.074 G  This 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  Discussion  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 a t 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 u s i n g 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 schools 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 ing 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 u s i n g 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 i n g methods  -90such 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 a t 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 component of V due to w i s 2.51%, and a t 20 knots, as i s p o s s i b l e with P r o c t o r ' s system, i t would be o n l y 0.78%.  L.W.  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 . 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 .  At 20 knots, the com-  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 movement 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 true.  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:  7u 2 + v 2 - 2uv  where u  l  cos  = 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 t o v clockwise (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 t o 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 t o u' from v w i l l be when  COSTT  $  =  ir,  i.e.  = -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 t o have s t r a t i f i e d estimates 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 surveys 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 o f 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 standard curve which i s non-  negative, unimodal and p o s s i b l y skewed.  I f the sample r e s u l t s a r e random  and a Pearson type I I I d e n s i t y describes 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 variance estimates o f the mean d e n s i t y ,  , from  equation 2.1, Skellam (1958) notes t h a t the variance w i l l be p r o p o r t i o n a l to the variance o f the number o f 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  -92no 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 o s c i l l a t o r y 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 z e , g, then the variance would be gX.  Any folding back by the targets would increase the variance in a  manner similar to aggregation. the size of the variance.  Also, any heterogeneities would increase  Effects which were in part dependent on target  speed and contributed to variance, would s i m i l a r l y be reduced in a search system where the search speed was much larger than that of the target, as discussed e a r l i e r in this section.  7.2  Pattern of School Distribution Choice of a model to describe the distribution of fish may be  entirely empirical, i . e . , that which gives best f i t to the data;  or the  distribution model may be chosen so as to describe some underlying environmental 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 distribution 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 distribution  -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 sets o f p o s t u l a t e s ( C a s s i e , 1962). Martin and K a t t i (1965) f i t t e d the P o i s s o n , Poisson with added zeroes, 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 sets o f b i o l o g i c a l data.  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 while the Poisson with added zeroes provided a good f i t to only a few sets 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 , while the Poisson and Poisson with added zeroes provided 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  with 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 length 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 miles 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 tempera 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 occurrence, methods o f 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  -94methods 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 -  ing 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 extent 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 occurrence, 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 occurrence, 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 choice 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 schools 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 schools 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 disappear 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 sampling.  