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Adaptive management of renewable resources with uncertain dynamics Smith, Anthony David Miln 1979

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ADAPTIVE MANAGEMENT OP RENEWABLE RESOURCES WITH UNCERTAIN DYNAMICS by ANTHONY DAVID MILN SMITH B.Sc.(Hons), University of Adelaide, 1974 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Zoology) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1979 © Anthony David Miln Smith, 1979 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of ~Z^ouo£ - V The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date i i ABSTRACT A range o f adapt ive p o l i c i e s i s app l i ed to the management of s imulated f i s h s tocks based on two s imple models of s tock dynamics, the R i cker model and the Schaefer model. Unce r ta in ty about s tock dynamics i s represented as un ce r t a i n t y about the parameters o f these models. The p o l i c i e s tes ted i nc lude an a c t i v e adapt ive p o l i c y where management con t ro l opt ions are chosen tak ing i n t o account the unce r t a i n t y i n the parameter es t imates ; a range o f pass ive adapt ive p o l i c i e s , where c o n t r o l s are chosen assuming parameter es t imates are c o r r e c t , but the es t imates are updated from time to t ime; and a non-adapt ive p o l i c y where the i n i t i a l parameter est imates are assumed to be c o r r e c t and are never updated. Ana l y s i s o f the r eg ress i on problems f o r e s t ima t i ng the parameters o f the two models shows tha t a major f a c t o r determin ing un ce r t a i n t y about parameter es t imates i s the v a r i a b i l i t y i n observed va lues of the independent v a r i a b l e s i n the r eg r e s s i on . Where there i s more than one independent v a r i a b l e , c on t r a s t s between v a r i a b l e s are a l s o impor tant . Comparison of p o l i c y performances shows that the a c t i v e adapt ive p o l i c y always performs we l l r e l a t i v e to the opt imal p o l i c y (where the t rue stock parameters are known). The pass ive adapt ive p o l i c y w i th r egu l a r parameter est imate updat ing gene r a l l y performs very we l l but o c c a s i o n a l l y very poo r l y . Poor performance occurs when the data po in t s i n the reg ress i on problem are c l u s t e r ed c l o se to the apparent opt imal l e v e l s o f the independ-i i i ent v a r i a b l e s . In most o ther cases poor i n i t i a l parameter es t imates cause s u f f i c i e n t pe r tu rba t i ons i n c on t r o l s to c o r r e c t l y i d e n t i f y parameter va lues . The non-adapt ive p o l i c y gene ra l l y performs poo r l y , except i n the case o f the R i cke r model where observa t ions are a v a i l a b l e near the e q u i l i b r i u m stock s i z e . Th is i s due to the i n s e n s i t i v i t y o f the opt imal escapement to v a r i a t i o n s i n the p r o d u c t i v i t y o f the s tock . The good performance o f the a c t i v e adapt ive p o l i c y i s achieved a t the expense of shor t term performance, which i s s a c r i f i c e d to improve parameter e s t ima tes . In f requent updat ing of parameter es t imates and low s i g n i f i c a n c e at tached to new data are both shown to lead to marked d e t e r i o r a t i o n i n performance f o r the pass ive adapt ive p o l i c y . The major conc lus i on from the cases s tud i ed i s tha t good es t ima t i on (equ iva len t to good understanding about stock dynamics) and hence good p o l i c y performance requ i res s u f f i c i e n t v a r i a b i l i t y i n , and con t r a s t s between, the independent v a r i a b l e s i n the corresponding r eg ress i on problems. I t i s suggested tha t t h i s conc lu s i on can be extended to more general problems o f unce r t a i n t y about system dynamics in managing renewable resources prov ided that the problems can be s i m p l i f i e d to an understanding o f the key processes and u n c e r t a i n t i e s i n vo l v ed . The q u a n t i t i e s correspond ing to independent v a r i a b l e s in an app rop r i a te r eg ress i on problem can then be i d e n t i f i e d and appropr ia te exper imenta l management s t r a t e g i e s dev ised to d i s c r i m i n a t e between a l t e r n a t i v e hypotheses. i v TABLE OF CONTENTS Page Abs t rac t i i Table o f Contents i v L i s t o f Tables v i i i L i s t o f F igures i x Acknowledgements x i i i CHAPTER I 1.0 In t r oduc t i on 1 1.1 Management. Under Unce r t a i n t y 2 Uncer ta in ty about Ob jec t i ves 2 Uncer ta in ty about Options 3 Uncer ta in ty about Nature 4 Adapt ive Management 6 1.2 Problem Des c r i p t i on 7 1.3 Problem Background 10 1.4 Ou t l i ne o f Study 12 CHAPTER II 2.0 Contro l and Es t imat ion Theory 14 V Page 2.1 Problem Formulation ". 14 2.2 State Augmentation 16 2.3 Information State 16 2.4 Classes of Control Po l i c i es 17 2.5 The Dual Control Problem 18 2.6 Optimal Stochastic Control 19 2.7 Problems in the General Solution 20 2.8 Wide-Sense.Adaptive Dual Control 20 2.9 Requirements and Limitat ions 22 2.10 Nonlinear F i l t e r i ng and State Estimation 23 CHAPTER III 3.0 The Ricker Model 26 3.1 Parameter Estimation 28 3.2 Range of Control Po l i c ies Tested 31 3.3 Active Adaptive Pol icy Formulation 34 3.4 Active Adaptive Control Law 35 3.5 Method of Pol icy Evaluation 45 3.6 Pol icy Comparison : Ef fect of Pr ior Data 47 3.7 Pol icy Comparison : Case S> = 0.3, S g = 0.05 72 3.8 Pol icy Comparison : Case S = 0.1, S g = 0.05 91 3.9 Discussion 95 v i Page 3.10 Summary 99 CHAPTER IV 4.0 The Schaefer Model 101 4.1 Parameter Es t imat ion 104 4.2 Range o f Contro l P o l i c i e s Tested 110 4.3 A c t i v e Adapt ive P o l i c y Formulat ion 110 4.4 A c t i v e Adapt ive Contro l Law 113 4.5 Method o f P o l i c y Eva lua t i on 113 4.6 P o l i c y Comparison : E f f e c t of P r i o r Data 114 4.7 P o l i c y Comparison : Case PPK = 0 .5 , S £ = 0.2 128 .Case!a 131 Case b 135 Case c 141 Case d 157 Further Comments 162 4.8 A p p l i c a t i o n to a "Rea l " F i she ry 166 4.9 D i scuss ion 167 4.10 Summary 169 v i i Page CHAPTER V 5.0 D i s cuss i on and Summary 171 5.1 Value and L im i t a t i o n s o f the Study 171 5.2 General Lessons f o r Resource Management 173 5.3 P r a c t i c a l Adapt ive Management 178 5.4 D i r e c t i on s f o r Future Research 180 Fur ther Ana l y s i s o f Simple Models 180 Extens ion to More Complex Models 182 5.5 Summary 183 L i t e r a t u r e C i t ed — 185 Appendix I 189 Appendix I I , 191 v i i i LIST OF TABLES Table No. Page I Recurs ive l e a s t squares r eg ress i on a l go r i t hm .... 32 I I Comparison o f shor t term and long term performance . . . 76 I I I Comparison o f e s t ima t i on performances 80 IV E f f e c t of i n f requent update o f parameter es t imates . . . 94 V E f f e c t of reduc ing weight g iven to new data 96 VI Optimal d e t e r m i n i s t i c e f f o r t as a f un c t i on o f i n i t i a l popu la t ion s i z e — 117 VII Development i n v a r i a b i l i t y o f catch per u n i t e f f o r t and e f f o r t a n d . c o r r e l a t i o n between them..Case.a , . , . 138 VI I I Development i n v a r i a b i l i t y of catch per un i t e f f o r t and e f f o r t and c o r r e l a t i o n between them. Case b .. 146 IX Development i n v a r i a b i l i t y o f catch per u n i t e f f o r t and e f f o r t and c o r r e l a t i o n between them. Case c .. 154 X Development i n v a r i a b i l i t y of catch per u n i t e f f o r t and e f f o r t and c o r r e l a t i o n between them. Case d .. 163 i x LIST OF FIGURES F igure No. Page 1 A c t i v e adapt ive c on t r o l law: e f f e c t o f the number of years of i n i t i a l data and p r o d u c t i v i t y o f the stock 38 2 A c t i v e adapt ive con t ro l law: e f f e c t o f d i scount . r a te and obse rva t i on no ise var iance 41 3 A c t i v e adapt ive t r a j e c t o r i e s i n the in fo rmat ion s t a t e space 44 4 Comparison o f p o l i c y performances r e l a t i v e to the mean and standard dev i a t i o n o f p r i o r escapements. 49 5 Comparison of e s t ima t i on performances f o r the a parameter 52 6 Comparison o f e s t ima t i on performances f o r the & parameter 54 7 Comparison of e s t ima t i on performances f o r the optimum escapement E* 56 8 Escapements chosen i n r e l a t i o n to p r i o r data 59 9 P o l i c y performance i n parameter space: case S = 0.1 . 63 10 P o l i c y performance i n parameter space: case S ='0.3 . 65 11 P o l i c y performance i n parameter space: case S = 0.5 . 67 12 P o l i c y performance i n parameter space: case S^  = 0.7 . 69 13 P o l i c y performance i n parameter space: case $T = 0.9 . 71 X F igure No. Page 14 D i s t r i b u t i o n of p o l i c y performances r e l a t i v e to the opt imal p o l i c y 74 15 D i s t r i b u t i o n of percent improvement i n performance o f a c t i v e adapt ive p o l i c y r e l a t i v e to the pass ive adapt ive p o l i c y 78 16 Comparison o f escapements chosen by the a c t i v e and pass ive adapt ive p o l i c i e s 82 17 Comparison o f parameter es t imates achieved by a c t i v e and pass ive adapt ive p o l i c i e s 84 18 Comparison of parameter u n c e r t a i n t i e s f o r the a c t i v e and.pass ive adapt ive p o l i c i e s 86 19 Stock rec ru i tment da t a " f o r the a c t i v e adapt ive p o l i c y 88 20 Stock rec ru i tment data f o r the pass ive adapt ive pol i c y 90 21 Comparison o f cumulat ive, catches f o r the opt imal andvthe- a c t i v e «5nd3pass- i.Ye ' ; ;adapttve lpo l ic ies-. 93 22 Representat ion o f the reg ress i on problem f o r es t ima t i ng the parameters o f the Schaefer model . 108 23 Comparison of p o l i c y performance and es t ima t i on performance f o r optimum e f f o r t E* 116 24 Comparison o f e s t ima t i on performance f o r the r,k and c parameters of the Schaefer model 119 x i F igure No. Page 25 P o l i c y performance i n (q*,E*) space: case PPK = 0.125 123 26 P o l i c y performance i n (q*,E*) space: case PPK = 0.500 125 27 P o l i c y performance i n (q*,E*) space: case PPK = 0.875 127 28 D i s t r i b u t i o n o f p o l i c y performances r e l a t i v e to PPK . 130 29 Management . t ra jector ies i n (q,E) space: case a) 134 30 Comparison o f e s t ima te s . o f q* and E* f o r the a c t i v e and pass ive adapt ive p o l i c i e s : case a) 137 31 Comparison o f cumulat ive catches f o r the o p t i m a l , a c t i v e and pass ive adapt ive p o l i c i e s : case a) . . . 140 32 Management t r a j e c t o r i e s i n (q,E) space: case b) 143 33 Comparison of es t imates of q* and E* f o r the a c t i v e and pass ive adapt ive p o l i c i e s : case b) 145 34 Comparison of cumulat ive catches f o r the o p t ima l , a c t i v e and pass ive adapt ive p o l i c i e s : case b) . . . 148 35 Management t r a j e c t o r i e s i n (q,E) space: case c) 151 36 Comparison o f est imates of q* and E* f o r the a c t i v e and pass ive adapt ive p o l i c i e s : case c ) . 153 37 Comparison o f cumulat ive catches f o r the o p t i m a l , a c t i v e and pass ive adapt ive p o l i c i e s : case c) 156 38 Management t r a j e c t o r i e s i n [q,E) space: case d) 159 x i i F igure No. Page 39 Comparison o f est imates o f q* and E* f o r the a c t i v e and pass ive adapt ive p o l i c i e s : case d) 161 40 Comparison of cumulat ive catches f o r the o p t i m a l , a c t i v e and pass ive adapt ive p o l i c i e s : case d) . . . 165 x i i i ACKNOWLEDGEMENTS A number of people have g iven me a s s i s t ance dur ing the course of t h i s s tudy. I would e s p e c i a l l y l i k e to thank my.superv i so r , Car l Wa l te r s , f o r h i s support and encouragement and f o r h i s pe r t i nen t and t ime l y adv ice throughout the course o f the s tudy. Spec i a l thanks are a l s o due to John Pars low, who helped to e xp l a i n some o f the myster ies of s t o c h a s t i c con t ro l theo ry , and who put forward many va luab l e sugges t i ons . Among the others who prov ided suggest ions and c r i t i c i s m s I would p a r t i c u l a r l y l i k e to mention Ray Hi 1 born , Don Ludwig and Mike S t a l e y . I am indebted to Joan Anderson and Mary McGechaen f o r t yp ing the t e x t and to my w i f e C h r i s t i n e f o r prepar ing the f i g u r e s . During the course o f t h i s s tudy , f i n a n c i a l support was provided by the U n i v e r s i t y o f Ade la ide and by the Canadian Commonwealth Scho l a r sh i p Committee. Above a l l I would l i k e to thank my w i f e f o r her pat ience and understand ing. 1 CHAPTER I 1.0 INTRODUCTION Theo re t i c a l work i n na tu ra l resource management has concentrated l a r g e l y on the development and a n a l y s i s of s imple models f o r response to d i s tu rbances such as ha r ves t i ng . In most cases i t has been assumed tha t nature i s d e t e r m i n i s t i c w i th known parameters, or a t best tha t l e a r n i ng about unknown parameters should be a matter f o r research s t a f f s r a the r than an i n t e g r a l pa r t o f the management process i t s e l f . In p r a c t i c e , resource d e c i s i o n makers use such t h e o r e t i c a l r e s u l t s on ly to prov ide general g u i d e l i n e s ; they go f a r beyond the d i r e c t r e s u l t s by behaviour t ha t hedges aga ins t un ce r t a i n t y and, i n some cases , they t r e a t management a c t i ons as d e l i b e r a t e l y exper imenta l a c t i -v i t i e s . Recent developments i n s t o c h a s t i c opt imal con t ro l theory have now made i t po s s i b l e to more r e a l i s t i c a l l y ana lyse s i t u a t i o n s i n v o l v i n g unce r t a i n t y from a t h e o r e t i c a l po in t of v iew, thus b r i ng i ng theory and p r a c t i c e much c l o s e r t oge the r . Th is study addresses some problems posed by unce r t a i n t y i n the management o f renewable resources . In p a r t i c u l a r i t examines the problem o f how to choose ha rves t i ng s t r a t e g i e s f o r f i s h s tocks where there i s s ub s t an t i a l un ce r t a i n t y concern ing the nature o f the dens i t y dependent processes which determine stock p r o d u c t i v i t y . Before o u t l i n i n g the problem i n more d e t a i l , i t i s appropr ia te to present a general d i s cu s s i on o f the problems posed by unce r t a i n t y i n management, and po s s i b l e s t r a t e -2 g ies f o r dea l i ng w i th them. In t h i s way i t w i l l be po s s i b l e to p lace the s p e c i f i c problem cons idered here i n i t s more general con tex t . 1.1 MANAGEMENT UNDER UNCERTAINTY Many resource management problems can be viewed as problems in sequent ia l d e c i s i o n making. Watt (1968) was the f i r s t to po in t out t ha t a l l resource management problems can be viewed as problems i n o p t i m i z a t i o n . Three key elements to the d e c i s i o n making or o p t im i z a t i o n problem can be i d e n t i f i e d : 1. a goal or set of o b j e c t i v e s - the reason f o r management 2. a set o f a l t e r n a t i v e courses o f a c t i o n or opt ions - the t o o l s of management 3. some understanding of the nature o f the "system" being managed, o f t en summarized i n a model. The problem i s to use the understanding o f the system dynamics to choose the sequence o f management opt ions over t ime which best ach ieves the goa ls o f management. So l v i ng t h i s problem may be d i f f i c u l t even when a l l the elements are f u l l y s p e c i f i e d . In p r a c t i c e , u n c e r t a i n t i e s concern ing each of them makes the de c i s i on making process f a r more d i f f i c u l t . Unce r t a i n t i e s About Ob jec t i ves Ob jec t i ves can be i l l - d e f i n e d or even unknown, p a r t i c u l a r l y i n the e a r l y stages o f the development of a resource . They a l s o tend to change over t ime (H i l bo rn and Peterman, 1976) as percept ions about the problem change. Changes in the r e l a t i v e i n f l uence o f d i f f e r e n t i n t e r e s t groups may occur (spor t versus commercial versus Indian f i she rmen) , j u r i s d i c t i c a l boundaries may a l t e r (extens ion to 200 mi le coas ta l economic zones) o r there may be r e l a t i v e l y r ap i d changes in s o c i e t a l va lues (the sudden concern w i th whale conserva t ion i n recent y e a r s ) . The ontogeny of ob j e c t i v e s i n f i s h e r y management, from maximum sus ta ined y i e l d to max i-mum economic y i e l d to optimum sus ta i nab l e y i e l d (Roede l , 1975; L a r k i n , 1977), r e f l e c t s an expansion i n awareness of the scope o f the problem, from pure ly b i o l o g i c a l concerns to those i n c l ud i ng economic i n t e r e s t s and f i n a l l y broad s o c i e t a l goa l s . The q u a n t i f i c a t i o n of o b j e c t i v e s -- measuring p re fe rences , r i s k ave r s i on , , c o n f l i c t i n g o b j e c t i v e s , e va l ua t i ng t imestreams -- can a l s o be a d i f f i c u l t problem (Keeney, 1977; H i !born and Wa l t e r s , 1977; C la rk and B e l l , 1976). Obv ious ly , the way i n which ob j e c t i v e s are perce ived and quan t i f i e d w i l l i n many cases have a major bear ing on the types of management dec i s i on s which are made. Unce r t a i n t i e s About Options F a i l u r e to recogn ize the range of opt ions a v a i l a b l e i n a p a r t i c u l management s i t u a t i o n • c o n s t i t u t e s another source o f un ce r t a i n t y . Aga in , t h i s w i l l app ly p a r t i c u l a r l y to the e a r l y stages of development of a resource . A major c o n t r i b u t i o n to f i s h e r y management was to show tha t by c o n t r o l l i n g both f i s h i n g e f f o r t and age a t f i r s t cap tu re , g rea te r y i e l d s cou ld be obta ined than by c o n t r o l l i n g e i t h e r f a c t o r by i t s e l f (Beverton and H o l t , 1957). On the o ther hand, another problem i s posed when i t turns out to be imposs ib le to implement opt ions which were thought to be a v a i l a b l e . 4 P o l i t i c a l , s o c i a l , economic or j u r i s d i c t i o n a l f a c t o r s may impose severe r e s t r a i n t s on the range o f op t ions a v a i l a b l e to management. F a i l u r e to con t ro l f i s h i n g e f f o r t has been imp l i c a t ed as an important f a c t o r i n the dep l e t i on of severa l major f i s h s tocks (Skud, 1973). Th is problem o f l i m i t a t i o n s on con t ro l opt ions has been termed the " r e a c h a b i l i t y problem" (Walters and Hi 1 born, 1976). Unce r t a i n t i e s About Nature The task of management i s to choose between a range o f op t ions in order to ach ieve a des i r ed goa l . In order to do t h i s , some under-stand ing i s requ i red o f the dynamics o f the "system" being managed, and a l s o o f the e f f e c t which management ac t i ons w i l l have on the dynamics. The use o f the word "system" i s d e l i b e r a t e l y vague. For the management problem o f choosing harves t ra tes f o r f i s h e r i e s , an understanding o f the e f f e c t s o f e x p l o i t a t i o n on the stock dynamics i s r e qu i r ed . I t may a l so be important to understand the nature o f the f l e e t dynamics, the i ndus t r y as a whole, or even the s o c i a l context o f the i n du s t r y . The nature o f the "system" cons idered w i l l i n par t be determined by the scope o f the goals of the p a r t i c u l a r management problem or by whether the management problem i s a t a p o l i t i c a l , s t r a t e g i c o r t a c t i c a l l e v e l . Whatever the system under c o n s i d e r a t i o n , l a ck o f understanding of i t s dynamics and response to po s s i b l e management ac t i ons c o n s t i t u t e s another source o f unce r t a i n t y in the d e c i s i o n making process . Two types o f unce r t a i n t y can be d i s t i n g u i s h e d : 5 • 1. r educ i b l e un c e r t a i n t y , which i s recogn ized and which may to a c e r t a i n extent be r e so l v ab l e by appropr i a te management a c t i o n s , and 2. i r r e d u c i b l e u n c e r t a i n t y , which i s not recogni:zed:or which,reven i f r e cogn i zed , i s not r e so l v ab l e by appropr i a te management a c t i o n s . Some examples may serve to i l l u s t r a t e the d i s t i n c t i o n . Management s t r a t e g i e s f o r many exp l o i t e d popu la t ions r e l y on the opera t i on of dens i t y dependent processes w i t h i n the popu la t ions tha t r e s u l t i n them being most p roduc t i ve a t popu la t i on s i z e s cons i de rab l y below un-exp l o i t e d l e v e l s . To e f f e c t i v e l y determine the nature of such p rocesses , the popu la t ion must be observed over a wide range of d e n s i t i e s . When the popu la t ion has been observed on ly over a narrow range of d e n s i t i e s , the unce r t a i n t y about the nature o f the processes may be q u a n t i f i e d e i t h e r as -uncer ta in ty about the c o r r e c t f unc t i ona l form o f the model d e s c r i b i ng the dens i t y dependent p rocess , o r as s t a t i s t i c a l un ce r t a i n t y concern ing the parameter va lues o f a p a r t i c u l a r model o f the dens i t y dependent process . In e i t h e r case , the unce r t a i n t y i s r e d u c i b l e , a t l e a s t i n p a r t , by managing the popu la t ion so tha t i t can be observed over a wider range o f d e n s i t i e s . I r r e du c i b l e u n c e r t a i n t i e s f a l l i n t o three types . F i r s t , there are u n c e r t a i n t i e s due to "random" v a r i a b i l i t y , such as the v a r i a b i l i t y i n rec ru i tment to f i s h s t o ck s . Such unce r t a i n t y might , i n p r i n c i p l e , be r e d u c i b l e , but i t has been shown t ha t i n many cases management de c i s i on s would not be a l t e r e d even i f such v a r i a b i l i t y cou ld be p red i c t ed (Wa l te rs , 1975). More r e c e n t l y , Ludwig (1979a) has shown t h a t . e n v i r o n -6 mental v a r i a b i l i t y can change opt imal stock l e v e l s i n cases where there are c on s t r a i n t s on changes in c on t r o l a c t i o n s . Second there i s the unce r t a i n t y which i s recogn ized but which cannot be reduced by t r i a l and e r r o r because the cos ts o f e r r o r are too g rea t . A lower th resho ld to dens i t y below which a popu la t i on w i l l go e x t i n c t may not be an unce r t a i n t y which should.be reduced by d r i v i n g the popu la t ion to tha t va lue . (On the o ther hand the p o s s i b i l i t y o f m u l t i p l e - e q u i l i b r i a phenomena ( H o l l i n g , 1973; Peterman, 1977) w i th undes-i r a b l e domains of s t a b i l i t y may suggest a s t r a t egy in which c e r t a i n popu la t ions are used as experiments to t e s t f o r such phenomena so tha t o ther popu la t ions may be be t t e r managed.) T h i r d , there ..is the t o t a l l y unexpected event which confounds p r e d i c t i o n . Such events may be in p r i n c i p l e unp red i c t ab l e , but i n many cases they are the r e s u l t o f f a i l u r e to recogn ize important v a r i a b l e s or de f ine the system broad ly enough (Jones and Wa l t e r s , 1976). Adapt ive Management Adapt ive management i s an approach to de c i s i on making which takes d e l i b e r a t e account of the sources of unce r t a i n t y o u t l i n e d above. In i t s most general form ( H o l l i n g , 1978), the adapt ive approach p re s c r i bes both a ph i losophy and a set o f p o l i c y des ign c r i t e r i a which on the one hand seek to reduce u n c e r t a i n t i e s as f a r as p o s s i b l e , and on the o ther hand recogn ize tha t they can never be e l i m i n a t e d , so t ha t p o l i c i e s should be designed which can absorb and bene f i t from un ce r t a i n t y . The s p e c i f i c des ign c r i t e r i a i nc lude the i d e n t i f i c a t i o n o f a broad range of p o s s i b l e ob j e c t i v e s and management op t i on s , the r e cogn i t i on o f 7 the in fo rmat ion va lue o f management a c t i o n s , and the m in im i za t i on o f the consequences of p o l i c y f a i l u r e . Robust p o l i c i e s should be r e l a t i v e l y i n s e n s i t i v e to changes i n o b j e c t i v e s , to f a i l u r e of implementat ion of p o l i c y - p r e s c r i b e d c o n t r o l s , and to unan t i c i pa ted responses o f the system. The adapt ive approach thus s t r i v e s to develop management systems which have the proper ty of r e s i l i e n c e - - a proper ty tha t a l l ows systems to absorb and make use o f change. 1.2 PROBLEM DESCRIPTION The problem under c ons i de ra t i on i n t h i s study can now be more f u l l y o u t l i n e d . A f i s h stock i s being managed whose dynamics are assumed to be adequate ly desc r ibed by a model r e l a t i n g stock s i z e to s tock p roduc t i on . Severa l such models have been descr ibed i n the f i s h e r i e s l i t e r a t u r e (Graham, 1935; Schae fe r , 1954; R i c k e r , 1954; P e l l a and Toml inson, 1969) and have been used as a bas i s f o r the management o f a number o f important commercial f i s h e r i e s . A l l desc r i be a dens i t y depen-dent r e l a t i o n s h i p between stock s i z e and product ion such tha t the stock i s most p roduc t ive a t some l e v e l below the unf i shed e q u i l i b r i u m . For the hypo the t i ca l stock being,.managed, the appropr i a te form o f the model i s assumed known, but the parameters o f the model are not known. These, however, may be est imated us ing data which become a v a i l a b l e through the ac tua l management o f the f i s h e r y . The nature o f the data w i l l depend on the type of f i s h e r y , but may inc lude catch and e f f o r t i n fo rmat i on o r observa t i ons on escapement and r e c ru i tmen t . The parameter es t imates based on such data w i l l g ene ra l l y be i n e r r o r , but t h e r e . i s the 8 po t en t i a l f o r improvement i n the est imates (equ iva len t to be t t e r under-standing of the nature of the dens i t y dependent processes r egu l a t i n g the popu la t ion) as management proceeds. The management goal i s to maximize long-term d iscounted catch and the management opt ions are harvest r a t e s , e f f o r t l e v e l s or l e v e l s of escapement which may be app l i ed p e r i o d i c a l l y . The aim of t h i s study i s to develop and eva lua te a range of ha rves t i ng s t r a t e g i e s appropr i a te to the management o f t h i s hypo the t i ca l s tock . The problem of managing a resource where there i s s t a t i s t i c a l un ce r t a i n t y concern ing the parameters of the model being used to desc r ibe the resource dynamics can be seen as a s i m p l i f i c a t i o n or spec i a l case of the more general problem o f management under un ce r t a i n t y . In t h i s ;case the management o b j e c t i v e i s known and not sub jec t to change. The range of po t en t i a l con t ro l a c t i ons i s known and i t i s assumed tha t i t w i l l be po s s i b l e to implement e x a c t l y any chosen con t ro l ac t ion . . A l l the unce r t a i n t y about the stock dynamics can be co l l apsed i n to unce r t a i n t y about the parameters of a model of known f unc t i ona l form. The problem, then, i s e s s e n t i a l l y one o f r educ i b l e unce r t a i n t y in our understanding o f the dens i t y dependent r e l a t i o n s h i p determin ing the stock dynamics. The des ign c r i t e r i o n o f the adapt ive management approach r e l evan t to t h i s problem i s the r e cogn i t i on of the in fo rmat ion value of management a c t i o n s . Th is in fo rmat ion va lue of the con t ro l a c t i on becomes apparent when i t i s r e a l i z e d tha t the source o f unce r t a i n t y i s the form o f a dens i t y dependent r e l a t i o n s h i p . S ince the sequence o f con t ro l a c t i ons w i l l determine the range o f d e n s i t i e s over which the response of the stock may be observed, i t i s c l e a r t ha t t h i s sequence w i l l determine how we l l the r e l a t i o n s h i p can be l ea rned , tha t i s , how a c cu r a t e l y the parameters can be determined. 