Optimal Harvest Policy

8/6/2019 Optimal Harvest Policy

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J . Math. Biology (1981) 12:265-293 ,Journal ofMa thema t i ca l131ologyby Springer-Verlag1981

Optimal Fishery Policy for Size-Specif ic ,Densi ty-Dependent Populat ion ModelsL o u i s W . B o t s f o rdUnivers i ty of Cal i fornia , Bodega M ar ine L abora tory , P . O. Box 247, Bodega B ay, CA 94923, US A

Ab st rac t . O p t im al har ves t po l icy i s der ived fo r a s ize -specif ic po pu la t io n m od elb a s e d o n t h e c o n t i n u i t y e q u a t i o n . I n t hi s m o d e l b o t h g r o w t h a n d r e c r u i t m e n tr a t e s can d ep en d o n e i t h e r t h e n u mb er o f i n d i v i d u a l s o f s p ec i f i c s i ze s i n t h ep o p u l a t i o n o r t h e l ev el o f f o o d av a i lab l e . N eces s a ry co n d i t i o n s f o r v a l u e s o ff i sh i n g p re s s u re an d s ize li m i t t h a t m ax i mi ze t h e p re s en t v a l u e o f t h e r e s o u rc ea r e o b t a i n e d a n d a r e i n t e r p r e t e d i n e c o n o m i c t e r m s . T h e g e n e r a l s o l u t i o no b t a i n ed h e re r ed u ces t o s o l u t i o n s o b t a i n ed p rev i o u s ly fo r s o m e s p ec ia l ca s e s :t h e s in g le age -c la s s mo d e l , an d t h e l i n ea r ag e -d ep e n d en t mo d e l . S o l u t i o n si n v o l v i n g co n s t an t f i s h e ry p o l i cy ax e s o u g h t f o r s ev e ra l d i f f e r en t , s p ec i f i cv e r s i o n s o f t h e g en e ra l mo d e l . I n e ach o f t h e s e v e r s i o n s a co n s t an t p o l i cys o l u t i o n is n o t o p t i ma l . T h i s i mp l i e s t h a t f o r a g en e ra l , r e a l is t ic mo d e l t h e p o l i cyt h a t m ax i mi zes p re s en t v a l u e i s a t i me -v a ry i n g o r " p u l s e f i s h i n g " p o l icy . T h et h eo re t i ca l an d p rac t i ca l i mp l i ca t io n s o f t h e r e s u l ts a r e d i s cus s ed in t h e l i g h t o fexis t ing resul ts .Ke y words : F i sh ery po l icy - Pu lse f i sh ing s ize specif ic m ode ls - Ag e-speci f icm o d e l s - F i s h e r y e c o n o m i c s

1 . I n t r o d u c t i o nI n s p i t e o f t h e f ac t t h a t i n d i v i d u a l g ro w t h , r ep ro d u c t i v e an d mo r t a l i t y r a t e sp r o f o u n d l y a f fe c t t h e d y n a m i c b e h a v i o r o f e x p lo i te d f is h p o p u l a ti o n s , t h e y a r es e l d o m ex p li c it ly in c l u d ed i n m o d e l s f o r an a l y s is o f o p t i ma l h a rv es t s t ra t eg ie s . F o rexam ple , the log is t ic m ode l , app l ied to f i sher ies .by G ra ha m (1935) , Sc hae fer (1954)a n d o t h er s , re f le c ts a c o m p e n s a t o r y d e c r e a se in p o p u l a t i o n g r o w t h r a t e a sp o p u l a t i o n s ize in c rea s es , b u t d o es n o t ex p l ic i tl y i n c l u d e th e d e n s i t y -d ep en d en tc h a n g e s i n g r o w t h , r e p r o d u c t i v e o r m o r t a l i t y r at e s t h a t a r e r e s p o n s ib l e f o r t h ed ec rea s e . A s eco n d ex am p l e is t h e w i d e l y u s ed m o d e l o f B ev e r t o n an d H o l t ( 1 95 7) .A l t h o u g h t h e y e x a m i n e d t h e e f fe c t s o f v a r i o u s d e n s i t y - d e p e n d e n t m e c h a n i s m s o n

Present address: Departmmat o f Wildlife an d Fisheries B iology, University o f California , Da vis, CA95616, USA

0303 - 6812/81/0012/0265/$05.80


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266 L.W. Botsford

o p t ima l p o l i c y o b t a in e d f r o m th e i r d y n a m ic p o o l m o d e l , a n a ly t i c a l s o lu t io n s o f t h i sp r o b l e m u s in g t h e ir mo d e l h a v e n o t i n c lu d e d d e n s i t y - d e p e n d e n c e ( c f. , G o h , 1 97 3;C la r k e t a l . , 1 9 7 3 ). A th i r d e x a m p le , t h e s t o c k - r e c r u i tm e n t m o d e l d e v e lo p e d b yRick e r (1954) , r e f lec ts the d ens i ty -de pen den t e f fec t o f s toc k on re c ru i tm ent level ,b u t d o e s n o t e x p l ic i tl y i n c lu d e t h e r e s p o n s ib l e b io lo g ic a l me c h a n i s m. A n a ly t i c a ls o lu ti o n s t o t h e o p t ima l h a r v e s t p r o b l e m u s in g t h is mo d e l d o n o t i n c lu d e mu l t i p l ey e a r s cl a ss e s n o r g r o w th a n d s u r v iv a l b e y o n d r e c r u i tm e n t . Se v e ra l w o r k e r s ( of.,W ai te r s , 1969; S i l liman , 1969; Bev er ton an d Ho l t , 1957) have com bine d a s tock-r e c r u i t m e n t m o d e l w i t h a m o d e l o f i n d iv i d u a l g r o w t h a n d m o r t a l it y , a n d e x a m i n e dr e s p o n s es t o d i f f e r e n t h a r v e s t p o l i c ie s t h r o u g h s imu la t i o n . H o w e v e r , t h e r e h a v eb e e n f e w a t te m p t s t o a n a ly t i c a l ly d e t e r m in e o p t ima l h a r v e s t p o l i c y u s in g mo d e l s o fs imi la r com plex i ty (a r ece n t exce p t ion is Ge tz , 1979).

T h e t o o l s n e c e s s a r y t o f o r m u la t e a m o r e r e a l i s ti c a n a lo g u e o f a f is h p o p u la t i o na r e a v a i la b l e i n c u r r e n t c o n s t r u c t s o f t h e o r e t i c a l b io lo g y . A g e - si ze s p e ci fi c mo d e l sdeve loped by S inko and St re i f e r (1967) r e f lec t spec i fica l ly the g rowth , m or ta l i ty an dr e p r o d u c t iv e r a te s o f i n d iv id u a ls i n a p o p u la t i o n . T h e i r mo d e l , a f o r m o f th ec o n t i n u i ty e q u a t io n , r ef le c ts t h e w a y t h a t g r o w t h , r e p r o d u c t i o n a n d m o r t a l i ty r a te si n fl u e n ce t h e n u m b e r o f a n im a l s a t e a c h s iz e a n d a g e a s a f u n c t io n o f t i m e . M o d e l so f d e n s i t y - d e p e n d e n t e f fe c t s i n g r o w th , m o r t a l i t y a n d r e p r o d u c t iv e r a te s ( a s s u m in gth e y a r e f o o d - r e l a t e d ) c a n b e b a s e d o n c o n c e p t s o f i n d iv id u a l b io e n e r g e ti c s (o f.,W a r r e n a n d D a v i s , 1 96 7; W a r r e n , 1 97 1; Pa lo h e imo a n d D ic k ie , 1 97 0).T h e w o r k d e s c r ib e d h e r e in i s a n a n a ly s i s o f o p t ima l h a r v e s t p o l i c y f o r a m o r er e al is ti c, g e n e r a l p o p u la t i o n m o d e l b a s e d o n t h e s e tw o k in d s o f mo d e l s . A f t e ro b t a in in g n e c e s s a r y c o n d i t i o n s f o r o p t im a l h a r v e s t i n t h e g e n e r a l m o d e l s p e ci fi cc a s e s a r e e x a m in e d in d e t a il . M o d e l s a s c o m p le x a s t h o s e u s e d h e r e a r e g e n e r a l ly n o tu s e d i n f i s h e r y a n a ly s i s d u e t o t h e l a c k o f a d e q u a t e d a t a t o c o m p le t e ly s p e c i fy t h e m .I n d e e d , t h e a m o u n t o f d a t a n e e d e d t o c o m p l e t e l y s p e c if y a c o m p l e x m o d e l o f aspec i f ic popula t ion- i s se ldom ava i lab le fo r a r ea l f i she ry . However , r a the r thana t t e mp t in g t o d e f in e o p tima l p o l i c y f o r a s p e c if ic p o p u la t i o n , t h e i n t e n t h e r e i s t oe x p o s e e c o n o m i c a s p e c t s o f p o p u l a t i o n b e h a v i o r t h a t a r e o c c l u d e d b y t h es imp l if ic a t io n s i n h e r e n t in o th e r mo d e l s . T h e s e m a y th e n s u g g es t a l te r e d s t r a te g i e sb a s e d o n c u r r e n t d a t a , o r n e w d a t a t h a t a r e v i ta l t o p r o p e r m a n a g e m e n t ~

So m e ma th e m a t i c a l l y s imi la r a n a ly s e s h a v e b e e n p u r s u e d i n t e r ms o f L e s li ema t r i x mo d e l s (1 94 5); h o w e v e r , r e s u lt s o b t a in e d a r e o f lim i t e d v a lu e d u e t o t h el i n e a ri t y o f t h e m o d e l ( s e e M e n d e l s s o h n , 1 9 7 6 , f o r a r e v i e w ) . Be c a u s e t h e m o d e l isl in e a r, o n ly a p o p u la t i o n w i th d o m in a n t e ig e n v a lu e (1 ) g r e a t e r t h a n o n e ( i.e ., a n o n -d e c r e a s in g p o p u la t i o n ) c o u ld y i e ld a s u s t a in e d h a r v e s t . T h i s m o d e l n o t o n ly y ie ld sunreasonab ly op t ima l po l icy bu t i s b io log ica l ly unrea l i s t ic in tha t i t r e f lec t s ap o p u la t i o n t h a t i f l e f t a l o n e w o u ld i n c r e a s e e x p o n e n t i a l l y . T h e o p t ima l h a r v e s tp r o b l e m h a s b e e n m a d e m a t h e m a t i c a l l y m o r e t r a c ta b l e b y i n c lu d i ng c o n s tr a i n ts(e.g ., no t r educ ing i be low on e o r o the r mo re gen e ra l l inea r cons t r a in t s ) (c fo , Ro r resa n d Fa i r , 1 97 5; Ro r r e s , 1 9 7 6 ) . T h e c r u c i a l lim i t a t i o n s o f t h is mo d e l a xe n o wc o m m o n ly re c o g n iz e d ( cf ., M e n d e l s s o h n , 1 97 6; B e d d in g to n a n d T a y lo r , 1 9 7 3;Re e d , 1 98 0)o T h e a p p r o a c h t a k e n h e r e i s b a s e d o n t h e b e l i e f t h a t m o r e u s e f u l r e s u lt sa r e o b t a in e d f r o m mo d e l s t h a t i n c lu d e e s s e n t i a l , b io lo g i c a l l y r e a l i s t i c n o n -l i n e a r i t i e s , t h a n a r e o b t a in e d f r o m l i n e a r mo d e l s w i th a r b i t r a r y , a u x i l i a r ycons t ra in t s .


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Optimal Size-Specific Fishery Policy 2672 . T h e M o d e lT h e m o d e l u s ed w as ch o s e n t o r ea l is t ic a l l y r e f le c t t h e d y n am i c b i o l o g i ca l b eh av i o ra n d i m p o r t a n t e c o n o m i c c h a r a ct e r is t ic s o f a n e x p l o i t e d p o p u l a t i o n . S i nc e g r o w t h ,r e p r o d u c t i v e a n d m o r t a l i t y r a t e s o f in d i v id u a l s i n a q u a t ic p o p u l a t i o n s u s u a l l yd ep e n d m o r e o n s i z e t h an o n ag e , an d i n d i v i d u a l s iz e is t h e v a r i ab l e o f e co n o m i ci m p o r t an ce , a co n t i n u i t y eq u a t i o n i n a s i z e v a r i ab l e w as ch o s en a s t h e b a s i c m o d e l .A r ev i ew o f r e l ev an t fe a t u r e s o f f is h an d c r u s t acea n p o p u l a t i o n s ( B o t s f o r d , 1 97 8)i n d i c a t e d t h a t t h e m o s t i m p o r t a n t d e n s i t y - d e p e n d e n t e f f e c t s i n c o m m e r c i a l l yi m p o r t a n t p o p u l a t i o n s a r e f e l t t h r o u g h m o r t a l i t y i n t h e y o u n g e r s t a g e s a n di n d i v i d u a l g r o w t h r a t e ( u s u a l l y o f j u v en i l e s) . T h es e r a t e s a r e t h e r e f o r e m o d e l ed a sd en s i t y - d ep en d en t .