A g r i d of continuous quadrats i s used, each s i d e o f the g r i d  -95c 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 density.  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 block 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 block 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 block 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. the type of d i s t r i b u t i o n used.  Variance estimates w i l l depend on  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 block size.  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 pattern. 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 prop 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:  -965ample length  Variance/mean r a t i o  5 miles 10 15 20 25  1.021 1.330 1.546 1.780 1.533  Greig-Smith concluded t h a t the s a f e s t procedure i s t o 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  exception 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 butions 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 mile 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  could 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 distribution variance. i s given by: S  2  Hogg and C r a i g (1968) show t h a t a b e t t e r estimate  =  S  l J"  1  +  2  "  r  j  where: S = c o r r e c t e d estimate o f standard d e v i a t i o n 2  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 mated S^.  underestv  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 o f 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  -97s 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 .  F o r 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 t o 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 t h e sample s i z e was s i m i l a r t o  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 contained no schools and a negative 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 confidence i n t e r v a l on estimates o f f i s h dens 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 _ required  D  r CI-j r £T~ - ^-I 1  2  where: D = d i s t a n c e covered i n r e f e r e n c e survey r  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 acous 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 t o 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 strata.  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 estimate o f v a r i a n c e , f o r each s t r a t a . standard e r r o r o f the mean d e n s i t y estimate w i l l be  The  -98-  Var (y  )  =  L  z  N  Var ( y j  2  ^  o —  where: ^st  =  m e a n  ^'  1S 1  density f ° ^ r  a  strata  =  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 density f o r stratum  (Cochran, 1963).  The variance of J ^ s  h.  depends only on the variances 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 population i n t o s t r a t a such that the s t r a t a were o f constant dens 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 information 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  (Merluccius  productus), 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 tracks of the survey. be i n v a l i d a t e d .  Hence attempts at optimum a l l o c a t i o n would  Cochran notes t h a t e r r o r i n stratum s i z e computation can  introduce s e r i o u s bias 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 stock assessment s t u d i e s i n the Black Sea used four s t r a t a , also p r o p o r t i o n a l l y a l l o c a t e d . f i d e n c e i n t e r v a l s , the variance estimates were 6.4%, of the s t r a t a estimates.  For 95% con-  11.4%, 10.4% and  9,4%  However, Thorne e t , a l . (1971) found the major  -99component o f v a r i a n c e i n the p o p u l a t i o n estimate was due t o c a l i b r a t i n g a c o u s t i c records with trawl c a t c h e s .  For a 95% confidence i n t e r v a l a +  50% range was obtained 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 Sherstnikov'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 value o f the expected school number. Entropy can be used however to provide 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 -  bution 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  >  -100i where  e  = z exp[-  e  y  fCxJ]  j=l  (7.2)  J  Wilson (1970) shows that that function f(x) which has maximum entropy minimizes the likelihood 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 r e a l i t y , the  form of f(x) can be changed, as long as i t s expected value is known.  