9 There a r e , t h e r e f o r e , two po t en t i a l e f f e c t s of a g iven con t ro l a c t i on (harvest r a t e , escapement, e t c . ) . On the one hand there i s the shor t - te rm payof f i n terms o f the catch which can be taken now. On the o ther hand there i s the i n fo rmat i on va lue which may improve the l o ng -term payof f . Un fo r tuna te l y , there i s o f ten a t r a d e - o f f between these b e n e f i t s , i n tha t l a rge pe r tu rba t i ons are u s ua l l y the most i n f o rma t i v e , but a l s o i nvo l ve h igher r i s k s and c o s t s , and in some cases a d r a s t i c reduc t ion i n shor t - te rm payo f f s . The quest ion of how to o p t i m a l l y balance present re turns aga ins t i n fo rmat ion va lue i s known as the dual con t ro l problem (Fe l 'dbaum, 1960-1961). The theory invo lved i s o u t l i n ed in Chapter I I , an approximate s o l u t i o n i s de s c r i bed , and t h i s s o l u t i o n i s used to develop harves t ing s t r a t e g i e s which a c t i v e l y seek to reduce u n c e r t a i n t i e s i n parameter e s t ima te s , to the extent j u s t i f i e d by expected improved performance. In the context o f management problems w i th r educ i b l e u n c e r t a i n t i e s about system responses, the adapt ive approach imp l i e s a process of l e a r n -ing about system responses through exper ience (Walters and Hi 1 born, 1978). A d i s t i n c t i o n has been made between a c t i v e and pass ive adapt i ve s t r a t e -g ies (Bar-Shalom and Tse, 1976). An a c t i v e adapt ive s t r a t egy assesses the in fo rmat ion va lue o f con t ro l a c t i o n s . A pass ive adapt ive s t r a t egy makes use of i n fo rmat ion which becomes a v a i l a b l e as management proceeds, but takes no account o f the f u t u r e a v a i l a b i l i t y of i n f o rma t i on . A non-adapt ive s t r a t egy makes no use of r e a l - t ime i n f o rma t i on . P o l i c i e s based o n . a l l three s t r a t e g i e s w i l l be app l i ed to the management problem cons idered in t h i s s tudy. 10 1.3 PROBLEM BACKGROUND Although unce r t a i n t y i s a pervas ive f ea tu re of a l l resource management problems, i n c l ud i ng f i s h e r i e s , r e cogn i t i on of the problems posed by unce r t a i n t y i n such s i t u a t i o n s has on ly r e c en t l y appeared in the r e l e van t l i t e r a t u r e . Recent reviews and b i b l i og raphy may be found in Ho l l i ng ' (1978) and Wal ters and H i l bo rn (1978). R e s t r i c t i n g a t t e n t i o n to the f i s h e r y l i t e r a t u r e , t h e o r e t i c a l i n t e r e s t i n the problem o f unce r t a i n t y about s tock dynamics and r e cogn i -t i o n o f the in fo rmat ion value of management a c t i ons i s o f very recent o r i g i n . A l l e n (1973) found t ha t opt imal escapements d i d not depend on environmental v a r i a b i l i t y i f the o b j e c t i v e was maximum sus ta ined y i e l d . Walters (1975) found tha t opt imal s t r a t e g i e s f o r Skeena R ive r sockeye salmon were r e l a t i v e l y i n s e n s i t i v e to judgmental unce r t a i n t y about the R i cker stock product ion parameter. The d i r e c t antecedents of t h i s study l i e i n the work of Walters and H i l bo rn (1976). They po inted out tha t the e s s e n t i a l l y pass ive adapt ive s t r a t e g i e s used i n f i s h e r y management do not guarantee tha t c o r r e c t parameter va lues w i l l be l ea rned . They suggested tha t such s t r a t e g i e s cou ld r e s u l t i n an e q u i l i b r i u m i n the f i s h e r y which i s ne i t he r opt imal nor product ive of the type of i n fo rmat ion requ i red to determine what i s op t ima l . They formulated very much the same problem cons idered here and used s t o c ha s t i c dynamic programming to determine opt imal con t ro l laws account ing f o r s t a t i s t i c a l un ce r t a i n t y i n parameter e s t ima tes . The s i t u a t i o n they cons idered was one i n which the parameters o f the R i cker s tock - rec ru i tmen t model were un ce r t a i n . However due to the "curse of d imens i ona l i t y " assoc i a ted w i th the op t im i z a t i o n technique they used, 11 they cou ld on ly cons ider the case of unce r t a i n t y in one parameter a t a t ime. The con t ro l laws generated were not t e s t ed o r compared w i th o ther po s s i b l e s t r a t e g i e s in terms of r e l a t i v e performance. S ince then a growing frequency o f p u b l i c a t i o n s i n the f i s h e r y l i t e r a t u r e suggests that r e cogn i t i on of the importance o f the problems posed by unce r t a i n t y about s tock dynamics, and the po t en t i a l i n f o rmat i on value o f con t ro l a c t i o n s , has been ach ieved . Walters (1977) presents techniques f o r eva l ua t i ng po s s i b l e bene f i t s o f d i f f e r e n t exper imental and non-experimental p o l i c i e s f o r enhancement of P a c i f i c salmon. Both Gul land (1977) and Ho l t (1977) advocate the use o f exper imental s t r a t e g i e s f o r marine resource management, and Ho l t suggests tha t d i f f e r e n t whale s tocks should be e xp l o i t e d a t d i f f e r e n t l e v e l s to help determine dens i t y dependent responses . , A l l e n and Kirkwood (1977) cons ide r the p r a c t i c a l p o s s i b i l i t i e s o f t h i s sugges t i on , and conclude tha t such experiments would be u n l i k e l y to produce usefu l r e s u l t s w i t h i n a reasonable per iod o f time g iven the no i s i nes s of the observat ions which are p o s s i b l e . S i l v e r t (1978) app l i ed the concepts of a c t i v e and pass ive adapt ive con t ro l to the problem of choosing between two a l t e r n a t i v e model forms and was ab le to c a l c u l a t e expected values o f a c t i v e adapt ive s t r a t e g i e s . Optimal de c i s i on s were found to be very s e n s i t i v e to d i scount ra tes used. Huang e t a l (1976) a l s o examined the "va lue of i n f o rma t i on " i n the context of unce r t a i n t y about model parameters. However they assumed t ha t research r a the r than the management process i t s e l f would be used to improve parameter e s t ima tes . Most r e c en t l y Ludwig (1979a) and Ludwig and Varah (1979) have exp lo red opt imal ha rves t i ng s t r a t e g i e s f o r randomly f l u c t u a t i n g resources and Ludwig (1979b) has developed con t ro l s t r a t e g i e s 1 2 which average over p o s t e r i o r p r o b a b i l i t y d i s t r i b u t i o n s o f parameter es t imates . Although t h e o r e t i c a l i n t e r e s t i s o f recent o r i g i n , severa l f i s h e r -ies agencies have recognized the po t en t i a l va lue of a c t i v e exper imentat ion f o r some t ime. In 1969 the agency r e spons i b l e f o r the management of eastern t r o p i c a l P a c i f i c tuna i n i t i a t e d a program o f d e l i b e r a t e inc reases in catch quotas, beyond apparent maximum sus t a i nab l e y i e l d , i n order to be t t e r determine the response o f the stock at h igher l e v e l s of e x p l o i t a -t i o n (IATTC, 1977). S i m i l a r experiments were conducted w i th h a l i b u t (Skud, 1976) and salmon (INPFC, 1962) in the 1950 's . Current academic i n t e r e s t i n the problem seems to be a case o f theory l agg ing behind implementat ion. 1.4 OUTLINE OF STUDY This study seeks to develop and ana lyse a range of ha rves t i ng s t r a t e g i e s appropr i a te to the management of f i s h s tocks w i th unce r ta in dynamics represented as s t a t i s t i c a l unce r t a i n t y in parameter es t imates o f models used to desc r ibe the stock dynamics. The s t r a t e g i e s w i l l range from a c t i v e l y adap t i ve , where c on t r o l s are chosen to d e l i b e r a t e l y improve parameter e s t ima tes ; through pass ive adap t i ve , where c on t r o l s are chosen assuming tha t cu r ren t parameter est imates are c o r r e c t but parameter es t imates change as i n fo rmat i on becomes a v a i l a b l e ; to non-adapt ive s t r a -t eg i e s which assume tha t i n i t i a l est imates are c o r r e c t and need not be mod i f ied on the bas i s of new in fo rmat ion Chapter II presents an o u t l i n e of opt imal s t o c ha s t i c con t ro l theory and shows how i t i s a p p l i c a b l e to the dual con t ro l problem. The 13 opt imal con t ro l problem i s shown to conta in both an e s t ima t i on and con t ro l problem. Some approximate s o l u t i o n s to these problems are d i s -cussed, and these are used to develop the s t r a t e g i e s to be tes ted in the seque l . In p a r t i c u l a r , a technique f o r deve lop ing approx imate ly opt imal a c t i v e l y adapt ive con t ro l p o l i c i e s i s d i s cus sed . Chapter I I I presents the main r e s u l t s of t h i s s tudy. The s p e c i f i c problem cons idered here i s . t h a t of managing a stock f o r which the major source of unce r t a i n t y i s i n the s tock - rec ru i tmen t r e l a t i o n s h i p . Th is i s the case f o r many salmon f i s h e r i e s . The R i cke r model ( R i c k e r , 1954) i s assumed to desc r ibe the r e l a t i o n s h i p adequate ly , but the t rue pa ra -meters are not known. These can be est imated i n i t i a l l y from p r i o r data a v a i l a b l e on.the s to ck - rec ru i tmen t r e l a t i o n s h i p , and subsequent ly from data prov ided by the-.ongoing management o f the stock i t s e l f . The per-formance of a range o f p o l i c i e s w i l l be t e s t e d , and i t w i l l be shown how the performance o f each p o l i c y depends very much on the p r i o r data a v a i l a b l e , or more s p e c i f i c a l l y , on the past e x p l o i t a t i o n h i s t o r y o f the f i s h e r y . Chapter IV extends the problem to cons i de ra t i on of o ther types o f models, i n p a r t i c u l a r the Schaefer model (Schae fer , 1954). Th is extends the problem to s i t u a t i o n s i n which the stock s i z e i s not d i r e c t l y observab le and the i n i t i a l s tock s i z e a t the beginn ing o f e x p l o i t a t i o n p lays a more important r o l e . ChapterV summarizes the main f i n d i ng s o f the study and cons iders the imp l i c a t i o n s o f the r e l a x a t i o n o f some o f the assumptions in the problem f o rmu l a t i on . The imp l i c a t i o n s o f t h i s study to more general problems i n resource management and to the general problem posed by unce r t a i n t y i n such s i t u a t i o n s i s d i s cus sed . 14 CHAPTER II 2.0 CONTROL AND ESTIMATION THEORY Th is chapter presents an o u t l i n e of opt imal s t o c ha s t i c con t ro l theory w i th p a r t i c u l a r re fe rence to the dual con t ro l problem. The not ion of s t a t e augmentation i s in t roduced to i n c lude the problem of unce r ta in parameters. Three c l a s s e s of con t ro l p o l i c y are d i scussed and the correspondence w i th the c l a s s e s o f adapt ive p o l i c i e s in t roduced i n chapter I i s i n d i c a t e d . The general s o l u t i o n to the dual con t ro l problem i s presented, problems i n the s o l u t i o n are d i scussed and an approximate s o l u t i o n which e x h i b i t s the dual e f f e c t o f con t ro l i s ou t -l i n e d . Th is s o l u t i o n i nvo l ves a non l i nea r f i l t e r i n g problem and some techniques f o r s o l u t i o n o f t h i s problem are d i s cu s sed . 2.1 PROBLEM FORMULATION Consider the system whose s t a t e x_(k) evo lves accord ing to the non l i nea r s t o c h a s t i c d i f f e r en c e equat ion x (k+ l ) = f [ x ( k ) , u ( k ) , k] + v(k) (2.1-1) k = 0, 1 , . . . , N-l where u_(-k) i s the con t ro l vec to r app l i ed a t t ime k and y_(k) i s the process no i s e . Observat ions y_(k) are a v a i l a b l e a t each s tage , and the 15 non l i nea r measurement equat ion i s y ( k ) =_h[x(k), k] + w(k) (2 .1-2) k = 1 , . . . , N where w(k) i s the measurement no i s e . The i n i t i a l s t a t e xjO) and the sequence o f process and measurement no ise terms are independent Gaussian vec to rs w i th x(0) ^ N ( x ( 0 / 0 ) , P ( 0 / 0 ) ) , v (k) % N(0, Q(k)) and w(k) ^ N(0_, R ( k ) ) . The more general f o rmu l a t i o n , where the no ise terms are not n e c e s s a r i l y a d d i t i v e nor are there any assumptions made about the form of the j o i n t p r o b a b i l i t y dens i t y of x j O ) , y_(k) and w(k) , i s o u t l i n e d in Bar-Shalom and Tse (1976). Consider a l s o the performance measure N-l J = E C U [x(N)] + S L. [ x ( k ) , u ( k ) , k]} (2 .1-3) N k=0 K ~ where the expec ta t i on E{-} i s taken over a l l under l y ing random q u a n t i t i e s . F i n a l l y cons ide r admissable c o n t r o l s o f the feedback type , u(k) = u(k , Y k , U k _ 1) (2 .1-4) where Y k = { y ( l ) , . . . , y ( k )} and Uk'1 = {u(0) u(k-l)}. The s t o ch -a s t i c con t ro l problem i s to f i n d the opt imal con t ro l sequence {u * ( k ) } , k = 0 , . . . , N-l which i s o f the form (2.1-4) and min imizes the performance measure (2.1-3) sub jec t to the dynamic c o n s t r a i n t s (2.1-1) and ( 2 . 1 - 2 ) . 2.2 STATE AUGMENTATION The problem as formulated above cons idered the problem o f c o n t r o l -l i n g a dynamic system wi th unce r ta in s t a t e . The problem cons idered in t h i s study i s one of unce r t a i n t y i n parameters. Th is c o n f l i c t i s e a s i l y overcome when i t i s recogn ized tha t the d i s t i n c t i o n between s t a t e s and parameters of dynamic systems i s l a r g e l y an a r t i f i c i a l one (Young, 1974). Parameters may be cons idered as s t a t i o n a r y or s l ow ly t ime va ry ing s t a t e v a r i a b l e s . Therefore when the con t ro l problem i s one of un ce r t a i n t y i n parameter e s t ima tes , the unce r t a i n parameters are s imp ly inc luded as s t a t e v a r i a b l e s . . The s t a t e o f the system i s now the augmented s t a t e and new dynamic equat ions are formulated in terms o f t h i s s t a t e . I f parameters are thought not to change over t ime , the dynamic equat ions f o r these s t a t e v a r i a b l e s are s imply formulated as d i f f e r e n c e equat ions w i th x(k+i) = x ( k ) . 2.3 INFORMATION STATE An in fo rmat ion s t a t e i s a quan t i t y which summarizes a l l the i n f o r -mation a v a i l a b l e about the cu r ren t s t a t e . I t can be thought of as a quan t i t y equ i va l en t to the observa t ion process Y , the known past c on t r o l k-1 h i s t o r y U and the a priori knowledge about the system s t a t e . The a priori knowledge i n t h i s case i s the p r i o r p r o b a b i l i t y d i s t r i b u t i o n of x (0) which i s g iven by the mean £(0/0) and e r r o r covar iance P (0 /0 ) . The dependence o f the in fo rmat ion s t a t e on a priori knowledge i s u s ua l l y suppressed n o t a t i o n a l l y . One po s s i b l e i n fo rmat i on s t a t e i s the combined 17 sequence P k 2 = ( Y k , U k _ 1 ) Another i n fo rmat i on s t a t e i s the c ond i t i o na l p r o b a b i l i t y dens i t y P k 2 = p ( x ( k ) / Y k , U k - 1 ) 2.4 CLASSES OF CONTROL POLICIES Three general c l a s se s o f s t o c ha s t i c con t ro l p o l i c i e s e x i s t . They are d i s t i n gu i s h ed by the extent to which they make use o f r e a l - t ime k k-1 in fo rmat ion ( i . e . , Y and U ) and by the extent to which they make use of the knowledge tha t f u tu re i n fo rmat ion w i l l be a v a i l a b l e . The three c l a s s e s are open- loop, open-loop feedback and c l o s ed - l o op . An open-loop p o l i c y makes no use o f rea l t ime i n f o rma t i on . The con t ro l i s chosen s imply as a f un c t i on of t ime , and on ly the p r i o r i n f o r -mation about the system s t a t e , together w i th knowledge of the system dynamics, are used to determine the sequence of con t ro l a c t i ons to be a p p l i e d . An open-loop feedback p o l i c y uses i n fo rmat ion which becomes a v a i l -ab le as con t ro l and observat ions of the system proceed. Rea l - t ime in fo rmat ion i s used to determine the con t ro l a c t i ons to be used. How-ever no subsequent feedback i s a n t i c i p a t e d in choosing con t ro l a c t i on s a t any t ime. A c l o sed- l oop p o l i c y a l so makes use of r e a l - t ime i n f o rma t i on . However, i n a d d i t i o n , i t a n t i c i p a t e s the f u tu re a v a i l a b i l i t y o f in fo rma-18 t i o n , tha t i s , i t knows tha t f u tu re observa t ions w i l l be made and tha t f u tu re l e a rn i ng i s p o s s i b l e . These three c l a s se s o f p o l i c y correspond d i r e c t l y to the three types of adapt ive s t r a t egy ou t l i n ed in chapter I . The open-loop p o l i c y i s a non-adapt ive s t r a t e g y , the open-loop feedback p o l i c y represents a pass ive adapt ive s t r a tegy and the c l osed- l oop p o l i c y corresponds to an a c t i v e l y adapt ive s t r a t egy . 2.5 THE DUAL CONTROL PROBLEM Fel'dbaum (1960-61) po inted out tha t there can be two e f f e c t s of app ly ing a con t ro l to a system. On the one hand the con t ro l w i l l a f f e c t the s t a t e o f the system. On the other hand i t may a l so a f f e c t the un-c e r t a i n t y about the s t a t e . In cons i de r i ng the performance of a p a r t i c u l a r p o l i c y , both e f f e c t s may have to be taken i n to account i n choosing con t ro l a c t i o n s . By a f f e c t i n g the s t a t e of the system, the con t ro l a f f e c t s shor t term b e n e f i t s . However, by a f f e c t i n g s t a t e unce r t a i n t y the con t ro l may a l so ac t to a f f e c t long-term b e n e f i t s . In assess ing the va lue o f a g iven con t ro l a c t i o n , i t s pos s i b l e i n fo rmat ion va lue must be taken i n t o account. C losed- loop con t ro l p o l i c i e s have the f ea tu re of assess ing the va lue of i n fo rmat ion by a n t i c i p a t i n g f u tu re feedback. I t can be shown tha t the g l o b a l l y opt imal p o l i c y f o r the s t o cha s t i c con t ro l problem i s of the c l o sed- l oop type (Sorenson, 1976). The opt imal c l o sed - l oop ' • i 19 con t ro l p o l i c y has the proper ty tha t the l e a rn i ng i s done on ly to the extent requ i red by the o v e r a l l performance. S ince l e a r n i ng might con-f l i c t w i th the con t ro l purpose, the opt imal c l o sed- l oop p o l i c y balances i t s l e a r n i ng and con t ro l e f f e c t s so as to minimize the o v e r a l l c o s t . 2.6 OPTIMAL STOCHASTIC CONTROL The opt imal con t ro l f o r the problem formulated above can be obta ined by app ly ing Be l lman 's P r i n c i p l e of Op t ima l i t y (Be l lman, 1961) i n the i n fo rmat i on s t a t e . Th is p r i n c i p l e s t a tes t h a t , whatever the present i n fo rmat i on s t a t e and prev ious dec i s i on s a r e , the remaining dec i s i on s must c o n s t i t u t e an opt imal p o l i c y w i th regard to the present i n fo rmat ion s t a t e . A p p l i c a t i o n of t h i s p r i n c i p l e y i e l d s a r e c u r s i v e system of equat ions which the opt imal con t ro l must s a t i s f y , the s t o ch -a s t i c dynamic programming equat ion: J C L 0 (N-k) = min E { L . [ x ( k ) , u ( k ) , k] + u(k) k " J C L 0 ( N - k - l ) / Y k , U k _ 1 } k = N - l , . . . , 0 CLO where J (N-k-1) i s the opt imal cos t - to -go f o r the l a s t N-k-1 stages k+1 k and i s a f un c t i o n of Y and U . The s o l u t i o n i s s t a r t ed by cons i de r i ng the f i n a l stage problem: J C L 0 ( 0 ) = E { L N [ x ( N ) ] / Y N , U N - 1 } I f the in fo rmat ion s t a t e i s represented by the c ond i t i o na l p r o b a b i l i t y dens i t y p { x ( k ) / Y k , U k _ 1 } ra the r than by ( Y k , l^ ' 1 ) then 20 the s o l u t i o n of the opt imal con t ro l problem a l so r equ i r e s a r e c u r s i v e r e l a t i o n f o r updat ing t h i s c ond i t i o na l d en s i t y . Th is i s a general non-l i n e a r f i l t e r i n g problem and i s s o l v ed , a t l e a s t i n theo ry , by app ly ing Bayes 1 r u l e . 2.7 PROBLEMS IN THE GENERAL SOLUTION Tse et al. (1973) have po inted out three problems in t r y i n g to so lve f o r the opt imal con t ro l law. F i r s t the in fo rmat ion s t a t e i s e i t h e r i n f i n i t e d imens i ona l , in the case tha t the c ond i t i o na l p r o b a b i l i t y den-s i t y i s used, or e l s e i s f i n i t e but grows w i th t ime , i f the sequence k k-1 (Y , U ) i s used. Second, the opt imal cos t - to -go assoc i a ted w i th the in fo rmat ion s t a t e i s g ene ra l l y not an e x p l i c i t f u n c t i o n . T h i r d , the de te rminat ion o f an opt imal con t ro l law f o r any po s s i b l e i n fo rmat ion s t a t e i s l i m i t e d by the "curse of d i m e n s i o n a l i t y " . 2.8 WIDE-SENSE ADAPTIVE DUAL CONTROL Using the same problem fo rmu la t i on as in 2 . 1 , Tse et al. (1973) and Bar-Shalom and Tse (1976) have developed an approx imat ion of the opt imal c l o sed- l oop s o l u t i o n to the opt imal con t ro l problem. They pro-posed t ha t a reasonable suboptimal approach, i n view of the three problems ou t l i n ed i n 2 .7 , would be t o : (1) approximate the in fo rmat ion s t a t e by the mean x ( k / k ) and covar iance P(k/k) of the c ond i t i o na l s t a t e , i n o rder to reduce the d imens i ona l i t y of the in fo rmat ion s t a t e and keep i t constant over t ime; (2) approximate : the opt imal cos t - to -go as soc i a t ed 21 wi th the in fo rmat ion s t a t e ; and (3) compute the con t ro l o n - l i n e . Th is avo ids the problem o f having to s pe c i f y a con t ro l law f o r any po s s i b l e i n fo rmat ion s t a t e which might a r i s e . The key to t h e i r approach i s the development of an approx imat ion CLO to the opt imal cos t - to -go J (N-k-1) which preserves i t s c l o sed- l oop f e a t u r e , tha t i s , which s t i l l a n t i c i p a t e s f u tu re observa t ions and assesses t h e i r va l ue . A very b r i e f o u t l i n e f o l l ows o f the way i n which t h i s approx imat ion i s ob ta ined . At every stage k and to each con t ro l u[k) there corresponds a p red i c ted s t a t e x_(k + 1/k) and covar iance P(k + 1 / k ) . To t h i s p red i c t ed s t a t e , a sequence o f c on t r o l s {u^ ( j ) } j = k + 1 , . . . , N-l i s at tached which de f ines a nominal s t a t e t r a j e c t o r y {x^ ( j ) } j = k + 1 , . . . , N. One way to ob ta in the nominal con t ro l sequence i s v i a the opt imal d e t e r -m i n i s t i c c o n t r o l . The cos t - to -go i s eva luated by expansion about t h i s nominal assuming an a dd i t i o n a l pe r tu rba t i on con t ro l a long the nomina l . The v a r i a t i o n to second order i s minimized in a c l o sed- l oop f ash ion and the approximate cos t - to -go J [ u j k ) ] a s soc i a ted w i th the i n i t i a l con t ro l a c t i on u jk) i s ob ta ined: J [u (k ) J = L k I x ( k / k ) , u(.k), kj + t r [ l ^ - P(k/k)] + h t r [K (k+l) P(k+l/k) N-l + h £ t r [ K o ( j + l ) Q( j) + A Q x x ( j ) P Q ( j / j ) ] j = k + 1 ' (2.8 -1) 22 where y „ > K and A are c a l c u l a t e d a long the nominal us ing the s e n s i -o o o,xx 3 t i v i t y equat ions f o r 3 f /3x , 3f/3u., 3L/3x_ and 3L/3u^, t ha t i s , the s e n s i t i v i t y of the s t a t e equat ions and the cos t to the s t a t e and con t ro l v e c t o r s . The exact formulae need not be presented here; they are r e a d i l y a v a i l a b l e in Bar-Shalom and Tse (1976). The po in t s to note about (2 .8-1) a r e : (1) the cos t i n c l udes both con t ro l performance and e s t ima t i on performance; (2) a con t ro l u{k) which reduces f u tu re unce r t a i n t y by reduc ing f u t u r e covar iance P Q ( J / J ) w i l l a l s o act to reduce the c o s t ; and (3) the f u tu re nominal con t ro l i s not a c t u a l l y app l i ed i n the f u t u r e but i s on ly used to approximate the opt imal c o s t - t o - go . The con t r o l u j k ) which min imizes the c o s t J.[i[(k)] can be found by an appropr ia te search procedure. Th is con t ro l u_* (k) i s then a p p l i e d , the obse rva t i on y (k+l) made, the i n fo rmat i on s t a t e updated to x_(k+l/k+l) and P ( k+l / k+l), and the whole procedure repeated. 2.9 REQUIREMENTS AND LIMITATIONS The requirements f o r a p p l i c a t i o n o f the wide sense adapt ive dual con t ro l a l go r i t hm a l s o de f i ne i t s l i m i t a t i o n s . Use o f t h i s a l go r i t hm requ i res t h a t : 1. o b j e c t i v e s are known and can be q u a n t i f i e d : 2. con t ro l opt ions are known and con t ro l vec tors are uncons t ra ined; 3. the system dynamics can be desc r i bed by a non l i nea r d i f f e r e n c e equa t i on ; 23 4. no ise processes are Gaussian- w i th known covar iance and are a d d i t i v e i n the dynamic and observa t i on equat ions; 5. the model d e s c r i b i ng the system dynamics i s o f the c o r r e c t f un c t i ona l form; and 6. an adequate non l i nea r f i l t e r can be found f o r updat ing s t a t e es t imates and covar i ance . The imp l i c a t i o n s o f some o f these l i m i t a t i o n s are d i scussed i n the f i n a l chapter . In p r a c t i c e , the major l i m i t a t i o n was found to be the assumption tha t c on t r o l a c t i on s are uncons t ra ined . In e f f e c t t h i s meant tha t p o l i c i e s based on t h i s a l go r i t hm were not s u f f i c i e n t l y caut ious i n app ly ing h igh harves t ra tes s i n ce the i m p l i c i t assumption was t ha t recovery from low stock s i z e s cou ld be achieved by app ly ing negat ive harves t r a t e s . Th is po in t w i l l be d i scussed f u r t h e r i n Chapter IV. There are severa l o ther supposed "dual c o n t r o l " a lgor i thms which have appeared i n the recent con t ro l theory l i t e r a t u r e . However they e i t h e r impose even s t r i c t e r l i m i t a t i o n s ( e . g . , l i n e a r system dynamics) o r e l s e use e s s e n t i a l l y t r i v i a l h e u r i s t i c s such as random pe r tu rba t i ons o f c on t r o l s (Jacobs and P a t c h e l l , 1972). Recent surveys o f the f i e l d are by Wittenmark (1975) and S a r i d i s (1977). 