D e n s i t y ~ e p c n d e n t e f f e c ts o n r e c r u i t m e n t a n d g r o w t h r a te s a r e m o d e l e d h e r e int w o w a y s : ( a ) f o o d - d e p e n d e n t a n d ( b ) p o p u l a t i o n - d e p e n d e n t . I n t h e f o r m e r -- in s ta n c e , re c r u i tm e n t a n d g r o w t h r a t e s d e p e n d o n t h e a m o u n t o f f o o d a v a i l a b l e t ot h e p o p u l a t i o n , w h i c h i s a s s u m e d t o d e p e n d o n l y o n t h e r a t e a t w h i c h f o o d i sp r o d u c e d o r m a d e a v a i l a b l e t o th e p o p u l a t i o n a n d t h e r a te a t w h i c h th e p o p u l a t i o nc o n s u m e s f oo d ~ T h e p r o d u c t i o n r a t e i s a s s u m e d t o b e c o n s t a n t a n d t h e t o t a lp o p u l a t i o n c o n s u m p t i o n r a t e i s t h e s u m o f c o n s u m p t i o n r a t e s o f e a c h in d i v i d u a l i nt h e p o p u l a t i o n .

I n t h e s im p l e r, p o p u l a t i o n - d e p e n d e n t c a s e in d i v id u a l re c r u i tm e n t a n d g r o w t hr a te s d e p e n d o n a w e i g h t e d s u m o f i n d iv i d u a ls o f e a c h s i z e p r e s e n t i n t h ep o p u l a t i o n . A n i m a l s o f d i f f e r en t si ze s i n t h e p o p u l a t i o n a r e w e i g h t ed ac co r d i n g t ot h e i r r e l a t iv e e ff ec t o n each d en s i t y - d ep en d en t r a t e . F o r ex am p l e , i f d en s i t y -d ep en d en t r e c r u i t m en t w e r e d u e t o can n i b a l i s m , t h e e f f ec t o f d i ff e r en t s iz ei n d iv i d u al s o n t h e r e c ru i t m e n t w o u l d b e m o d e l e d b y a w e i g h ti n g f u n c t io n t h a tr e f l e c ted t h e p r o p en s i t y o f an i n d i v i d u a l o f a c e r t a in s iz e t o c an n i b a l i z e t h e y o u n g( cf ., B o t s f o r d an d W i ck h am , 1 97 8)o

E q u a l i n i m p o r t an c e t o a r e a l i st i c b i o l o g i ca l m o d e l i s a r e a l is t ic d e s c r i p t i o n o ft h e e f f ec t o f f is h i ng o n a p o p u l a t i o n an d t h e eco n o m i c b en e fi t s d e r i v ed t h e r e f r o m .T h e r e l a t i o n s h i p b e t w een fr ac t i o n a l m o r t a l i t y r a t e d u e t o f i sh i n g an d t h e co s t o ff i sh i n g is c l a s si ca ll y a s s u m ed t o b e a d i r ec t p r o p o r t i o n , i n d ep en d en t o f p o p u l a t i o ns ize. F r ac t i o n a l m o r t a l i t y r a t e i s a s s u m ed t o b e p r o p o r t i o n a l t o f i s h in g e f f o r t w h i chi s i n t u rn a s s u m e d p r o p o r t i o n a l t o c o s t . I n p r a c t ic e e a c h o f t h e s e a s s u m p t i o n s m a yn o t h o l d d u e t o s u ch e f f ect s a s s ch o o l i n g b eh a v i o r o r i n ad eq u a t e d e f i n i t io n o f e f f o r t( c f. , R o t h s ch i l d , 1 97 7; M acC a l l , 1 9 7 6 ; C l a r k a n d M an g e l , 1 9 79 ). T h es e a s s u m p -t i o n s a l s o i g n o r e t h e cap i t a l i n v es t m en t n ece s s a r y t o m a i n t a i n f i s h i n g cap ab i l i t yev en w h en n o t a c t i v e l y f i sh i n g ( see C l a r k e t a l ., 1 97 9, f o r a d i s cu ss i o n an d an a l y s i s o ft hi s p ro b l e m ) . A l t h o u g h t h e s e p o t e n t i a l d r a w b a c k s a r e a c k n o w l e d g e d , th e y a r e n o td ea l t w i t h h e re . M ea n i n g f u l , p r ac t i c a l r e s u l ts c an b e o b t a i n e d u s i n g t h e l i n ea ra s s u m p t i o n a n d , i f t h is a s s u m p t i o n t h e n a p p e a r s t o b e o f c r u ci a l i m p o r t a n c e , c a s e sf o r w h i ch i t d o es n o t h o l d can b e t h en t r e a t ed .

I n ad d i t i o n t o t h e p r eced i n g co n s i d e r a t i o n s w h i ch a r i s e i n m o s t f i s h e r y9o p t i m i z a t io n p r o b le m s , t h e m o d e l s u s e d h e r e r e q u i r e d e s c ri p ti o n o f t h e d e p e n d e n c eof fi sh ing e f fo r t on i nd iv idua l s ize . S i ze se l ec t i v i t y i n fi sh ing depen ds o n m esh s i ze i nn e t s, e s cap e p o r t s i n t r ap s a n d o n b o a r d s o r t i n g i n s o m e o t h e r c a s e s . M es h s i ze i nn e t s an d e s cap e p o r t s i n t r ap s s e l ec t f o r an i m a l s ab o v e a c e r t a i n s i ze w h e r ea s


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268 L.W. Botsfordo n b o a r d s o r t i n g i s m o re f lexib le . T h e co s t o f f is h i n g each s ize is u s u a l ly eq u a l a n dt h e co s t i s u s u a l l y n o t i n c rea s ed b y i n c rea s i n g t h e r an g e o f s iz es f is h ed . Th e re fo re ,wh i le s ize spec i f i c m odels a l low the f l ex ib i l ity o f a d i f fe ren t f i sh ing e f fo r t a t ea chs ize, t h e p ro b l em s e n co u n t e red i n m o d e l i n g t l~ e a s s o c i a ted co s t a r e co m p l ex . In t h ew o rk d e s c r i b ed h e re , a co m p ro m i s e b e t ween e f fo r t b e i n g co m p l e t e l y s i z e -d ep en d en tan d co m p l e t e l y s i z e - in d ep en d en t i s u s ed . B r i e fl y , co n s t an t e f fo r t is ap p l i ed o v e ro p t i m a l s iz e r an g es , w i t h t h e co s t o f e f fo r t i n d ep e n d en t o f s iz e r an g e .

F i s h in g e f fo r t , E ( t ) , i s d e f i n ed t o b e t h e f r ac t i o n a l m o r t a l i t y p e r u n i t o f t i m e a tt i m e t . Th i s e f fo r t is ap p l i ed o n l y o v e r a s p ec i fi ed s ize r an g e . Th e l o wer an d u p p e rb o u n d s o n t h e S iz e r a n g e a t t im e t a r e d e n o t e d b y m ~ ( t ) an d mu( t) , respec t ive ly . ( Int h e fo l l o wi n g co n t ro l an a l y s i s t h e n o t a t i o n r e f l e c ts o n l y t h i s s i n g le siz e r an g e b e i n gf i s h ed ; t h e p o s s i b il it y o f t wo o r m o re u n c o n n ec t e d r an g es is d i scu s sed la t e r ). Th e ex -v ess e l v a l u e o f a f i sh o f s iz e m a t t i m e t i s d en o t ed p (m , t ) . Th e r a t e a t w h i ch r ev en u esa re g e n e ra t ed d u e t o f i sh i n g is t h en

~mu(t)A ( t) - E ( t ) [ n ( m , t )p ( m , t ) d m , ( I )ml(t)w h e r e n ( m , t ) d m i s t h e n u m b er o f in d i v i d u a l s b e t ween s izes m a n d m + d m a t t i m e t .No t e t h a t f r ac t i o n a l m o r t a l i t y ( f i s h i n g e f fo r t ) i s co n s t an t o v e r t h e s i z e r an g e( rn~( t) , m ~( t ) ) . I n m o s t p r a c t ic a l s i tu a t io n s t h e c o s t o f a u n i t o f e f fo r t is i n d e p e n d e n to f e f fo r t. Th e co s t o f f is h i n g i s t h e re fo re a s s u m ed t o b e p ro p o r t i o n a l t o e f fo r t . Th ep r o p o r t i o n a l i t y c o n s t a n t K r e p r e s e n ts t h e p r o d u c t o f t h e c o s t p e r u n i t n o m i n a le f fo r t an d t h e am o u n t o f r e a l e f fo r t p e r u n i t n o m i n a l e f fo r t (e .g ., d o l l a rs p e r b o a t -d a y t i m e s f ra c t i o n a l m o r t a l i t y r a t e p e r b o a t - d a y ) . T h e r a t e a t w h i c h r ev e n u e s a r es p en t t h r o u g h f i sh i n g is t h en

is(t) -----K E ( t ) . (2 )T h e c o m p l e t e m o d e l c a n n o w b e e x p r e s s e d i n t e r m s o f t h e o p t i m a l h a r v e s t

p r o b l e m : t o c h o o s e t h e f u n c t i o n s E ( t ) , m ~ ( t ) a n d m u ( t ) s o t h a t t h e p r e se n t v a l ue o ft h e r e s o u rce ,

J ' = - f 2 e - t t [ J ; ( t ) - j s ( t ) ] d t

f : _ + ,e - n ' E ( t ) L Jm,,,) p ( m , t ) n ( m , t ) d m (3 )is m ax i m i zed s u b j ec t to t h e co n s t r a i n t

0


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Optima l Size-Specific Fishery Policy 269The c ons t ra in t t ha t de f ines the pop u la t io n i s a s ize - spec i fi c , con t inu i ty eq ua t ionp o p u l a t i o n m o d e l ,On(m, Q 0. . . . [ g ( m , t ) n ( m , t ) ] - [ d ( m ) + E ( t ) ] n ( m , t ) ; m t < m < m . , (5a)Ot Om

O- - - [ 9 ( m , t ) n ( m , t ) ] - d ( m ) n ( m , t) ; m ~< mt, m >t m~, (5 b)Omw h i c h r e q u i r e s b o u n d a r y c o n d i t i o n s

g(mo , t )n (mo , t ) ---- R i [ n ( ' , t ) , t ] ,n (m , O) = no (m) ,

w h e r e

(6)(7 )

0(m, t ) = the g ro w th ra t e o f an ind iv idu a l o f s ize m a t t ime td ( m ) = t h e f r a c t io n a l m o r t a l i t y r a t e o f a n i n d i v i d u a l o f si ze m

m o = t h e s iz e a t wh i c h y o u n g e n t e r t h e m o d e l sp a c e (i .e ., a r er e c r u i t e d t o t h e p o p u l a t i o n )

L2 = b ( m ) n ( m , t ) d in . (9b)Fo r l i n e a r r e p r o d u c t i o n

a n dR ~ [ n ( . , t ) , t ] = t h e r e c r u i t m e n t r a t e a t s iz e d e n s i t y f u n c t i o n n ( - , t ) a n d

t i m e t ( f o r th e i t h v e r s i o n o f t h e m o d e l ).T h e Fa s t t e rm i n t h e p o p u l a t i o n m o d e l d e sc r ib e s t h e e ff e c t o f g r o w t h r a t e o n s i zed e n s i t y f u n c t i o n n a n d t h e s e c o n d t e r m d e sc r i b e s t h e e f f ec t o f m o r t a l i ty . F r a c t i o n a lm o r t a l i t y r a t e i n c lu d e s f i sh in g m o r t a l i t y o n l y f o r s i ze s b e t we e n m r ( t ) a n d m , ( t ) . T h eb o u n d a r y c o n d i t io n s ( 6 ) a n d ( 7 ) d e sc r i b e b e h a v i o r a t t h e s iz e b o u n d a r y ( m = m e )fo r a l l t ime , and the t ime b ou nd ar y ( t = 0 ) fo r a ll si ze . A l tho ug h no t exp l i c it l yr e f le c t e d i n t h e n o t a t i o n u se d , b o t h g r o w t h r a t e g a n d r e c r u i t m e n t r a t e R ( c a nd e p e n d o n e i t h e r f o o d l e v e l F ( t ) ( f o o d - d e p e n d e n t ) o r t h e o t h e r m e m b e r s i n t h ep o p u l a t i o n ( p o p u l a t i o n - d e p e n d e n t ) . I n t h e l a tt e r c a se t h e se r a te s d e p e n d o ne f f e ct i v e p o p u l a t i o n s iz e , wh i c h i s d e f i n e d b yf.( t ) - - c ( m ) n ( m , t ) d m (8)

o

w h e r e c ( m ) i s a w e igh t ing fun c t ion re f l ec t ing the re l a t ive e f fec t o f an an im a l o f si ze mo n g r o w t h o r r e c r u i t m e n t r a t e .Se v e r a l v e rs i o n s o f R~ a r e c o n s i d e r e d i n t h e f o l l o wi n g a n a l y si s . F o r c a se s inwh i c h r e c r u i tm e n t i s a s su m e d i n d e p e n d e n t o f a d u l t s to c k , r e c r u it m e n t i s a s su m e d t obe spec i f i ed