If  the expected value is not known then s t a t i s t i c a 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 directly 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 f  where e = Lagrangian m u l t i p l i e r  1  1  (x) +  + e f m  m  (x)]}  -101(Jaynes, 1957),  Such an expression would enable other v a r i a b l e s , such as  temperature, s a l i n i t y , plankton, 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 o f 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 vessel 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 species 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 t o 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 a t 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 vessel 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 vessel 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 considered i n i n t e r p r e t a t i o n o f catch results. 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 of 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 , there  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.  Also  s i m i l a r i t i e s or d i s s i m i l a r i t i e s i n behavioural 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 movement 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 could 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  -103found by the In  search v e s s e l . development o f C r a i g and Graham's (1965) data, 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 When x i s below some determined t h r e s h o l d l e v e l , then a bene-  may take.  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 confronted 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 of 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 of o p t i o n s .  From estimates of p o s s i b l e l o s s e s the best branch of 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 of catch per e f f o r t assume independence of f i s h i n g e f f o r t .  I f the  manner i n which co-operation 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 concentrat 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 of v a r i a b l e degrees of 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 ives of 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,  'experts',  -104s 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 reaching 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 of 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 returns 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 of r e l a t e d d e c i s i o n s i n a management situation.  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  of 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 . Gull 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 science and t h a t with the r a p i d development of f i s h e r i e s technology, management d e c i s i o n s are r e q u i r e d long before 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 could be made.  To date, up to f o u r years may  occur  -105between 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 well 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 choices f o r the  f i s h stock s i z e can be explored and t h a t choice 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. G u l l a n d (1971) suggests t h a t any course o f a c t i o n which o f f e r s at l e a s t a 51% chance of success i s a c c e p t a b l e i n a f i s h e r i e s management s i t u a t i o n and should be used i n the absence o f more d e f i n i t e i n f o r m a t i o n .  It  seems probable t h a t Bayesian methods based on s u b j e c t i v e f e e l i n g s and sample data as a v a i l a b l e may provide the b e s t c u r r e n t b a s i s f o r such d e c i s i o n s .  -106-  LITERATURE CITED A l v e r s o n , D.L.  1971.  appraisal.  Manual o f methods f o r f i s h e r i e s r e s o u r c e survey and  P a r t 1.  Survey and c h a r t i n g o f f i s h e r i e s r e s o u r c e s .  FAO F i s h . Tech. Pap. 102, 80p. Anon  1972a.  North Sea t r a w l i n g on t h e wane a t Grimsby.  J_n: F i s h i n g  News, No. 3060. Feb, 18. Anon. 1973. Best. E.A.  F i s h e r i e s a c o u s t i c methods. 1964.  Nature  244(5412):134.  I d e n t i f y i n g P a c i f i c Coast f i s h e s from echo-sounder  r e c o r d i n g s , pp. 413-414, Modern f i s h i n g gear o f the world. News Books. Beverton, R.J.H. and S.J. H o l t . - 1957. populations.  