2.10 NONLINEAR FILTERING AND STATE ESTIMATION I t was s t a t ed i n s e c t i on 2.6 tha t the non l i nea r f i l t e r i n g problem i s so lved by a p p l i c a t i o n of Bayes 1 r u l e to determine the c ond i t i o na l k k-1 p r o b a b i l i t y dens i t y p{x(k)/Y , U }. Th is prov ides a l l the in fo rmat ion 24 about t he : s t a t e i d e r i v ab l e from the Observations'?*' From the c ond i t i o na l d en s i t y , any type o f es t imate about the s t a t e (minimum va r i an ce , maximum l i k e l i h o o d , e t c . ) can be determined. For l i n e a r , Gauss ian, quad ra t i c systems, the c ond i t i o n a l dens i t y i s complete ly s p e c i f i e d by the mean\x(k/k) and covar iance mat r i x P ( k / k ) . These s t a t i s t i c s are g iven by the Kalman f i l t e r equat ions (Kalman, 1960; Kalman and Bucy, 1961). However f o r non l i nea r systems there does not gene ra l l y e x i s t a f i n i t e se t o f parameters which cha r a c t e r i z e s the c ond i -t i o n a l dens i t y f u n c t i o n . Approximate s o l u t i o n s must be found. These are reviewed i n Sorenson (1976) and Gelb (1974). The most common approximat ion i s the use of the extended Kalman f i l t e r . Th is s p e c i f i e s equat ions f o r s t a t e est imate and e r r o r covar iance update f o r a system de f ined by the dynamic and obse rva t i on equat ions of the form o f (2.1-1) and ( 2 . 1 - 2 ) , so i t i s obv i ous l y appropr i a te f o r use w i th the wide sense adapt ive dual con t ro l a l go r i t hm . At each time step the system equat ions f_ and h_ are expanded i n a Tay lo r s e r i e s about the most recent s t a t e e s t ima tes , t r un ca t i ng a f t e r the f i r s t two terms. The r e s u l t i n g l i n e a r i z e d equat ions can be used d i r e c t l y i n the Kalman f i l t e r equat ions . The v a l i d i t y of t h i s approach depends on the q u a l i t y of the l i n e a r approx imat ion , which maybe poor i f the i n i t i a l s t a t e est imates are; poor. There i s no guarantee o f convergence and the mat r i x P(k/k) i s not n e c e s s a r i l y an accurate i n d i c a t i o n of the covar iance mat r i x o f the e s t ima t i on e r r o r s . Two extens ions o f t h i s approach are the second order Kalman f i l t e r , which i nc ludes more terms in the Tay lo r s e r i e s expansions o f f_ and h^ , and the s t a t i s t i c a l l y l i n e a r i z e d f i l t e r which approximates f and h^  by 25 s t a t i s t i c a l l y opt im ized power s e r i e s . The computat ional burden i s g r ea t e r , but Gelb (1974) suggests t h a t ' t h e performance advantages may make these approaches wor thwh i le . Non l inear f i l t e r s of each o f these three types were developed and app l i ed to the parameter e s t ima t i on problems f o r the models used i n chapters I I I and IV. None were found to perform s a t i s f a c t o r i l y and the techniques desc r ibed i n each o f the chapters were used f o r the parameter and s t a t e e s t ima t i on problems encountered. 26 CHAPTER I I I 3.0 THE RICKER MODEL A model which has been used as a bas i s f o r the management of a number o f salmon s tocks i s the R i c ke r model ( R i c k e r , 1954). Th i s model desc r ibes a r e l a t i o n s h i p between parent stock s i z e (the escapement from the f i s h e r y ) and the number o f t h e i r progeny which re tu rn from the ocean a generat ion l a t e r (the rec ru i tment to the f i s h e r y ) . Th is prov ides the bas i s f o r a very s imple model of salmon stock dynamics: R(t+1) = S ( t ) exp(a-e S ( t ) + v ( t ) ) (3.0-1) S ( t ) = R(t) - C( t ) where R i s r e c ru i tmen t , S i s escapement, C i s c a t c h , a and g are para-meters o f the model, v i s an environmental "no i se " term, normal ly 2 d i s t r i b u t e d w i th zero mean and var iance , and t denotes generat ion t ime. Walters and H i l bo rn (1976) j u s t i f y the assumption tha t v ( t ) i s normal ly d i s t r i b u t e d by not ing tha t " e xp (v ( t ) ) can be viewed as a random s u r v i v a l f a c t o r r e s u l t i n g from severa l independent and m u l t i p l i c a t i v e environmental f a c t o r s ope ra t i ng i n s e r i e s . Thus v ( t ) represents the sum o f severa l random f a c t o r s and should be normal ly d i s t r i b u t e d by the cen-t r a l l i m i t theorem." Peterman (1979) presents emp i r i ca l evidence tha t v ( t ) i s normal f o r severa l salmon s t o ck s . Given the management ob j e c t i v e of maximiz ing the sum o f d iscounted catches over t ime , i . e . , 27 max ? C( t ) • [ l / U + d ) ] * t=0 where d i s the d i scount ra te per gene ra t i on , the opt imal c on t r o l p o l i c y can be shown to be a f i x e d escapement p o l i c y ( C l a r k , 1976). Th is means tha t the opt imal con t ro l a c t i o n a t any time t i s to a l l ow tha t catch from the rec ru i tment which a l lows the optimum number o f spawners S* to escape the f i s h e r y . Thus the opt imal con t ro l p o l i c y i s : C( t ) = R(t) - S* (R( t ) > S*) = 0 (R( t ) < S*) The determinat ion o f the optimum escapement S* i s s t r a i gh t f o rwa rd g iven the parameters a and 3 o f the model. I t i nvo lves the s o l u t i o n of the t ranscendenta l equat ion: (1 - 3 • S*) exp(a -r 3 • S*) = 1+d I t should be po inted out tha t t h i s i s s t r i c t l y c o r r e c t on ly i n the case tha t ay2 - 0 (Ludwig, 1979a). In the problem we are c on s i d e r i n g , however, the parameters o f the stock are not known. Observat ions o f the rec ru i tment are assumed to be a v a i l a b l e , w i th the observa t i on model be ing: R o b s ( t ) = R(t) exp(w(t)) (3 .0-2) where w i s the observa t i on "no i se " or e r r o r and i s assumed to be zero 2 mean w i th var iance a , . With past observa t ions o f rec ru i tment and know-w ledge of past escapements, es t imates o f the parameters o f the stock can be obta ined by us ing a s u i t a b l e curve f i t t i n g procedure ( s e c t i on 3 .1 ) . 28 The problem addressed i n t h i s chapter i s to what ex tent s t a t i s t i c a l un-c e r t a i n t y i n such est imates should be cons idered i n dec id ing on con t ro l p o l i c i e s f o r the management o f t h i s hypo the t i ca l s tock . 3.1 PARAMETER ESTIMATION I t has been shown (Dahlberg, 1973) how the parameter e s t ima t i on problem f o r the R i cke r model can be cas t i n the form of a l i n e a r r eg re s -s i o n . Combining (3.0-1) and ( 3 . 0 - 2 ) : R o b s ( t + l ) = S ( t ) e x p(a-6 • S ( t ) + v ( t ) + w( t+ l ) ) Th is can be r ew r i t t en l n ( R o b s ( t + l ) / S ( t ) ) = a - e • S ( t ) + v ( t ) + w(t+l) . which i s a l i n e a r reg ress i on of the form y-j = 1 + e i (3 .1-1) where y. = l n ( R o b s ( t + l ) / S ( t ) ) , x J = [1 - S ( t ) ] and a = [a g ] T . With the assumptions tha t have been made about the no ise processes , the e r r o r term e^  i n the reg ress i on i s o f the c o r r e c t form, zero mean w i th var iance 2 2 2 2 a = a + a . The o v e r a l l va r i ance a can be est imated from examining e v w e a r e s i d ua l s i n reg ress ions of ln (R/S) aga ins t S from e x i s t i n g s tock -rec ru i tment da ta . However the p a r t i t i o n i n g o f the var iance between the environmental and observa t i on no ise processes i s more d i f f i c u l t . Prov ided tha t the matr ix x^x^ i s non-s ingu la r ( tha t i s , the elements of _x must be l i n e a r l y independent) then the l e a s t squares para-29 meter es t imates w i th k observat ions f o r the l i n e a r r eg ress i on model (3 .1-1) are g iven by k T -1 k where P. = [ £ x_. x_. ] and b. = Z x^ . y . . The e r r o r covar iance i= l 1 1 i = l 1 1 matr i x o f the es t imates i s P*k = % P k k k k 2 Let z n = Z S . , z 9 = Z 1 n ( R . . 1 / S . ) J z Q = Z S . and k 1 i= l 1 L i = l 1 + 1 1 6 i = l 1 z 4 = _Z S. • l n ( R . + 1 / S . . ) . Let d = k • z 3 - z^ . Then the R i cke r para-meter es t imates and the elements o f the e r r o r covar iance mat r i x are g iven by a = ( z 3 • z 2 - z 1 • z 4 ) / d 3 = ( z x • z 2 - k • z 4 ) / d ? 2 = a ' ; • z Q / d a: e 3 2 0 o = 0 „ - z , / d a 3 e 1 2 2 . . , a B = 0 e • k/d Expressed i n t h i s way, the e f f e c t o f the data on the parameter un-c e r t a i n t y i s not obv ious . However the elements of P*^  can a l s o be expressed in terms o f three f a ce t s o f the da ta : the mean escapement S, 2 the var iance of the escapement S<. , and the number of data po in t s k. Not ing tha t = k • S, and z 3 = (k-1) • S $ 2 + k • S 2 , the parameter u n c e r t a i n t i e s can be r ew r i t t en as: 30 a .2 a a 2 e S 2 a a e 2 1 2 a (k-1) s 2 s • a e These formulae seem i n t u i t i v e l y reasonab le . The unce r t a i n t y i n the a parameter w i l l be lowest when the mean of the observed escapements i s low, s ince a i s the i n t e r c ep t of the r eg ress i on at zero escapement. S ince -6 i s the s lope o f the r e g r e s s i o n , the unce r t a i n t y in the 3 pa ra -meter does not depend on the mean escapement but i s reduced as the var iance in the escapement i n c r ea se s . As expected, unce r t a i n t y decreases as the number of data po in t s inc reases and as the r eg ress i on e r r o r va r i ance (equ iva l en t to var iance i n environmental no ise and obse rva t i on no i se) decreases. The l e a s t squares s o l u t i o n presented above requ i r e s a l l the past observa t i ons and va lues o f independent v a r i a b l e s . The s o l u t i o n can a l s o be presented in a r e cu r s i v e format (Young, 1974). Wal ters and H i l bo rn (1976) have shown how t h i s can be app l i ed to parameter e s t ima t i on f o r the R i cker model. The bene f i t o f t h i s approach i s t h a t , as each new observa-t i o n i s made, the new parameter es t imates and covar iance mat r i x can be obta ined d i r e c t l y from the prev ious es t imates and cova r i ance . A l l the in fo rmat ion from the past data i s c o l l a p sed i n t o the i n fo rmat i on s t a t e 31 P*^)- This approach i s summarized in Table I . The r e cu r s i v e approach to parameter e s t ima t i on w i l l be requ i red f o r a p p l i c a t i o n o f the wide sense adapt ive dual con t ro l a l g o r i t hm , s ince i t s p e c i f i e s the way i n which the e r r o r covar iance mat r i x (the unce r t a i n t y about the parameters) w i l l change over t ime in r e l a t i o n to po s s i b l e fu tu re con t ro l a c t i ons (escapements) a long the nominal t r a j e c t o r y d i s c u s s -ed in s e c t i on 2.8. The r e cu r s i v e r eg ress i on equat ions a r e , i n f a c t , i d e n t i c a l to the Kalman f i l t e r equat ions when the f i l t e r i s i n i t i a l i z e d w i th a d i f f u s e p r i o r . The equat ions generate r e cu r s i v e maximum l i k e l i h o o d e s t ima tes , which are i d e n t i c a l to l e a s t squares est imates when normal e r r o r s are assumed. 3.2 RANGE OF CONTROL POLICIES TESTED The range o f p o l i c i e s to be app l i ed to the management o f the s imulated salmon stock inc ludes the f o l l o w i n g : 1. The opt imal p o l i c y (OP) where the t rue parameters o f the stock are known. The performance o f t h i s p o l i c y serves as an upper bound on the performance of any p o l i c y which might be t e s t e d . 2. An a c t i v e adapt ive p o l i c y (AAP) based on the wide sense adap-t i v e dual con t ro l a l go r i t hm descr ibed in s e c t i on 2.8. The con t ro l a c t i on chosen at each stage i s a f un c t i on both o f the parameter es t imates and of the unce r t a i n t y in those e s t ima te s , as measured i n the e r r o r covar iance ma t r i x . The fo rmu la t i on requ i red f o r the a p p l i c a t i o n o f t h i s p o l i c y i s presented in the next s e c t i o n . 3. Var ious types o f pass ive adapt ive p o l i c i e s (PAP). These are 32 Given a l i n e a r r eg ress i on o f the form yn- = x.'.a + en- i = 1 , 2 , - — , k where en- ~ N(0, ? e 2 ) , the r e cu r s i v e l e a s t squares r eg res s i on a l go r i t hm f o r the parameter vec to r a i s as f o l l o w s : Ik= § k _i - - k k { * k ik-i - y k } P k = p k - l " -kk h ' P k - 1 where P i s the parameter e r r o r covar iance ma t r i x . a Q and P Q must be s p e c i f i e d . Tab le I . Recurs ive l e a s t squares r eg ress i on a l g o r i t hm . 33 open-loop feedback p o l i c i e s and have the property tha t the con t ro l a c t i o n chosen a t any stage i s a f un c t i on on ly o f the a v a i l a b l e parameter e s t i -mates. The con t ro l chosen i s the opt imal d e t e r m i n i s t i c c o n t r o l , tha t i s , i t i s assumed tha t the parameter es t imates a v a i l a b l e a t any time rep res -ent the t rue parameters o f the stock and the s t o c h a s t i c v a r i a t i o n v ( t ) can be ignored . These p o l i c i e s a l l make., use of i n fo rmat ion which becomes a v a i l a b l e as the con t ro l process proceeds, but vary in the way in which a v a i l a b l e data i s used to update parameter es t imates . S p e c i f i c a l l y , they vary accord ing t o : a. the number o f data po in ts which are a l lowed to accumulate between parameter est imate updates; and b. the extent to which each, new data po in t on the s to ck - rec ru i tmen t r e l a t i o n s h i p . i s g iven i t s due weight i n r e v i s i n g the parameter e s t ima tes . Th is i s v a r i ed by i n t r oduc i ng a m u l t i p l i c a t i v e f a c t o r GF to the f i l t e r g a i n , 0 < GF < 1. The lower the va lue o f GF, the l e s s weight i s g iven to each new data po i n t . In the s eque l , un less o therwise s t a t e d , a pass ive adapt ive p o l i c y w i l l r e f e r to a p o l i c y which chooses the opt imal d e t e r m i n i s t i c con t ro l a t each s tage , and which updates parameter es t imates a t each stage g i v i n g f u l l weight to the new i n f o rma t i on . 4. A non-adapt ive p o l i c y (NAP). Th is i s an open-loop p o l i c y , tha t i s , i t makes no use o f r e a l - t ime in fo rmat ion and chooses the same escapement each year based on the i n i t i a l parameter e s t ima tes . (Note tha t t h i s p o l i c y i s equ i va l en t to the pass ive adapt ive p o l i c y w i th GF = 0 or w i th the number o f years between parameter es t imate updates being i n f i n i t e ) . 34 3.3 ACTIVE ADAPTIVE POLICY FORMULATION To develop an a c t i v e adapt ive p o l i c y based on the wide sense adap-t i v e dual con t ro l a l g o r i t hm , the problem must be formulated in the form presented i n se c t i on 2 .1 . Th is r equ i r e s s p e c i f i c a t i o n o f the s t a t e and obse rva t i on equat ion and a s u i t a b l e performance measure. Let C ( t ) = H(t) • R ( t ) , t ha t i s , H( t ) i s the harvest r a t e or p ropor t i on o f the rec ru i tment caught. Then (3.0-1) can be r ew r i t t e n as Augmenting the s t a t e as d i scussed in s e c t i on 2.2 and r ede f i n i n g the c o n t r o l , l e t x ^ k ) = I n R ( t ) , x 2 ( k ) = a, x 3 ( k ) = 3 and u(k) = l n ( l - H ( t ) ) . Then the s t a t e equat ions become For the obse rva t i on equa t i on , l e t y ( k ) = l n R ( t ) . Then from (3.0-2) In R(t+1) = l n S ( t ) + a - 3 S ( t ) + v ( t ) = In [ R ( t ) ( l - H ( t ) ) ] + a - 3 [R(t) ( l - H ( t ) ) ] + v ( t ) x ^ k + l ) x 2 ( k + l ) x 3 ( k + l ) X j ( k ) + u ( k ) + x 2 ( k ) - x 3 ( k ) e x p ( X l ( k ) + u ( k ) ) + v ( k ) x 2 ( k ) (3 .3-1) x 3 ( k ) y (k ) = X l ( k ) + w(k) (3 .3-2) The performance measure to be minimized was o f the form N-l J = E{L M [x (N)] + I L. [ x ( k ) , u ( k ) , k]} IN k=0 K 35 Noting tha t C( t ) = R( t ) • H(t) = exp( X l ( l<) ) • [1 - exp(u(k) ) ] the performance measure f o r t h i s problem can be s p e c i f i e d as L N [ x (N) ] = I j E x ^ N ) - 1 2 ] 2 (3 .3-3) L k [ x ( k ) , u ( k ) , k] = - e x p ( x 1 ( k ) ) [ l - e x p ( u ( k ) ) ] [ l / ( l + d ) ] k where 1^  i s a parameter tha t s p e c i f i e s the importance of being near a. f i n a l va lue 1,, at the end time N. The con t ro l a c t i on a v a i l a b l e at each stage i s the catch C( t ) or harvest r a te H ( t ) , 0 < H(t) < 1. For the con t ro l a c t i on u(k) de f ined i n t h i s f o rmu l a t i o n , the set o f con t ro l op t ions i s {u (k ) / - » x u(k) < 0} (3.3-4) With the fo rmu la t i on s p e c i f i e d in (3.3-1) to (3.3-4) the a c t i v e adapt ive p o l i c y can be implemented. 3.4 ACTIVE ADAPTIVE CONTROL LAW Although i t i s not po s s i b l e to ob ta in the con t ro l law f o r a l l pos-s i b l e combinat ions of parameter es t imates and covar iance ma t r i c e s , due to the curse o f d imen s i o na l i t y , i t i s of i n t e r e s t to determine the nature of the a c t i v e adapt ive con t ro l f o r a subset o f the po s s i b l e i n fo rmat ion s t a t e s . In p a r t i c u l a r , i t i s of i n t e r e s t of know how the con t ro l should change i n r e l a t i o n to the unce r t a i n t y in the parameter e s t ima tes . I t was shown i n s e c t i on 3.1 tha t t h i s unce r t a i n t y can be represented by three 36 f a ce t s o f the p r i o r data — the mean escapement S, the var iance i n the 2 escapement S<* and the number o f data po in ts k. Th is s e c t i on w i l l show how the con t ro l a c t i on should change i n r e l a t i o n to these three p rope r t i e s o f the da ta . A number o f sets o f s t ock - rec ru i tmen t data were generated by s i m u l a t i o n , each set being s p e c i f i e d by the mean and var iance of the escapement and by the number o f data po i n t s . The same random set of environmental and obse rva t i ona l no ise terms was used f o r each se t of da ta . The parameter va lues and no ise va r i ance l e v e l s used to generate the nominal set o f data were based on va lues obta ined from s tock - rec ru i tmen t data f o r Skeena R iver sockeye salmon (F. Wong, pers . comm.). From these data s e t s , parameter est imates and e r r o r covar iance mat r i ces were obta ined us ing the batch l e a s t squares e s t ima t i on procedure descr ibed i n se c t i on 3 .1 . However these est imates and covar iance matr i ces cou ld not be used d i r e c t l y i n the dual con t ro l a l g o r i t hm , s ince t h i s r equ i res s t a t e es t imates and e r r o r covar iance matr i ces f o r the s t a t e de f ined i n se c t i on 3 .3 . A method was requ i red to t rans form the d i s t r i b u -t i o n of parameter es t imates i n t o a d i s t r i b u t i o n of s ta te es t imates f o r the augmented s t a t e . The method f o r t h i s t rans fo rmat ion i s presented i n Appendix I . From the s t a t e est imate and e r r o r covar iance obta ined i n t h i s way, the a c t i v e adapt ive con t ro l a c t i on can be c a l c u l a t e d . The r e s u l t s are shown i n F igure 1 and F igure 2, which p l o t the percent inc rease or decrease i n escapement chosen by the a c t i v e adapt ive p o l i c y r e l a t i v e to tha t which would be chosen by the pass ive adapt ive p o l i c y f o r the same parameter e s t ima tes . 37 F igure 1. A c t i v e adapt ive con t ro l law: e f f e c t o f the number o f years o f i n i t i a l data and p r o d u c t i v i t y of the s tock . Numbers represent percent inc rease or decrease i n escapement chosen by the a c t i v e adapt ive p o l i c y r e l a t i v e to tha t which would be chosen by the pass ive adapt ive p o l i c y , a) Nominal' c a se : . t en 2 years i n i t i a l d a t a , a = 2.0, d i scount ra te = 0%, a = 0.04, 2 v a = 0.04; b) twenty years i n i t i a l da ta ; c) a = 1.0; w d) a = 3.0 . S = mean and S<. = standard d ev i a t i o n o f the p r i o r escapements. 38 -•»"! -3 8 -3 0 -2 3 -1 8 -1 It -11 -9 -it 5 -3 8 -29 -2 2 -17 -1 3 -1 1 -9 -it6 -37 -2 8 -2 1 -1 6 -1 2 -10 -8 -t* 8 -37 -2 e -2 1 -15 -1 1 -9 -7 -It9 -3 8 -29 -22 -1 5 -1 1 -8 -6 1 0 0 -"tt -3 5 -2 5 -1 5 -9 -5 -•t 1 0 0 1 0 0 -it7 -2 8 -8 -2 -1 -0 1 0 0 1 0 0 81 1*7 1 6 7 it 2 1 0 0 1 0 0 lt5 2 it 1 3 8 1 0 0 9 it 2 9 a) 0-10 -3 8 -2 6 -1 7 - 1 1 -8 -6 -5 -„ -3 8 -2 5 -1 6 -1 1 - 8 - 5 - "t - 3 -3 8 - 2 * - 1 5 -1 0 -7 -5 - u - 3 -3 8 -2 3 -1 it -9 -6 -•t - 3 -2 -3 9 -2 It -1 "t -7 - it - 3 -2 -2 - U l -2 8 -1 3 - 5 -2 -2 - 1 - 1 -5 1 -36 -6 -0 -0 -0 -0 - 1 1 0 0 6 5 1 5 5 2 1 0 0 1 0 0 3 0 1 2 5 2 1 1 0 0 1 5 2 b) 0.10 -52 -"t8 -itO - 3 3 -27 -22 -1 8 -1 6 -5 3 -<t7 -39 -31 -2 5 -2 0 -17 -1 it -53 -"16 -38 -30 -2 it -2 0 -1 6 -1 3 1 0 0 -it6 -3 8 -3 1 -2 5 -20 -1 5 -12 1 0 0 10 0 -it 3 -36 -2 8 -2 1 - 1 "t -1 1 1 0 0 1 0 0 1 0 0 87 - 3 5 -2 1 -1 1 -7 1 0 0 1 0 0 1 0 0 1 0 0 69 1 8 it 2 1 0 0 1 0 0 10 0 89 6 1 3 5 1 8 9 1 00 1 0 0 1 0 0 6 8 •tit 2 8 1 0 0 1 0 0 1 0 0 -37 -3 1 -2 it -1 6 -1 it - 1 1 -9 -7 -3 8 -31 -2 3 - 1 7 -1 3 -1 0 -8 -7 -it 0 -3 1 -2 3 -17 -1 2 -9 -7 -6 -"t2 -31 -2 2 - 1 6 -12 -9 -7 -5 -it it -3 1- -2 2 - 1 5 -1 1 - 8 -6 -It -•t 6 -3 3 -2 3 -1 6 - 1 0 -6 - it -3 -50 - t o -2 8 -15 -7 - 3 -2 -1 1 0 0 1 0 0 -30 it 2 1 1 1 1 0 0 7 5 3 3 1 it 7 it 1 0 0 . it 0 1 6 c) d) 39 The r e s u l t s f o r the nominal case shown i n F igure l a seem i n t u i t i v e l y reasonab le . I f the mean escapement has been low and the var iance in escapement s m a l l , the best a c t i on i s to a l l ow a high escapement. I f the mean escapement has been h i gh , a low escapement should be chosen. At in te rmed ia te l e v e l s o f past escapement, e i t h e r a very high or very low escapement should be chosen. In a l l cases where the v a r i a b i l i t y i n escape-ment has been low, the aim i s to choose an escapement we l l ou t s i de the range o f p r ev i ou s l y observed escapements. As the v a r i a b i l i t y i n escape-ment i n c r ea se s , so the amount of probing by the a c t i v e adapt ive p o l i c y can be seen to decrease. The e f f e c t of i n c r eas i ng the number of data po in ts from 10 to 20 i s shown i n F igure l b . Th is o f course w i l l reduce the i n i t i a l unce r t a i n t y about the parameters and the con t ro l law responds by reduc ing the pe r t u r -bat ion to the escapement. However the pe r tu rba t i on i s s t i l l s ub s t an t i a l i f the i n i t i a l v a r i a b i l i t y i n escapement i s low. The e f f e c t of the apparent p r o d u c t i v i t y o f the stock i s shown i n F igures l c and Id . In both cases the covar iance matr i ces are the same as i n the nominal case , but the est imates of the a parameter (the stock p r o d u c t i v i t y ) are lower and h igher r e s p e c t i v e l y . The r a t i o a/g o f the parameter es t imates i s the same as i n the nominal ca se , so the e q u i l i b r i u m stock s i z e s are the same. The r e s u l t s seem to i n d i c a t e tha t i f the stock i s thought to be of low p r o d u c t i v i t y then the response of the stock at high l e v e l s of escapement should be exp l o r ed , un less the mean escapement has been very high r e l a t i v e to the est imated e q u i l i b r i u m stock s i z e . On the other hand i f the p r o d u c t i v i t y i s apparent ly high (F igure Id) the best s t r a t egy i s to t r y to improve the est imate of t h i s p r o d u c t i v i t y . 40 F igure 2. A c t i v e adapt ive con t ro l law: e f f e c t o f d i scount ra te and observa t i on no ise va r i ance . Numbers represent percent inc rease or decrease in escapement chosen by the a c t i v e adapt ive p o l i c y r e l a t i v e to tha t which would be chosen by the pass ive adapt ive p o l i c y , a) Nominal case: ten years i n i t i a l d a t a , a = 2 .0 , d i s count ra te = 0%, a 2 = 0.04, a 2 =0.04; V w b) d i scount ra te = 10%; c) a 2 = 0.16; d) a 2 = 0.36. "S = ' ' w ' w mean and S<- = standard dev i a t i o n o f the p r i o r escapements. 41 - I t * - 3 8 - 3 0 - 2 3 - 1 8 - U - 1 1 - 9 - 4 5 - 3 8 - 2 9 - 2 2 - 1 7 - 1 3 - 1 1 - 9 - 4 6 - 3 7 - 2 8 - 2 1 - 1 6 - 1 2 - 1 0 - 8 - 4 8 - 3 7 - 2 8 - 2 1 - 1 5 - 1 1 - 9 - 7 - 4 9 - 3 8 - 2 9 - 2 2 - 1 5 " 1 1 - 8 - 6 1 0 0 - 4 4 - 3 5 - 2 5 - 1 5 - 9 - 5 - 4 1 0 0 1 0 0 - 1 7 - 2 8 - 8 - 2 - 1 - o 1 0 0 1 0 0 8 1 4 7 1 6 7 4 2 1 0 0 1 0 0 4 5 2 4 1 3 8 1 0 0 9 it 2 9 0 . 1 0 - 4 4 - 3 8 - 3 0 • - 2 3 - 1 8 - 1 t - 1 1 - 9 - i. s - 3 7 - 2 8 - 2 1 - 1 6 - 1 3 - 1 0 - e - 4 6 - 3 7 - 2 7 - 2 0 - 1 5 - 1 1 - 9 - 7 - 4 7 - 3 6 - 2 6 - 1 9 - 1 it - 1 0 - 8 - 6 - " . 8 - 3 6 - 2 6 - 1 8 - 1 3 - 9 - 6 - 5 - 5 0 - 3 9 - 2 9 - 1 9 - 1 1 - 6 -"t - 3 1 0 0 1 0 0 - 3 7 " 1 7 - I. - 2 - 1 - o 1 0 0 1 0 0 6 6 2 9 1 0 5 3 1 1 0 0 9 1 3 e 2 0 1 1 6 1 0 0 8 4 2 6 0 . 1 0 a) b) - 4 4 - 3 8 - 3 0 - 2 3 - 1 7 - 1 2 - 9 - 7 - 4 5 - 3 8 - 2 9 - 2 1 - 1 6 - 1 1 - 8 - 6 - 4 7 - 3 8 - 2 9 - 2 0 - 1 it - 1 0 - 7 - 5 -it 8 - 3 9 - 2 9 - 1 9 - 1 2 - 8 _ c - 3 - 5 1 - 4 2 - 3 0 - 1 8 - 1 0 - 5 - 3 - 1 1 0 0 -it 8 - 3 3 - 1 5 - 6 - 2 0 1 1 0 0 1 0 0 . 3 it 0 It 3 3 2 ; 1 0 0 1 0 0 8 2 2 8 1 2 7 5 1 0 0 1 0 0 1.7 2 3 1 3 8 1 0 0 9 it 3 3 - i t it - 3 8 - 2 9 - 2 0 - 1 4 - 9 - 5 - 3 - 4 5 - 3 8 - 2 8 - 1 8 - 1 2 - 7 - 4 - 2 - i t 7 - 3 9 - 2 7 - 1 7 - 9 - 5 " 2 . - 0 - 4 9 -•t 0 - 2 6 - 1 4 - 7 - 2 - o 1 1 0 0 - • t 2 - 2 5 - 1 0 - 3 0 2 3 1 0 0 - 4 8 - 2 1 - 3 2 4 4 4 1 0 0 1 0 0 7 2 1 1 9 7 6 6 1 0 0 1 0 0 6 6 2 2 1 3 9 7 6 1 0 0 1 0 0 4 5 2 2 1 4 1 0 1 0 0 9 1 3 5 D) 42 The e f f e c t of the d i scount r a t e i n choosing con t ro l a c t i ons i s shown in F igure 2b. The form of the con t ro l law i n r e l a t i o n to the mean and var iance of the past escapements i s not very d i f f e r e n t from the nominal (F igure 2a) w i th no d i scount r a t e . The ex tent to which the escapement i s perturbed from the pass ive adapt ive con t ro l i s reduced s l i g h t l y , and there i s some i n d i c a t i o n tha t the con t ro l i s l e s s w i l l i n g to exp lo re i n the d i r e c t i o n of h igher escapements. These w i l l i n vo l ve low catches which w i l l be more heav i l y pena l i zed when d i s coun t i ng i s i n e f f e c t . The e f f e c t of the est imated l e v e l o f observa t i on no ise var iance i s shown i n F igures 2c and 2d. The s e n s i t i v i t y o f the con t ro l to t h i s l e ve l i s important to determine, s ince i t i s assumed tha t the p r o b a b i l i t y d i s t r i b u t i o n s o f the no ise processes are known, when i n f a c t they must be est imated and may be in e r r o r . The form of the con t ro l law seems to be r e l a t i v e l y i n s e n s i t i v e to v a r i a t i o n s in the assumed no ise var iance l e v e l s . S ince environmental and obse rva t i ona l no ise are i n d i s t i n g u i s h a b l e i n the s to ck - rec ru i tmen t r e l a t i o n s h i p , the con t ro l should a l so be i n s e n s i t i v e to assumptions about the l e ve l o f environmental no ise va r i ance . I t should not matter how the o v e r a l l var iance i s p a r t i t i o n e d between the two sources . Before proceeding to an eva lua t i on of the performance of the three types of p o l i c i e s , i t i s of i n t e r e s t to note how the a c t i v e adapt ive p o l i c y r e s u l t s i n a t r a j e c t o r y through (S , S<.) space. Two such t r a j e c t o r i e s are shown i n F igure 3 s t a r t i n g from the same i n i t i a l v a r i a b i l i t y i n escapement a t two d i f f e r e n t l e v e l s of mean escapement. At each po in t "sampled" by the a c t i v e adapt ive p o l i c y , the percent inc rease or decrease i n escapement r e l a t i v e to the pass ive adapt ive escapement i s p l o t t e d . 