R t = R ( t ) . (9a )


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270 L.W . BotsfordF o r f o o d - d e p en d e n t re c r u i tm e n t

R 3 = R 2 f [ F ( t ) ] . (9c)F o r p o p u l a t i o n - d e p e n d e n t r e c r u i t m e n t

R 4 = R 2 f c [ C ( t ) ] . (9d )F i n a l l y t h e co n s t r a i n t t h a t d e f i n e s b eh av i o r o f fo o d l eve l F ( t ) is

d F / d t - - ~ + / d t - w ( m , N t ) ) n ( m , t ) d m ( 1 0 )0

w h i c h re q u i re s a b o u n d a r y c o n d i t i o nr(O ) = Fo (11)

w h e r ew(m, F ( t ) ) - . t h e c o n s u m p t i o n r a t e o f a n i n d iv i d u al o f s iz e m a t f o o d

leve l F( t ) ,d F + / d t = t h e r a t e a t w h i c h f o o d i s m a d e a v a il a bl e t o t h e p o p u l a t i o n( a s s u m e d c o n s t a n t) .

Th i s ex p re s s i o n r ef l ec t s t h e r a t e o f i n c rea s e o f fo o d av a i l ab l e t o t h e p o p u l a t i o n ,w h i c h is a ss u m e d t o b e t h e s u m o f a p o s it iv e c o m p o n e n t d u e t o p r o d u c t i o n a n d an e g a ti v e c o m p o n e n t r e fl e c ti n g t o t a l f o o d c o n s u m p t i o n b y t h e p o p u la t io n ~

Th u s , t h e co n t ro l p ro b l em i s b a s i ca l l y t o m ax i m i ze t h e p re s en t v a l u e o f t h ere s o u rce s u b j ec t t o o n e o r t wo co n s t r a i n t s : o n e b e i n g a s i z e -s p eci fi c co n t i n u i t ye q u a t i o n m o d e l w i t h v a r i o u s r e p r o d u c t i v e b o u n d a r y c o n d i t i o n s , a n d t h e o t h e rb e in g t h e f o o d e q u a t i o n w h e n a p p l i ca b le . I n g e n e r al , g r o w t h a n d r e p r o d u c t i v e r a t e sc a n d e p e n d o n f o o d l ev e l o r o t h e r m e m b e r s o f t h e p o p u l a ti o n .

To av o i d p o t en t i a l co n fu s i o n d u e t o ad d i t i o n a l n o t a t i o n , i n t h e fo l l o wi n gd e r iv a t io n s b o t h g r o w t h a n d r e c r u i t m e n t r a t e s in t h e m o d e l a r e a s s u m e d t o d e p e n do n f o o d l ev e l F ( t ) [ i . e . , i - - 3 in Eq . (6 ) ( see Eq . 9c) ] and g i s de no ted b y g (m, F) .Ho w ev e r , f in a l r e s u lt s a r e g i v en fo r s ev e ra l d i f f e r en t c a s e s o f i n t e r e st , an d t h ereq u i r ed ch an g es i n t h e d e r i v a t i o n a r e o b v i o u s .

3. Necessary Condit ions for Optimal HarvestT h e a p p r o a c h t o d e r i v in g n e c e s s a ry c o n d i t i o n s is b a s e d o n a s t a n d a r d a p p r o a c h t oa p p li e d o p t im a l c o n t r o l o f l u m p e d p a r a m e t e r s y s t em s ( cf ., B r y s o n a n d H o , 1 96 9, o rC i t ro n , 1 96 9)o Th e co n t ro l p ro b l em can b e d i s c re t iz ed i n t h e s iz e v a r i ab le a n dt r e a t e d a s a n a c t u a l lu m p e d p a r a m e t e r s y s t e m w i t h a n a l o g o u s r e su l ts ( cf ., B o t s f o rd ,1 97 6) . Th e l u m p ed p a ra m e t e r ap p ro ach i s b a s i ca ll y t o ad j o i n co n s t r a i n t s t o t h e co s tfu n c t i o n a l , ch o o s e v a l u e s fo r m u l t ip l i e r s ( ad j o i n t v a r i ab le s ) t h a t c au s e v a r i a t i o n sd u e t o v a r i a t i o n s in s t a t e v a r i ab le s t o b e ze ro , an d c h o o s e t h e co n t ro l v a r i ab le s s ot h a t t h e v a r i a t io n d u e t o v a r i a t i o n s i n t h e co n t r o l v a r i ab l e a r e ze ro .Th e d e r i v a t i o n o f n eces s a ry co n d i t i o n s fo r o p t i m a l co n t ro l o f t h i s s y s t em isd es c r i b ed i n Ap p en d i x A . Th e r e s u l t i n g ad j o i n t eq u a t i o n s a r e :


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Opt imal Size-Speci f i c Fi shery P o l icyO 2 , ( m , t ) . ~ 02 , (m, t )0 - - - - - ~ - - - [ a ( m ) + E ( t ) ] 2 , ( m , t ) - a r m , ~ ) " ~m

- 2 . (mo , t ) b ( m ) f [ F ( t ) ] + 2 F ( t )w ( m , F ) ;O2,(m, t ) = d ( m ) 2 , ( r n , t ) - g ( m , F ) a 2 . ~ m , t)0-----~ 2,(mo, t ) b ( m ) f [ F ( t ) ]

+ 2 v ( t ) w ( m , F ) ; m > m , ( t ) , m < m r (t ),d 2 F = _ n ( m , t ) O 0 F ) 0 2 . ( m , t ) . . . O w ( m , F )d t o O m - z r ( t ) . ~

- 2 . ( too, t ) b ( m ) f ' [ F ( t ) ] J dra .N e c e s s a r y c o n d i t i o n s f o r c o n t r o l v a r i a b l e s rn t a n d m u a r e

e - r t E ( t ) p ( m )m r ( t )


8/29

27 2

Th e de rivat ion of the expression

6 [ f s i i : ; P ( m ) n ( m , t ) d m - K ]=

,Iml(O

- - e 6 ' 2 v ( t ) w ( m ,F ) + e~'2.(mo, ) b ( m ) f [ F ] } d i n ,f rom d S / d t = 0 i s descr ibed in A ppendix B. Derivat ion of

L . W . Bo t s f o r d

(18)

1 d m ~E ( t ) = - - d - ~ I G ( m . , t ) n ( m . , t ) ( [ - ~ - - g ( m u , F ) )

- G (m ,,t ) n ( m ,,t ) (- ~ - o ( m ,: _ b " ) )- - ~ 1 f ' ' ~ I F ) O G ( m ' t ) O G ( m ' t ) 19 6 K a . ,, ~ o n ( m , t ) g(m , ~ + 0------~-- d ( m ) G ( m , ) d i n , (1 9 )

wh ere G(m, t) is defined to be

G ( m , t ) = - o ( m , F ) d ~ _ p ( m ) d ( m ) - e a '2 F ( ) w ( m , F )+ e 6 '2 . ( m o , t ) b ( m ) f [ F ] - 6 p ( m ) , (20)

from d Z S / d t ~ - - 0 i s a l so descr ibed in Ap pendix B. S inc eE (t ) would no t appear inthis expression for the case 6 = 0, the problem is not o f fi rst orde r singularity in thatcase. This special case is not pursued further here.The necessary condi t ions for opt imal i ty der ived thus far are for fo od-d epen dentrecrui tmen t and grow th rate. Th e de rivat ion is only slightly different i f ei ther ofthese is mode led as popu la t ion depende nt . I f grow th ra te i s pop ula t ion-depend entra ther than food-d epend ent the term~,. m'0 1 . ( m ' , t ) 0 0 ( , t )c ( m ) n ( m ' , ) ~ O ~ d in '

o

will appear in add it ion to 2 r ( t) w ( m , F ) whe rever the lat ter appears in the adjo intequa tions (12a, b) an d the con dit ions for a singular arc (18, 19). If re crui tm ent ispopula t ion-dependent a ther than food-depend ent , he appropria te express ions forRi (i .e . , R4 in E q. (9d)) wil l lead to appeara nce o f the term2 . ( m o , t ) [ b ( m ) f , [ C ( t ) ] + c ( m ) f ' c [ C ( t ) ] f . ~ b ( m ' ) n ( m ' , t ) d m " ]

instead of2.(too, t ) b (m ) [ F ( ) ]

wherever the lat ter appears in th e ad joint equations and co ndit ions for a singular


9/29

Optimal Size-Specific Fishery Policy 273a r c . I f n e i t h e r r ec r u i t men t n o r g r o w t h r a t e a r e m o d e l e d a s f o o d d ep en d e n t , t h ed ep en d en ce o n F i n t h e mo d e l an d t h e ad j o i n t eq u a t i o n w i l l n o t b e p r e s en t .

I n ca s e s f o r w h i ch t h e g r o w t h r a t e d o es n o t d e p en d o n F , t h e p a r t i a l d i f fe r en t ia leq u a t i o n ( 1 2) fo r t h e ad j o i n t v a r i ab l e 2 ,( m , t ) c an b e s o l v ed u s i n g th e m e t h o d o fch a r ac t e r is t ic s . A l s o , a l t h o u g h t h e ca s e o f f o o d d ep e n d e n t s u r v i v a l o f y o u n g i s u s edi n t h e d e r i v a t i o n , f o r t h e a l t e r n a t e ca s e i n w h i ch s u r v i v a l o f th e y o u n g d ep en d si n s t ead o n t h e n u m b er o f o ld e r an ima l s , t h e r ep r o d u c t i v e t e r m i n t h e p a r ti a ld i f f e r en t i a l eq u a t i o n f o r t h e ad j o i n t v a r i ab l e w o u l d b e r ep l aced b y a t e r m t h a td ep en d s o n t h e s i ze d en s i t y n ( m , t ) . Th er e f o r e , a l t h o u g h t h e ex p r e s s i o n s d e r i v edh e r e h o l d f o r t h e l a t t e r c a s e, t h e f ac t th a t t h e ac t u a l s o l u t i o n f o r t h e ad j o i n t v a r i ab l ed e p e n d s o n t h e d e p e n d e n t v a r i a b le n ( m , t ) m u s t b e k e p t in m i n d. T h e d e r i v a t io n o ft h e s o l u t i o n is d e s c r i b ed i n A p p en d i x C . 2 ~ (m, t ) ( f o r a g r o w t h r a t e t h a t d e p en d s o ns ize o n l y ) can b e ex p r e s s ed i n t e r ms o f ag e s r a t h e r t h an s ize m a s

X ( s , t o ) = ~ ( 0 ,. -~ + t o ) [ ~ ( z ) f [ F ( z + to)] o'(s---'ff-- 2r(~ +/ o )~ (z ,F (~ + to)) a--~-s)d*, s > s u ( t ) (21a)

f ?(s , to) = [p(~)E(~ + to )e - ~ ' 1 7 6 + ~ ( 0 ,~ + t o ) f [ F ( ~ + to)]~(~)- ,~ F (~ + t o l ~ ( z , F ( T + t o ) l ] ~ d ~ + J ~ ( s ~ , o ) ~ ( s u )~ ( s ) '

s t ( t )


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274 L. W, Botsford

--- ex p - [~( x)] d x ; s < s t ( t ) , s > s ~ (t ) (22)SO

where if(s) ~ d ( m ( s ) ) .The o rd ina ry d i f fe ren t ia l equ a t ion fo r t h e ad jo in t va r i ab le 2F cai a be so lved in aw a y s i m il a r to t h e m e t h o d o f so l u t i o n f o r e q u a t i o n (2 1). T h e so l u t i o n ( wh i c h is a l soi n Ap p e n d i x C ) i s2 , ( O = f , (m ,s ) [ O . C m o ,_ _ f(s))2.(m, s ) ]

O w ( m , F ( x ) ) , _ _ ] _- ( 2 ( m o , s ) b ( m ) f ' [ F ( s ) ] ) d m e x p [ - ~ f , , o n ( m , x ) - ~ -- f a m a x j a s .