On the dynamics o f e x p l o i t e d f i s h  F i s h . I n v e s t . S e r . I I . 29, 533p.  Bui l i s , H.R. and J.A. Benigno. veillance.  1970„  F i s h d e t e c t i o n through a e r i a l s u r -  FAO Tech* Conf. F i s h F i n d i n g , Purse S e i n i n g and Aimed  Trawling.  No. 78, 13p.  Burbank, A. and R. Douglas.  1969.  F i s h e r i e s f o r e c a s t i n g systems:  study o f t h e Japanese f i s h e r i e s f o r e c a s t i n g system. Laboratory. C a s s i e , R.M.  a  Aerosciences  TRW Systems. 99900-6865-R0-00, 60p.  1962.  Frequency d i s t r i b u t i o n models i n the ecology o f plankton  and o t h e r organisms. Cochran, W.G.  Fishing  1964.  J . Anim. E c o l . 31:65-92.  Sampling t e c h n i q u e s , 2nd e d i t i o n .  John Wiley &  Sons, New York. 413p. Cove, J . J .  1973.  Hunters, t r a p p e r s , and gathers o f the sea:  t i v e study o f f i s h i n g s t r a t e g i e s . 249-259.  a compara-  J . F i s h . Res. Bd. Canada 30:  -107C r a i g , W.L. and J . J . Graham.  1961.  Report on a c o o p e r a t i v e  preseasoned  survey o f the f i s h i n g grounds f o r a l b a c o r e (Thunnus germo) i n the Eastern North P a c i f i c , 1959. C r u t c h f i e l d , J.A.  1972.  C a l i f . F i s h and Game 47(l):73-85.  Economic and p o l i t i c a l o b j e c t i v e s i n f i s h e r y  management, pp. 74-89 I_n: World f i s h e r i e s p o l i c y , B. R o t h s c h i l d , editor. Danskin, J.M. Dreaver, G.H.G.  Univ. Washington P r e s s , S e a t t l e .  1960.  272pp.  A theory o f reconnaissance, I.  1970.  Op. Res. 10:285-299.  The use o f e l e c t r o n i c aides i n t r a w l i n g .  FAO Tech,  Conf. F i s h F i n d i n g , Purse S e i n i n g and Aimed T r a w l i n g , No. 30. 13p, F i s h e r , R.A., A.S. Corbet, and C.B. W i l l i a m s .  1943.  The r e l a t i o n s between  the number o f s p e c i e s and the number o f i n d i v i d u a l s i n a random sample o f an animal p o p u l a t i o n .  J . Anim. E c o l . 12:42-58.  Gerhardsen, T.S., A. Borud, P. P e t t e r s e n and J.M. S a r l a n d . echo sounding systems. Greenwood, A.J. and D. Durand.  1972.  Scientific  Simrad B u l l . No. 1. 20p. 1955.  The d i s t r i b u t i o n o f length and com-  ponents o f the sum o f n_ Random U n i t V e c t o r s .  Ann. Math. S t a t .  26:233-246. Green, R.E., W.F. P e r r i n and B.F. P e t r i c k . seine f i s h e r y .  1970.  The American tuna purse  FAO Tech. Conf. F i s h F i n d i n g , Aimed Trawling and  Purse S e i n i n g , No. 86. 28p. Greig-Smith, P. Gulland, J.A.  1964. 1971.  Q u a n t i t a t i v e p l a n t ecology.  Butterworth.  Science and f i s h e r y management.  246p.  J . Cons. i n t .  E x p l o r . Mer. 33(3):471-477. .  1972b. Population dynamics i n world f i s h e r i e s .  WSG 72-1,  Univ. Washington Press, S e a t t l e . 335p. Gumbel, E . J . , J.A. Greenwood, and D. Durand.  1953.  The c i r c u l a r normal  -108distribution.  Theory and t a b l e s .  Am. S t a t . Assoc. J . 48(261):  131-152. Gurland, J .  1957.  Some i n t e r r e l a t i o n s h i p s among compound and g e n e r a l i z e d  distributions. Haldane, J.B.S.  1941.  Eugenics Hearn, D.J.  B i o m e t r i k a 44:265-268. The f i t t i n g o f binomial d i s t r i b u t i o n s .  Ann.  11:179-181.  1970.  A f i s h counter f o r use i n commercial f i s h e r i e s .  Tech.  Conf. Fish F i n d i n g , Purse S e i n i n g and Aimed T r a w l i n g F l l FF/70/54. 9p. Hodder, V.  Unpub. MS.  The c l a s s i c a l model and i t s i m p l i c a t i o n s . 41p.  Hogg, R.V. and A.T. C r a i g . 2nd e d i t i o n . Jaynes, E.T.  1957.  Rev.  1968.  I n t r o d u c t i o n t o mathematical s t a t i s t i c s ,  M a c m i l l a n , New York.  383p.  Information theory and s t a t i s t i c a l mechanics.  Phys.  106:620-630.  Johannesson, K.A. and G.F. Losse.  1973.  Some r e s u l t s o f observed abundance  e s t i m a t i o n s o b t a i n e d i n s e v e r a l UNDP/FA0 r e s o u r c e survey p r o j e c t s . Symp. A c o u s t i c Methods i n F i s h e r i e s Research No. 3, 77p, Rome. WS/P9180. Johnson, H.M. and L.W. P r o c t o r . mercial f i s h i n g . Industry. Jones, R. :  1959.  1970.  New approaches t o sonar f o r com-  Proc. Conf. Automation and Mech. i n the F i s h i n g  Canada F i s h . Rept.  15:607-617.  A method o f a n a l y s i s o f some tagged haddock r e t u r n s .  