43 F igure 3. A c t i v e adapt ive t r a j e c t o r i e s i n the in fo rmat ion s t a t e space. Numbers bes ide po in ts on t r a j e c t o r i e s represent percent i nc rease or decrease i n escapement chosen by the a c t i v e adapt ive p o l i c y r e l a t i v e to t ha t which would be chosen by the pass ive adapt ive p o l i c y . 45 For t r a j e c t o r y A, s t a r t i n g from low mean p r i o r escapement, the i n i t i a l con t ro l cho ice i s to inc rease the escapement r e s u l t i n g i n a s l i g h t inc rease in S but a s ub s t an t i a l inc rease i n S<~. The subsequent increase i n escapement causes another inc rease i n S and S^, b r i ng i ng the t r a j e c t o r y i n to the reg ion where decreases i n escapement are chosen. However by t h i s t ime the v a r a b i l i t y i n escapement i s great enough tha t on ly smal l pe r tu rba t i ons are i nd i c a t ed and the a c t i v e adapt ive p o l i c y e s s e n t i a l l y becomes pas s i ve . For case B, where the mean p r i o r escapement has been h ighe r , the t rend i s towards decreas ing S and i n c r ea s i ng S^- In t h i s case the decrease i n S i s not r ap id enough to enter a reg ion where the d i r e c t i o n of pe r tu rba t i on i s reversed before the v a r i a b i l i t y i n escapement has inc reased to an extent where f u r t h e r pe r tu rba t i ons are not cons idered wor thwh i le . 3.5 METHOD OF POLICY EVALUATION The aim o f t h i s study i s to compare p o l i c i e s , based on a range o f adapt ive s t r a t e g i e s , f o r t h e i r e f f e c t i v ene s s in managing a resource w i th unce r ta in dynamics. The method f o r eva l ua t i on o f p o l i c y performance i s as f o l l o w s . An i n i t i a l d i s t r i b u t i o n o f the parameter est imates i s s p e c i f i e d . - - T that i s , the vec to r o f the est imates themselves [a g] and the assoc i a ted 46 e r r o r covar iance mat r i x P . From t h i s i n i t i a l d i s t r i b u t i o n , a c tua l pa ra -meter va lues are chosen at random and these become the t rue parameter values f o r a p a r t i c u l a r s i m u l a t i o n . From these same i n i t i a l c o n d i t i o n s , and us ing the same se t o f r an -dom environmental and obse rva t i ona l no ise p e r t u r ba t i o n s , each p o l i c y i s app l i ed to the management of the " s tock" and the f i s h e r y i s s imulated over a number o f y e a r s . Where appropr i a te to the p o l i c y , parameter es t imates are updated us ing the r e cu r s i v e l e a s t squares es t imato r d i scussed in s e c t i on 3 .1 . Each p o l i c y ' s con t ro l op t i on i s to choose an escapement a t each s tage , and a record i s kept o f the cumulat ive d iscounted ca tch f o r each p o l i c y . Two performance measures are obta ined f o r each p o l i c y f o r each s i m u l a t i o n . The f i r s t measure, PM<-, i s the sum of the d iscounted catches over 20 generat ions o f the s tock , p lus the d iscounted f i n a l s tock s i z e . The second measure, PM L , i s the sum of the d iscounted catches over the 20 generat ions p lus the expected f u tu re va lue of the ca tch which would be obta ined a f t e r the f i n a l year of s i m u l a t i o n . Th is expected f u t u r e va lue o f catch i s c a l c u l a t e d by assuming tha t the parameter es t imates obta ined by the end of the s imu l a t i on per iod are used to determine the con t ro l p o l i c y from tha t time on w i th no f u r t h e r l e a rn i ng ( i . e . , an open-loop p o l i c y a f t e r the 20th gene ra t i on ) . Where the d i scount ra te i s z e r o , a t ime hor i zon o f 50 generat ions i s s e t . The va lues f o r PM<- and PM^ can be cons idered as measures of the shor t - te rm and long-term performance of a p a r t i c u l a r p o l i c y . R e l a t i v e measures of the performance of each of the adapt ive p o l i c i e s were obta ined by express ing PNL and PM, as a percentage of the corresponding 47 values f o r the opt imal p o l i c y w i th known parameters. These r e l a t i v e mea-sures are denoted P0<. and PO^. The expected value o f a p o l i c y f o r a p a r t i c u l a r i n i t i a l d i s t r i b u t i o n of parameter es t imates i s obta ined by s imu l a t i ng a l a r ge number o f t imes , each t ime choos ing the t rue parameter va lues f o r the s imu l a t i o n a t random from the p r i o r d i s t r i b u t i o n o f the es t ima tes . 50 s imu la t i ons were deemed s u f f i c i e n t to determine the expected va l ue . The expected va lues f o r a p o l i c y are denoted EV<, and EV^ and the expected va lues as a percentage of the expected va lue o f the opt imal p o l i c y are denoted E0 C and E0, . 3.6 POLICY COMPARISON: EFFECT OF PRIOR DATA The r e l a t i v e performances o f the three bas i c types of adapt i ve p o l i c y - a c t i v e adap t i v e , pass ive adapt ive and non-adapt ive - were compared across a range o f d i s t r i b u t i o n s of i n i t i a l parameter e s t ima te s . The range of d i s t r i b u t i o n s t es ted was determined by the mean and va r i ance o f the escapement i n the data a v a i l a b l e p r i o r to commencing implementat ion of each p o l i c y . The i n i t i a l data sets were generated in the same fash ion as descr ibed in s e c t i on 3.4. The major r e s u l t f o r p o l i c y a n a l y s i s o f the R i cke r model i s summari-zed in F igure 4. Th is shows the r e l a t i v e expected long-term performance E0^ of the three p o l i c i e s across a range of mean p r i o r escapements and f o r three d i f f e r e n t l e v e l s of p r i o r v a r i a b i l i t y i n escapement. I t can be seen tha t the mean o f the p r i o r escapements has a l a rge e f f e c t on the r e l a t i v e performances o f the three p o l i c i e s . In each case , the performance o f the non-adapt ive p o l i c y improves s t e a d i l y as the mean 48 F igure 4. Comparison of p o l i c y performances r e l a t i v e to the mean and standard d ev i a t i o n o f . p r i o r escapements. 50 p r i o r escapement i n c r eases . The pass ive adapt ive p o l i c y performs very we l l f o r very low and medium to high p r i o r escapements, but performs poor ly at medium to low l e v e l s of escapement. The a c t i v e adapt ive p o l i c y does we l l across the whole range o f p r i o r escapements. The e s t ima t i on performance o f the three p o l i c i e s , as measured by the square root o f the mean squared e r r o r i n the est imates of the a and 3 parameters as we l l as in the est imate of the optimum escapement S*, i s shown in F igures 5, 6 and 7: The e r r o r s in the est imates are measured a t the end of each 20 generat ion s i m u l a t i o n , and the mean i s over the 50 s imu la t i ons f o r each p r i o r data s e t . Both the a c t i v e and pass ive adapt ive p o l i c i e s es t imate the a pa ra -meter q u i t e ; we l l (F igure 5 ) , though the pass ive adapt ive e s t ima t i on i s worst a t medium-low l e v e l s o f p r i o r escapement. The i n c r ea s i ng e r r o r i n the a es t imate w i th the non-adapt ive p o l i c y s imply r e f l e c t s the i n c reas i ng unce r t a i n t y about the a parameter as the mean p r i o r escapement inc reases ( se c t i on 3 .1 ) . The e r r o r i n the est imate o f the 6 parameter (F igure 6) f o r the non-adapt ive p o l i c y i s independent o f the mean p r i o r escapement but reduces as the v a r i a b i l i t y i n p r i o r escapement i n c r ea se s , again r e f l e c t i n g the p rope r t i e s of unce r t a i n t y in 6 ( s e c t i on 3 . 1 ) . The poor es t imate of the 6 parameter at medium-low l e v e l s of p r i o r escapement w i th the pass ive adap-t i v e p o l i c y accounts f o r i t s poor est imate o f S*.at t ha t l e v e l , which in tu rn accounts f o r i t s poor performance. The performance of the three p o l i c i e s corresponds c l o s e l y to the es t ima t i on performance in determin ing S*, as i t should do. 51 F igure 5. Comparison of e s t ima t i on performances f o r the a parameter. S = mean and S<.= standard dev i a t i o n of p r i o r escapements. 52 53 F igure 6. Comparison of e s t ima t i on performances f o r the 6 parameter. S = mean and S = standard dev i a t i o n of p r i o r escapements. V 55 F igure 7. Comparison of e s t ima t i on performances f o r the optimum escapement S*. S = mean and = standard dev i a t i o n of p r i o r escapements. 57 The exp lana t i on of the r e s u l t s summarized in F igures 4 to 7 i s s t r a i g h t f o rwa r d . The improved performance of the non-adapt ive p o l i c y as the mean escapement inc reases i s due to the f a c t tha t (1) the est imate of the e q u i l i b r i u m stock s i z e a / 3 improves as the mean escapement • -> i n c r ea se s , and (2) the optimum escapement S* i s r e l a t i v e l y i n s e n s i t i v e to the p a r t i c u l a r l e v e l s o f a and g prov ided t h e i r r a t i o a / 3 i s known. The est imates of both a and 3 are very poor f o r the non-adapt ive p o l i c y a t high mean escapement, ye t the es t imate of S* and the r e s u l t i n g p e r f o r -mance are good. The good performance of the pass ive adapt ive p o l i c y f o r both low and high p r i o r escapements i s due to the f a c t t h a t , by f o l l ow i ng the d e t e r m i n i s t i c p o l i c y , the escapement chosen w i l l be we l l ou t s i de the range o f observed p r i o r escapements. Only when the d e t e r m i n i s t i c escape-ment f a l l s w i t h i n the range of the p r i o r escapements w i l l t h i s s e l f -generat ing pe r tu rba t i on by the pass ive adapt ive p o l i c y f a i l to opera te . Th is po in t i s exemp l i f i ed i n F igure 8- which p l o t s the s t o c k - r e c r u i t -ment data f o r the three cases S" = 0 . 1 , 0 .3 , 0.5 f o r S<~ = 0.05.' The s o l i d l i n e s through the data sets represent the best f i t o f the R i cker model and the broken l i n e s represent a l t e r n a t i v e po s s i b l e forms of the model P A P which f i t the data almost equa l l y w e l l . The escapement S which would be chosen by the pass ive adapt ive p o l i c y based on the best es t imate i s p l o t t ed in each case . I t i s ev ident tha t on ly in the case S" = 0.3 does t h i s escapement f a l l w i t h i n the p r i o r range o f escapements. The escape-ments chosen in the other two cases would stand a good change of d i s c r i -minat ing between the a l t e r n a t i v e model forms. 58 F igure 8. Escapements chosen in r e l a t i o n to p r i o r da ta . S o l i d curve through each data se t i s l i n e o f best f i t . Broken l i n e s represent a l t e r n a t i v e l i n e s o f good f i t . V e r t i c a l l i n e s i n d i c a t e escapements chosen by a c t i v e and pass ive adapt ive p o l i c i e s . 2.4 1.8 12 0.6 / / A 60 The escapement S which would be chosen by the a c t i v e adapt ive p o l i c y i s a l so p l o t t ed in F igure 8-. I t can be seen tha t t h i s p o l i c y always chooses escapements ou t s i de the p r i o r observed range and at a l e v e l which would e f f e c t i v e l y d i s c r i m i n a t e between the a l t e r n a t i v e model forms ( represen t ing d i f f e r e n t parameter e s t ima t e s ) . Th is proper ty of the a c t i v e adapt ive p o l i c y accounts f o r i t s high l e v e l o f performance across the e n t i r e range o f mean p r i o r escapements. The r e l a t i v e performances P0^ of the three p o l i c i e s i n the i n d i v i -dual s imu la t i ons over the p r i o r p r o b a b i l i t y d i s t r i b u t i o n s o f the parameter est imates are shown in F igures 9, 10,. 11, 12 and 13. Each p l o t i n para-meter space i s f o r a p a r t i c u l a r p r i o r d i s t r i b u t i o n determined by the mean o f the p r i o r escapements (the var iance and number of data po in t s are f i x e d ) . The e f f e c t o f the mean p r i o r escapement on the d i s t r i b u t i o n of es t imates i s ev i den t . The broken l i n e through each d i s t r i b u t i o n r ep re -sents the set of combinat ions o f a and 3 parameters f o r which the opt imal d e t e r m i n i s t i c escapement, based on the mean of the p r i o r e s t ima tes , i s a l so op t ima l . The performance of the non-adapt ive p o l i c y improves as the mean p r i o r escapement inc reases because the p r i o r d i s t r i b u t i o n of the est imates comes to ove r l ap more and more w i th t h i s d i s t r i b u t i o n o f parameters f o r which the non-adapt ive escapement i s o p t ima l . The other f ea tu re to note in t h i s set o f f i g u r e s i s the s o l i d l i n e through each d i s t r i b u t i o n . Th is represents the se t of combinat ions o f a and 6, or the set of rec ru i tment cu rves , which p r ed i c t the same rec ru i tment f o r the d e t e r m i n i s t i c escapement as p red i c t ed by the i n i t i a l e s t ima tes . I f the t rue parameters o f the s imulated stock f a l l on t h i s 61 Exp lanat ion f o r f i g u r e s 9,10,11,12 and 13, Each f i g u r e shows the d i s t r i b u t i o n o f pa i r s o f t rue parameters chosen from a p a r t i c u l a r p r i o r d i s t r i b u t i o n of parameter es t imates . The mean p r i o r escapement v a r i e s between f i g u r e s but the standard d ev i a t i o n i s f i x e d (S<- = 0 .05 ) . Each po in t i n the f i g u r e represents a p a r t i c u l a r s imu l a t i on and the performance o f the three adapt ive p o l i c i e s r e l a t i v e to the opt imal performance i s i n d i c a t e d . Open po in t s represent cases i n which the a c t i v e adapt ive p o l i c y achieved be t t e r than 90% of the opt imal performance. For c l osed po in t s the a c t i v e adapt ive performance was between 80% and 90%. C i r c l e s represent cases where the performance o f both the pass ive adapt ive and non-adapt ive p o l i c i e s was g rea te r than 90%. T r i ang l e s are cases where the pass ive adapt ive performance was g rea te r than 90% and the non-adapt ive performance was between 80% and 90%. Squares represent a l l o ther cases and the numbers bes ide each i n d i c a t e the performances o f the pass ive adapt ive and non-adapt ive p o l i c i e s r e s p e c t i v e l y . For example 92 i n d i c a t e s that the pass ive adapt ive performance was g rea te r than 90% and the non-adapt ive performance was between 20% and 30%. The broken l i n e represents the se t o f parameter va lues f o r which the opt imal d e t e r m i n i s t i c escapement chosen on the bas i s o f the i n i t i a l parameter es t imates would a l s o be o p t ima l , the s t r a i g h t l i n e represents the se t o f parameter values which would generate the same d e t e r m i n i s t i c rec ru i tment as tha t p red i c t ed by the i n i t i a l e s t ima tes , a t the opt imal d e t e r m i n i s t i c escapement chosen on the bas i s o f those es t ima tes . 62 F igure 9. P o l i c y performance i n parameter space: case S = 0.1 . For exp lana t i on of f i g u r e see page 61. 63 Figure 10. P o l i c y performance i n parameter space f o r exp lana t i on of f i g u r e see page 61. 66 F igure 11. P o l i c y performance i n parameter space: case S = 0.5 . For exp lana t i on of f i g u r e see page 61. 67 3 68 F igure 12. P o l i c y performance i n parameter space: case S = 0.7 . For exp lana t i on o f f i g u r e see page si. 69 .6 70 t F igure 13. P o l i c y performance i n parameter space: case S = 0.9 For exp lana t ion o f f i g u r e see page 61. 7 1 8 72 l i n e , then the p o l i c y which chooses the d e t e r m i n i s t i c escapement w i l l observe the rec ru i tment i t expects based on i t s cu r ren t e s t ima tes . I t w i l l t he re f o re not a l t e r those est imates and w i l l cont inue to choose the same escapement and to get the same p red i c t ed response. The pass ive adapt ive p o l i c y i s l i k e l y to perform l e a s t we l l when the p r i o r d i s t r i b u -t i o n of the es t imates corresponds w i th t h i s l i n e . Th is can be seen to occur i n F igure 10. Not ice a l s o t ha t the pass ive adapt ive p o l i c y on ly performs poor l y when the t rue parameters f a l l at the extremes o f the d i s t r i b u t i o n . In p a r t i c u l a r , the pass ive adapt ive p o l i c y on ly performs very poor l y when the e q u i l i b r i u m stock s i z e i s much l a r g e r than the i n i t i a l es t imates i n d i c a t e . 3.7 POLICY COMPARISON: CASE S = 0 .3 , S £ = 0:05 The g rea tes t d i f f e r en c e in performance between the a c t i v e and pass ive adapt ive p o l i c i e s occurs where the pass ive adapt ive escapement chosen f a l l s w i t h i n the range o f the p r i o r escapements. For the case where the p r i o r v a r i a b i l i t y i n escapement was a t the l e v e l S<* = 0 .05, t h i s occurred f o r a mean p r i o r escapement o f S = 0 .3 . Th is s e c t i on w i l l examine more c l o s e l y the r e s u l t s of i n d i v i d u a l s imu la t i ons f o r t h i s case. The f i r s t po in t to note i s tha t the a c t i v e adapt ive p o l i c y always performs we l l r e l a t i v e to the opt imal p o l i c y where the t rue parameters are known. The frequency d i s t r i b u t i o n of the long-term performance P0^ o f each p o l i c y r e l a t i v e to the opt imal i s shown in F igure 14. The mean performance of each p o l i c y i s h i gh , but the a c t i v e adapt ive p o l i c y always 73 Figure 14. D i s t r i b u t i o n o f p o l i c y performances r e l a t i v e to the opt imal p o l i c y . 7-1 1: 90 0 N Mean Active Adaptive Policy 96.0757 90.4361 Pass ive Adaptive Policy Q Non-Adaptive Policy 89.5127 80 70 J23 60 \ 50 40 30 Percentage of Optimal Policy P0 L 75 performs we l l (80% of the opt imal or b e t t e r ) whereas the o ther two p o l i c i e s o c c a s i o na l l y ' p e r f o rm very poor l y (as low as 30% of the opt imal p o l i c y ) though t h i s does not happen very f r e quen t l y . The next po in t to note i s t ha t a l though the a c t i v e adapt ive p o l i c y does we l l i n the l ong- te rm, i t does so a t the expense o f shor t - te rm performance (Table I I ) . The a c t i v e adapt ive p o l i c y must s a c r i f i c e sho r t -term payof f (catch) in order to improve the parameter es t imates and so do be t t e r i n the long- term. Th is i s g ene r a l l y the case i n dual c on t r o l problems. The pass ive adapt ive and non-adapt ive p o l i c e s do equa l l y we l l (or poo r l y ) i n the shor t - te rm and the l ong- te rm. The other i n d i c a t i o n tha t the a c t i v e adapt ive p o l i c y s a c r i f i c e s shor t - te rm performance i s tha t the i n t r o du c t i o n of d i s coun t i ng i n the o b j e c t i v e f unc t i on reduces the performance r e l a t i v e to the opt imal performance. The next i tem o f i n t e r e s t i s the improvement in performance o f the a c t i v e adapt ive p o l i c y over the pass ive adapt ive p o l i c y in i n d i v i d u a l cases . The q u a n t i t i e s of i n t e r e s t , PI<- and PI^, represent the shor t - te rm and long-term percentage improvement i n performance o f the a c t i v e adapt ive p o l i c y over the pass ive adapt ive p o l i c y r e l a t i v e to the performance of the pass ive adapt ive p o l i c y . The frequency d i s t r i b u t i o n of PI^ over the 50 s imu la t i ons i s shown in F igure 15. I t i s ev ident t ha t the mode of the d i s t r i b u t i o n i s near z e r o , tha t i s , i n the ma j o r i t y of cases the performance o f the two p o l i c i e s i s very s i m i l a r . However the d i s t r i b u t i o n i s a l s o skewed. Negat ive va lues are of small magnitude but p o s i t i v e va lues are o c c a s i o n a l l y qu i t e l a r g e . Th is means tha t the a c t i v e adapt ive p o l i c y never performs much worse than the pass ive adapt ive p o l i c y , and o c c a s i o n a l l y i t performs very much b e t t e r . 76 a) AAP PAP NAP 20 YEARS 94.4 84.7 84.2 (92.9) (85.1) (84.9) 50 YEARS 96.2 84.5 83.7 (94.6) (84.7) (84.3) AAP vs PAP PAP vs NAP 20 YEARS 11.4 0.6 (9.3) (0.2) 50 YEARS 13.8 1.0 (11.7) (0.4) c) AAP PAP NAP 48* 28 21 Table I I . Comparison of shor t term and long term performance, a) mean performance as a percentage of mean performance o f the opt imal p o l i c y ; b) percentage improvement of p o l i c y x over p o l i c y y r e l a t i v e to the mean performance o f p o l i c y y ; c) number o f s imu l a t i ons (out o f 50) f o r which the r e l a t i v e long term performance exceeds the r e l a t i v e shor t term performance. *denotes s i g n i f i c a n t e f f e c t (at 5% s i g . l e v e l ) . Resu l t s shown i n brackets are f o r the case where the d i scount ra te i s 5%. 77 F igure 15. D i s t r i b u t i o n o f percent improvement i n performance of a c t i v e adapt ive p o l i c y r e l a t i v e to the pass ive adapt ive p o l i c y . Upper f i g u r e shows d e t a i l o f l e f t hand po r t i on o f lower f i g u r e . 12 4 -100 5.0 0 50' 100 P e r c e n t I m p r o v e m e n t PI, 16 12 M e a n : 13.85 4 V 20 0 20 AO 60 80 100 120 140 160 180 200 Pe r cen t I m p r o v e m e n t PI. 79 The average improvement of 13.85% turns out not to be s i g n i f i c a n t l y d i f f e r e n t from zero over the 50 s i m u l a t i o n s . Re f e r r i ng back to F igure 14, i t i s ev ident tha t even the non-adapt ive p o l i c y achieves 90% of the o p t i -mal performance in the l a rge p ropor t i on o f cases . Comparing the performance of the a c t i v e and pass ive adapt ive p o l i c i e s over the s imu la t i ons where the performance of the non-adapt ive p o l i c y i s l e s s than 90% of the o p t i m a l , the improvement in performance o f the a c t i v e adapt ive p o l i c y i s h i gh l y s i g n i f i c a n t . The e s t ima t i on performance of the three p o l i c i e s i s presented i n Table I I I . The es t ima t i on performance f o r the a c t i v e adapt ive p o l i c y i s s i g n i f i c a n t l y b e t t e r than tha t f o r the pass ive adapt ive p o l i c y , which does no be t t e r than the non-adapt ive p o l i c y . The e f f e c t which the a c t i v e adapt ive p o l i c y has in reduc ing para-meter unce r t a i n t y and hence improving long-term performance i s exemp l i f i ed in the d e t a i l e d ana l y s i s of a p a r t i c u l a r s i m u l a t i o n . F igure 16 shows the sequence o f escapements chosen by the a c t i v e and pass ive adapt ive p o l i c i e s , together w i th the a c t i v e adapt ive p e r t u r ba t i o n . The range o f escapements chosen by the a c t i v e adapt ive p o l i c y i s obv i ous l y much g rea t e r . The e f f e c t o f t h i s sequence o f escapements on the parameter es t imates f o r each p o l i c y are shown i n F igure 17. The i n i t i a l e s t ima t i on e r r o r i s qu i t e l a r g e , but r a p i d l y improves w i th the a c t i v e adapt ive p o l i c y . The parameter est imates f o r the pass ive adapt ive p o l i c y do not change much over t ime, and show no s ign of approaching the c o r r e c t va l ues . The para-meter u n c e r t a i n t i e s (elements o f the parameter e r r o r covar iance mat r i x ) are shown in F igure 18. F igures 19 and 20 p l o t the s t o ck - r e c ru i tmen t data f o r the a c t i v e and pass ive adapt ive p o l i c i e s r e s p e c t i v e l y . Both the p r i o r data and the AAP PAP NAP c  0.19* 0.32 0.33 3 0.63* 1.05 1.03 s 5 0.08* 0.43 0.43 Table I I I . Comparison o f e s t ima t i on performances. Tabled are the square root o f the mean squared e r r o r s i n the est imates o f the two parameters and the optimum escapement S. * denotes a s i g n i f i c a n t d i f f e r en c e (at.5% s i g . l e v e l ) i n e s t ima t i on performance between p o l i c i e s . 81 F igure 16. Comparison o f escapements chosen by the a c t i v e and pass ive adapt ive p o l i c i e s . L ine OP represents opt imal escapement. Escapement Chosen CP c n O ho C P cn oo co 5 "0 > o > -o "0 00 r o 83 F igure 17. Comparison of parameter es t imates achieved by a c t i v e and pass ive adapt ive p o l i c i e s . Parameter E s t i m a t e s 6! C O 3 cn cn CD c o i i • i i i c n ro O cn —t tb> XI C > m > c > m > 9~° TI 00 85 F igure 18. Comparison o f parameter u n c e r t a i n t i e s f o r the a c t i v e and pass ive adapt ive p o l i c i e s . U n c e r t a i n t y i n P a r a m e t e r E s t i m a t e s CO 3 <x> LP cn oo co p In cn o cn CO o CO cn o u -cn A <A o, > > 5 13 5 ~0 5 00 ure 19. Stock rec ru i tment data f o r the a c t i v e adapt ive p o l i c y . Dots represent the i n i t i a l data se t and crosses represent data obta ined by the p o l i c y . The s o l i d curve represents the t rue stock rec ru i tment r e l a t i o n s h i p and the broken curve represents the est imated r e l a t i o n s h i p a f t e r 10 y ea r s . RECRUITMENT P r* r-* N cn k) oo oo co 89 F igure 20. Stock rec ru i tment data f o r the pass ive adapt ive p o l i c y . Dots represent the i n i t i a l data se t and crosses represent data obta ined by the p o l i c y . The s o l i d curve represents the t rue stock rec ru i tment r e l a t i o n s h i p and the broken curve represents the est imated r e l a t i o n s h i p a f t e r 10 y e a r s . RECRUITMENT 91 data which become a v a i l a b l e through management o f the stock are p l o t t e d . The s o l i d l i n e represents the t rue s t o ck - re c ru i tmen t r e l a t i o n s h i p and the broken l i n e represents the est imated s tock - rec ru i tmen t r e l a t i o n s h i p a f t e r management f o r 10 genera t i ons . These l i n e s correspond c l o s e l y f o r the a c t i v e adapt ive p o l i c y because the data po in t s span a wide range o f escapement l e v e l s . However the escapements chosen by the pass ive adapt ive p o l i c y a l l l i e w i t h i n the range o f the p r i o r escapements and the c o r r e c t form o f the r e l a t i o n s h i p i s never l ea rned . The cumulat ive d iscounted catch over time f o r the a c t i v e adap t i ve , pass ive adapt ive and opt imal p o l i c e s i s compared i n F igure 21. Although the a c t i v e adapt ive p o l i c y very q u i c k l y l ea rns the c o r r e c t parameter va l ue s , i t performs very poor ly over the f i r s t few stages and does not surpass the pass ive adapt ive p o l i c y u n t i l the s i x t h s tage . However by the 20th generat ion i t s performance compares very favourab ly w i th the opt imal p o l i c y , whereas the pass ive adapt ive p o l i c y has f a l l e n we l l beh ind. A f t e r 50 generat ions the improvement i n performance o f the a c t i v e adapt ive p o l i c y over the pass ive adapt ive p o l i c y i s 45.6%. 