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Op timal Size-Specific Fishery Policy 275t e rp re t a t i o n s r e f e r t o t h e f ac t t h a t t h e ad j o i n t v a r i ab l e r ep re s en t s th e ch an g e i n t o ta lfu t u re r ev en u es fo r a u n i t ch an g e i n i ts co r r e s p o n d i n g s t a t e v ~ ir iable .

E q u a t i o n s ( 21 a - c ) c a n b e i n t e rp r e t e d s im i l a rl y f o r th e a d j o i n t v a r i a b l e 2, . T h i sad j o i n t v a r i ab l e i s t h e t o t a l f u t u re v a l u e o f an i n d i v i d u a l o f ag e s , in c l u d i n g t h e v a l u ed u e t o f i s h in g r e v e n u e s , t h e v a l u e d u e t o r e p r o d u c t i o n a n d t h e v a l u e d u e t o f o o d n o tea t en . Th e ad j o i n t v a r i ab l e 2 , t h u s rep re s en t s t h e ch an g e i n fu t u re v a l u e o f t h er e s o u r c e i n c u r r e d b y r e m o v i n g a n i n d i v i d u a l o f a g e s. N o t e t h a t 2F , t h e m a r g i n a lv a l u e o f a u n i t o f fo o d l ev el, ap p e a r s i n t h e t e rm re f l ec ti n g t h e fu t u re v a l u e o f fo o dn o t co n s u m e d . S i m i l a r l y ~ '(0, t ) , t h e m arg i n a l v a l u e o f an i n d i v i d u a l o f ag e ze ro ,ap p ea r s i n t h e t e rm re f l ec t in g t h e fu t u re r ep ro d u c t i v e v a lu e .

Th e a d j o i n t v a r i ab l e 2~, c a n b e i n t e rp re t ed s i m i l a r l y . Th e t wo t e rm s i n t h e i n n e ri n te g r a l [ E q . ( 2 3 )] a r e e a c h a p r o d u c t o f th e a m o u n t t h a t a c h a n g e i n F a f f e ct s t h ed e n s i ty fu n c t i o n s ( g r o w t h a n d s u r vi v a l o f y o u n g ) a n d t h e a m o u n t t h a t t h e s e a ff e c tfu t u re v a l u e ( t h ro u g h 2 , ). Th es e t e rm s , b e i n g fu n c t i o n s o f si ze , a r e i n t eg ra t ed o v e ra l l s iz e. T h e i n t eg ra l o f t h e r e s u l t o v e r a l l f u t u re t i m e i s m u l t i p l ied b y a d ec rea s i n gex p o n en t i a l . Th i s f ac t o r r e fl e c t s t h e f ac t t h a t t h e re i s a f r ac t i o n a l ch an g e i n 2 r w i t ht im e t h a t is p r o p o r t i o n a l t o t h e s e n s it iv i ty o f c o n s u m p t i o n r a t e t o F .

T h e d i f fe r e n t ia l e q u a t i o n s d e s c r ib i n g t h e r a t e o f c h a n g e o f t h e a d j o i n t v a r i ab l e sw i t h ti m e c a n b e i n t e r p r e t e d a s m i n u s t h e r a t e o f d e p r e c i a t io n o f c a p it a l ( D o r f m a n ,1 96 9; C l a rk , 1 9 7 6 ) . A l o n g t h e o p t i m a l p a t h t h is m u s t eq u a l t h e s u m o f t h eco n t r i b u t i o n t o t h e r a t e a t wh i ch p ro f it s a r e r ea l iz ed an d t h e r a t e a t w h i ch t h e v a l u eo f t h e s t o ck i s en h an c ed (D o r fm an , I96 9) . I n t h e f i rs t d i f f e r en t i a l eq u a t i o n fo r 2 ,,( 12 a) , t h e t h i rd t e rm o n t h e f i g h t -h an d s i d e i s t h e co n t r i b u t i o n t o p ro f i t s wh i l e t h er e m a i n d e r o f t h e t e r m s a n d a ll t e r m s i n ( 12 b) a r e c o n t r i b u t i o n s t o v a l ue ( t h r o u g hfu t u re p ro f i ts ) . I n t h e d i f f e r en t i a l eq u a t i o n fo r 2 ~, (1 3) , t h e re a r e n o co n t r i b u t i o n s t op r o fi ts s i n c e f o o d is n o t h a r v e s t e d o r p r o v id e d . T h e r a t e o f c h a n g e o f 2~ t h e r e f o r eco n t r i b u t e s t o v a l u e o n l y .I n t e r p r e t a t i o n o f t h e c o n d i t i o n f o r s i n g u l a r c o n t r o l ( d S / d t = 0 ) c an b e b e s td e s c r i b ed t h ro u g h d e f i n i t i o n o f n e t b i o v a l u e V

V =- ( m ) n ( m , t ) d m - K . (27)T h i s d e f i n i t i o n i s s i m i l a r t o t h e d e f i n i t i o n o f n e t b i o v a l u e f o r t h e s i n g l e a g e - c l a s s ,B c v c r t o n - H o l t m o d e l i n C l a r k e t al. ( 1 9 7 3 ). N e t b i o v a l u e i s t h e n e t r e v e n u e s g a i n e dp e r u n i t f i s h in g e f fo r t p e r u n i t t i m e . T h e l e f t - h a nd s i d e o f t h e e q u a t i o n d e r i v e d f r o md S / d t = 0 [ E q . ( 1 8 ) ] i s 5 V . T h e r i g h t - h a n d s i d e o f t hi s q u a t i o n i s d V / d t d u e t o n o n -f i s h i ng c a u s e s ( i.e., e m o v a l b y f i s h i n g i s n o t i n c l u d e d a s a d e c r e a s e i n V ) . T h e f ir stt e r m i n c l u d e s t h e i n c r e a s e i n v a l u e d u e t o g r o w t h , t h e s e c o n d t h e d e c r e a s e i n v a l u ed u e t o m o r ta l i ty , t h e t h i r d t h e d e c r e a s e i n v a l u e d u e t o f o o d c o n s u m e d , a n d t h e l as tt h e i n c r e a s e i n v a l u e d u e t o r e p r o d u c t i o n . S i n g u l a r c o n t r o l t h u s m a i n t a i n s t h en a t u r a l r a t e o f c h a n g e o f n e t b i o v a l u e w i t h t i m e e q u a l t o t h e r a t e o f g r o w t h o f t h em o n e t a r y e q u i v a l e n t o f n e t b i o v a l u e a t in te re st a t e 6 (i,e., V / d t = 6 V ) . T h i s m a k e ss e n s e i n t u i t iv e l y . I f d V / d t w e r e g r e a t e r t h a n 6 V , b i o v a l u e w o u l d b e i n c r e a s i n g a t af as te r a t e t h a n a n e q u i v a l e n t i n v e s t m e n t , h e n c e it o u l d b e b e s t t o p o s t p o n e f i s h in ga n d l et it g r o w . I f d V / d t w e r e l e s s t h a n 5 V , t h e r e s o u r c e s s h o u l d b e f i s h e d a t t h em a x i m u m r a t e a n d t h u s b e c o n v e r t e d t o c a pi t al w h i c h c a n i n c r e a s e i n v a l u e fa s te r.


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276 L.W . BotsfordAs sum ing Emax i s l a rge e nou gh , t he s t r a t egy desc r ibed a s f i sh ing a t E ,~az w hendV/d t > 6 V i s equ iva len t t o s t ay ing on the l i ne dV/d t = 6 V .

T h e o p t i m a l v a l u e s o f c o n t r o l v a r i a b le s m~ a n d m ~ a r e su c h t h a t t h e c u r r e n tm a r k e t v a l u e o f an i n d i v i d u a l a t e i t h e r b o u n d a r y s iz e e q u a l s t h e t o t a l f u t u r e v a l u e t ob e d e ri v e d f r o m t h a t i n d i v id u a l . N o t e t h a t t h e e x p r e ss i o n f o r d V / d t [ t h e r i g h t - h a n ds i d e o f E q . ( 1 8 ) ] d o e s n o t i n c lu d e a n y t e r m s d e sc r i b in g t h e f l o w o f v a l u e d u e t og r o w t h o f i n d i v id u a l s a c r o s s t h e f ish in g b o u n d a r i e s . T h i s i s d u e t o t h e f a c t t h a t t h ec o n d i t i o n s f o r o p t i m a l v a l u e s o f th e b o u n d a r i e s p r e sc r ib e t h a t t h e v a l u e o f a nind iv idua l a t t he s i ze bo un da ry i s ze ro ( i. e., t he su m o f i t s cu r re n t a nd fu tu r e va lue i sz e ro ) . A s e c o n d c o n se q u e n c e o f t h e s i ze li m it c o n d i t i o n s i s t h a t t h e c u r r e n t m a r k e tva lue o f a l l i nd iv idua l s w i th in the rang e o f sizes to be f i shed i s g rea t e r t han the i rf u t u r e v alu e~ T h e d e c i s i o n c o n c e r n i n g h o w m u c h t o f ish e a c h s i ze o f i n d iv i d u a l s c a nt h u s b e v i e we d a s a t wo - s t e p p r o ce s s~ T h e c u r r e n t m a r k e t v a l u e o f t h e f i sh a t p r e se n tm us t be g rea t e r t han i t s fu tu re va lue i f l e f t unf i shed , i n o rd e r fo r it t o be wi th in thef i shab le range . Once i t i s wi th in the f i shab le range , t he dec i s ion conce rn ing howm u c h f i shi n g p r e s su r e sh o u l d b e a p p l i e d i s t h e n b a se d o n t h e t o t a l n u m b e r w i t h i nthe f i shab le range . The l a t t e r dec i s ion i s based on cons ide ra t ions d i scussed ea r l i e rrega rd ing E( t ) . T h e se d e c i s i o n s a re , o f c o u r se , n o t m a d e i n d e p e n d e n t l y s i n c e t h econ t ro l l aw fo r t he s i ze l imi t s depen ds o n the fu tu re va lue o f fi sh ing e f fo r t , t hec o n t r o l l a w f o r f i sh in g e f f o r t d e p e n d s o n t h e c u r r e n t v a l u e o f s iz e l im i t s, a n d t h e s iz el imi t con t ro l s a l so a f fec t t he cond i t ions fo r s ingu la r con t ro l .

4 . S o lu t ion s to S evera l C asesT h e : n e c e s sa ry c o n d i t i o n s f o r o p t i m a l h a r v e s t c a n b e so l v e d f o r s e v e r a l d if f e r e n tc a se s o f b o t h t h e o r e ti c a l a n d p r a c t ic a l i n t e re s t. A n a l y ti c a l so l u t io n s c a n b e o b t a i n e df o r t h e f ir s t t wo c a se s, wh e r e a s n u m e r i c a l t e c h n iq u e s a r e r e q u i r e d f o r t h e r e m a i n i n gcases. The f i r st ca se , the dyn am ic po o l m ode l (Beve r ton and H ol t , 1957), i s so lved tod e m o n s t r a t e t h a t t h e g e n e r a l so l u t i o n o b t a i n e d h e r e yi e ld s t h e s a m e r e su l t s a s th o seob ta ined p rev ious ly by Cla rk e t a l . (1973) and o the r s fo r a s imple mode l . These c o n d c a se is so l v e d t o r e a f f ir m t h e l a c k o f r e a li sm a n d p r a c t ic a l v a l u e o b t a i n e df rom so lu t ions to a l i nea r age*spec i fi c po pu la t ion mo de l . Th i s so lu t ion am pl i f ie s t hec o m m e n t s o f M e n d e l s so h n ( 1 9 7 6 ) a n d R e e d (1 97 9) b y p re se n t in g t h i s p r o b l e m i n anew l igh t . The rema in ing cases a re new so lu t ions app l i cab le to f i she ry anda q u a c u l t u r e p r o b l e m s .