Cons. i n t . E x p l o r . Mer. 25:58-72. Khintchin, A.I.  1957.  Dover Pubs. Kodera, K.  1970.  Mathematical foundations o f i n f o r m a t i o n theory.  120p. Aimed bottom and midwater t r a w l i n g techniques o f  -109Japanese f a c t o r y s t e r n t r a w l e r s .  FAO Tech. Conf. F i s h F i n d i n g ,  Purse S e i n i n g and Aimed T r a w l i n g , No. 47. Koopman, B.O.  1956a.  18p.  The theory of search, I.  Kinematic bases.  Op.  Res. 4(3):324-340. .  1956b.  The theory of search, I I . Target d e t e c t i o n .  Op.  Res. 4(5):503-531. .  1957.  The theory of search, I I I . Optimum d i s t r i b u t i o n of  searching e f f o r t . Lahore, H.W.  and D.W.  Lyttle.  Op. Res. 5(5):613-623. 1970.  An echo i n t e g r a t o r f o r use i n the  e s t i m a t i o n of f i s h p o p u l a t i o n s . 70-1. L a r k i n , P.A.  Washington Sea Grant Publ. No.  38p. 1972.  A c o n f i d e n t i a l memorandum i n f i s h e r i e s s c i e n c e s .  Jjn:  World F i s h e r i e s P o l i c y , B.J. R o t h s c h i l d , e d i t o r . Univ. Washington P r e s s , S e a t t l e . 282p. Lee, A.J.  1952.  The i n f l u e n c e o f hydrography on the Bear I s l a n d cod f i s h e r y .  Cons. i n t . E x p l o r . Mer 6:433. L i n d l e y , O.V.  1965a.  I n t r o d u c t i o n to p r o b a b i l i t y and s t a t i s t i c s from a  Bayesian viewpoint.  Part I. P r o b a b i l i t y .  Cambridge Univ. P r e s s ,  Cambridge. 259p. ._  1965b.  I n t r o d u c t i o n to p r o b a b i l i t y and s t a t i s t i c s from a  Bayesian viewpoint. Cambridge. Magnuson, J . J .  1966.  P a r t I I . Inference.  Cambridge Univ. Press,  292p. A comparative  study of the f u n c t i o n of continuous  swimming by scrombrid f i s h e s . A b s t r a c t , Proc. 11th Pac. S c i . Cong. 7, 15p. 1970.  H y d r o s t a t i c e q u i l i b r i u m of Euthynnus a f f i n i s , a  p e l a g i c t e l e o s t without a gas bladder.  Copeia 1:56-85.  -na.  Undated Ms.  Comparative study of continuous swimming among  scombrid fishes.  12p. + tables.  Martin, D.C and S.K. K a t t i .  1965.  F i t t i n g of some contagious d i s t r i b u -  tions to some available data by the maximum likelihood method. Biometrics 21:34-48. McKenzie, M.K.  1964.  The d i s t r i b u t i o n of tuna in relation to oceano-  graphic conditions. Minne, J . F .  1970.  Proc. M.Z. Ecol. Soc. 11:6-10.  New Dutch experience in two-boat midwater trawling  with medium-sized s t e r n , trawlers, and prospects of combinationarrangements for trawling and seining.  FAO Tech. Conf. Fish  Finding, Purse Seining and Aimed Trawling, No. 36. Mood, A.M. and F.A. G r a y b i l l .  1963.  s t a t i s t i c s , 2nd e d i t i o n . Nakamura,  E.L.  1967.  5p.  Introduction to the theory of  McGraw-Hill.  443p.  A review of f i e l d observations on tuna behaviour.  FAO Fish Rept. No. 62(2):59-68. -  .  1968.  Visual acuity of two tunas, Katsuwonis pel amis and  Euthynnus a f f i n i s . Neyman, J .  1949. fish.  Nickerson, T.B.  Copeia 1:41-48.  On the problem of estimating the number of schools of  Univ. C a l i f . Publ. S t a t i s t i c s 1(3):21-36. 1970.  fishing systems. and R.G. Dowd.  Systems analysis in the design and operation of Canad. F i s h . Rept. 15:11-16. 1973.  Design and operation of survey patterns  for demersal fishes using the computerized echo counting system. ICES/FAO Symp. Acoustic Methods in Fisheries Research.  Bergen.  No. 19. 6p. Nishimura, M. 1963. Investigation of tuna behaviour by f i s h finder. Proc. World S c i . Meeting B i o l . Tuna and Related Species. FAO Fish Rep. No. 6. 3:1113-1123. Paloheimo, J . E .  1971a.  On a theory of search.  Biometrika 58 (1), 61-75  .  1971b.  A stochastic theory of search:  predator-prey relations. P a t i l , G.P. and S.W. Joshi.  Math. Bio. S c i . 12:105-132.  1968.  discrete distributions. Pearson, K.  1946.  implications for  A dictionary and bibliography of Oliver and Boyd, Edinburgh. 268p.  Tables of the incomplete r-function.  Biometrika.  164p. Pella. J . J .  1969.  fishery. Pielou, E.C.  A stochastic model for purse seining in a two-species J . Theoret. B i o l . 22:209-226.  1969.  An introduction to mathematical ecology.  John Wiley  & Sons, New York. 286p. Pirenne, M.H.  1962.  H. Davson, editor. Raiffa, H.  1968.  In:  Visual acuity.  The eye, 2. The visual process.  Academic Press, New York.  Decison analysis.  Richardson, H.R. and L.D. Stone.  