3.8 POLICY COMPARISON: CASE S = 0 . 1 , S£ = 0.05 This case represents the s i t u a t i o n i n which t h e r e , i s the g rea tes t d i f f e r en c e i n performance between the pass ive adapt ive and non-adapt ive p o l i c i e s . Th is s e c t i on w i l l compare the performance o f a range o f pass ive adapt ive p o l i c i e s as d i scussed i n s e c t i on 3.2. The e f f e c t o f the number o f stages between parameter est imates..updates i s shown in Table IV. The performance decreases as the frequency o f parameter es t imate updates i s reduced, but the d i f f e r en c e i s not s i g n i f i -92 F igure 21. Comparison o f cumulat ive catches f o r the opt imal p o l i c y and the a c t i v e and pass ive adapt ive p o l i c i e s . Cumulative Catch GO NYBU SO,. oc 3 s i 96.9 0.12 0.54 0.13 2 96.9 0.11 0.55 0.10 4 96.0 0.12 0.50 0.10 8 93.9 0.12 0.50 0.19 16 87.7 0.12 0.50 0.66 32 84.5 0.12 1.11 0.59 Table IV. E f f e c t of i n f requen t update o f parameter e s t ima tes . NYBU = number o f years between parameter es t imate updates; SO. i s the^average performance r e l a t i v e to the opt imal performance s , 3 and I are the mean e r r o r s i n the est imates o f the pa ra-meters and optimum escapement. 95 cant i f the frequency i s every 8 stages or b e t t e r . The e f f e c t on the e s t ima t i on performance i s i n c o n s i s t e n t , and i s on ly c l e a r l y adverse i n the case where there i s on ly one parameter es t imate update (NYBU = 16). Th is r e f l e c t s the range o f escapements over which observat ions w i l l have been made. The e f f e c t o f the extent to which each new data po in t i s g iven i t s f u l l weight i n r e v i s i n g parameter est imates i s shown i n Table V. The performance decreases as the data i s g iven l e s s and l e s s we ight . However a s i g n i f i c a n t d i f f e r en c e in performance from the p o l i c y which g ives f u l l weight to the data i s not obta ined u n t i l the weight i s reduced to one s i x t een th the normal weight (GF = 0.0625). The performance in e s t ima t i ng the opt imal escapement S* corresponds c l o s e l y w i th the average performance ob ta ined . The r e s u l t s o f t h i s s e c t i on suggest tha t i t i s best to update para-meter est imates as each new data po in t becomes a v a i l a b l e , g i v i n g f u l l weight to the data in r e v i s i n g the e s t ima tes . The frequency of update and weight g iven to the data must be reduced s u b s t a n t i a l l y before a s i g n i f i c a n t "expected" decrease in performance w i l l occur . However i n i n d i v i d u a l s cases the reduc t i on i n performance may be s u b s t a n t i a l . 3.9 DISCUSSION The major conc lus i on from comparison o f the range o f adapt ive s t r a t e -g ies f o r managing a stock w i th unce r ta in R i cker s t o ck - rec ru i tmen t dynamics i s as f o l l o w s . The most important f a c t o r determin ing the type o f ' h a r v e s t -ing s t r a t egy which should be chosen i s how the past escapements are d i s -t r i b u t e d w i th respect to the best est imate o f the s to ck - rec ru i tmen t r e l a -GF s o L '.• c  s : 1 96.9 0.12 0.54 0.13 1/2 97.1 0.11 0.46 0.13 1/4 96.7 0.11 0.42 0.17 1/8 95.6 0.11 0.43 0.22 1/16 92.5 0.11 0.53 0.44 1/32 89.4 0.11 0.72 0.53 1/64 87.3 0.11 0.88 0.56 0 84.5 0.12 1.11 0.59 Table V. E f f e c t o f reduc ing weight g iven to new da ta . GF = ga in f a c t o r , the m u l t i p l i c a t i v e f a c t o r on the p r e d i c t i o n e r r o r ga in i n the r eg r e s s i on ; SO,;, i s the average performance r e l a t i v e to the optimal, performance; £, 3 and s-are the mean e r r o r s i n the est imates o f the parameters and optimum escapement. 97 t i o n s h i p based on those escapements and the rec ru i tments observed from them. Noting tha t the past escapements a l s o represent the past con t ro l o p t i o n s , the conc lus i on i s t ha t the best s t r a t egy to choose now depends very much on what s t r a tegy was used to manage the stock in the pas t . The r e s u l t s i n s e c t i on 3.6 show tha t the pass ive adapt ive p o l i c y does worst when the p r i o r v a r i a b i l i t y i n escapement i s small and the escapement which the p o l i c y chooses based on the p r i o r data f a l l s w i t h i n the range o f escapements a l ready observed. For a l l o ther cases , where the escapement chosen f a l l s ou t s i de the p r i o r range, the pass ive adapt ive p o l i c y does we l l because i t i s e s s e n t i a l l y generat ing i t s own p e r t u r b a t i o n s , which improve parameter es t imates by a c c i den t . The p r o b a b i l i t y tha t the escapement chosen by the pass ive adapt ive p o l i c y w i l l f a l l w i t h i n the range of the p r i o r observed escapements may seem s m a l l , but t h i s i s not n e c e s s a r i l y the case . I f the p o l i c y which generated the p r i o r escapements i s a l so a pass ive adapt ive p o l i c y , then i t i s qu i t e l i k e l y tha t the escapements chosen w i l l have been over a narrow range and w i l l correspond to the opt imal de te rmin i s t i c -escapement based on the est imates generated by the p o l i c y i t s e l f . In o ther words, the nature of a pass ive adapt ive p o l i c y i s to generate a set of s tock -rec ru i tment data over a narrow range of escapements which p r ed i c t tha t the optimum escapement f a l l s w i t h i n tha t range. The sequence o f escapements generated by the pass ive adapt ive p o l i c y in the example in s e c t i on 3.7 (see F igure 18) demonstrates' t h i s po in t n i c e l y . The suggest ion by Walters and H i l bo rn (1976) tha t a pass ive adapt ive p o l i c y can lead to an e q u i l i b r i u m "which i s ne i t he r :op t ima l nor p roduc t ive of the type o f i n fo rmat ion requ i red to determine what i s 98 op t ima l " seems to be born ou t , at l e a s t on the bas i s o f these s imu l a t i o n s . Of course when p r i o r data are very s ca r ce , as in the development o f a new f i s h e r y , f o l l ow i ng a t rue pass ive adapt ive p o l i c y may r e s u l t i n parameter est imates which are s ca t t e r ed w ide ly enough to r e s u l t i n a wide range o f escapements being chosen. However such techniques are r a r e l y app l i ed i n the e a r l y stages o f development, and in many o l de r f i s h e r i e s the data on e a r l y e x p l o i t a t i o n i s miss ing or u n r e l i a b l e . S ince adopt ion of a pass ive adapt ive s t r a tegy can and apparent ly does lead to poor performance, c ons i de r a t i on of an a c t i v e l y adapt ive s t r a t egy in managing w i th unce r ta in stock dynamics seems warranted. The expected performance o f the a c t i v e p o l i c y compared favourab ly w i th the opt imal p o l i c y across the e n t i r e range of mean p r i o r escapements (F igure 4 ) . Furthermore, i t s performance i n any p a r t i c u l a r case a l so compared favourab ly (F igure 14). The maximum expected improvement in performance o f the a c t i v e adapt ive p o l i c y over the pass ive adapt ive p o l i c y was on ly 14%. However i n i n d i v i d u a l cases the improvement was as high as 180% (F igure 15). Th is improvement was, however, obta ined at the expense o f some reduc t i on i n y i e l d in the shor t - te rm (Table II and F igure 21) . Comparison w i th the performance of the pass ive adapt ive and non-adapt ive p o l i c i e s leads to c ons i de r a t i on of how the a c t i v e adapt ive p o l i c y might be improved. For example, the very good performance o f the non-adapt ive p o l i c y a t h igh l e v e l s of mean p r i o r escapement ( r e l a t i v e to the apparent e q u i l i b r i u m stock s i z e ) suggests tha t probing by the a c t i v e adapt ive p o l i c y in such s i t u a t i o n s i s not wor thwh i le . However t h e ' " -a c t i v e adapt ive con t ro l law shows tha t s i g n i f i c a n t pe r tu rba t i ons are s t i l l chosen (F igure l a ) . Although the a and 6 es t imates are v a s t l y improved 9 9 (F igures 5 and 6 ) , the est imate of the opt imal escapement S* i s not (F igure 7) and there i s no improvement in long-term performance to outway the l o s s in shor t - te rm y i e l d . The a c t i v e adapt ive p o l i c y i s apparent ly a s s i gn ing a va lue to i n f o rmat i n where there i s none. I t has not p rope r l y assessed the i n s e n s i t i v i t y o f the opt imal p o l i c y to uncer-t a i n t y i n the a parameter. S i m i l a r l y , f o r very low or medium to high l e v e l s of p r i o r escape-ment, the pass ive adapt ive p o l i c y performs we l l because in these s i t u a t i o n s the escapements chosen are i n f o rmat i ve anyway. Aga in , the a c t i v e adapt ive con t ro l law i n d i c a t e s t ha t l a rge pe r tu rba t i ons over and above those gener-ated by the pass ive p o l i c y are chosen-. The va lue o f these a dd i t i o n a l pe r tu rba t i ons may be doubt fu l when weighed aga ins t the shor t - te rm cos ts i ncur red through po s s i b l e l o s s i n y i e l d s . A l l these cons i de ra t i ons suggest t h a t , a t l e a s t f o r the problem cons idered in t h i s chap te r , an improved a c t i v e adapt ive s t r a t egy would i nvo l ve l a rge pe r tu rba t i ons on ly where pass ive adapt ive escapements f a l l c l o se t o , or w i t h i n the range o f , escapements a l ready observed i f the range has been s m a l l . Strong pe r tu rba t i ons towards low escapements to improve the est imate of the a parameter w i l l g ene ra l l y not be wor thwh i le . 3.10 SUMMARY 1. The a c t i v e adapt ive pol icy-, based on the wide sense adapt ive dual con t ro l a l g o r i t hm , compares favourab ly w i th the opt imal p o l i c y over a wide range of p r i o r d i s t r i b u t i o n s of parameter e s t ima tes . The pe r f o r -mance i s good both i n an expected sense and f o r i n d i v i d u a l s i t u a t i o n s . 100 2. The pass ive adapt ive p o l i c y compares favourab ly w i th the opt imal p o l i c y except i n s i t u a t i o n s where the escapement chosen f a l l s w i t h i n the range o f escapements a l ready observed, and t h i s range i s narrow. I t i s suggested tha t these s i t u a t i o n s w i l l be l i k e l y to occur when the p r i o r data i s a l s o the r e s u l t o f a pass ive adapt ive management s t r a t egy . In such cases , the performance of the pass ive adapt ive p o l i c y can be poor and the performance advantages of an a c t i v e l y adapt ive ha rves t i ng s t r a t egy cons i de rab l e . 3. The r e l a t i v e l y good performance of the non-adapt ive p o l i c y f o r mean p r i o r escapements c l o se to the apparent e q u i l i b r i u m stock s i z e suggests t ha t the opt imal p o l i c y f o r the R i cke r model i s r e l a t i v e l y i n sen -s i t i v e to unce r t a i n t y i n the a parameter. Th is suggests tha t e f f o r t s to improve est imates o f the e q u i l i b r i u m stock s i z e a/6 are more l i k e l y to prove wor thwh i le . 4. Sound management of a s tock whose dynamics can be adequate ly desc r ibed by the R i cke r s tock - rec ru i tmen t model r equ i r e s tha t the response of the stock be observed over an adequate ly wide range of escapements. The management de c i s i on s which should be made in the present w i l l be determined very l a r g e l y by what management dec i s i on s have been made in the pas t . 101 CHAPTER IV * 4.0 THE SCHAEFER MODEL The R i cker s tock - rec ru i tment model can be used as a s imple r ep res -en ta t i on of the dynamics o f a number o f commerc ia l ly important f i s h s t o c k s , notab ly the anadramous spec i e s . However f o r most s t o c k s , t h e . dynamics which are of importance to managing the stock i nvo l ve more than j u s t the r e l a t i o n s h i p between stock and rec ru i tmen t . Th is i s p a r t i c u l a r l y t rue f o r s tocks where a s i g n i f i c a n t par t o f the o v e r a l l p roduct ion i s due to growth o f i n d i v i d u a l s a f t e r sexual ma tu r i t y . A l s o , f o r most s t o c k s , d i r e c t observat ions on adu l t spawning numbers and subsequent rec ru i tment to the stock are s imply not a v a i l a b l e . Indeed the s i z e of the stock i s not d i r e c t l y observab le a t a l l . For many wor ld f i s h e r i e s the on ly s t a t -i s t i c s from which to i n f e r something about the dynamics of the s tocks i nvo lved are records of annual catch and e f f o r t . These cons i de ra t i ons have l ed to the development of a number of "product ion f u n c t i o n " models o f f i s h stock dynamics, where growth, r e c r u i t -ment and na tura l m o r t a l i t y are a l l c o l l ap sed i n t o a s i n g l e product ion quan t i t y which i s a f unc t i on of s tock s i z e . Examples of the use of such product ion models may be found i n Graham (1935), Schaefer (1954, 1957), P e l l a and TomTinson (1969) and Fox (1970). The model which has been most w ide ly app l i ed i n f i s h e r y management i s the s imple l o g i s t i c or Schaefer model (Schae fe r , 1957) and i t i s t h i s model which w i l l be used as a p ro to -type f o r development o f adapt ive p o l i c i e s i n t h i s chapter . 102 Although o r i g i n a l l y presented as a d i f f e r e n t i a l equa t i on , the Schaefer model has f r equen t l y been expressed as a d i f f e r en c e equat ion (Walters and Hi 1 born, 1976; Uh le r , 1978; H i l b o r n , 1979): where N i s popu la t ion s i z e ( i n numbers or b iomass) , C i s c a t c h , r and k are parameters o f the model (corresponding to growth ra te and c a r r y i n g capac i t y ) and v i s an environmental "no i se " term, normal ly d i s t r i b u t e d 2 w i th zero mean and var iance a . The s ub s c r i p t t denotes t ime, gene ra l l y a year o r a f i s h i n g season. The f i n a l term v t N^ i s used to represent s t o c h a s t i c v a r i a b i l i t y i n the growth f u n c t i o n . The assumption tha t the s t o c h a s t i c growth term should be o f t h i s form i s d i f f i c u l t to j u s t i f y t h e o r e t i c a l l y . However Fox (1971) suggests tha t t h i s i s a reasonable assumpt ion, argu ing tha t " i t i s e a s i e r to conceive tha t under e q u i l i b r i u m cond i t i on s a popu la t i on w i l l f l u c t u a t e more r a d i c a l l y near i t s env i ronmenta l l y l i m i t e d maximum s i z e than at sma l l e r s i z e s under constant e x p l o i t a t i o n . " H i l bo rn (1979) i m p l i c i t l y assumes t h i s form f o r the s t o c h a s t i c term i n h i s " l i n e a r " e s t ima t i on scheme. In many f i s h e r i e s the management r egu l a t i on i s v i a f i s h i n g e f f o r t l i m i t a t i o n r a the r than catch quotas , so tha t the r e l a t i o n s h i p between catch and e f f o r t must be s p e c i f i e d . One of the most common assumptions i s tha t catch i s l i n e a r l y r e l a t e d to both e f f o r t and stock s i z e , t ha t i s where E i s e f f o r t , c i s a parameter, c a l l e d the c a t c h a b i l i t y c o e f f i c i e n t , and w. i s a s t o c h a s t i c term normal ly d i s t r i b u t e d w i th zero mean and var iance N T + 1 = N t + r N t (1 - N t /k ) - C t + v ^ (4.0-1) c E t N t + w t (4.0-2) 103 ao-.' Combining (4.0-1) and (4.0-2) r e s u l t s i n an express ion f o r the stock w dynamics o f the form N t + 1 = N t + r N t (1 - N t/k) - c + - w t (4 .0-3) I t should be po inted out tha t t h i s i s not a very r e a l i s t i c d e s c r i p t i o n o f the dynamics of a f i s h stock sub jec t to e x p l o i t a t i o n . Fox (1975) d i s cusses a number o f assumptions which are requ i red f o r the use o f product ion models i n cont inuous form ( d i f f e r e n t i a l equa t i ons ) . His comments apply i n the d i s c r e t e case a l s o . These are tha t "(1) the model i s being app l i ed to a c l osed s i n g l e u n i t popu l a t i on , (2) the concept of e q u i l i b r i u m cond i t i ons app l i e s to the popu la t ion under a n a l y s i s , and (3) the age-groups being f i s hed have remained, and w i l l cont inue to remain, the same." Furthermore, use of data from the t r a n s i t i o n a l s t a t e s o f a popu la t ion under e x p l o i t a t i o n r equ i r e s the a dd i t i o n a l assumptions tha t "both (1) time lags i n processes a s soc i a t ed w i th popu la t i on change and (2) dev i a t i on s from the s t ab l e age s t r u c t u r e a t any popu la t i on l e v e l have n e g l i g i b l e e f f e c t s on the product ion r a t e " ( a f t e r Fox, 1975). The d i s c r e t e form (4.0-3) assumes i n a dd i t i o n tha t net product ion i s a f unc t i on o f stock s i z e a t on ly one time o f the year or season, and tha t t h e . e f f o r t i s app l i ed d i r e c t l y to tha t popu la t i on s i z e o n l y , but not to the product ion from i t . In a d d i t i o n , the catch cE^N^ can be g rea te r than the popu la t ion s i z e to which the e f f o r t i s a p p l i e d . Despi te these l i m i t a t i o n s , product ion models have been and are being used as a bas i s f o r r e gu l a t i n g many commercial f i s h e r i e s . As in the prev ious chapter , the management o b j e c t i v e cons idered i s the maximizat ion of the sum o f d iscounted catches over t ime , tha t i s , 104 max E C. [ l / ( l + d ) r t=0 z where d i s the d i scount r a t e . Th is leads to express ions f o r the optimum popu la t ion s i z e N* and the optimum e q u i l i b r i u m e f f o r t E* of the form N* = k • ( r - d ) / 2 r E* = (r+d)/2c Given a popu la t ion s i z e N^, the opt imal de t e rm in s t i c con t ro l i s to choose E^  such tha t N t + 1 i s equal to N*, w i th the c on s t r a i n t tha t E t > 0. Thus determinat ion o f the opt imal e f f o r t a t each stage r equ i r e s knowledge o f the popu la t ion s i z e N and the parameters r, k and c. Sec t ion 4.1 shows how these may be est imated from past data on catch and e f f o r t . The r e s t o f the chapter presents an ana l y s i s of the e f f e c t s of s t a t i s t i c a l unce r t a i n t y in the parameters o f the Schaefer model on the performance of a range o f adapt ive con t ro l p o l i c i e s . S ince the problem cons idered i n t h i s chapter i s a t l e a s t an order o f magnitude more " d i f f i c u l t " than tha t cons idered i n the prev ious chap te r , the a n a l y s i s presented i s not as exhaus t i ve . Th is chapter r ep re sen t s , then, a p a r t i a l survey of some of the problems assoc i a ted w i th the use of catch and e f f o r t data to regu la te f i s h s tocks i n an adapt ive f a s h i o n . 4.1 PARAMETER ESTIMATION ; A number o f methods can be used to es t imate product ion model para-meters from catch and e f f o r t da ta . Reviews o f these techniques may be found i n R i cker (1975) and Fox (1975). A d i s cu s s i on of the importance of 105 the s t a t i s t i c a l model used may be found i n Fox (1971). More r e c e n t l y , Schnute (1977) has i n ve s t i g a t ed some t h e o r e t i c a l aspects o f improving est imates f o r the Schaefer model. Uhler (1978) has i n ve s t i g a t ed the source o f b ias i n est imates f o r the Schaefer model a r i s i n g from the " e r r o r s i n v a r i a b l e s " r eg ress i on problem. H i l bo rn (1979) has analysed the performance of three d i f f e r e n t e s t ima t i on proce-dures f o r the Schaefer model i n the context o f comparing con t ro l systems which use catch and e f f o r t da ta . A technique used in t h i s study to ob ta in parameter es t imates f o r the Schaefer model from catch and e f f o r t data f o l l ows c l o s e l y on one of the techniques analysed by both Uhler and H i l b o r n . Use i s made o f the assumption in (4.0^-2) tha t the catch per un i t o f e f f o r t q t i s a measure of popu la t ion s i z e , tha t i s \ - C t / E t = c * N t ( 4 ' 1 " 1 ) In t h i s s tudy, i t w i l l be assumed tha t there i s no s t o c h a s t i c v a r i a t i o n i n the catch term, t ha t i s , w^  =0 f o r a l l t . Although u n r e a l i s t i c , t h i s assumption means tha t the e r r o r s i n v a r i a b l e s problem reported by Uhler (1978) i s avo ided. Th is avo ids the problem o f b ias i n the es t ima tes . S u b s t i t u t i n g (4.1-1) i n (4.0-3) y i e l d s the f o l l ow i ng equat ion i n q, V l = q t + r q t ( 1 ~ q t / c k ) " c E t q t + v t q t and rear rang ing terms < V l - q t > / q t = r - c t q t " c E t + v t ( 4 - 1 " 2 ) 106 which i s a l i n e a r r eg ress i on of the form y . = x/ a + e. where y . = ( q t + 1 - q t ) / q t , x J = [1 - q t - E t ] and a = [r r/ck c ] T . Again i t can be seen tha t the e r r o r term e^  i n the reg ress i on i s o f the 2 2 co r r e c t form w i th zero mean and var iance a = a . e v The two independent v a r i a b l e s i n the reg ress i on equat ion ( 4 . 1 - 2 ) are the catch per un i t e f f o r t q^ . and the e f f o r t E .^ The es t ima t i on problem i s i l l u s t r a t e d i n F igure 22 ( a f t e r H i l b o r n , 1 9 7 9 ) . Given observa t ions i n (q^, E^ ) space, the problem i s to est imate the plane of best f i t through them. The i n t e r c ep t o f t h i s plane on the obse rva t i on ax i s w i l l be the est imate of r, and the s lopes along the l i n e s at q t = 0 and E t = 0 de f ine -c and - r / c k . I t i s ev ident tha t to ob ta in good est imates of a l l parameters there must be observat ions over a range o f both catch per un i t e f f o r t and e f f o r t . In add i t i o n there must be a con t r a s t i n pa i red va lues of the independent v a r i a b l e s , so tha t observa t i ons are obta ined at both high and low values of catch per un i t e f f o r t f o r both high and low va lues of e f f o r t . I f a l l the observat ions f a l l a long a s t r a i g h t l i n e i n (q^, E t ) space, even though there be a wide range o f both q and E va l ue s , est imates cannot be obta ined as there are an i n f i n i t e number of planes which can be . ' f i t t e d through a s t r a i g h t l i n e . Fo l lowing the arguments presented i n chapter I I I , there are now s i x p r ope r t i e s of the data which w i l l determine the unce r t a i n t y i n the para-meter, estimates,, f o r t h i s reg ress ion problem ( i . e . , the -parameter covar iance matr ix has s i x d i s t i n c t e lements) . These p rope r t i e s a re : the mean q~ 2 — of the catches per u n i t e f f o r t ; the var iance S of the CPUEs; the mean E 107 F igure 22. Representat ion o f the reg ress i on problem f o r e s t ima t i ng the parameters of the Schaefer model. The plane de f ined by the po in ts A,B and C represents the m u l t i p l e r eg ress i on p lane. The independent v a r i a b l e s are ca tch per u n i t e f f o r t (q) and e f f o r t ( E ) . The p o s i t i o n o f the opt imal catch per un i t e f f o r t and e f f o r t (q*,E*) i s shown. r ,k and c are the parameters o f the Schaefer model. ( A f t e r H i l b o r n , 1979j). 108 109 and variance S,- in the e f f o r t s ; the corre lat ion r :• between catch per unit e f fo r t and e f f o r t ; and the number of observations n. Expanding the regression equations for the parameter uncertaint ies expressed in terms of these components, the fol lowing results are obtained (the para-meter vector [r r/ck c] i s rewritten as [a 3 Y ! for convenience): E 2 _ n ( q j 2 - q 2 E 2) . (n- l) S 2 ( n - l ) 2 S q 2 S 2 I + n ( n - l ) S , ( i - C ) q E (h-1) Sr ( 1"V> (n- l )S t ( 1 - V E } ( n - l ) S , E r _°JL ( n - l ) S q S E ( l " r n ) q E a .r • ay ( n - l ) S t q r RE_ ( n - l ) S q S E ( l - ^ q E ; ) PY - r , qE ( n - l ) S q S E (1-r ) v qE ; 110 Aga in , the formulae f o r the u n c e r t a i n t i e s are i n t u i t i v e l y reason-ab l e . Re fe r r i ng to F igure 22 and the above equat ions , i t can be seen tha t unce r t a i n t y about r (or a ) w i l l be reduced when both q~ and E are low. Th is imp l i e s a s i t u a t i o n where the popu la t ion s i z e i s low and the popu la t ion i s recover ing w i th l i t t l e or no ha rves t , which w i l l g ive a good est imate o f the growth r a t e . Uncer ta in ty i n both r/ck (k) and c ( y ) depend l a r g e l y on the var iances i n q and E r e s p e c t i v e l y , s ince the parameters i n quest ion are the s lopes a long these axes. Not i ce tha t 2 the term 1/(1 - r qE) appears in a l l the equa t i ons , i n d i c a t i n g t ha t as the c o r r e l a t i o n between catch per un i t e f f o r t and e f f o r t approaches one, so the un c e r t a i n t i e s become i n f i n i t e l y l a r g e . I t w i l l be shown in a l a t e r s e c t i on how the var ious con t ro l p o l i c i e s ac t to exp lo re the (q , E) space of the independent v a r i a b l e s i n the r e g r e s s i o n . 4.2 RANGE OF CONTROL POLICIES TESTED The range of con t ro l p o l i c i e s t es ted w i l l be the same as i n the prev ious chapter (see sec t i on 3 .2 ) . However, the e f f e c t of i n f requent parameter est imate updat ing and low we ight ing on recent data w i l l not be exp l o red . 4.3 ACTIVE ADAPTIVE POLICY FORMULATION In order to s pe c i f y the s t a t e equat ion f o r the Schaefer model, use i s made of the suggest ion by Schnute (1977) tha t catch per un i t e f f o r t can be thought of as po t en t i a l catch per u n i t e f f o r t . The s t a t e equat ion w i l l t he re fo re be w r i t t e n i n terms of q . , the po t en t i a l catch I l l per un i t e f f o r t . Using equat ion (4.0-1) and s u b s t i t u t i n g q t / c f o r N^ , the f o l l ow i ng equat ion i s ob ta ined: V i = q t + r q t - r \ / c k - c E t q t + v t q t Augmenting the s t a t e vec to r and de f i n i n g the c o n t r o l , l e t x^(k) = q^, x^ Ck) = r, Xg (k ) = r /ck , x^(k) = c, and u(k) = E .