Single A#e ClassThe f t r s t ca se invo lves op t im a l ha rv es t o f a s ing le age c l a ss o f iden t i ca l s ize wi th n odens i ty -d epen den t e f fec t s ( s imi la r t o the mode l o f Bever ton and H ol t , 1957). S incethe re is on ly a s ing le age c l a ss w e can se t to = 0 in b o th the ad jo in t and s t a t ee q u a t i o n s . S i n ce sp e c if y i n g b o t h s iz e a n d t im e i s r e d u n d a n t i n t hi s c a se , t h e n u m b e rof an ima l s a t t ime t i s de f ined to be N ( t ) ( w i th N( 0 ) = No ) . S i m i la r ly b o t h g r o wt ha n d m o r t a l i t y r a t e s c a n b e e x p r e s se d a s f u n c t io n s o f t im e o n l y . B a se d o n a so l u t i o nto the s ize spec if ic con t inu i ty e qu a t io n ob ta in ed in Bo t s fo rd (1978) , N ( t ) c a n b eexpressed a s


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Optimal S ize-SpecificFishery Policy 277

N ( t ) = ~ N o e x p l - f ' o ( ~ ( s ) + E ( s) ) s ] . (28)T h e e x p r e s s i o n f o r t h e a d j o i n t v a r i a b l e c o n t a i n s o n l y t h e fi sh in g t e r m ; f r o m t h e~ d j o i n t e q u a t i o n ( 2 1 a - c ) a n d f ish i n g s iz e l im i t c o n d i t i o n s ( 1 4 a , b ) , a s su m i n g t h a t

the re i s on ly one so lu t ion to the l a t t e r , t h i s cond i t ion can be expressed a sm ( t , )= , le - ' ( '- '" g ( z ) m ( z ) e x p l - , i ( f f ( s )+ E ( s )) d s ] d z , (29)

w he re t~ i s the f i shing a ge l imi t a nd p ( t ) = k m ( t ) .Fr o m ( 1 5 ) , S ( t ) c a n b e e x p r e s se d a s

S ( t ) = e - ~ * k m ( t ) N ( t ) - K e - ~ ' - N ( t ) A ( t ) . (30)F r o m ( 1 8 ) , d S / d t = 0 impl i e s; 6 k m ( t ) N ( t ) ~ 6 K = N ( t ) k o ( t ) - N ( t ) k m ( t ) d ( t ) (31)w h i c h c a n b e s o l v e d f o r N ( t ) .

6 KN * ( t ) = (32)k [ g ( t ) - r e ( t ) d ( t ) - 6 m ( t ) ]T h i s i s a s l i g ht ly m o r e g e n e r a l v e r s i o n o f C l a r k ' s e x p r e s s i o n f o r t h e s a m e p r o b l e m[Eq . (8 .37) in Cla rk , 1976] .T h e o p t i m a l c o n t r o l so l u t i o n c a n b e d e t e r m i n e d f r o m t h e se e x p r e ss i o n s . A tt = O , S ( t ) i s a s su m e d t o b e n e g a t i v e in a ll ca se s o f i n te r es t , a n d t h e c o n t r o l E ( t ) isze ro . Th e m in im um s i ze l imi t i s r eached a t t = t~ ; how ever , f i sh ing does no tnecessa r i ly beg in . The f i sh ing p ress ure E ( t ) i s s t il l zero bec au se a t t = f i, f i shing s izel imi t con t ro l i nd ica t e s tha t t he f i r s t and th i rd t e rms o f S ( t ) a r e e q u a l , h e n c e S ( t ) iss til l nega t ive (excep t i n the case K = 0 ). Wh en N ( t ) reaches N * ( t ) s i n g u l a r c o n t r o lbeg ins . F i sh ing p ressure i s ad jus t ed so tha t N ( t ) = N * ( t ) , a s su m i n g t h e r e q u i r e dva lue is no t l a rge r t h an Em~,. S ince 2 ( 0 in the th i rd t e rm o f S ( t ) i s ze ro a t t he t imef i sh ing i s d i sco n t inued , i t is d i sco n t inue d w hen the f i r s t two t e rms a re equa l . Thus ,w h e n N ( t ) h a s d e c r e a se d t o K / k m ( t ) , f i sh ing i s d i scon t inued . Seve ra l d i f fe ren tc o n t r o l h i s t o r i e s c a n a r i s e f o r v a r i o u s p a r a m e t e r v a l u e s . T h e se a r e d i s c u s se d i nCla rk (1976) , C la rk e t a l . (1973) , and Gol i (1973) , and the re fo re a re no t r epea tedhe re ( see Cla rk , 1976 , fo r g rap h ica l r e presen ta t ion s o f t he va r ious cases ) .

L i n e a r M o d e lFo r t h e s e c o n d c a se , th e s t a t e e q u a t i o n f o r t h e p o p u l a t i o n i s t h e c o n t i n u i t y e q u a t i o nw i t h t h e l i n e a r r e p r o d u c t i v e b o u n d a r y c o n d i t i o n a n d w i t h o u t d e n s i t y - d e p e n d e n tg r o w t h r a t e. S i n c e g r o w t h r a t e i s n o t d e n s i t y d e p e n d e n t , t h e s iz e -sp e ci fi c c o n t i n u i t yequ a t ion i s equ iva len t t o the age -spec if i c con t inu i ty equa t ion . Th e l a tt e r , wh enc o m b i n e d w i t h a l in e a r r e p r o d u c t i v e b o u n d a r y c o n d i t i o n , is e q u i v a l e n t t o a r e n e wa le q u a t i o n wh i c h i s t h e c o n t i n u o u s - t i m e , c o n t i n u o u s - a g e a n a l o g u e o f th e L e s li ema t r ix (Les l i e , 1945) . For ~b de f ined a s

q ~ b = f ~ b ( a ) a ( a ) d a , (33)


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Optimal Size-SpecificFishery Policy 279o p t i m a l u n t i l t = T . F o r t h e s ak e o f s i m p li c it y , t h is d i s cu s s io n h a s t h u s f a r n o tinvo lved the d i sc ou n t r a t e f i: I f6 i s l a rge , t he " in i t i a l c on d i t i on " on ,~ ( i. e. , i t s va lu e a tt = t o ) i s s m a l l , h en ce S eq u a l s z e r o s o o n e r . T h u s 6 ac t s i n t h e s am e w ay b u t i no p p o s i t i o n t o t h e d o m i n a n t r o o t o f th e r e n e w a l e q u a t i o n f o r ~(0 , t) . T h e b e h a v i o r ist h e r e f o r e d e t e r m i n ed b y t h e d i f f e r en ce b e t w een t h em .

A g a i n , t h e se q u a l i ta t i v e c o n c l u si o n s a r e b a s e d o n t h e d o m i n a n t b e h a v i o r o f th ea d j o i n t v a r i a b l e , a n d s o m e e x c e p t i o n s c o u l d p o s s i b l y b e f o u n d . H o w e v e r , t h eg en e r a l r e s u l t h a s in t u i t iv e ap p ea l . I f o n e h a s a s t o ck t h a t c an i n c r ea s e ex p o n en t i a l l ya t a r a t e g r e a t e r t h an t h e d i s co u n t r a t e , t h e b e s t h a r v e s t s t r a t eg y i s t o l e t i t i n c r ea s eu n t i l j u s t b e f o r e t h e en d o f t h e p l an n i n g p e r i o d , t h en h a r v es t . T h i s b e h av i o r iss i m i la r t o t h a t o b s e r v ed f o r o t h e r ag e - s p ec if ic , l i n ea r m o d e l s ( c f. , M en d e l s s o h n ,1976 , Ror res , 1976 , and Reed , 1979) .

More Realistic CasesA n a l y t i c a l s o l u t i o n s t o t h e r em a i n i n g ca s e s d o n o t ap p e a r p o s s i b l e . S i n ce n u m er i ca ls o l u t i o n s i n v o l v i n g o r d i n a r y d i f f e ren t i a l eq u a t i o n s f o r s i z e - d is c r e te v e r s i o n s o f t h ea d j o i n t v a r i a b l e (2 ) a n d s t a t e v a r i a b le ( n) w o u l d l e a d to a t w o - p o i n t b o u n d a r y v a l u ep r o b l e m o f h i g h d i m e n s i o n , t h e y a r e n o t a t t e m p t e d h e re . I n s te a d a s im p l e r f o r m o fs o l u t i o n i s s o u g h t f o r e ach ca s e .I n s o m e s i m i l a r c o n t r o l p r o b l e m s t h e s o l u t i o n a p p r o a c h e s a s i n g u l a r a r c ,r em a i n s o n t h e s i n g u l a r a r c f o r a n o n - ze r o p e r i o d o f t im e , t h en l e av es t h e s i n g u l a ra r c b e f o r e t h e en d o f t h e t i m e h o r i zo n ( T ) ( B r y s o n an d H o , 1 96 9) . W h i l e o n t h es i n g u l a r a rc , n ece s s a r y co n d i t i o n s f o r o p t i m a l i t y a r e t h a t : ( 1 ) t h e s w i t ch i n g f u n c t i o nb e ze ro , (2 ) t h e t i m e d e r i v a t i v e o f t h e s w i t ch i n g f u n c t i o n b e ze r o , (3 ) t h e ad j o i n teq u a t i o n s b e s a t i s f i ed , an d ( 4 ) t h e s t a t e eq u a t i o n s b e s a t i s f i ed . F o r t h e r em a i n i n gcases , the po ss ib i l i ty o f a s ing ular so l u t io n on ( t l , t2) (0 ~< t l < t2 ~< T) wi l l bee x a m i n e d .T o f u r t h e r s i m p l i f y t h e p r o b l em , t h e s o l u t i o n s o u g h t w il l b e o n e in w h i ch t h es t a te v a r ia b l e s a r e a t e q u i l i b ri u m a n d c o n t r o ls a r e c o n s t a n t w i t h t i m e . T h e o p t i m a lh a r v es t s o l u t i o n f o r t h e l o g i s t i c m o d e l ( C l a r k , 1 9 7 6 ) i n v o l v e s a s i n g u l a r a r cp r eced ed an d f o l l o w ed b y n o n - s i n g u l a r p e r i o d s . T h e p o p u l a t i o n l ev e l i s co n s t an tw i t h t i m e t h r o u g h o u t t h e s i n g u l a r a r c . A c o n s t an t f i s h e r y p o l i cy (i .e ., co n s t an t s i zel im i t s an d f i sh i n g p r e s s u r e ) w o u l d b e ea si e r t o i m p l em e n t t h an a t i m e - v a r y i n gp o l i cy . B ecau s e o f th e ex i s t en ce o f a co n s t an t - p o l i cy s o l u t i o n i n t h e l o g i s ti c c a s e an di t s d e s i r ab i l i t y f r o m a p r ac t i c a l s t an d p o i n t , s i n g u l a r s o l u t i o n s i n w h i ch co n t r o lv a r i a b l e s a r e c o n s t a n t a n d t h e p o p u l a t i o n i s a t e q u i l i b r i u m a r e s o u g h t h e r e .

T h e a p p r o a c h t a k e n f o r e a c h s o l u t i o n i s t o e x p r e ss t h e n e c e ss a r y c o n d i t i o n s i nag e - s pec i fi c f o r m , t h en p r e s en t t h e a t t em p t ed s i n g u l a r -eq u i l i b ri u m s o l u t i o n s i nt e rm s o f a c o m m o n l y u s e d n u m e r i c a l p r o c e d u r e . T h i s p r o c e d u r e c o n s i s ts b a s i c a ll yo f d e t e r m i n i n g v a l u e s o f co n t r o l s t h a t s a t i s fy t h e ad j o i n t eq u a t i o n , t h e s t a t ee q u a t i o n s a n d t h e c o n d i t i o n t h a t t h e t im e d e r i v a ti v e o f t h e s w it c hi n g f u n c t io n b eze r o , th en s eek i n g a b o u n d a r y c o n d i t i o n ( a t t2 ) o n t h e ad j o i n t v a r i ab l e s s o t h a t t h es w i t ch i n g fu n c t i o n i s z e r o o n t h e s i n g u l a r a r c. ( T h i s p r o ced u r e can b e f o l lo w ed , fo rex am p l e , i n d e t e r m i n i n g t h e o p t i m a l co n t r o l i n th e l o g is t ic h a r v e s t p r o b l em . ) T h i sp r o ced u r e i s f o l l o w ed a s a m n em o n i c d ev i ce t o d em o n s t r a t e h eu r i s t i c a l l y t h a t


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2 80 L . W . B o t s f o r d

s i n g u la r e q u i l i b r iu m c o n t r o l s c a n n o t s a t is f y n e c e s sa ry c o n d i t i o n s f o r o p t i m a l i t y f o rthe rema in ing cases .For the s imples t case , cons t an t spec i f i ed rec ru i tment , t he s i ze l imi t cond i t ion(14) can be expressed a s

e - ~ ' k m ( s 3 = X ( sl , t ) (35)w h e r e f r o m (2 1 a - c )

~ ( s l , t ) e - 6+ E k f + 2 - ' o= - - m ( . c )a ( . c )e -6 ( , - , , ) d ~a ( s , ) , ,e-aZk f r - + o ( t o + ~ ) m ( z ) a ( ~ ) e - + ( ' - !+ 2 -,o )) d z (36)

wh ere t ime t = to + s l, t he time o f b i r th p lus cur ren t age an dp (s ) = k i n ( s ) . T h e f i rs tt e rm app l i e s to the s ingu la r a rc an d the seco nd to the in t e rva l ( t2 , T ) . The co nd i t iond S / d t = 0 I -Eq . (18) ] can be expressed a s

[ ; i ~ I f + ~~ k m ( z ) a ( r ) d~ - K := R~a(~)k[~(z) - m(~) ~(~)] dT (37 )I Iwh e r e we h a v e u se d t h e f a c t t h a t a t e q u i l i b r iu m n ( a , t ) = R + a ( a ) , wh e r e R , i s t h esp e c if ie d c o n s t a n t r e c r u i t m e n t v a l u e . F r o m E q . (1 5) th e c o n d i t i o n S ( t ) = 0 c a n b ewr i t t en

f o ;+ o- 6 'R ~ k m ( - c ) a ( z ) d z ~ - K e - 6 ' = R + a ( z ) ~ ( z , t ) d z (38)in wh ich ,~(z, t ) can be exp ressed by Eq . (36) wi th st r ep laced by ~ .