Pp: 175-195.  Addison Wesley, 309p.  1971.  Operations analysis during the  underwater search for scorpion.  Naval Res. Log. Quart. 18(2):  141-157. Rohlf, J.F. and R.R. Sokal. and Co.  1969.  1955.  Hawaiian waters in 1953.  Saetersdal, G.  Observations of skipjack schools in Spec. S c i . Rept. 147.  U.S. Fish  31 p. 1973.  Assessment of unexploited resources.  Mgmt. and Devel. S a i l a , S.B.  W.H. Freeman  253p.  Royce, W.F. and R. Otsu.  Wildl.  S t a t i s t i c a l tables.  1969.  FAO FI:FMD/73/5-33.  Tech. Conf.  lip.  Some applications of observations and experiments on  fish behaviour in designing fishing gear and devising tactics with suggestions for future studies.  FAO Fish Rept. 63:45-58.  -112-  _ and J.M. Flowers.  1967.  Elementary a p p l i c a t i o n s of search  theory to f i s h i n g t a c t i c s as r e l a t e d to some aspects o f f i s h behaviour.  FAO F i s h . Rept. 2(62) :343-355.  1968.  Movements and behaviour o f b e r r i e d female l o b s t e r s  d i s p l a c e d from o f f s h o r e areas to Narragansett Bay, Rhode I s l a n d . J . Cons. perm. i n t . E x p l o r . Mer. 31(3):342-351. 1969.  Toward a g e n e r a l i z e d model o f f i s h m i g r a t i o n .  Trans,  Am. F i s h . Soc. 98(3):582-588. Sams, M.  1970.  Southeastern P a c i f i c a i r c r a f t a s s i s t e d purse s e i n i n g .  FAO Tech. Conf. F i s h F i n d i n g , Purse S e i n i n g and Aimed T r a w l i n g . No. 9. S e c k e l , G.R.  6p.  1963.  C l i m a t i c c o n d i t i o n s and the Hawaiian s k i p j a c k f i s h e r y .  Proc. World S c i . Conf. B i o l . Tunas and Related S p e c i e s .  FAO F i s h  Rept. 3(6):1201-1208. Shannon, C.E.  1948.  The mathematical theory o f communication.  Bell  System. Techn. J . 27:379-423. Shenton, L.R.  1949.  On the e f f i c i e n c y of the method o f moments and  Neyman's type A d i s t r i b u t i o n . S h e r s t n i k o v , D.N.  1968.  Biometrika 36:450-454.  Some mathematical models f o r the search o f f i s h .  Opyt. p r i n o n e n i y a matematicke s k i l c h metodov y rybolchozyaystuennylch I s s l e t o v a n i y a c h (TRUDY) 20:56-98.  T r a n s l . Foreign  Languages D i v i s i o n , Sec. S t a t e , Canada. Skellam, J.G.  1951.  Random d i s p e r s a l i n t h e o r e t i c a l p o p u l a t i o n s .  Biometrika 38:196-218. ;  .  1958.  The mathematical  t r a n s e c t s i n animal ecology. Southward, T.R.E.  1965.  foundations u n d e r l y i n g the use of B i o m e t r i c s 14:385-400.  E c o l o g i c a l Methods.  Methuen.  391p.  -113S t e i n b e r g , R.  1970.  Two boat bottom and midwater t r a w l i n g .  Conf. F i s h F i n d i n g , Purse S e i n i n g and Aimed T r a w l i n g . Stone, D.L. and J.A. Stanshine.  1971.  No. 14.  12p.  Optimal search using u n i n t e r -  rupted c o n t a c t i n v e s t i g a t i o n . T e p l i t s k i i , V.A. and L.Z. S h e i n i s .  FAO Tech.  SIAM J . Appl. Math. 20(2) :241-263.  1970.  Mathematical models f o r the  complex planning o f the deployment of f i s h i n g f l e e t s on f i s h i n g grounds.  Rybnoe Khozyaistvo No. 1. Moskva. 80-82.  (Bureau o f  Commercial F i s h e r i e s T r a n s l a t i o n ) . Thomas, M.  1949. in Ecology.  A g e n e r a l i z a t i o n of Poisson's Binomial L i m i t f o r use Biometrika 36:18-25.  Thorne, R.E., J.E. Reeves and A.E. M i l l i k a n .  1971.  E s t i m a t i o n o f the  P a c i f i c hake (Merluccius productus) p o p u l a t i o n i n Port Susan, Washington, using an echo i n t e g r a t o r .  J.Fish.Res.Bd.  Canada  28(0):1275-1284. Tucker, D.G. and V.G. Welsby. purposes.  1964.  Sector scanning sonar f o r f i s h e r i e s  Modern F i s h i n g Gear of the World I, pp: 367-370.  F i s h i n g News Books. 607p. Watson, G.S. and E.J. W i l l i a m s .  1956.  On the c o n s t r u c t i o n o f s i g n i f i c a n c e  t e s t s on the c i r c l e and the sphere. Wilson, A.G. Wolfe, D.C.  1970. 1971.  Biometrika 43:344-352.  Entropy i n urban and r e g i o n a l m o d e l l i n g . P e l a g i c f i s h survey:  schools i n inshore waters. Yudovich, Y.B. and A.A. B a r a l .  1968.  (Trans!. by I s r a e l Prog.). Science Foundation.  260p.  Pion.  seasonal a v a i l a b i l i t y o f f i s h  Tasmanian F i s h . Res.  5(2):2-11.  E x p l o r a t o r y f i s h i n g and s c o u t i n g U.S. Dept. I n t e r i o r and National  166p,  

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