^ Then the s t a t e equat ions become X l ( k + 1 ) = x ^ k ) + x 2 ( k ) X l ( k ) - x 3 ( k ) x 1 ( k ) 2 - x 4 ( k ) u ( k ) x 1 ( k ) + v(k)Xl(k) x 2(k+l) = x 2 ( k ) x 3(k+l) = x 3 ( k ) (4 .3-1) x 4(k+l) = x 4 ( k ) The observa t ion equa t i on , w i th y (k ) as the observed catch per u n i t e f f o r t at t ime t , i s y (k ) •= x ^ k ) + w(k) (4.3-2) where i n the present study w(k) = 0 f o r a l l k. In t h i s case , as w i th the R i cker model, on ly parameter es t imates and covar iances are obta ined from the da ta . The method o f t rans forming these parameter d i s t r i b u t i o n s i n to d i s t r i b u t i o n s f o r the s t a t e vec to r de f ined above i s presented i n Appendix I I . The performance measure, as be fo re , i s o f the form N-l J =• E{L w [x (N) ] + z L. [ x ( k ) , u ( k ) , k]} IN k=0 K 1.12 Noting tha t C t = q t E t the components o f the above express ion are s p e c i f i e d as L M [ x ( N ) l = n A x A H ) - J U 2  N ~ 1 1 2 (4 .3-31 L k [ x ( k ) , u ( k ) , k] = - X l ( k ) u ( k ) [ l / ( l + d ) ] k " The f e a s i b l e c on t r o l s i n t h i s case are e f f o r t ra tes w i th the c o n s t r a i n t 0 < E^.<;oo> so tha t the se t of p o t en t i a l c on t r o l s i s {u(k)/0 < u(k) < «.} (4 .3-4) I t turns out tha t the c o n s t r a i n t on the con t ro l ( e f f o r t ) such tha t i t must be p o s i t i v e poses a l i m i t a t i o n on the a p p l i c a b i l i t y of the wide sense adapt ive a l go r i t hm . As d i scussed i n Chapter I I , the a l go r i t hm assumes no c on s t r a i n t s on con t ro l a c t i o n s . Although t h i s was p o t e n t i a l l y a problem i n the R i cker model f o rmu l a t i o n , i t r e su l t ed i n no cases o f unreasonable con t ro l a c t i ons being chosen. However w i th the Schaefer model f o rmu l a t i o n , a number of cases occurred i n which the adapt ive a l go r i t hm chose con t ro l a c t i ons which e xp l o i t e d the popu la t i on a t a high e f f o r t l e ve l even though the est imates i nd i c a t ed i t was a l ready over -e xp l o i t e d and the d e t e r m i n i s t i c con t ro l was zero e f f o r t . Such a p o l i c y i s understandable i f p o t en t i a l negat ive e f f o r t s can be app l i ed i n the fu tu re to r e c t i f y problems o f severe o v e r e x p l o i t a t i o n . Th is problem l ed to a c on s t r a i n t being p laced on the use of the adapt ive a l go r i t hm such tha t the range over which i n i t i a l con t ro l a c t i ons were exp lored was l i m i t e d 113 by the ex tent to which they invo lved zero e f f o r t s being app l i ed a long the nominal t r a j e c t o r y . 4.4 ACTIVE ADAPTIVE CONTROL LAW A f u l l p resen ta t i on of the con t ro l law f o r the a c t i v e adapt ive p o l i c y w i l l not be attempted i n t h i s case. The dimension of the problem ( s i x components of the data set determin ing unce r t a i n t y r a t he r than three i n the case of the R i cker model) prec ludes the comprehensive e xp l o r a t i o n presented i n the prev ious chapter . However, some fea tu res of the a c t i v e adapt ive con t ro l law f o r the Schaefer model w i l l emerge in succeeding s e c t i o n s . In p a r t i c u l a r the e f f e c t which the a c t i v e adapt ive con t ro l has i n break ing up c o r r e l a t i o n between catch per u n i t e f f o r t and e f f o r t , and i n exp l o r i ng the (q , E) space, w i l l be noted. 4.5 METHOD.OF POLICY EVALUATION The method, f o r p o l i c y eva l ua t i on and the no ta t i on f o l l ow those used i n Chapter I I I (see s e c t i on 3 . 5 ) . In t h i s case , ten years o f i n i t i a l data were p rov ided . These were generated by s p e c i f y i n g the mean and var iance o f the e f f o r t to be app l i ed over the i n i t i a l ten y ea r s . The i n i t i a l popu la t ion s i z e i n each case was tha t f o r which the mean e f f o r t used represented the e q u i l i b r i u m e f f o r t at tha t popu la t ion s i z e and f o r the parameters chosen to generate the da ta . Thus low var iance in e f f o r t g ene ra l l y r e su l t e d i n a f a i r l y narrow range o f popu la t i on s i z e s (hence catches per u n i t e f f o r t ) being exp lo red . By generat ing i n i t i a l 114 parameter d i s t r i b u t i o n s in t h i s way, on ly a po r t i on of the po t en t i a l s i t u a t i o n s which might a r i s e i n rea l f i s h e r i e s were exp l o red . 4.6 POLICY COMPARISON: EFFECT OF PRIOR DATA The r e l a t i v e performances o f the three types o f adapt ive p o l i c y across the range o f d i s t r i b u t i o n s o f i n i t i a l parameter es t imates are shown i n F igure 23a. P l o t t ed are the expected va lues across each p r i o r d i s t r i b u t i o n as a percentage of the expected va lues of the opt imal p o l i c y , over a range o f i n i t i a l s tock s i z e s expressed as a f r a c t i o n of the upper e q u i l i b r i u m stock s i z e . The parameters used to generate the i n i t i a l data sets were r = 0 . 5 , k = 2.0 and c = 0 . 1 . The va lue f o r a was se t v a t 0 .1 . Discount ra te was ze ro , and the time hor i zon was 50 yea r s . The e f f e c t o f the i n i t i a l s tock s i z e (which determines the mean of the p r i o r e f f o r t l e v e l s ) i s not c l e a r . Both the a c t i v e and pass ive adapt ive p o l i c i e s seem to perform equa l l y we l l across mean p r i o r e f f o r t s and both perform we l l r e l a t i v e to the opt imal p o l i c y . There i s a suggest ion of a s l i g h t decrease i n performance (not s t a t i s t i c a l l y s i g n i -f i c a n t ) where the i n i t i a l (and a l so approx imate ly the mean) p r i o r stock s i z e i s a t one h a l f c a r r y i n g c apa c i t y . In t h i s case the i n i t i a l parameter es t imates were such tha t the d e t e r m i n i s t i c e f f o r t l e v e l chosen i n the f i r s t year f e l l w i t h i n the range o f the p r i o r e f f o r t l e v e l s . In a l l o ther cases the i n i t i a l e f f o r t chosen was we l l ou t s i de the p r i o r range (Table V I ) . The non-adapt ive p o l i c y does poor ly across a l l p r i o r e f f o r t l e v e l s , w i th no c l e a r t rend apparent. The e s t ima t i on performance of the three p o l i c i e s i s p l o t t e d i n F igures 23b and 24. I t i s c l e a r t ha t there i s a d i s t i n c t improvement i n 115 Figure 23. Comparison o f p o l i c y performance and e s t ima t i on performance f o r optimum e f f o r t E*. Po in ts are p l o t t e d aga ins t the mean popu la t i on s i z e i n the i n i t i a l data se t r e l a t i v e to the c a r r y i n g capac i t y k. EO^ i s the mean long term performance as a percentage of the opt imal performance. E* i s the mean e r r o r i n e s t ima t i ng the optimum e f f o r t . 1 100 .125 . 25 . 5 . 75 .875 In i t ia l Population S ize as Fraction of k PPK 117 INITIAL MEAN POPULATION SIZE MEAN EFFORT TO GENERATE INITIAL DATA SET ESTIMATED POPULATION SIZE OPTIMAL DETERMINISTIC EFFORT 0.25 0.50 1.00 1.50 1.75 4.375 3.750 2.500 1.250 0.625 0.26 0.51 1.00 1.50 1.76 0.0 0.0 2.50 4.59 4.92 I n i t i a l parameter es t imates i n a l l cases are: r = 0.50 k = 2.00 c = 0.10 0.10 2.50 -Table V I . Optimal d e t e r m i n i s t i c e f f o r t as a f un c t i on of i n i t i a l popu la t ion s i z e s . 118 F igure 24. Comparison of e s t ima t i on performance f o r the r ,k and c parameters o f the Schaefer model, po in t s are p l o t t e d aga i n s t the mean popu la t i on s i z e in the i n i t i a l data set r e l a t i v e to the c a r r y i n g capac i t y k. r ,k and c are the mean e r r o r s i n the est imates o f the r e spec t i v e parameters. 119 120 parameter es t imates and est imates of optimum e f f o r t l e v e l s by both the a c t i v e and pass ive adapt ive p o l i c i e s r e l a t i v e to the non-adapt ive p o l i c y . The est imates by the a c t i v e adapt ive p o l i c y are a l i t t l e be t t e r than by the pass ive adapt ive p o l i c y , but again the d i f f e r en c e i s not s t a t i s t i c a l l y s i g n i f i c a n t . There were very few cases of e s t ima t i on f a i l u r e by e i t h e r p o l i c y . Some p a r t i c u l a r cases w i l l be d i scussed i n the next s e c t i o n . F igures 25, 26 and 27 i l l u s t r a t e more c l e a r l y the e f f e c t the p r i o r d i s t r i b u t i o n s have on p o l i c y performance. The p l o t s are i n ( q , E) space and each po in t represents a combinat ion of optimum catch per un i t e f f o r t q* and optimum e f f o r t E* chosen at random from each o f three of the p r i o r d i s t r i b u t i o n s . These f i g u r e s i n d i c a t e the r e l a t i v e performance P0^ obta ined by each p o l i c y f o r each (q* , E*) combinat ion. A l so p l o t t ed are the mean ( q , E) o f each p r i o r d i s t r i b u t i o n and the optimum (q* , E*) based on the mean o f the d i s t r i b u t i o n o f i n i t i a l e s t ima tes . Several i n t e r e s t i n g f ea tu res emerge. The f i r s t concerns the exp lana-t i o n f o r the p a r t i c u l a r d i s t r i b u t i o n of po in t s p l o t t e d . Re f e r r i ng back to F igure 22, i t i s ev ident tha t i n ( q , E) space (q* , E*) f a l l s midway along the l i n e AB j o i n i n g the i n t e r c ep t s of the plane w i th the q and E axes. Th is l i n e represents the sequence o f combinat ions o f q t and E t at which the observa t i on (q^+.j i ^ t ^ t W 1 '^ be z e ro , tha t i s , no change i n popu la t ion s i z e . S ince the p r i o r d i s t r i b u t i o n s i nvo lved here represent cases o f approximate e q u i l i b r i u m wi th l i t t l e change i n popu la t i on s i z e , the mean (q~, E) o f the p r i o r data should l i e approx imate ly on the l i n e corresponding to AB f o r each combinat ion (q* , E*) chosen at random from the p r i o r d i s t r i b u t i o n of est imates based on the da ta . T h i s , i n f a c t , occurs i n F igures 25, 26 and 27. 121 Exp lanat ion f o r f i g u r e s 25,26 and 27. Each f i g u r e shows the d i s t r i b u t i o n o f pa i r s (q*,E*) o f va lues f o r opt imal catch per un i t e f f o r t and opt imal e f f o r t based on the sets of parameters chosen from a p a r t i c u l a r p r i o r d i s t r i b u t i o n o f parameter e s t ima tes . The mean p r i o r popu la t i on s i z e r e l a t i v e to the c a r r y i n g c apa c i t y (PPK) f o r the data which generated each d i s t r i b u t i o n i s i nd i c a t ed on each f i g u r e c a p t i o n . Each po in t i n the d i s t r i b u t i o n represents a p a r t i c u l a r s imu l a t i on and the performance of the three adapt ive p o l i c i e s r e l a t i v e to the opt imal performance i s shown. The performance o f the non-adapt ive p o l i c y i s i n d i c a t ed as f o l l ow s : 90% ( . ) ; 80% ( x ) ; 70% ( + ) ; 60% ( A ) ; 50% ( O ) ; 40% ( • ) ; 30% ( • ) ; 20% ( •:•) ; 10% ( A ) . The performance o f the a c t i v e and pass ive adapt ive p o l i c i e s i s shown next to each po i n t , e .g . 75 i n d i c a t e s tha t the performance o f the a c t i v e adapt ive p o l i c y was between 70% and 80% and tha t o f the pass ive adapt ive p o l i c y between 50% and 60%. A l l unnumbered po in t s represent cases i n which both p o l i c i e s achieved be t t e r than 90% of the opt imal performance. The symbol O shows the opt imal (q*,E*) f o r the mean of the i n i t i a l es t imates and * shows the mean (q,iT) o f the catch per un i t e f f o r t and the e f f o r t i n the i n i t i a l data set which generated each d i s t r i b u t i o n . 122 F igure 25. P o l i c y performance i n (q*,E*) space: case PPK = 0.125 For exp lana t i on o f f i g u r e see page 121. 123 .44 r .40 .36 .32 .28 .24 .20 •16 .12 .08 .04 \ 98-V^ A 0 98 • • • • • 881 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 E* 124 F igure 26. P o l i c y performance in (q*,E*) space: case PPK = 0.500 Po in ts l a b e l l e d a,b,c and d i n d i c a t e cases examined in a l a t e r s e c t i o n . For exp lana t i on of f i g u r e see page 121. 125 .44 40 .36 .32 .28 .24 .20 .16 .12 .08 .04 '98 c 96* '98 98 75 78 89 • A O • 77 d 98 . a •0 1 2 3 4 5 6 7 8 9 10 11 .12 13 14 15 E* 126 F igure 27. P o l i c y performance in (q*,E*) space: case PPK = 0.875 For exp lana t i on of f i g u r e see page 121. 127 .A A .AO .36 .32 .28 .24 .20 .16 .12 .08 .04 ' 78 • 7 6 4 B 8 D 8 9 89. a yt6 89 » 88 9.8 o A A I - - • * » * ' ~* 1 2 3 A 5 6 7 8 9 10 11 12 13 1A 15 E* 128 Fo l lowing on from t h i s e xp l o r a t i o n i t i s c l e a r tha t the p o s i t i o n o f (q , E) r e l a t i v e to (q* , E*) o f the p r i o r est imates w i l l determine to what extent the va lues f o r E* randomly se l e c t ed from the p r i o r d i s t r i b u -t i o n w i l l be below the E* p red i c ted by the e s t ima tes . Thus in F igure 25 cf i s much lower than q* and E i s about twice E* so t ha t the minimum va lue which any E* can take i s not much l e s s than E*. On the o ther hand, i n F igure 27 q i s much h igher than q*, E i s about one quar te r o f E* and many randomly chosen values of E* l i e below E*. S ince the performance o f the non-adapt ive p o l i c y w i l l be c l o s e l y t i e d to the dev i a t i o n of the t rue E* from E*, i t should perform more poor l y i n the l a t t e r case , wh ich , i n f a c t , i t does. The l a s t po in t which i s ev ident from these f i g u r e s i s tha t the a c t i v e and pass ive adapt ive p o l i c i e s not on ly perform we l l i n an expected sense, they a l so perform wel l over near l y a l l s imu l a t i on t r i a l s . The d i s t r i b u t i o n of va lues of PO^ f o r the three p o l i c i e s are shown i n F igure 28. Very r a r e l y does e i t h e r adapt ive p o l i c y perform poo r l y . Some o f the cases in which they do perform poor l y are i n v e s t i g a t ed i n the next s e c t i o n . 4.7 POLICY COMPARISON: CASE PPK = 0 .5 , S £ = 0.2 Th is s e c t i on w i l l exp lore i n more d e t a i l some of the s imu la t i ons shown as po in t s i n F igure 26. These are a l l cases i n which the i n i t i a l e f f o r t chosen by the pass ive adapt ive p o l i c y i s w i t h i n the range o f the e f f o r t s a l ready chosen. Severa l s i t u a t i o n s corresponding to d i f f e r e n t . stages i n the development of a f i s h e r y w i l l be i n v e s t i g a t e d . A l so cases i n which e i t h e r o r both the a c t i v e or pass ive adapt ive p o l i c i e s f a i l e d 129 F igure 28. D i s t r i b u t i o n of p o l i c y performances r e l a t i v e to PPK. PPK i s the mean popu la t ion s i z e i n the i n i t i a l data se t r e l a t i v e to the c a r r y i n g capac i t y k. Frequencies are p l o t t e d aga ins t p e r c en t i l e s o f performances r e l a t i v e to the opt imal performance. AAP PAP 130 NAP 50 r 40 30 20 10 PPK =0.125 9 8 7 6 5 4 3 2 1 9 8 7 6 5 4 3 2 1 9 8 7 6 5 4 3 2 1 50 40 30 20| 10 PPK =0.50 9 8 7 6 5 4 3 2 1 9 8 7 6 5 4 3 2 1 9 8 7 6 5 4 3 2 50 40 30 20 10 PPK =0.875 9 8 7 6 5 4 3 2 1 9 8 7 6 5 4 3 2 1 9 8 7 6 5 4 3 2 1 131 to est imate (q* , E*) w i th reasonable accuracy w i l l be exp l o red . The case l a be l s correspond to the po in ts l abe l ed i n F igure 26. Case a Th is represents a s i t u a t i o n which might be encountered in the e a r l y stages o f development d f a f i s h e r y . The i n i t i a l es t imate i s tha t r = 0 .5 , k = 2 .0 , c = 0 . 1 , q* = 0.1 and E* = 2 .5 . The mean i n i t i a l e f f o r t l e v e l from the p r i o r data i s 2.55 and the catches per un i t e f f o r t have f l u c t u a t ed around a l e ve l o f 0.10. The es t imate i s tha t the popu la t ion s i z e i s 1.00 and the d e t e r m i n i s t i c e f f o r t to use i s 2.50. However the " r e a l " s i t u a t i o n i n the f i s h e r y i s as f o l l o w s : r = 0.84, k = 2.42 and c = 0.046. The t rue q* i s . 0.056 and E* = 9.08. The cu r ren t popu la t i on s i z e i s 2 .31 , t ha t i s , c l o se to the c a r r y i n g c apa c i t y . Ev i den t l y the f i s h e r y i s s eve re l y underexp lo i ted but the i n i t i a l est imates suggest tha t i t i s a t MSY (maximum sus t a i nab l e y i e l d ) . The i n i t i a l data po in t s i n ( q , E) space, together w i th the t r a j e c t o r i e s t raced by the a c t i v e and pass ive adapt ive p o l i c i e s are shown in F i gu re .29 . I t i s ev ident tha t both p o l i c i e s even tua l l y a r r i v e i n the neighbour-hood o f (q* , E*). However the a c t i v e adapt ive p o l i c y a r r i v e s sooner and exp lores a much l a r g e r reg ion o f ( q , E) space than the pass ive adap-t i v e p o l i c y . The l a t t e r seems to be trapped i n the reg ion o f the i n i t i a l data se t o f the f i r s t f i v e y e a r s , e ven tua l l y break ing out and f i n d i n g i t s way (though not d i r e c t l y ) to (q* , E*). The a c t i v e adapt ive p o l i c y , on the o ther hand, breaks away immediately from the reg ion de f ined by the i n i t i a l d a t a , though i n i t i a l l y i n the wrong d i r e c t i o n . I t f i r s t exp lores reg ions o f low (zero) e f f o r t , then inc reases the e f f o r t to high l e v e l s , d r i v i n g the t r a j e c t o r y out and around (q* , E*) and even tua l l y 132 Exp lanat ion f o r f i g u r e s 29,32,35 and 38. Each f i g u r e shows the i n i t i a l set o f pa i r s of observed ca tch per u n i t o f e f f o r t and e f f o r t (the open c i r c l e s ) and the subsequent pa i r s of observed va lues (q,E) obta ined by the a c t i v e adapt ive p o l i c y (+) and the pass ive adapt ive p o l i c y (x) over the f i r s t ten years o f s imu l a t i o n . Success ive po in ts f o r each p o l i c y are l i n k e d to show t r a j e c t o r i e s over t ime. The b lack s t a r i n d i c a t e s the optimum va lues (q*,E*) f o r catch per u n i t e f f o r t and e f f o r t i n each case . 133 Figure 29. Management t r a j e c t o r i e s in (q,E) space: case a ) . For exp lana t i on of f i g u r e see page 132. 134 E 135 s e t t l e s c l o se to i t . In the process i t per turbs the de t e rm in s t i c e f f o r t f i r s t downward, then upward i n order to exp lo re a wide range of ( q , E) va lues . The sequence o f es t imates (q* , E*) obta ined by each p o l i c y i s shown i n F igure 30. The a c t i v e adapt ive p o l i c y l ea rns the t rue va lues q u i c k l y and w e l l . The pass ive adapt ive p o l i c y takes longer to break away from the i n i t i a l est imates and then l ea rns them l e s s w e l l , perhaps because i t exp lores a sma l l e r po r t i on o f the (q , E) space. The success ive 2 2 2 va lues of S q , S- and r q E " f o r each p o l i c y are shown i n Table V I I . The var iances i n both the catch per u n i t e f f o r t and e f f o r t i nc rease much more r a p i d l y w i th the a c t i v e adapt ive p o l i c y , and to a h igher va l ue . 2 The eventual c o r r e l a t i o n ">s h igher f o r the a c t i v e adapt ive p o l i c y , though n e i t h e r . i s h igh enough to s e r i o u s l y a f f e c t the es t ima tes . The cumulat ive catch f o r the two p o l i c i e s p lus the opt imal p o l i c y i s shown i n F igure 31. As i n many cases w i th the R i cke r model, the a c t i v e adapt ive p o l i c y s a c r i f i c e s shor t - te rm catch to improve parameter es t imates and hence long-term c a t c h . However both p o l i c i e s ach ieve be t t e r than 90% of the performance of the opt imal p o l i c y over the 50 year time ho r i z on . The non-adapt ive p o l i c y achieves on ly 46%. Case b This case represents a s i t u a t i o n where the stock i s i n i t i a l l y o ve r exp l o i t ed and has been f o r some t ime. The i n i t i a l es t imates are the same as i n case a , and indeed the i n i t i a l est imates are the same i n a l l the cases t es ted i n t h i s s e c t i o n . The t rue parameters o f the stock are r = 0.83, k = 2.24 and c = 0.289, w i th q* = 0.324 and E* = 1.43. 136 F igure 30. Comparison o f es t imates o f q* and E* f o r the a c t i v e and pass ive adapt ive p o l i c i e s : case a ) . Optimum catch per un i t e f f o r t q* and optimum e f f o r t E* are a l so shown. • 11 r .OA • •03 • .02 • .01 • 0 1 2 3 A 5 6 7 8 9 10 10 2 1 0 1 2 3 A 5 6 7 8 9 10 T i m e . 138 STAGE q 2 AAP q c 2 PAP q «.2 AAP • b E c 2 PAP * E r 2 AAP R 2 P A P 1 0.0001 0.0001 0.0636 0.0636 0.0110 0.0110 2 0.0001 0.0001 0.6413 0.0574 0.0119 0.0117 3 0.0001 0.0001 1.0254 0.0565 0.1723 0.0009 4 0.0001 0.0001 1.2831 0.0526 0.1397 0.0001 5 0.0001 0.0001 8.2997 0.0715 0.0508 0.0004 6 0.0004 0.0001 10.9239 0.0759 0.1144 0.0029 7 0.0005 0.0001 10.3418 0.3976 0.0461 0.0066 8 0.0006 0.0001 13.8850 1.6736 0.1152 0.1486 9 0.0006 0.0001 14.6008 1.6425 0.1740 0.0546 10 0.0006 0.0001 15.8309 5.5417 0.2316 0.0463 20 0.0007 0.0005 15.8987 10.0540 0.4121 0.4319 Table V I I . Development i n v a r i a b i l i t y o f catch per un i t e f f o r t and e f f o r t and c o r r e l a t i o n between them. Case a . 139 Figure 31. Comparison o f cumulat ive catches f o r the o p t i m a l , a c t i v e and pass ive adapt ive p o l i c i e s : case a ) . 141 The cu r ren t popu la t i on s i z e i s 0.37 so the stock i s badly o v e r e xp l o i t e d . The i n i t i a l data po in ts and t r a j e c t o r i e s f o r t h i s case are shown, in F igure 32. In t h i s case the i n i t i a l a c t i on by the a c t i v e adapt ive p o l i c y (to swi tch o f f the harves t ) i s the c o r r e c t one, and t h i s p o l i c y a l lows the stock to recover from a low popu la t i on s i z e r i g h t up to the unf i shed e q u i l i b r i u m stock s i z e . At t h i s p o i n t , the harvest r a te i s increased and the stock i s d r i ven down to the neighbourhood o f (q* , E*). Once again the a c t i v e adapt ive p o l i c y exp lores a l a r g e , r e g i o n o f ( q , E) space. The pass ive adapt ive p o l i c y spends l e s s time near the o r i g i n a l data se t i n t h i s case and exp lores a cons ide rab le po r t i on of ( q , E) space on i t s way to (q* , E*), though again much l e s s than the a c t i v e adapt ive p o l i c y . A l so i t does not a l l ow the stock to recover as q u i c k l y as i t shou ld . The sequence of (q* , E*) est imates shown in F igure 33 i n d i c a t e tha t the pass ive adapt ive p o l i c y l ea rns about (q* , E*) more r a p i d l y than i n case a , and i t seems to l e a rn q* more r a p i d l y than the a c t i v e adapt ive p o l i c y , though i t lags we l l behind i n e xp l o r i ng the space (Table V I I I ) . The sequence o f cumulat ive catches (F igure 34) show tha t again the a c t i v e adapt ive p o l i c y s a c r i f i c e s shor t - te rm f o r long-term ga i n . Both p o l i c e s f i n a l l y ach ieve 95% o f the opt imal performance, wh i l e the non-adapt ive p o l i c y scores on ly 50%. Case c The s i t u a t i o n i n t h i s case i s somewhat s i m i l a r to case b i n tha t the stock i s i n i t i a l l y o v e r e xp l o i t e d . The t rue parameters a r e , however, 142 Figure 32. Management t r a j e c t o r i e s in (q,E) space: case b ) . For exp lana t i on o f f i g u r e see page 132. E 144 Figure 33. Comparison o f es t imates o f q* and E* f o r the a c t i v e and pass ive adapt ive p o l i c i e s : case b ) . Optimum catch per un i t e f f o r t q* and optimum e f f o r t E* are a l s o shown. Time 146 STAGE c-2 AAP * q c2 PAP q c2 AAP o E C2 PAP 3 E r2 AAP r 2 PAP 1 0.0001 0.0001 •'• 0.0636 0.0636 0.0110 0.0110 2 0.0001 0.0001 0.6413 0.0574 0.0119 0.0117 3 0.0005 0.0001 1.0254 0.0565 0.3848 0.0324 4 0.0021 0.0002 1.0851 0.0520 0.3435 0.0314 5 0.0055 0.0002 1.1327 0.5138 0.3632 0.1495 6 0.0122 0.0003 1.2950 0.7133 0.4648 0.0262 7 0.0256 0.0015 1.4072 0.6725 0.4919 0.0263 8 0.0396 0.0026 1.3261 0.7293 0.2499 0.1304 9 0.0377 . 0.0050 1.2487 0.7702 0.2494 0.2249 10 0.0360 0.0084 1.2048 0.7456 0.2569 0.2070 20 0.0259 0.0124 0.8004 0.5617 0.2588 0.3034 Table V I I I . Development i n v a r i a b i l i t y o f catch per un i t e f f o r t and e f f o r t and c o r r e l a t i o n between them. Case b. 147 F igure 34. Comparison o f cumulat ive catches f o r the o p t i m a l , a c t i v e and pass ive adapt ive p o l i c i e s : case b ) . Cumulative Catch 149 qu i t e d i f f e r e n t , being r = 0.20, k = 5.29, and c = 0.068 w i th q* = 0.179 and E* = 1.49. The i n i t i a l s tock s i z e i s 1.58. The i n t e r e s t i n g f ea tu re i n t h i s case i s tha t the pass ive adapt ive p o l i c y f a i l s bad ly a t l e a r n i n g (q* , E*). The i n i t i a l data and t r a j e c t o r i e s f o r t h i s case are shown i n F igure 35. Two fea tu res are o f i n t e r e s t . F i r s t , the pa t te rn o f e x p l o r a t i o n by the a c t i v e adapt ive p o l i c y i s qu i t e d i f f e r e n t than i n case b. The p o l i c y inc reases the harvest r a t e before the catch per un i t e f f o r t reaches q*. However i n t h i s case the pe r tu rba t i on i s s t i l l towards reduced e f f o r t as the d e t e r m i n i s t i c con t ro l c a l l e d f o r an even h igher e f f o r t than was chosen. However the next e f f o r t chosen i s d e l i b e r a t e o v e r e x p l o i t a t i o n ; The second f ea tu re i s the t r a j e c t o r y of the pass ive adapt ive p o l i c y which never dev ia tes very f a r from the i n i t i a l space exp l o red , and even tua l l y s e t t l e s down we l l w i t h i n i t (a l a t e r par t o f the t r a j e c t o r y , not shown). The sequence of est imates (q* , E*) i s shown in F igure 36. Not i ce tha t the pass ive adapt ive est imates tend to d r i f t i n the wrong d i r e c t i o n , . from the i n i t i a l e s t ima tes , and tha t the a c t i v e adapt ive es t imate f o r q* goes negat ive i n one year but q u i c k l y r e cove r s . The s t a t i s t i c s f o r S_ , M 2 2 Sj: and r r^.-.. are presented in Table IX.and the cumulat ive catches i n F igure 37. A f t e r ten years the cumulat ive catch f o r the pass ive adapt ive p o l i c y i s a c t u a l l y much h igher than f o r the opt imal p o l i c y . However the stock has been d r i ven to very low l e v e l s (5% of k, the un f i shed e q u i l i -br ium). A f t e r 20 years the performance o f the pass ive adapt ive p o l i c y r e l a t i v e to the opt imal po l i cy ; i s 80%, and a f t e r 50 years i t i s on ly 67%. At t h i s stage the a c t i v e adapt ive p o l i c y had achieved 96% of the opt imal performance. 150 Figure 3 5 . Management t r a j e c t o r i e s in ( q , E ) space: case c ) . f o r exp lana t i on o f f i g u r e see page 1 3 2 . 151 152 Figure 36. Comparison of est imates o f q* and E* f o r the a c t i v e and pass ive adapt ive p o l i c i e s : case c ) . Optimum catch per un i t e f f o r t q* and optimum e f f o r t E* are a l s o shown. 