Assu m i n g t h a t t h e v a l u e s o f E a n d s t c o u l d b e c h o se n t o s a t i s fy t h e s iz e li m i tc o n d i t i o n ( 3 5 ) a n d t h e c o n d i t i o n d S / d t = 0 (37) , t he va lues o f ~ a long the endbo un da ry ( t = t2 ) t ha t y i e ld S ( t ) = 0 o n t h e s i n g u la r a r c w o u l d t h e n b e so u g h t . F o rt h e c a se c o n s i d e r e d h e r e t h e l a s t c o n d i t i o n c a n n o t b e s a t i s f i e d f o r t h e f o l l o wi n gr e a so n .T h e a d j o i n t v a r i a b l e ( f u t u r e v a l u e ) c o n t a i n s t wo t e r m s : ( 1 ) t h e t o t a l v a l u eob ta ined by f i sh ing o f t he ag e c l a ss fo r age.4 g rea t e r t han s+ a t t he s ingu la r f i sh ingp r e s su r e u p t o t i m e t2 a n d (2 ) t h e t o t a l v a l u e o b t a i n e d f r o m f ish i ng t h a t s a m e a g e

OmOX

AGE

I

- - - - - / / ' - S - - - I - - -characteristic

~ - t l " ' r / ~ ' t 2 T2-OmoTIME

F i g . 2 . A g e v e r s u s t i m e o v e r t h e p l a n n i n gp e r i o d f r o m t i m e z e r o t o t i m e 7". F o r s i n g u l a re q u i l ib r i u m c o n t r o l t o b e p o s s i b l e o n ( t t , t 2)t h e v a l u e s o f a l l i n d i v i d u a l s ( a d j o i n t v a r i a b l e )m u s t d e p e n d o n h o w t h e i r a g e c la s s is f i s h e df o r t > t v S i n c e i n d i v i d u a ls b o r n a t a n yt < t 2 - ~ w i l l n o t b e a l i v e a t t 2 ( e . g . , a ni n d i v id u a l b o r n a t t 3 s i n g u l a r e q u i l i b r i u mc o n t r o l i s n o t p o s s i b l e o n ( t t , t 2)


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Optimal S ize-SpecificFishery Policy 281c la s s f r o m t i m e t z t o T a t f i sh i n g p r e s su r e E r~x o r z e r o . A ssu m i n g t h a t i n d i v i d u a l s d ono t l ive bey on d a m ax im um age , a r . a , fo r a l l t ime t < t z " o-max, no i nd iv idua l scur re n t ly a l i ve w il l su rv ive t o be f ished a t t > t2 (F ig . 2 ). H en ce t he v a lue o f X(a, t ) i si n d ep en d en t o f J [( a, tz ), an d c o n seq u en t l y d ep e n d s o n ! y o n v a l u es o f E an d s l. T h u s ,e v e n t h o u g h v a l u e s o f E a n d s~ m a y b e f o u n d t h a t s a t is f y d S / d t = 0 and the s i ze l imi tc o n d i t i o n , t h e c o n d i t i o n S ( t ) = 0 c an n o t b e s a t i s f i ed ( ex c ep t f o r t u i t o u s l y ) f o rc o n s t an t v a l u e s o f E an d s~.B ec au se o f th i s i t ap p e a r s t h a t s i n g u l a r - eq u il i b ri u m c o n t r o l w i th c o n s t an t v a l u e so f c o n t r o l v a r i ab l e s i s n o t p o s s i b le . F u r t h e r m o r e , f r o m t h is b eh a v i o r o f th e ad j o i n tv a r i ab le , o n e w o u l d su sp ec t t h a t so m e so r t o f p u ls e f ish i n g in w h i c h t h e p e r i o d o f t h ep u l se w as l e ss t h an t w i ce t h e m ax i m u m ag e o f an i n d i v i d u a l w o u l d b e o p t i m a l . T h i ssu p p o r t s t h e c o n j ec t u r e o f C l a r k ( 1 97 6 ) c o n c e r n i n g a s i m i la r p r o b l em ( r ec r u i tm en ta t d i sc re t e t imes) .F o r t h e s ec o n d c a se , p o p u l a t i o n - d ep e n d e n t r ec r u i tm en t , t h e ex p r e s s i o n f o r t h ead j o i n t v a r i ab l e i s t h e s am e a s ( 3 5 ) ex c ep t f o r an ad d i t i o n a l t e r m [ f r o m t h ed i scuss ion fo l l owing Eq . (20) ] .

X (s l , t ) = e - ~'E k ~ t2 - , o m ( z ) a ( z ) e - ~ " - ~ ') . d za ( s , ) 3 , ,

e-ttk f T - t o- - " E ( t o + " c ) m ( ' r )a ( z ) e - e~ ' - l '2 - ' ~ d z+ a ( ~ - t o) ~ , , - , o~ r - , o a ( O - . ,+ ~(0, t + to)-7-~-~ b ( t ) f i [ C .] - c ( t ) f ~ [ C . ] R , ~ b ] dt (39 )* /$ ! u~..Jo

w h e r e f ' , [ C , ] i s th e d e r i v a t i v e o f f ~ w i t h r e sp ec t t o i ts a r g u m en t ev a l u a t ed a tequ i l i b r ium, ~b i s def ined by Eq . (33 ) and R, i s t he equ i l i b r ium rec ru i tmen t l eve l .T h e t e r m c o n t a i n i n g ~ ( : ) r e f le c t s t h e p o s i t iv e i n fl u en c e o f f ec u n d i t y o n r ec r u i t m e n tw h i l e th e t e r m c o n t a i n i n g ? ( : ) r e fl e c ts th e n eg a t i v e i n f lu en c e o f d en s i t y - d ep en d en te f f ec ts o n r ec r u i tm en t .T h e c o n d i t i o n d S / d t = 0 wi l l a l so have an add i t i ona l t e rm

[ ; ; 3 ;R , k m ( ' c ) a ( z ) d z - K = R , a ( ' c ) k [ ( z ) - m(z)3(z)] dzI 1+ ,t(0,~)e%(~)E~(~)f~[C,]

I

- ~ ( z ) f ; [ C , ] R , qr~b] d z . (40)T h i s ad d i t i o n a l t e r m ap p ea r s s i m i l a r b u t d i ff e rs f r o m t h e ad d i t i o n a l t e r m i n E q . ( 3 9 )in t ha t t he i n t eg ra l i n (40) i s ove r age a t cur ren t t ime wh i l e t he i n t eg ra l i n (39 ) i s ov erage a t fu tu re t ime ( i . e . , a long a charac t e r i s t i c ) .T h e ap p r o ac h t o a s i n g u l a r - eq u il i b ri u m so l u t i o n i n th i s c a se i s t o c h o o se v a l u eso f E , sl , ~[(0, t ) an d X(~, t2) (V z), tha t sat is fy d S / d t = O , S ( t ) = 0, the s ize l imi tc o n d i t i o n a n d t h e ad j o i n t eq u a t i o n . W e a s su m e t h a t f o r an y v a l u e o f ~ '(0 , t ) w e c ancho ose va lues o f E and s~ tha t sa t i s fy d S / d t = 0 [Eq . (40) ] and the s i ze l imi tcon di t i on [Eq . (35 ) w i th ~(s~, z ) exp resse d by Eq . (39 ) ] . F ro m the con di t i on S ( t ) = 0


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2 8 2 L . W . B o t s fo r d

[Eq . (38) wi th X( r , t ) e xpresse d b y Eq . (39) wi th "c sub s t i t u t ed fo r s t ] , t he t imed e p e n d e n c e o f ~ '(0 , t ) m u s t b e o f t h e f o r m C e - ~ ' wi t h C a c o n s t a n t . Ho we v e r , f r o mt h e e x p r e s s i o n f o r X( 0, t ) d e r i v e d f r o m t h e a d j o i n t c o n d i t i o n s

-6, Ek ~ u - , o~ .(0 , t ) = e a ( s l ) J , , m ( r ) a ( z ) e - ~ * - s ' ~ d zf , _ , k [ , - , o

+ a ( t2 - t o) . l u - , o E ( t o + T ) m ( r ) a ( r ) e - ~ * - ~ ' 2 - ' ~ d r

; 7 0I ( 0 ,r + to)a(r ) [G(z)f~[C .] - C--(r )f ;EC.]R.e~] dr . (41)A .so lu t ion o f t he fo rm J [(0, t ) - - C e - 6 t , f o r t < tz - a ~ w o u l d b e

.... I m ( r ) a ( r ) e - ~ * - ~ ~ d xX ( O , t ) = e - 6 ' a ( s l ) d ~ , (42)f ? oI - e -%(r )E~; ( r ) f~EC,] - e ( r ) f ' E C , ] R , ~ b ]

S i nc e a ( r ) i s z e r o f o r r > a ~ x t h is e x p r es s i o n d e te r m i n e s t h e v a l u e o f t h e c o n s t a n t C .S o l u t io n o f t h e c o n d i t io n S ( t ) = 0 f o r t h e c o n s t a n t C r e q u i r e d t o s e t t h e sw i t c h in gf u n c t i o n t o z e r o o n t h e s i n g u la r a r c sh o ws t h a t i t is - no t th e s a m e a s t h e c o n s t a n t Cr e q u i r e d b y th e a d j o i n t e q u a t i o n . T h e r e f o r e i t a p p e a r s t h a t t h e se c o u l d n o t b o t h b esa t i sf ied on ( t l , tz ) .

Thus fo r t h i s p rob lem a l so a s ingu la r -equ i l ib r ium i s no t poss ib l e , bas i ca l lyb e c a u se v a lu e s o f t h e a d j o i n t v a r i a b l e o n t h e s i n g u la r a r c a r e n o t r e la t e d t o v a l u e s a tthe end o f t he a rc d ue to the l imi t ed l i fe t ime o f i nd iv idua l s i n the po pu la t ion .Because o f t h i s , it aga in ap pea rs tha t a fo rm o f pu l se f i sh ing , wi th pe r iod s l e ss t hantwice the max im um l i fe t ime o f f ish wi ll be the op t im a l po l i cy .