153 Time 154 STAGE _ 2 AAP q 2 PAP q cz AAP ( q <2 PAP * E 2 AAP r 2 PAP 1 r z 1 0.0001 0.0001 0.0636 0.0636 0.0110 0.0110 2 0.0001 0.0001 0.6413 0.0574 0.0119 0.0117 3 0.0001 0.0001 1.0254 0.0565 0.1614 0.0075 4 0.0002 0.0001 1.2831 0.0519 0.3100 0.0050 5 0.0004 0.0001 2.6544 0.1358 0.1407 0.0125 6 0.0004 0.0001 2.4790 0.1721 0.1456 0.0181 7 0.0004 0.0001 2.6457 0.1752 0.0906 0.0345 8 0.0004 0.0001 2.7549 0.2093 0.0385 0.0108 9 10 0.0007 0.0001 2.8599 0.6392 0.0199 0.0088 20 0.0010 0.0002 2.4833 1.2008 0.0032 0.0741 Table IX. Development i n v a r i a b i l i t y o f catch per un i t e f f o r t and e f f o r t and c o r r e l a t i o n between them. Case c. 155 F igure 37. Comparison o f cumulat ive catches f o r the o p t i m a l , a c t i v e and pass ive adapt ive p o l i c i e s : case c ) . Cumulative Catch 157 Case d The f i n a l case cons idered here i s one i n which both the a c t i v e and pass ive adapt ive p o l i c i e s f a i l to est imate (q* , E*) c o r r e c t l y . The s i t u a t i o n i s somewhat s i m i l a r to case a i n t ha t the stock i s i n i t i a l l y unde rexp lo i t ed . The parameter va lues f o r the s imu l a t i on are r = 0 .51 , k = 2.29, and c = 0.054, w i th q* = 0.062 and E* = 4 .77. The i n i t i a l s tock s i z e i s 1.99, t ha t i s , c l o se to unf i shed e q u i l i b r i u m . The i n i t i a l data and t r a j e c t o r i e s are shown i n F igure 38. The a c t i v e adapt ive p o l i c y seems to exp lo re the (q , E) space qu i t e w e l l , a l though there i s on ly one obse rva t i on a t high l e v e l s o f e f f o r t . The f i n a l par t o f the t r a j e c t o r y w i th zero e f f o r t and stock s i z e i n c r ea s i ng from a f a i r l y low l e v e l should ensure tha t r i s we l l e s t ima ted , and a l so r /ck . The pass ive adapt ive p o l i c y again spends qu i t e a long t ime near the o r i g i n a l data space and exp lo res much l e s s space than the a c t i v e adap-t i v e p o l i c y . F igure 39 shows the sequence o f est imates (q* , E*). The a c t i v e adapt ive p o l i c y achieves a good es t imate s t r a i g h t away but t h i s p rogress-i v e l y d e t e r i o r a t e s u n t i l the est imate a f t e r 10 years i s very c l o se to the i n i t i a l e s t imate . Examination o f the ac tua l parameter est imates shows tha t r, c and r/ck are a l l underest imated and these es t imates never improve much though they change i n va lue r e l a t i v e to one another i n the wrong d i r e c t i o n . t o est imate e i t h e r q* or E*. The f i n a l var iances i n the est imates taken from the covar iance mat r i x are qu i t e low i n d i c a t i n g tha t the parameters should have been learned b e t t e r . E v i den t l y the p a r t i c u l a r random sequence i nvo l ved l ed to the poor es t imates . 158 Figure 38. Management t r a j e c t o r i e s i n (q,E) space: case d ) . For exp lana t i on o f f i g u r e see page 132. 159 E 160 F igure 39. Comparison o f es t imates o f q* and E* f o r the a c t i v e and pass ive adapt ive p o l i c i e s : case d ) . Optimum catch per un i t e f f o r t q* and optimum e f f o r t E* are a l so shown. 0 1 2 3 4 . 5 5 7 8 9 10 Time 162 Table X presents the in fo rmat ion on v a r i a t i o n in the independent v a r i a b l e s . The va lue o f r f o r the a c t i v e adapt i ve p o l i c y i s h igh r e l a t i v e to tha t f o r the pass ive adapt ive p o l i c y . Th is may have con t r i bu ted to the poor .es t imates . The cumulat ive catch i s shown i n F igure 40. The f i n a l performance r e l a t i v e to the opt imal p o l i c y was 74% and 79% f o r the a c t i v e and pass ive adapt ive p o l i c i e s r e s p e c t i v e l y . Fur ther Comments There were a number o f o ther cases in which e i t h e r or both the a c t i v e and pass ive adapt ive p o l i c i e s performed poor l y r e l a t i v e to the t rue opt imal p o l i c y . These i n v a r i a b l y i nvo lved cases i n which t r a n s i t i o n a l parameter e s t ima tes , p a r t i c u l a r l y very low est imates o f c, l ed to the cho ice o f very l a r ge con t ro l e f f o r t s which drove the stock very low and which subsequent ly i nvo lved very long recovery per iods w i th no ha rves t . Th is cou ld be p a r t i c u l a r l y severe when the t rue growth r a t e r was a l so low. These cases serve to under l i ne the extent to which the s imple models o f both f i s h and p a r t i c u l a r l y e f f o r t "dynamics depart from what occurs i n the " r ea l wo r l d " . The assumption here i s tha t cho ice o f e f f o r t i s t o t a l l y f l e x i b l e , so tha t i t may inc rease or decrease ten o r one hundred f o l d from one year to the next;. In p r a c t i c e , there are u s ua l l y very severe r e s t r i c t i o n s on the a b i l i t y o f management to modify r egu l a t i on s from one year to the next , o r on f l e e t s to respond q u i c k l y enough to very l a r ge inc reases i n a l l owab l e quotas. These po in ts w i l l be d i scussed f u r t h e r i n the conc lud ing chapter . 163 STAGE ^2 AAP q . o2 PAP q <;2 AAP b E cz PAP 3 E 2 AAP r r 2 PAP 1 0.0001 0.0001 0.0636 0.0636 0.0110 0.0110 2 0.0001 0.0001 0.6413 0.0574 0.0119 0.0117 3 0.0001 0.0001 1.0254 0.0565 0.0628. 0.0203 4 0.0002 0.0001 8.2879 0.0519 0.2996 0.0181 5 0.0007 0.0001 8.2362 0.9587 0.2415 0.2220 6 0.0009 0.0002 8.1195 0.8942 0.2778 0.0778 7 0.0011 0.0003 7.9632 0.8366 0.3118 0.0654 8 0.0011 0.0004 7.7840 1.2463 0.3394 0.0071 9 0.0011 0.0003 7.5926 1.4457 0.3557 0.0270 10 0.0010 0.0003 7.3960 1.3753 0.3425 0.0237 20 0.0007 0.0002 5.0051 1.0861 0.3468 0.0392 Table X. Development in v a r i a b i l i t y o f catch per un i t e f f o r t and e f f o r t and c o r r e l a t i o n between them. Case d. 164 Figure 40. Comparison of cumulat ive catches f o r the o p t i m a l , a c t i v e and pass ive adapt ive p o l i c i e s : case d ) . Cumulative Catch NJ w o t> o o 3' fD 166 4.8 APPLICATION TO A "REAL" FISHERY In a study a l ready c i t e d , H i l bo rn (1979) app l i ed a range, o f e s t i -mation techniques coupled w i th d i f f e r e n t pass ive adapt ive con t ro l systems to a s imulated f i s h e r y w i th r e l a t i v e l y complex dynamics (age-dependent natura l m o r t a l i t y and f e cund i t y , lagged r e c ru i tmen t , e t c . ) . He found tha t f o r h i s "Model A" s t o ck , a long l i v e d , slow growing spe c i e s , a l l the c on t r o l - e s t ima t i o n schemes performed poor ly when s t a r t i n g from cond i -t i on s o f o v e r e x p l o i t a t i o n , and i n many cases the e s t ima t i on schemes were unable even to ob ta i n p o s i t i v e parameter e s t ima tes . He designed an "adapt ive probing p o l i c y " to exp lo re the (q , E) space and used the data generated as i n i t i a l c ond i t i ons f o r one o f h i s con t ro l p o l i c i e s . In t h i s case the parameter es t imates improved d r a m a t i c a l l y and the r e s u l t i n g performance was very c l o se to o p t ima l . He suggested tha t a c t i v e adapt ive management p o l i c i e s cou ld prove most worthwhi le i n t h i s type of s i t u a t i o n . An attempt was made to apply the a c t i v e adapt ive p o l i c y o u t l i n e d i n s e c t i on 4.3 to the f i s h e r y model developed by H i l b o r n , f o r the cases in which h i s pass ive adapt ive con t ro l p o l i c i e s f a i l e d . The a c t i v e adapt ive p o l i c y d i d manage to ach ieve p o s i t i v e parameter es t imates but gene ra l l y performed poor ly and f a i l e d to recogn ize tha t the stock was badly o v e r e xp l o i t e d . The main problem seemed to be w i th the e s t ima t i on scheme, which was the one ou t l i n ed i n s e c t i on 4.1 and the same as H i l b o r n ' s " l i n e a r e s t ima to r " . Apparent ly t h i s es t imator does not work we l l when the assumptions i m p l i c i t i n i t are v i o l a t e d . Uh ler (1978) has po inted out tha t t h i s es t imato r w i l l g ive b iased r e s u l t s when there are s t o c h a s t i c e r r o r s i n the ca tch f u n c t i o n . In general the r parameter was badly overest imated and the est imates suggested tha t the stock was underexp lo i t ed . 167 ra the r than o v e r e x p l o i t e d , . l ead ing the a c t i v e adapt ive p o l i c y to probe towards h igher r a the r than lower e f f o r t s . The other problem was w i th the adapt ive a l go r i t hm i t s e l f . In some cases i t chose high e f f o r t l e v e l s even when the est imates i n d i c a t ed tha t the stock was badly over -e x p l o i t e d . Ev i den t l y f u r t h e r work on a p p l i c a t i o n of the adapt ive a l go r i t hm to t h i s problem i s j u s t i f i e d , g iven the poor performance o f the pass ive adapt ive p o l i c i e s . P o s s i b l e d i r e c t i o n s f o r f u r t h e r research i n t h i s area w i l l be o u t l i n e d i n the conc lud ing chapter . 4.9 DISCUSSION The problem o f managing a stock f o r which on ly catch and e f f o r t data are a v a i l a b l e i s a d i f f i c u l t one. In the s tock / rec ru i tmen t problem cons idered in the prev ious chap te r , the stock cou ld be managed so as to ach ieve a known and des i red s tock s i z e . In the more complex case , not on ly i s the cu r ren t s tock s i z e unknown, but the e f f e c t of app ly ing the known con t r o l - - - ' : r e f f o r t -- i s a l s o un ce r t a i n . Yet , f o r the range o f s i t u a t i o n s cons idered so f a r , the pass ive adapt i ve p o l i c y seems to perform be t t e r than i t d i d i n managing the s imp le r s tock and rec ru i tment f i s h e r y . C e r t a i n l y the occas iona l very poor performances by t h i s p o l i c y were f a r more ra re and apparent ly not as severe . The exp lana t i on f o r t h i s l i e s perhaps i n the very f a c t tha t the e f f e c t of a g iven con t ro l a c t i on i s un ce r t a i n . With the Ricker. model, smal l changes in the magnitude o f a or g do not gene ra l l y l ead to very l a r ge changes i n the opt imal d e t e r m i n i s t i c escapement. In f a c t the opt imal 168 p o l i c y was found to be f a i r l y i n s e n s i t i v e to these parameters. However w i th the Schaefer model smal l changes i n the parameters, p a r t i c u l a r l y i n the es t imate o f the c a t c h a b i l i t y c o e f f i c i e n t c , can lead to qu i t e s i g n i f i c a n t changes i n the e f f o r t l e v e l s chosen. These smal l pe r tu rba -t i o n s to the e f f o r t can a l s o lead i n turn to s i g n i f i c a n t changes i n s tock s i z e and thus to changes i n the o the r independent v a r i a b l e i n the r eg re s -i o n , the catch per un i t e f f o r t . In three of the four cases presented i n s e c t i on 4.7 the pass ive adapt ive p o l i c y spent a s ub s t an t i a l per iod o f t ime c l o s e to the reg ion exp lored by the p r i o r con t ro l sequence. In most cases the p o l i c y g radua l l y moved f u r t h e r away from t h i s r e g i o n , and su c ce s s i v e l y l a r g e r changes i n the parameter es t imates l ead to s u c c e s s i v e l y l a r g e r dev i a t i on s i n ( q , E) space. The q u a l i t a t i v e a n a l y s i s of the reg ress i on problem i n s e c t i o n 4.1 i nd i c a t ed some requirements o f the independent v a r i a b l e s , catch per un i t e f f o r t and e f f o r t , i n o rder to reduce un ce r t a i n t y i n the parameter e s t i -mates. These inc luded a range o f q and E va l ue s , s u f f i c i e n t c on t r a s t i n them, and perhaps low e f f o r t l e v e l s a t low stock s i z e s to es t imate r. Although the pass ive adapt i ve p o l i c y exp lored the (q , E) space inadver -t e n t l y , i t g ene ra l l y managed to ob ta in reasonably good parameter e s t ima tes , suggest ing t h a t , when the model which generates the data i s o f the c o r r e c t f un c t i ona l form, the requirements on the d i s t r i b u t i o n o f po in t s "sampled" i n ( q , E) space are not p a r t i c u l a r l y r e s t r i c t i v e . The a c t i v e adapt ive p o l i c y , which sampled the space f a r more thorough ly , d i d not i n general s u b s t a n t i a l l y improve i t s parameter e s t ima tes , and i t f r e quen t l y su f f e red i n shor t term performance * I t would appear then tha t w i th t h i s model, and f o r the range o f s i t u a t i o n s cons i de red , a pass ive p o l i c y i s 169 qu i t e adequate f o r managing a stock based on catch and e f f o r t s t a t i s t i c s . Two se ts o f r e s u l t s suggest tha t t h i s l a s t statement should be heav i l y q u a l i f i e d . F i r s t , the performance of the non-adapt ive p o l i c y was un i fo rmly poor, suggest ing t ha t w i th poor parameter es t imates there i s the po t en t i a l f o r very poor performance. Second, the r e s u l t s obta ined by H i l bo rn (1979) suggest t h a t , when the model used i s more complex and the assumptions of the es t ima t i on technique are v i o l a t e d , problems can a r i s e i n the es t ima t i on of parameters which are not apparent when the model i s o f the f unc t i ona l form assumed by the es t imato r and the assumptions are net . E v i d en t l y , f u r t h e r research i s r e qu i r e d , both .to extend the range o f s i t u a t i o n s f o r which the s imple model i s ana lysed , and to extend the a n a l y s i s to s i t u a t i o n s i n which more complex dynamics are i nvo l ved . 4.10 SUMMARY 1. Ana l y s i s o f the es t ima t i on problem o f f i t t i n g the Schaefer model to catch and e f f o r t data i nd i c a t e s the need f o r a range of obser-va t i ons i n both the independent v a r i a b l e s , catch per un i t e f f o r t and e f f o r t , and the avoidance of a high c o r r e l a t i o n between them. 2. The performance o f both the a c t i v e and pass ive adapt ive p o l i c i e s over a range of mean p r i o r e f f o r t s compares f avourab ly w i th the performance of the opt imal p o l i c y where the t rue parameters are known. On the o ther hand, the non-adapt ive p o l i c y performs poor l y across t h i s range. 170 3. I t i s suggested that, for the passive adaptive po l i cy , small changes in parameter estimates can lead to successively larger changes in control e f for ts which in most cases are su f f i c i en t to cor rect ly deter-mine parameter values. 4. Conclusion 3 should be qua l i f i ed to the extent that the analysis only applies to s i tuat ions where the correct"funct ional form of the model i s known, and where the assumptions inherent in the estimation technique used are not v io lated (the analysis i s not at a l l robust to v io la t ions in basic assumptions). 171 CHAPTER V 5.0 DISCUSSION AND SUMMARY The purpose o f t h i s study has been to address some o f the problems posed by unce r t a i n t y i n the management of renewable resources . The approach has been to t e s t the performance o f a range o f adapt ive management s t r a t e g i e s app l i ed to s imulated f i s h e r i e s based on two very s imple models of s tock dynamics. The hope was to ga in i n s i g h t s , from the a n a l y s i s of such s imple views of the wo r l d , i n t o more general problems o f managing unce r ta in dynamic resources . Th is chapter w i l l assess the va lue and l i m i t a t i o n s of the approach taken i n t h i s s tudy. Suggest ions w i l l be made as to general l essons which may be drawn from the s p e c i f i c problems s t u d i e d , and a s p e c i f i c c r i t e r i o n f o r good adapt ive management w i l l be proposed. Some p r a c t i c a l aspects of a c t i v e adapt ive management w i l l be d i scussed and some suggest ions made as to f u r t h e r avenues f o r research in t h i s a rea . F i n a l l y the major r e s u l t s and conc lus ions from t h i s s tudy w i l l be b r i e f l y summarized. 5.1 VALUE AND LIMITATIONS OF THE STUDY The major l i m i t a t i o n on the cho i ce of models f o r a n a l y s i s , and hence on the scope o f the s tudy, was the a b i l i t y to fo rmulate adequate parameter e s t ima t i on techn iques , and as soc i a t ed measures o f unce r t a i n t y regard ing the e s t ima tes , f o r non l i nea r models. The i n i t i a l cho ice o f model f o r the a n a l y s i s was the general p roduct ion model ( P e l l a and Tom!inson, 1969) f o r 1 7 2 the case where the on ly in fo rmat ion a v a i l a b l e i s catch and e f f o r t d a t a . Th is model i s somewhat more r e a l i s t i c than the Schaefer model, i n t ha t i t a l l ows f o r skewed product ion f unc t i on s w i th opt imal popu la t i on s i z e s we l l above or below one h a l f the c a r r y i ng c apa c i t y . Several non l i nea r e s t ima t i on techniques were tes ted (extended Kalman f i l t e r , second order f i l t e r ) , but none were found to perform s a t i s f a c t o r i l y . Chapters I I I and IV show how the parameter e s t ima t i on problems f o r both the R i cke r and the Schaefer model can be formulated as l i n e a r r eg ress i ons and the appendices i n d i c a t e how the d i s t r i b u t i o n s o f parameter es t imates can be transformed to s t a t e est imate d i s t r i b u t i o n s . The e s t ima t i on problem f o r most non l i nea r models cannot be l i n e a r i z e d in t h i s f a s h i o n . H i l bo rn (1979) has suggested tha t p o t en t i a l c on t r o l s t r a t e g i e s f o r f i s h e r i e s should be tes ted i n i t i a l l y aga ins t complex s imu la t i ons and then su c ce s s i v e l y in l abo ra t o r y and rea l f i s h e r i e s . In t h i s s tudy, the s t r a t e g i e s have been tes ted aga ins t s imu la t i ons based on very s imp le models. Moreover the form o f the model used to generate the data has been of the same form as tha t assumed by the con t ro l and es t ima t i on a l go r i t hms , and the va r i ous assumptions o f these a lgo r i thms have been met. A l though u n r e a l i s t i c , the bene f i t o f t h i s approach i s t ha t any f a i l u r e s by the pass ive adapt ive p o l i c y cannot be a t t r i b u t e d to f a i l u r e s i n the assumptions o f the techniques used. They would have to represent more fundamental f a i l u r e s o f these p o l i c i e s to cope w i th unce r t a i n t y about the dens i t y dependent processes r egu l a t i n g the popu l a t i on . Any advantage of a c t i v e adapt ive behaviour under these cond i t i on s should on ly be inc reased when the rea l wor ld i s more complex and the va r i ous assumptions o f the e s t ima t i on techniques are not met. Th is i s borne out by Hi 1 born ' s r e s u l t s , where the r e l a t i v e performance 173 of h i s pass ive adapt ive p o l i c i e s f o r managing s tocks us ing catch and e f f o r t data was i n general much poorer than the r e s u l t s i n my study would i n d i c a t e . The models used in t h i s a na l y s i s a l s o represent a good t e s t o f the po t en t i a l advantages of an a c t i v e adapt ive s t r a t egy from another po in t of v iew. I t i s we l l known tha t there i s no dual e f f e c t o f con t ro l f o r l i n e a r systems (known parameters l i n e a r l y coup l i ng unknown s t a t e s ) ; tha t i s , no advantage to a c t i v e adapt ive behaviour (Bar-Shalom and Tse, 1976). Moreover, Ludwig (1978) has shown tha t any p o t en t i a l advantage to an a c t i v e adapt ive approach l i e s i n the second d e r i v a t i v e o f the product ion f u n c t i o n . The sharper the peak i n t h i s f u n c t i o n , the more poor ly w i l l a p o l i c y perform which mainta ins the stock away from the opt imal l e v e l . The product ion curves f o r both models analysed i n t h i s study are qu i t e f l a t , so the po t en t i a l f o r bene f i t i n d e l i b e r a t e probing to improve parameter es t imates i s not l i k e l y to be h i gh . The f a c t tha t the a c t i v e adapt ive p o l i c y does appear to do be t t e r and tha t a pass ive adapt ive p o l i c y can sometimes do very poor ly suggests tha t the bene f i t to a c t i v e probing would be even h igher where the stock dynamics are more h i g h l y non l i n ea r . 5.2 GENERAL LESSONS FOR RESOURCE MANAGEMENT The quest ion now a r i s e s as to how the lessons learned i n t h i s study can be extended to more general problems o f unce r t a i n t y i n renewable resource management. F i r s t , the more important conc lus ions from t h i s study should be rev iewed. The ana l y s i s of s tock rec ru i tment r e l a t i o n s h i p s i n chapter I I I i nd i c a t ed the need f o r observat ions of rec ru i tment over a reasonably wide 174 range of escapements. The need f o r v a r i a b i l i t y i n escapement was i n d i c a t ed from the q u a l i t a t i v e a n a l y s i s o f the r eg ress i on problem i n s e c t i on 3 .1 . In t h i s case there i s a s i n g l e independent v a r i a b l e in the r e g r e s s i o n , s tock s i z e , and parameter unce r t a i n t y depends on the d i s t r i b u t i o n o f po in t s a t which observa t ions are a v a i l a b l e a long the ax i s o f t h i s v a r i a b l e . The f a c t tha t understanding of a dens i t y dependent process r equ i r e s observa t ions of the response o f the stock over a wide range of d e n s i t i e s i s obvious but has not been w ide l y emphasized. Another l esson to emerge from chapter I I I i s t ha t the pass ive adapt ive p o l i c y gene ra l l y performs much b e t t e r than the non-adapt ive p o l i c y . In most cases er rors i n parameter est imates l ead to cons ide rab le changes i n p o l i c y over t ime, thereby i m p l i c i t l y probing a wide range o f escapements. In f a c t , t rue pass ive adapt ive p o l i c i e s are r a r e l y app l i ed i n p r a c t i c e . Inf requent updat ing of parameter est imates and low s i g n i f i c a n c e at tached to new in forma-t i o n r e s u l t i n p o l i c i e s changing on ly very s l ow ly over t ime , i f a t a l l . Resu l t s show tha t such p o l i c i e s w i l l perform we l l on ly i n cases where observa t ions are a v a i l a b l e near the e q u i l i b r i u m stock s i z e . Ana l y s i s of the use of catch and e f f o r t data i n chapter IV extended the scope of the conc l u s i on s . In t h i s case the r eg ress i on problem invo lved two independent v a r i a b l e s , one being a s t a t e v a r i a b l e (catch per un i t e f f o r t ) and the other a con t ro l v a r i a b l e ( e f f o r t ) . Ana l y s i s of parameter unce r t a i n t y i nd i c a t ed the need, not on ly f o r v a r i a b i l i t y i n the observed va lues o f each independent v a r i a b l e , but a l s o f o r c on t r a s t s between the two. I t would seem, then , t ha t a general l e sson f o r managing systems where unce r t a i n t y about dynamics can be ca tego r i zed as parameter un ce r t a i n t y i s a 175 need f o r v a r i a b i l i t y i n , and con t r a s t s between, observed va lues of the independent v a r i a b l e s i n the corresponding reg ress i on problems. The use o f systems a n a l y s i s and con t ro l t h e o r e t i c f o rmu la t i ons does not bypass t h i s very o l d admonit ion from bas i c s t a t i s t i c s . In f a c t , many cases o f r educ i b l e unce r t a i n t y about system dynamics can be expressed as problems of s t a t i s t i c a l un ce r t a i n t y in the parameters (or s t a t e s ) o f an app r op r i a t e l y formulated model. Even the case o f unce r t a i n t y about the c o r r e c t f un c t i ona l form of an app rop r i a t e model can be posed as a parameter e s t ima t i on problem where some of the parameters are p r o b a b i l i t i e s tha t each o f a range o f p o t en t i a l models i s the c o r r e c t one (Smallwood, 1968). Al though the posing o f such problems as cases o f parameter, un ce r t a i n t y prov ides a use fu l conceptual framework to app l y , t h i s approach i s l i m i t e d by the a b i l i t y . t o fo rmulate robust non l i nea r e s t ima to r s . C l e a r l y there i s a need f o r the development o f fundamental research i n t h i s a r ea . The conc lus i on tha t good performance by management r equ i r e s v a r i a b i l i t y and con t r a s t i n the independent v a r i a b l e s of the appropr i a te r eg ress i on problem needs some q u a l i f i c a t i o n . F i r s t , performance can be qu i t e i n s e n s i t i v e to some types o f unce r t a i n t y so tha t good est imates of a l l parameters may not be r equ i r ed . For example, the opt imal e q u i l i b r i u m e f f o r t f o r the Schaefer model i s independent of the k parameter. Second, i t i s not c l e a r to what extent the above cond i t i ons are s u f f i c i e n t f o r good est imat ion: 'and- ' performance, p a r t i c u l a r l y f o r cases where n o n l i n e a r i t i e s are more impor tant . The independent v a r i a b l e s i n the r eg ress i on problem f o r the general produc-t i o n model are the same as f o r the Schaefer model, but the r eg ress i on problem i s c l e a r l y qu i t e d i f f e r e n t . I f the peak i n the product ion f un c t i o n can be a t e i t h e r very high or very low s tock s i z e s , the importance of 176 observ ing the response o f the stock over a wide range of s tock s i z e s i s g r ea t e r . A b i l i t y to ana lyse such problems would l i k e l y y i e l d f u r t h e r i n s i g h t s i n t o requirements f o r good adapt i ve management. Even i n cases where the formal a n a l y s i s i s not p o s s i b l e , and where covar iance mat r i ces can not be generated, the r e s u l t s from t h i s s tudy i n d i c a t e some general f ea tu res which are l i k e l y to be of importance i n managing unce r ta i n dynamic systems. In chapter IV, management was presented as a t r a j e c t o r y through the space de f ined by the independent v a r i a b l e s in the r e g r e s s i o n . The po in t s sampled on the t r a j e c t o r y can.be thought of as experiments performed by each p o l i c y at d i f f e r e n t s tock s i z e s and w i t h d i f f e r e n t con t ro l measures ( t reatments) i n e f f e c t . Viewed i n t h i s way, the problem o f p lann ing an e f f e c t i v e management s t r a t egy i s s imply the problem o f des ign ing a good set of experiments to d i s t i n g u i s h between a l t e r n a t i v e hypotheses (sets of parameter v a l u e s ) . In order to pursue adapt ive management as exper imenta l de s i gn , there i s the c l e a r need f o r r e l a t i v e l y s imple d e s c r i p t i o n s of the dynamics of the system being managed. A l a r ge number of s t a tes and c on t r o l s would r equ i r e a p r o h i b i t i v e l y l a rge number of sample po in t s to ensure s u f f i c i e n t v a r i a b i l i t y i n and contrast , between a l l the v a r i a b l e s . There i s an obvious requirement f o r an i n i t i a l a n a l y s i s to i d e n t i f y sources of un ce r t a i n t y and the key processes i n vo l v ed , and to c r e a t i v e l y c o l l a p s e the problem i n t o a r e l a t i v e l y s imple model which captures e s s e n t i a l f ea tu re s o f the dynamics. From such a model, the q u a n t i t i e s corresponding to independent v a r i a b l e s i n a r eg ress i on problem can be i d e n t i f i e d , and management p o l i c i e s developed to sample t h i s space. Of course w i thout some means of formal a n a l y s i s , the t r adeo f f between cu r ren t performance and reduc t i on i n unce r t a i n t y i s d i f f i c u l t to quan t i f y . 