Fo r t h e t h i rd c a se , f o o d d e p e n d e n t r e c r u i tm e n t , t h e a d j o i n t c o n d i t i o n f o r 2F a n dt h e s t a te e q u a t i o n f o r f o o d v a r i a b l e F a r e a d d i t i o n a l n e c e s sa r y c o n d i t i o n s . T h e s i zel imi t cond i t ion i s express ed by E q . (35) and the cond i t ion S ( t ) = 0 is expressed byE q . ( 3 8 ) e a c h w i t h t h e a p p r o p r i a t e e x p r e s si o n f o r X t a k e n d i r e c tl y f r o m E q s .(2 1 a -c ) ( i. e. , t h i s is t he case used in the de r iva t ion o f necessa ry cond i t ions) .B e c a u se t h e y a r e q u i t e s i m i l ar t o t h o se f o r t h e p r e v i o u s c a se , n e c e s sa r y c o n d i t i o n sfo r t h is case a re no t r ep ea ted he re . Al tho ugh these express ions invo lve 2F( t) , the i rs imi l a ri ty to th e p re v iou s case is appa ren t a f t e r f ac to r ing X(0 , t ) = C e - 6 ' f ro m 2~,( t)to ob ta in 2 k ~ 2 F ( t ) / C e -~ t . T h e sa m e a r g u m e n t a s wa s f o l l o we d in t h e p r e c e d i n gc a se c a n n o w b e f o l lo w e d in t h is c a se w i t h o n e e x c e p ti o n . R a t h e r t h a n t h e e n d r e su ltb e i n g t wo d i f f e r e n t e x p r e s s io n s f o r t h e c o n s t a n t C , we c o u l d a t t e m p t t o a d j u s t t h ev a l u e o f 2~, ( wh i c h a p p e a r s i n e a c h ) so t h a t t h e y wo u l d b e e q u a l. Ho we v e r , t h i sso lu t ion requ i re s ;t~, t o be cons tan t wi th t ime . Ex am ina t ion o f Eq . (23) sho w s tha t i ti s n o t c o n s t a n t . He n c e f o r t h i s c a se a l so a s i n g u l a r so l u t i o n d o e s n o t a p p e a rpossible .For each o f t hese cases on ly a s ing le s i ze l imi t ( a l ower l imi t ) has beenc o n s i d er e d . T h e r e m a y b e i n s ta n c e s i n wh i c h t wo o r m o r e s i ze li m it c o n d i t i o n s c a n


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Optima l Size-Specific Fishery Policy 283b e s a t i s f i ed . H o w ev e r , t h i s p o s s i b i l i t y d o es n o t s eem t o r em o v e t h e b a s i ci m p ed i m en t t o o b t a i n i n g a s i n g u l a r -eq u i l i b ri u m s o l u t i o n . A ch an g e i n th e l im i t s o fi n t eg r a t i o n i n S ( t ) = 0 an d an ad d i t i o n a l t e r m i n ~ [(0 , t ) d o n o t a l l o w a co m m o ns o l u t i o n t o b o t h .

A l t h o u g h - t h e y h a v e n o t b e e n i n v e s t i g a t e d i n d e t a i l , i t a p p e a r s t h a t t h e s a m ed i f fi cu l ti e s w o u l d b e en co u n t e r ed i n a t t em p t i n g a s i n g u l a r eq u i l ib r i u m s o l u t i o n f o rt h e r em a i n i n g ca s e s (i .e ., t h o s e i n v o l v in g d en s i t y - d e p en d en t g r o w t h r a te ) . A te q u i l ib r i u m , g r o w t h r a t e w o u l d b e c o n s t a n t w i t h t im e a n d t h e n e c e s sa r y c o n d i t i o n sw o u l d b e s i m i l a r t o t h o s e f o r t h e ca s e s d e s c r i b ed h e r e .

5 . D iscuss ionF o r t h e a s s u m e d f o r m s o f th e f i s hi n g c o s t f u n c t i o n a n d h a r v e st i n g c o n t ro l s ,n ece s s a r y co n d i t i o n s f o r ~ 0p ti m al h a r v es t h av e b een d e v e l o p ed f o r a g en e r a lp o p u l a t i o n n i o d e l t h a t r e f le c ts a w i d e r a n g e o f p o p u l a t i o n d y n a m i c s a n d b e h a v i or .T h es e co n d i t i o n s i n d i ca t e t h a t w h en m u l t i p l e ag e o r s i z e - c l a s s e s an d d en s i t y -d e p e n d e n t g r o w t h o r r e c r u it m e n t r a t e a r e e x p li c it ly in c l u d e d i n p o p u l a t i o n m o d e l so p t i m a l p o l i cy i s q u i t e d i f f e r en t t h an f o r s i m p l e , l e ss re a l is t ic m o d e l s .F r o m a p r a c t i c a l p o i n t o f v ie w t h e m o s t i n tr ig u i n g a s p e c t s o f t h e r e s u l t so b t a i n ed h e r e a r e t h e i m p l i ca t i o n s o f t h e l a ck o f co n s t an t f i s h e r y p o l icy s o l u t io n s .T h i s r e s u l t d i f f e r s s u b s t an t i a l l y f r o m s t an d a r d s o l u t i o n s t o t h e o p t i m a l f i s h e r yp o l i cy p r o b l em o b t a i n ed w i t h t h e l og i s ti c m o d e l . T h e l a t t e r s o l u t i o n s i n c lu d e as i n g u l a r a r c o f co n s t an t p o l i cy a t an eq u i l i b r i u m l ev e l ( c f. , C l a r k , 1 97 6; C l i f f an dV i n cen t , 1 9 7 3 ; G o h , 1 96 9/ 19 70 ). T h e l o g i s ti c m o d e l i s w i d e l y r eg a r d ed a s ana d e q u a t e p o p u l a t i o n m o d e l f r o m w h i c h g e n e ra l , r o b u s t c o n c l u s io n s c a n b e d r a w n .I t r e f l e c t s t h e s e l f - i n h i b i t i n g b eh av i o r t h a t l i m i t s p o p u l a t i o n l ev e l , b u t d o es n o tex p l i c it l y i n c l u d e t h e s p ec i f i c d en s i t y - d e p en d e n t m ech an i s m s r e s p o n s i b l e f o r th i sb e h a v i o r o f th e i r a g e - d ep e n d e n c e . T h e r e s u lt s o b t a i n e d h e r e d e m o n s t r a t e t h a t w h e nt h es e m ech a n i s m s a r e ex p l i c it ly i n c l u d ed i n a p o p u l a t i o n m o d e l , t h e r eb y m ak i n g i tm o r e r ea l i st i c, a co n s t an t eq u i l i b r i u m p o l i cy i s n o l o n g e r o p t i m a l .T h e d i f f e r en ce i n r e s u lt s s t em s f r o m t h e f ac t t h a t i n t h e s i m p l e , s in g le s t a t ev a r i a b l e c a s e t h e c o n d i t i o n d S / d t = 0 a n d t h e s o lu t i o n t o t h e a d j o i n t e q u a t i o n a r ei d en t i c a l a t e q u i l i b r i u m , w h e r ea s i n t h e ca s e s d i s cu s s ed h e r e in t h is i s n o t t ru e .B eca u s e o f t h is , a s d i s cu s s ed in t h e p r ev i o u s s ec t i o n , S ( t ) = 0 a n d d S / d t = 0 c a n n o tb o t h b e s a t i sf i ed o n t h e s i n g u l a r a r c . A l t h o u g h t h e s e r ea s o n s f o r t h e d i f f e ren t r e su l t sa r e n o t p a r t i cu l a r l y s a t i s fy i n g in t u i ti v e l y , i t i s n o t d i f f i cu l t t o a cce p t t h e f ac t t h a t o n eco u l d d o b e t t e r b y h a r v es t in g a p o p u l a t i o n t o a l o w l ev el , th en l e t ti n g it g r o w rap i d l yt o a h ig h l ev e l w i t h a m i n i m u m o f d en s i t y - d e p en d e n t i n t e rf e r en ce , t h an b ym a i n t a i n i n g a co n s t an t l ev el an d r em o v i n g b i o v a l u e a t a co n s t a n t r a t e ( e s pec i a ll yw h en o n e co n s i d e r s t h a t co s t s a r e i n cu r r ed o n l y when ac t i ve ly f i sh ing ) .T h e p r ac t i c a l i m p l i ca t i o n s o f t h e r e su l t s o b t a i n e d a r e q u i t e s en s i ti v e t o s o m e o ft h e a s s u m p t i o n s o n w h i c h t h e a n al y si s w a s b a s e d . F o r e x a m p l e , th e c o s t f u n c t io n a lacco u n t s o n l y f o r v a r i ab l e co s t s . I n c l u s i o n o f f i x ed co s t s b y ad d i n g t h e co s t o fm a i n t a i n i n g c a p i t a l e q u i p m e n t t h r o u g h p e r i o d s o f n o f i s hi n g c o u l d c h a n g e r e s ul ts .O n t h e o t h e r h a n d , t h e f ac t t h a t b o a t s t h a t f i sh s ev e r a l s p ec i es co u l d s w i t ch f r o m o n et o t h e o t h e r a t d i f f e r en t t i m es w o u l d d ec r ea s e t h i s e f f ec t . O t h e r f a c t o r s t h a t co u l d


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Op timal Size-Specific Fishery Policy 285T h a t i s, w h e n th e p r o d u c t iv e c a p a b i l i ty o f t h e p o p u la t i o n is le ss t h a n t h e d i s c o u n tr a t e , t h e o p t ima l p o l i c y f o r t h e s o le o w n e r is t o c o n v e r t t h e r e s o u r c e t o c a p i t a l ( s eealso W att , 1968, p . 125).N e c e s s a r y c o n d i t i o n s o b t a in e d h e r e a r e s imi l a r , b u t t h e y e x p l i c i t l y d iv id eg e n e r a l b io lo g i c a l g r o w th i n to i ts a c tu a l a g e - o r s iz e -s p e ci fi c c o m p o n e n t s . O p t im a lle ve ls a s d e t e r m in e d f r o m th e l o g i st ic mo d e l d e p e n d b io lo g ic a l l y o n t h e r e l a t io n s h ipb e t w e e n t h e i n tr in s i c ra t e o f i n c r e a s e o f t h e p o p u l a t i o n ( r) a n d t h e m a x i m u mpo pula t ion leve l (K) . Th is r e la t ionsh ip fo r the log is tic equa t io n i s a spec i f ic on e tha t ,whi le i t possesses ce r ta in g ross s imi la ri t ie s to beh av io r o f ac tu a l pop ula t ion s , i s no tb a s e d o n s p e c i f ic g r o w th , r e p r o d u c t iv e o r mo r t a l i t y r a t e s in t h e p o p u la t i o n . T h ei m p r o v e m e n t p r o v i d e d b y t h e w o r k p r e s e n t e d h e r e i s t h a t r e s u l t s a r e b a s e d o ne x p li c it c o n s id e r a t i o n o f h o w b io lo g i c a l p o p u la t i o n g r o w th o c c u r s .