177 Walters and H i l bo rn (1976) have proposed a method f o r e va l ua t i ng a l t e r n a t i v e management opt ions to assess t h i s t r a d e o f f . The c o n t r i b u t i o n o f the present study i s to help i n the i d e n t i f i c a t i o n of l i k e l y cand idate p o l i c i e s f o r exper imenta l management schemes. One c o r o l l a r y to the need f o r high con t r a s t between independent v a r i a b l e s i s the r e a l i z a t i o n tha t an incrementa l approach to exp l o r i ng the response of the system i s h i gh l y counter adap t i ve . The no t ion tha t e f f o r t l e v e l s i n a deve lop ing f i s h i n g should be increased s l ow ly in the face of unce r t a i n t y seems eminent ly reasonab le . Th is w i l l r e s u l t i n a gradual r educ t i on in the stock s i z e , so tha t e ven tua l l y the v a r i a b i l i t y i n observed va lues o f both the e f f o r t and catch per e f f o r t w i l l be h i gh . Un f o r t una te l y , i t w i l l a l s o r e s u l t i n a very high (negat ive) c o r r e l a t i o n between e f f o r t and ca tch per e f f o r t . By a l l ow i ng on l y smal l incremental changes i n the c o n t r o l , the system e s s e n t i a l l y remains c l o se to the e q u i l i b r i u m e f f o r t at each stock s i z e . The r e s u l t i n g t r a j e c t o r y i n the space de f ined by the independent v a r i a b l e s i s a s t r a i g h t l i n e (the l i n e AB i n F igure 22 ) . I t has been suggested tha t p o l i c i e s i n renewable resource management f r equen t l y lead to confounding o f e f f e c t s . The i n t r odu c t i o n of sea lamprey to the Great Lakes r e su l t e d in an inc rease i n f i s h i n g e f f o r t to e x p l o i t the commercial f i s h s tocks i n the lakes i n a n t i c i p a t i o n of t h e i r demise. Th is has l ed to an i n a b i l i t y to d i s t i n g u i s h between the e f f e c t s of the lamprey and the f i s h i n g i n the subsequent d e c l i n e o f the s tocks (Car l Wa l t e r s , pe rs . comm.). The need to avo id c o r r e l a t i o n between key v a r i a b l e s , which leads to the confounding of e f f e c t s and i n a b i l i t y to d i s c r i m i n a t e between h y p o t h r eses , emerges as a key c r i t e r i o n which should be added to the l i s t o f 178 c r i t e r i a f o r good adapt ive management o u t l i n e d in chapter I . 5.3 PRACTICAL ADAPTIVE MANAGEMENT A general f ea tu re of a c t i v e adapt ive management i s the s a c r i f i c e of shor t term performance i n an e f f o r t to improve long term performance. In the examples cons idered in t h i s s tudy , a c t i v e adapt ive p o l i c i e s f r equen t l y i nvo lved very low catches over severa l i n i t i a l t ime pe r i od s . In p r a c t i c e , there may be cons i de rab l e d i f f i c u l t y i n conv inc ing the e x p l o i t e r s o f the resource tha t such p o l i c i e s are warranted. The d e s i r a b i l i t y of a c t i v e adapt ive behaviour has been shown to be very s e n s i t i v e to s o c i a l d i scount ra tes ( S i l v e r t , 1978). A r e l a t e d problem i s the f requent need f o r wide f l u c t u a t i o n s i n con t ro l va lues (harves t r a t e s ) from year to y e a r . Th is presupposes a g rea te r degree of f l e x i b i l i t y i n cho ice of management a c t i ons than i s o f ten the c a se . • Wal ters and H i l bo rn (1976) have proposed a method f o r dea l i ng w i th con t ro l problems where con t ro l v a r i a b l e s are cons t ra ined i n the extent to which they can vary from one d e c i s i o n po in t to the next . C l a r k and Be l l (1976) have quest ioned the appropr ia teness o f a s imple d i s count ra te f o r assess ing time streams w i th w ide ly f l u c t u a t i n g va l u e s . Severe c on s t r a i n t s on changes i n con t ro l a c t i ons from one per iod to the next cou ld make i t d i f f i c u l t to ach ieve the requ i red con t r a s t s i n v a r i a b l e s d i s cussed i n the prev ious s e c t i o n . I n a b i l i t y to implement des i r ed con t ro l a c t i ons can, i n some cases , l ead to v a r i a b i l i t y i n independent v a r i a b l e s and p o t e n t i a l l y to be t t e r unders tand ing. At a recent workshop, Dave Schutz (management b i o l o g i s t , .179 F i s h e r i e s and Marine S e r v i c e , Canada) noted tha t i n a b i l i t y to e f f e c t i v e l y con t ro l escapements has l ed to cons ide rab le v a r i a b i l i t y i n spawning data f o r B r i t i s h Columbia salmon s t o cks . He quest ioned tha t f u r t h e r , a c t i v e d i s t u r b -ance would be worth the t r oub l e they would cause him i n dea l i ng w i th i n du s t r y . I t i s t rue tha t a number o f s tocks have been observed over a wide range of escapements. However many o f the observat ions a t h igh and low escapements, were obta ined i n the 1950s or e a r l i e r . Changes i n the composi-t i o n or b i o l o g i c a l c h a r a c t e r i s t i c s o f the s t o c k s , and/or changes i n the environment would e f f e c t i v e l y mean tha t the parameters of the s tock r e c r u i t -ment r e l a t i o n s h i p s have changed s i n ce those observa t i ons were made. Recent observa t ions have tended to c l u s t e r more c l o s e l y t oge the r , and nearer the l e v e l of apparent opt imal escapement. Th is i s p a r t i c u l a r l y t rue o f key systems such as the F rase r , Skeena and R ivers -Smi ths I n l e t . Be t t e r w i t h i n season con t ro l systems have made i t po s s i b l e to h i t t a rge t escapements even more c l o s e l y ; f o r example, Washington S ta te salmon escapements are now being very t i g h t l y c o n t r o l l e d (Sam Wr ight , pe r s . comm.). Th is reduc t i on i n v a r i a b i l i t y i n recent data cou ld become important i f , i n f a c t , the parameters of the system have been changing over t ime. A f u r t h e r po in t should be made concern ing the po t en t i a l uses o f random v a r i a b i l i t y i n the da ta , whether due to na tura l f l u c t u a t i o n s and c y c l e s in stock abundances or to i n a b i l i t y to con t ro l escapements or e f f o r t . Whi le i t i s t rue t ha t we owe much of our understanding o f s tock dynamics to such natura l or man-induced, v a r i a b i l i t y , i t cannot be r e l i e d upon to prov ide the range or c on t r a s t s requ i red to e f f e c t i v e l y d i s c r i m i n a t e between a l t e r n a -t i v e hypotheses about s tock dynamics. The a c t i v e adapt ive p o l i c i e s developed 180 i n t h i s study are not s imply random pe r t u r ba t i o n s , they are h i gh l y d i r e c t -i ona l i n c ha r a c t e r . Random probing p o l i c i e s of va r i ous types have been t e s t e d , and were found to perform un i f o rm ly worse than pass ive adapt ive p o l i c i e s (Car l Wa l t e r s , pers . comm.). A f i n a l p r a c t i c a l po in t i s tha t natura l r e p l i c a t i o n i n space can be o f great va lue i n deve lop ing a c t i v e l y adapt ive o r exper imenta l management p o l i c i e s f o r renewable resources . Many e xp l o i t e d popu la t ions c on s i s t o f a number of d i s t i n c t substocks which i n some cases can be e xp l o i t e d indepen-den t l y . Th is can g r e a t l y inc rease the r a t e a t which experiments can be performed, thus shor ten ing the a c t i v e adapt ive phase. I t a l s o reduces the cos t a s soc i a t ed w i t h experiments i n v o l v i n g low harvest r a t e s , as o ther s tocks can s imu l taneous ly be e xp l o i t e d a t much h igher r a t e s . 5.4 DIRECTIONS FOR FUTURE RESEARCH Th is s e c t i on presents some b r i e f suggest ions on d i r e c t i o n s f o r f u r t h e r research on adapt ive s t r a t e g i e s f o r managing unce r ta in dynamic resources . The f i r s t par t o u t l i n e s some e l abo ra t i on s o f the cu r ren t approach, i n v o l v i n g f u r t h e r a n a l y s i s of the s imple models used i n t h i s s tudy. The second par t d i s cusses some pos s i b l e extens ions us ing more complex dynamic models. Fur ther Ana l y s i s of Simple Models The e f f e c t o f the l e v e l o f v a r i a b i l i t y i n the dynamic and observa-t i o n models should be exp l o red . For both models, the parameter unce r t a i n t y i s d i r e c t l y p ropo r t i ona l to the e r r o r va r i ance i n the r e g r e s s i o n . Inc reas-ing t h i s l e v e l i s l i k e l y to lead to q u a n t i t a t i v e but not q u a l i t a t i v e changes i n the pe r tu rba t i ons by the a c t i v e adapt ive p o l i c y . I t might a l s o r e s u l t i n more f requent cases of poor performance by the pass ive adapt ive p o l i c y , as 181 g rea te r v a r i a b i l i t y i n independent v a r i a b l e s would be requ i r ed to l e a rn parameter va lues c o r r e c t l y . The p o l i c i e s developed i n t h i s study should be t e s t ed aga i n s t s imu la t i ons us ing more complex models o f s tock dynamics. Uh ler (1978) has shown that se r i ous b iases in parameter es t imates f o r the Schaefer model can r e s u l t when s t o c h a s t i c terms appear i n the ca tch equa t i on . H i l b o rn (1979) found tha t pass ive adapt ive p o l i c i e s us ing catch and e f f o r t data and app l i ed to complex s imu la t i ons performed much more poo r l y than was i n d i c a t ed i n t h i s s tudy. The importance of the assumptions about the form o f the v a r i a b i l i t y i n the dynamic and obse rva t i on equa t i ons , as we l l as the e f f e c t o f smal l s t r u c t u r a l changes i n the model, needs to be i n v e s t i g a t e d . There i s a need to i nco rpo ra te pena l ty f unc t i ons on negat ive e f f o r t and stock s i z e s in the cos t f u n c t i o n f o r the wide sense adapt ive a l g o r i t hm . This would he lp to avo id the problems of o v e r e x p l o i t a t i o n a t low stock s i z e s mentioned i n chapter IV. I t would a l s o make the a c t i v e adapt ive p o l i c y more cau t i ous about app ly ing h igh harvest ra tes when un ce r t a i n t y i s very h igh . P o l i c i e s should be developed f o r the i n i t i a l s tages i n the e x p l o i t a -t i o n o f a r e sou r ce , where l i t t l e o r no p r i o r i n f o rmat i on i s a v a i l a b l e . Resu l t s from t h i s study suggest t ha t i t may be po s s i b l e to ob ta i n nea r l y opt imal development s t r a t e g i e s o f an open loop na tu re , tha t i s , independent of feedback from the data which become a v a i l a b l e over the f i r s t few yea r s . The important th ing i n i t i a l l y i s to develop so t ha t the requ i red v a r i a b i l i t y and c on t r a s t s i n the data are obta ined a t an e a r l y s tage . Management p o l i c i e s should a l s o be exp lo red f o r systems whose c h a r a c t e r i s t i c s change over t ime. S lowly changing parameters can be 182 modelled as increments i n elements o f the covar iance mat r i x (Young, 1974). Increas ing the covar iance elements pushes the system back i n t o the domain ( i n the in fo rmat ion s t a t e v a r i a b l e s ) where a c t i v e probing becomes the opt imal behav iour . Thus changing parameters w i l l imply more f requent or p e r s i s t e n t ep isodes o f exper imenta l p rob ing . The e f f e c t o f c on s t r a i n t s on changes i n con t ro l v a r i a b l e s from one time pe r i od to the next should be i n v e s t i g a t e d . However a more p ress ing concern i s f o r a more comprehensive ana l y s i s f o r the Schaefer model . The e f f e c t of l e v e l s o f v a r i a b i l i t y i n both e f f o r t and catch per e f f o r t should be exp lo red . In p a r t i c u l a r , the importance o f the degree o f c o r r e l a t i o n between the two v a r i a b l e s in the r eg res s i on remains to be i n v e s t i g a t e d . Extens ion to More Complex Models The range o f models f o r which adapt ive s t r a t e g i e s can be exp lored i s c u r r e n t l y r e s t r i c t e d , not by the l i m i t a t i o n s o f s t o c h a s t i c con t ro l theory , but by the a b i l i t y to develop robust techniques f o r e s t ima t i ng the parameters of non l i nea r models. There i s a c l e a r need f o r the development o f e s t ima-t i o n techniques which w i l l a l l ow more r e a l i s t i c models o f stock dynamics to be ana lyzed . The va lue of being a c t i v e l y adapt ive should be g rea te r w i th more h i gh l y non l i nea r models, and the a n a l y s i s of such models would l i k e l y lead to f u r t h e r i n s i g h t s i n t o requirements f o r good adapt ive management. A p p l i c a t i o n of r e cu r s i v e non l i n ea r es t ima to r s such as extended Kalman f i l t e r s might prov ide some c l ues to requirements f o r good e s t ima t i o n . Using such e s t ima t o r s , the s e n s i t i v i t i e s 8P/9u_ and 9P/9x_ of the u n c e r t a i n t i e s about the parameters ( represented i n the covar iance mat r i x P) to v a r i a t i o n s in the con t ro l and s t a t e vec to rs u^  and x cou ld be exp l o red . 183 ..The e f f ec t s " , on e s t ima t i on o f - d i f f e r e n t observa t ion models cou ld a l s o be examined. 5.5 SUMMARY 1. A range of adapt ive p o l i c i e s was app l i ed to the management of s imulated f i s h e r i e s based on two s imple models o f stock dynamics, the R i cke r s tock rec ru i tment model and the Schaefer product ion model. Unce r ta in ty about s tock dynamics was represented as unce r t a i n t y i n the parameter es t imates o f a model of known f un c t i ona l form. 2. The po t en t i a l advantage of a c t i v e adapt ive behaviour ( d e l i b e r a t e prob ing to improve parameter es t imates ) was not expected to be high f o r e i t h e r model, s i nce p r o d u c t i v i t y i s not p a r t i c u l a r l y s e n s i t i v e to stock s i z e i n e i t h e r case. The cho ice of models f o r a n a l y s i s was l i m i t e d by the a b i l i t y to formulate s u i t a b l e parameter e s t ima t i on techn iques . 3. In most cases the a c t i v e and pass ive adapt ive p o l i c i e s performed very we l l r e l a t i v e to the opt imal p o l i c y , where the t rue parameters were known. However i n c e r t a i n cases the performance of the a c t i v e adapt ive p o l i c y was c l e a r l y supe r i o r to t ha t o f the pass ive adapt ive p o l i c y . These s i t u a t i o n s arose when the p o l i c i e s were app l i ed to. s tock rec ru i tment data and the apparen t l y opt imal escapement based on the parameter es t imates f e l l w i t h i n the range o f escapements a l r eady observed. Examinat ion of the r eg ress i on problem i nd i c a t e s the need f o r s u f f i c i e n t v a r i a b i l i t y i n observed va lues o f escapement. In most cases e r r o r s i n parameter es t imates f o r the pass ive adapt ive p o l i c y caused i t to choose escapements over a s u f f i c i e n t l y wide range to c o r r e c t l y determine parameter va l ue s . 184 4. The a c t i v e adapt ive p o l i c y performed c o n s i s t e n t l y we l l r e l a t i v e to the opt imal p o l i c y f o r both models. However i t d i d so a t the expense of shor t term performance so t ha t h igh s o c i a l d i s count r a te s would negate any advantage of a c t i v e probing to improve parameter e s t ima tes . 5. Ana l y s i s of the reg ress i on problem f o r both models suggests tha t good parameter e s t ima t i on (equ iva len t to the a b i l i t y to c l e a r l y d i s t i n g u i s h between a l t e r n a t i v e hypotheses concern ing stock dynamics) r equ i r e s both v a r i a b i l i t y i n and con t r a s t between the observed va lues o f the independent v a r i a b l e s i n the r eg r e s s i on . Th is i s proposed as a c r i t e r i o n f o r good adapt ive management i n more general problems of managing unce r ta i n dynamic resources , where the q u a n t i t i e s corresponding to independent v a r i a b l e s i n an appropr i a te r eg res s i on problem must be i d e n t i f i e d . P o l i c i e s which advocate incrementa l behaviour i n changing con t ro l a c t i ons are shown to r e s u l t i n high c o r r e l a t i o n between independent v a r i a b l e s , and t he re f o r e confounding of e f f e c t s . 185 LITERATURE CITED A l l e n , K.R. 1973. The i n f l uence o f random f l u c t u a t i o n s in the s tock -rec ru i tment r e l a t i o n s h i p on the economic re tu rn from salmon f i s h e r i e s . Cons. I n t . Exp lo r . Mer . , Rapp. 164: 350-359. A l l e n , K.R. and G.P. Kirkwood. 1977. Is exper imenta l management of whale s tocks p r a c t i c a b l e ? Manuscr ip t , CSIRO D i v i s i o n o f F i s h e r i e s and Oceanography, C r o n u l l a , A u s t r a l i a . Bar-Shalom, Y. and Tse, E. 1976. Concepts and methods i n s t o c h a s t i c c o n t r o l . In Contro l and Dynamic Systems, V o l . 12, C.T. Leondes, ed. Academic P ress , New York. Be l lman, R. 1961. Adapt ive Contro l Processes: A Guided Tour. P r ince ton U n i v e r s i t y P ress , P r i n ce t on , New Je r sey . Bever ton, R.J.H.- and S . J . Ho l t . 1957. On the dynamics of e x p l o i t e d f i s h popu l a t i ons . U.K. Min. A g r i c . F i s h . , F i s h . I n ve s t . , Se r . 2 19: 1-533. C l a r k , C W . 1976. Mathematical Bioeconomics: The Optimal Management  o f Renewable Resources. Wi ley I n t e r s c i e n c e , New York. C l a r k , W.C. and D.E. B e l l . 1976. Inter tempora l i n d i c a t o r e v a l u a t i o n . I n s t . Anim. Res. E c o l . , Un iv. B r i t i s h Columbia, Working Paper W-7 Vancouver, Canada. Dahlberg, M.L. 1973. S tock-and-recru i tment r e l a t i o n s h i p s and optimum escapements o f sockeye salmon s tocks o f the Ch ign ik Lakes, A l a s ka . Cons. In t . Exp lo r . Mer . , Rapp. 164: 98-105. Fel 'dbaum, A.A. 1960-1961. Theory o f dual con t ro l I - IV. Autom. Remote  Contro l USSR, 21: 1240-49, 1453-65; 22: 3-16, 129-43" (In Russian) Fox, W.W. J r . 1970. An exponent ia l s u r p l u s - y i e l d model f o r o p t im i z i n g e xp l o i t e d f i s h popu l a t i ons . Trans. Amer. F i s h . Soc. 99: 80-88. Fox, W.W. J r . 1971. Random v a r i a b i l i t y and parameter e s t ima t i on f o r the gene ra l i z ed product ion model. F i s h . B u l l . (U .S . ) 69 (3 ) : 569-580. Fox, W.W. J r . 1975. F i t t i n g the gene ra l i z ed stock product ion model by l eas t - squa res and e q u i l i b r i u m approx imat ion . F i s h . B u l l . (U.S.) 73(1) : 23-37. Ge lb, A. ( e d . ) . 1974. App l i ed Optimal E s t ima t i on . MIT P ress , Cambridge, Mass. 186 Graham, M. 1935. Modern theory of e x p l o i t i n g a f i s h e r y , and a p p l i c a t i o n to North Sea t r a w l i n g . J . Cons. I n t . Exp lo r . Mer. 10(3): 264-274. Gu l l and , J .A . 1977. Goals and ob j e c t i v e s o f f i s h e r y management. F.A.O. F i s h . Tech. Pap. No. 166 H i l b o r n , R. 1979. A comparison o f f i s h e r i e s con t ro l systems tha t u t i l i z e catch and e f f o r t da ta . Can. J . F i s h . Aquat i c S c i . 1 (In press) H i l b o r n , R. and R.M. Peterman. 1975. Changing management o b j e c t i v e s . In P a c i f i c Salmon Management f o r Peop l e , ' P .V . E l l i s , ed . Un ive r -s i t y of V i c t o r i a P ress , V i c t o r i a , Canada. H i l b o r n , R. and C . J . ;Wa l t e r s . 1977. D i f f e r i n g goals o f salmon management on the Skeena R i v e r . J . F i s h . Res. Board Can. 34(1): 64-72. H o l l i n g , C.S. 1973. Re s i l i e n c e and s t a b i l i t y o f e co l og i c a l systems. Ann. Rev. E c o l . Sy s t . 4: 1-23 H o l l i n g , C.S. ( e d . ) . 1978. Adapt ive Environmental Assessment and  Management. John Wi ley & Sons, New York. Ho l t , S . J . 1977. Whale management p o l i c y . Rep. I n t . Whal. Comm. 27: 133-137. Huang, C C , I .B . Ve r t i n sky and N.J. Wi l imovsky. 1976. Optimal c on t r o l s f o r a s i n g l e spec ies f i s h e r y and the economic va lue o f r e sea r ch . J . F i s h . Res. Board Can. 33: 793-809. Inter-Amer ican T rop i ca l Tuna Commission. 1977. Annual Report f o r 1976. IATTC, La J o l l a , C a l i f o r n i a . I n t e rna t i ona l North P a c i f i c F i s he r i e s Commission. 1962. The e x p l o i t a t i o n , s c i e n t i f i c i n v e s t i g a t i o n , and management o f salmon (genus Oncorhynchus) s tocks on the P a c i f i c coast o f the Uni ted S ta tes i n r e l a t i o n to the Abs ten t ion P r ov i s i on s o f the North P a c i f i c F i s h e r i e s Convent ion. INPFC, B u l l e t i n 10: 1-160. Jacobs, O.L.R. and J.W. P a t c h e l l . 1972. Caut ion and probing i n s t o c h a s t i c c o n t r o l . I n t . J . Cont. 16: 189-199. Jones, D.D. and C . J . Wa l t e r s . 1976. Catastrophe theory and f i s h e r i e s , r e g u l a t i o n . J . F i s h . Res. Board Can. 33(12): 2829-2833. Kalman, R.E. 1960. A new approach to l i n e a r f i l t e r i n g and p r e d i c t i o n problems. J . Bas i c Eng. 82D: 35-45. Kalman, R.E. and R.S. Bucy. 1961. New r e s u l t s i n l i n e a r f i l t e r i n g and p r e d i c t i o n theory . J . Bas i c Eng. 83D: 95-108. 187 Keeney, R.L. 1977. A u t i l i t y f unc t i on f o r examining p o l i c y a f f e c t i n g salmon on the Skeena R i ve r . J . F i s h . Res. Board Can. 34(1): 49-63. Lark. in, P.A. 1977. An ep i taph f o r the concept o f maximum sus ta ined y i e l d . Trans. Amer. F i s h . Soc. 106(1): 1-11. Ludwig, D. 1978. Harvest s t r a t e g i e s which account f o r unce r t a i n t y i n parameters. Manuscr ip t , Department o f Mathematics, U n i v e r s i t y of B r i t i s h Columbia, Vancouver, Canada. Ludwig,.D. 1979a. Optimal ha rves t i ng o f a randomly f l u c t u a t i n g resource I: A p p l i c a t i o n o f pe r tu rba t i on methods. SIAM J . App l . Math. (In press) Ludwig, D. 1979b. I n t e r p r e t a t i o n o f incoherent da ta : a d e c i s i o n -t h e o r e t i c approach to salmon, management. Manuscr ip t , Department of Mathematics, U n i v e r s i t y of B r i t i s h Columbia, Vancouver, Canada. Ludwig, D. and J .M. Varah. 1979. Optimal ha rves t i ng o f a randomly f l u c t u a t i n g resource I I : Numerical methods and r e s u l t s . SIAM J . App l . Math. (In press) P e l l a , J . J . and P.K. Toml inson. 1969. A gene ra l i z ed stock product ion model. Inter-Amer. Trop. Tuna Comm., B u l l . 13(3): 419-496. Peterman, R.M. 1977. A s imple mechanism tha t causes c o l l a p s i n g s t a b i l i t y reg ions i n e xp l o i t e d salmonid popu l a t i ons . J . F i s h . Res. Board  Can. 34(8) : 1130-1142. Peterman, R.M. 1979. D i s t r i b u t i o n o f marine s u r v i v a l r a tes o f P a c i f i c salmon. Manuscr ip t , Department o f Zoology, Simon Fraser U n i v e r s i t y , Vancouver, Canada. R i c ke r , W.E. 1954. Stock and rec ru i tmen t . J . F i s h . Res. Board Can. 11: 559-623. Roedel , P.M. ( e d . ) . 1975. Optimum sus ta ined y i e l d as a concept i n f i s h e r i e s management. Amer. F i s h . S o c , Spec. Pub l . No. 9 , Washington, D.C. S a r i d i s , G.N. 1977. Se l f -O rgan i z i ng C o n t r o l ' o f S t o cha s t i c Systems. Cont ro l and Systems Theory, V o l . 4 , J .M. Mendel, ed . Marcel Dekker I n c . , New York. Schaefer , M.B. 1954. Some aspects o f the dynamics o f popu la t ions important to the commercial marine f i s h e r i e s . Inter-Amer. Trop.  Tuna Comm., B u l l . 1 (2 ) : 25-56. Schaefer , M.B. 1957. A study o f the dynamics of the f i s h e r y f o r y e l l ow -f i n tuna i n the eastern t r o p i c a l P a c i f i c Ocean. Inter-Amer.  Trop. Tuna Comm., B u l l . 2 ( 6 ) : 245-285. 188 Schnute, J . 1977. Improved est imates from the Schaefer product ion model: t h e o r e t i c a l c on s i d e r a t i o n s . J . F i s h . Res. Board Can. 34(5) : 583-603. S i l v e r t , W. 1978. The p r i c e o f knowledge: f i s h e r i e s management as a research t o o l . J . F i s h . Res. Board Can. 35(2) : 208-212. Skud, B.E. 1973. Management o f the P a c i f i c h a l i b u t f i s h e r y . J . F i s h .  Res. 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Econ. , Res. Pap. No. 23, Vancouver, Canada. Wa l t e r s , C . J . 1975. Optimal harvest s t r a t e g i e s f o r salmon in r e l a t i o n to environmental v a r i a b i l i t y and unce r ta in product ion parameters. J . F i s h . Res. Board Can. 32: 1777-1784. Wa l t e r s , C . J . 1977. Management under un ce r t a i n t y . In P a c i f i c Salmon  Management f o r People, D.V. E l l i s , e d . , U n i v e r s i t y o f V i c t o r i a P ress , V i c t o r i a , Canada. Wa l te r s , C . J . and R. H i l b o r n . 1976. Adapt ive con t ro l o f f i s h i n g systems. J . F i s h . Res. Board Can. 33(1) : 145-159. Wa l t e r s , C . J . and R. H i l b o r n . 1978. Eco l og i ca l o p t im i z a t i o n and adapt ive management. Ann. Rev. Eco'T. Sy s t . 9: 157-188. Watt, K.E.F. 1968. Ecology and Resource Management: A Quan t i t a t i v e  Approach. McGraw-H i l l , New York. Wittenmark, B. 1975. S t o chas t i c adapt ive con t ro l methods: a survey. In t . J . Cont. 21(5): 705-721... Young, P. 1974. Recurs ive approaches to time s e r i e s a n a l y s i s . I n s t .  Math. A p p l . , B u l l . 10(5/6): 209-224. 189 APPENDIX I The problem i s to convert the d i s t r i b u t i o n of parameter es t imates f o r the R i cker model i n t o a d i s t r i b u t i o n o f s t a t e e s t ima te s , where the s t a t e i s de f ined as x = [InR a g ] ' The d i s t r i b u t i o n of the parameter es t imates i s determined by the mean a = [ a g] and the covar iance mat r i x P . — a Not ing tha t the s t a t e equat ion R = S exp { a - g S + v} can be r ew r i t t en as InR = InS + a - g • S + v The s t a t e est imate can be w r i t t e n as InR InS + a -: 3S + V a a 0 g g 0 = 1 -S ; a + InS 1 .0 g 0 0 1 0 '= T a + c + d The covar iance o f x, P , can then be c a l c u l a t e d A P = T P T' + Q x a where 0 0 0 0 0 0 191 A P P E N D I X I I The problem i s to convert the parameter est imate vec to r a = [a e y ] ' and covar iance mat r i x P i n t o s t a t e es t imate x = [q a — a and s t a t e e r r o r covar iance mat r i x P . A Noting tha t the s t a t e equat ion can be w r i t t e n V i = q t + a q t - e q t - Y E t q t + v t q t The s t a t e es t imate can be expressed as e t l q + a q - e q - YEq ,+ • q v a 0 3 0 - Y 0 q + q - E q a + V. q 0 l 0 0 6 0 0 0 1 0 Y 0 0 0 0 1 0 = S + T a + v Making the assumption tha t the unce r t a i n t y about q i s zero (the prev ious catch per u n i t e f f o r t has been observed w i thout e r r o r ) , then P can be c a l c u l a t e d as 192 P = T P T' + x a where 2 a V 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 

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