T h e r e i s, h o w e v e r , a p r a c t i c a l d i ff e r e n c e b e tw e e n th e r e su l t s o b t a in e d h e r e a n dth o s e o b t a in e d u s in g t h e l o g i s t i c mo d e l . T h e " b io v a lu e " c o n s id e r e d i n s e t t i n gexplo i ta t ion r a te inc ludes on ly those ind iv idua ls be tween the f i sh ing s ize l imi t s ,w h e r e a s i n t h e l o g is t ic m o d e l i t i n c lu d e s a ll in d iv id u a l s in t h e p o p u la t i o n s . T h u s ,o p t ima l p o l i c y d e r iv e d u s in g t h e l o g i s t i c mo d e l w o u ld d i f f e r s i g n i f i c a n t ly f r o mo p t i m a l p o l i c y f o r t h e s a m e p o p u l a t i o n u s in g t h e m o d e l s a n a l y z e d h e r e.T h e w a y i n w h i c h o p t i m a l v a l u e s o f c o n t ro l v a r i a bl e s a c h ie v e t h e o p t i m a lb a l a n c e b e t w e e n c u r r e n t a n d f u t u r e p r o f i t s , a n d h o w g r o w t h , r e p r o d u c t i v e ,mo r t a l i t y , a n d d i s c o u n t r a t e s a f f e c t t h i s b a l a n c e c a n b e mo s t e a s il y s e e n i n t h en e c e s s a r y c o n d i t i o n f o r s i n g u la r c o n t r o l , dS/dt = 0 [E q . (18) ] , bu t is a l so a f fec tedby the s ize l imi t con d i t io n [Eq . (14)] . Fo r a ny f ixed va lues o f mr an d m~, E i sa d j u s t e d s o t h a t t h e " p o p u l a t i o n l e v el " (a c t u a ll y t h e p r e se n t m a r k e t v a l u e o fa n ima l s b e tw e e n mt a n d m~) e q u a l s t h e r a t i o o f g r o w th r a t e ( RH S o f E q . (1 8)) t o t h ed i s c o u n t r a t e ( 6). T h e o p t ima l s iz e lim i ts a s s u r e t h a t o n ly i n d iv id u a l s w h o s e c u r r e n tm a r k e t v a lu e ( L H S o f E q . (1 4)) is g r e a t e r t h a n t h e t o t a l v a lu e d e r iv e d f r o m th e m inth e f u tu r e ( RH S o f E q . (1 4)) a r e f is h e d .T h e o p t im a l s e l e c ti o n o f a g e o r s iz e o f c a p tu r e a s d e t e r m in e d b y t h e s iz e lim i tc o n d i t i o n i s a ls o r e l a t e d t o e x i s t in g t h e o r y . S i ze l im i t s a r e d e t e r min e d b y c u r r e n tm a r k e t v a lu e re l a t iv e t o t o t a l f u tu r e v a lu e t o b e d e r iv e d f r o m a n a n im a l a l iv e a t t h a ts iz e. F u tu r e v a lu e i n v o lv e s c o n s id e r a t i o n o f d i r e c t v a lu e t h r o u g h f i s h in g , r e p r o -d u c t i o n , d e n s i t y - d e p e n d e n t ef fe c ts a n d f o o d c o n s u m e d .A l t h o u g h t h e y a r e n o t o f t e n e x p li ci tl y i n c l u d e d i n f o r m u l a t i o n o f f i s h e ry p o li c y,m o s t o f t h e s e c o m p o n e n t s o f v a l u e o f a n i n d i vi d u a l h a v e a t l e as t b e e n q u a l i ta t iv e l yc o n s id e r e d i n ma n a g e me n t . Fo r e x a mp le , t h e s t r a t e g y o f f i s h in g a s h e a v i ly a sp o s s ib l e w i th o u t d e s t r o y in g t h e r e p r o d u c t iv e a b i l i ty o f t h e s t o c k i n a s e n s e c o n s id e r sr e p r o d u c t iv e v a lu e o f in d iv id u a l s . A l s o , c o n s id e r a t i o n o f t h e v a lu e o f f o o d n o t e a t e nb y f i s h t h a t a r e r e m o v e d b y h a r v e s t w a s e m p h a s i z e d b y P a l o h e i m o a n d D i c k i e(1 97 0). A r e m a r k a b le e x a mp le o f p r io r c o n s id e r a t i o n o f t h e i d e a s e x p r e s s e d h e r e is as h o r t , c o n c i s e a n a ly s i s b y R . H . M a c A r th u r t h a t l e a d s t o a c o n s id e r a t i o n o fr e p r o d u c t iv e v a lu e t h a t i s sim i l ar t o t h a t e x p r e s s e d h e r e ( M a c A r th u r , 1 96 0).S t a r t i n g f r o m F i s h e r ' s d e f in i t io n o f re p r o d u c t iv e v a lu e o f a n i n d iv id u a l o f a c e r t a ina g e ( F i s h e r, 1 9 5 8 ) ( i d e n ti c a l t o t h e re p r o d u c t iv e c o m p o n e n t o f t h e a d jo in t v a r i a b l ea t t h e s a me a g e ) , h e r e a c h e d t h e c o n c lu s io n t h a t a n a g e s e l e c t i v e p r e d a to r( h a r v e s t e r ) s h o u ld t a k e a n ima l s w i th t h e l a r g e s t r a t i o ( v a lu e t o p r e d a to r /r e p r o d u c t iv e v a lu e ).


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2 86 L . W . B o t s f or dC o m p a r i s o n o f c u r r e n t m a r k e t v a l u e w i t h d i re c t f u t u r e v a l u e d u e t o f i sh i ng i nse t t ing an ag e l imi t a l so r e su l ted f ro m th e ana lys i s o f Cla rk e t a l . (1973) fo r a s ing lea g e cla ss . A g a in r e l a ti v e v a lu e o f f u tu r e c a t c h e s w a s d e s c r ib e d t h r o u g h th e d i s c o u n tr a t e . Wh e n th e d i s c o u n t r a t e i s z e r o , ma x imu m e f f o r t i s a p p l i e d a t t h e a g e o fma x imu m b io ma s s . T h i s i s t h e c l a s s i c a l c a s e c o n s id e r e d b y Be v e r to n a n d H o l t( 19 57 ) a n d i n v o lv e s e q u a l c o n s id e r a t i o n o f c u r r e n t a n d f u tu r e v a lu e s. A s t h ed i s c o u n t r a t e i n c r e a s e s f r o m z e r o , t h e a g e o f f i rs t c a p tu r e d e c r e a s e s . A s d i s c o u n t r a t einc reases , the re i s le ss an d less con s ide ra t ion o f fu tu re va lue . S imi la r va r ia t ions ind iscount r a te wi l l p roduce s imi la r changes in the lower s ize l imi t in the r e su l t so b t a in e d h e r e .H o w e v e r , t h e r e s u lt s a r e c o n c e p t u a l l y m o r e s im i la r t o t h o s e o b t a i n e d b y G o h(1 97 3), w h o t r e a t e d t h e c a s e f o r w h ic h f is h in g c o s t s w e r e z e r o a n d e f f o r t w a s a t a

l im i ti n g m a x im u m v a lu e . T h e g r e a t e r s im i l a ri t y i s d u e t o t h e f a c t s t h a t i n t h e o p t ima lp o l i c y d e r iv e d h e r e , t h e s i ze l im i t c o n d i t i o n d o e s n o t i n v o lv e c o s t o f f is h in g , a n d t h ev a lu e o f f is h in g p r e s s u re i s i n a s e n s e sp e c i fi e d b y a n o th e r n e c e s s a r y c o n d i t i o n( a s ~ a t = 0 ) .

A p o s s ib l e t h e o r e t i c a l i n t e r p r e t a t i o n o f t h e r e s u l t s o b t a in e d h e r e i s a c o m-b in a t i o n o f e x i s ti n g th e o r i e s o f o p t ima l a d ju s tm e n t o f s e lf - c o mp e n s a t i n g b io log i c a lr a te s wi th ex is t ing theor ie s o f op t im a l se lec tion o f f ish ing age o r s ize limi t s. Th us ,t h e s e re s u lt s c o mb in e t h e p e r s p e c t iv e o f t h e lo g i st ic o r p r o d u c t io n mo d e l w i th t h epe r spec t ive o f the s ing le age c la ss mod e ls .W h i le t h e s e c o m p a r i s o n s a i d c o n c e p t i o n o f t h e n e w r e su l ts , t h e r e a r e i m p o r t a n td i ff e re n c e s . T h e r e s u lt s d i f f e r f r o m th o s e o b t a in e d f r o m th e l o g is t ic e q u a t io n i n t h a ton ly ind iv idua ls wi th in the s ize l imi t s would be cons ide red in se t t ing f i sh ingpressure . They d i f f e r f rom the s ing le age c la ss r e su l t s in tha t the s ize l imi t s a ree f fe c t iv e ly s e t a t t h e v a lu e s c o r r e s p o n d in g t o t h e c a s e o f z e r o f i s h in g c o s t a n dspecif ied f ishin g level.I n s u mma r y , t h e r e s u l t s o b t a in e d h a v e b o th t h e o r e t i c a l a n d p r a c t i c a l imp l i -c a t i o n s f o r f i s h e r y r e s e a r c h . O n th e t h e o r e t i c a l s id e , i n t e r p r e t a t i o n o f t h e n e c e s s a r yc o n d i t i o n s f o r o p t ima l i t y e x t e n d s a n d u n i f ie s p r e v io u s r e s u l ts . O n th e p r a c t i c a l s id e ,t h e i n c r e a s e d b io lo g ic a l r e a l i sm o f t h e mo d e l s u s e d a p p a r e n t ly l e a d s t o t ime - v a r y in gf i sh e r y p o l ic y . T h i s re s u l t m a y c h a n g e a s t h e f is h in g c o s t a n d m a r k e t p r i c e mo d e l sa r e ma d e m o r e r e a li s ti c . T h e e x a c t fo r m s o f s o lu t i o n s a n d t h e i n f lu e n c e o f t h e s ee c o n o m i c a s s u m p t io n s o n t h e m a r e c u r r e n t l y b e in g p u r s u e d .Acknowledgements. I a m i n d e b t e d t o A . L . S u e r f o r h e r r e v i e w o f t h i s m a n u s c r i p t . T h i s w o r k i s a ne x t e n s i o n o f p a r t o f m y P h . D . d i s s e r t a t i o n a t t h e U n i v e r s i ty o f C a l i f o r n i a , D a v i s . I a m g r a t e f u l t o W . A .G a r d n e r , A . J . K r e n n e r a n d H o E . R a n c h f o r t hV , r a d v i c e a n d a s s i s ta n c e i n t h a t e n d e a v o r . F i n a l l y , Iw o u l d l i k e t o t h a n k C o l i n C l a r k f o r h i s c o m m e n t s o n t h i s w o r k .

T h i s w o r k i s a r e s u lt o f r e s e a r c h sp o n s o r e d b y N O A A O f fi c e o f S e a G r a n t , D e p a r t m e n t o fC o m m e r c e , u n d e r G r a n t # N O A A - M 0 1 - 1 8 4 R / F 5 2 . T h e U . S . G o v e r n m e n t i s a u t h o r iz e d to p r o d u c ea n d d i s tr i b ut e r e p r in t s f o r g o v e r n m e n t a l p u r p o s es n o t w i t h s t a n d in g a n y c o p y r i g h t n o t a t i o n t h a t m a ya p p e a r h e r e o n .Appendix AN e c e s s a r y C o n d i t i o n sA d jo in in g c o n s t r a in t s t o J y i e ld s t h e a u g m e n te d p e r f o r m a n c e f u n c t io n a l


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' = { e - 6 ' E [ f . . ,P " d m - + J - , L a m; [ . o . a l r - r n o O .' 0 2 , d n 2 . + d m + - d n 2 . +- - n ~ J d m+ n g o m a t J . ] . ~ k O m

+ 2 F - - -~ - - ,- 2 r w n d m + F - - ~ - + 2 . ( m o , t ) R i d t - r 2 . ( m , T ) n ( m , 7 " )o o

- 2 . ( m, O)n(m, 0 ) ] d m - [ 2 ~ ( T ) F ( T ) - , t d 0 ) F ( 0 ) ] . ( A - 5 )T h e v a r i a t i o n i n J w i t h re s p e c t t o a v a r i a t i o n i n E i s

r E o r - . )J = A E e -6~ p n d m - - +. 1 0 L \ ~ l m l


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288 L .W . Botsford

t r F { ' , / 0 2 . a , ~ . ] d mA n g ~ - - d 2 .+ J o L J . o ~ , am + ~ - / /+ J . A a m T i -J+ J = , \ a m d 2 . - E 2 , + - ~ + e -6 'E p d m

f ' ] ( 'r ' ' [ - ('~ F o g- - - -+ 2 . ( m o , ) V . ( R ) - 2 r o ' a ~ a, + j o L J , . . L e ~ " ~ mr i o2.(too, t l~-ff + - ~ - j - A n ( m , T ) 2.(m, T )d m _--- A F(T )2r(T)

+ A i nu ~ [ E( t) p (m u , t ) e - " n ( m ~ , t ) - E ( t) 2 .( m ~ , t) n (m u , t ) ] act

+ Am z ~oo [E( t )p(mz, t )n(m l , t )e -6 ' - E( t )2n(m t , t )n(mt , t ) ] d twh ere for Rt g iven by (9a)

for R2 given by (9b)

for R3 given by (9c)

V.(RO = 0 ,

L, , ( R 2 )= b ( m ) d i n,f ' b ( ) d m f [ F (,,(R2 ) = ra t) ]

and for R , g iven by (9d)V . ' R , ' = f ~ { b ( m ) f . [ : i c ( m g " ( m ' . t ) a m]

( A - 6 )

(A-7a)

(A-7b)

( A - T c )

+ f c ( m ' ) n ( m ' , t) d m c ( m ) b ( m ' ) n ( m ' ,t ) d m ' d in . (A-7d)o

Setting the coefficients of the va riat ion in state var iable F ( t ) an d n(m, t ) equal tozero yields the ad joint e quation s (12a, b , 13). Sett ing th e coeff icients of the variat ionin co ntrol v ar iables mz an d mMequal to zero y ie lds the contro l equat ions (14a and b).Th e coeff icient o f the variat ion in con trol var iable E ( t ) is the switching function,S ( t ) [Eq . ( I5) ] . Boundary condi t ions are ob ta ined by se t t ing var ia t ions inbou nda ry values to z ero (17a, b).


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O ptima l Size-SpecificFisheryPolicy 289Append ix BS i n g u la r C o n t r o lT o d e t e r m i n e d S / d t , d i f f e ren t ia t ing (15 ) y ie lds

d S 6 e _ ,, f m " " [ p d m .- - = - - p ( m ) n ( m , t ) d m + e - 6 ' ( m . ) n ( m . , t ) - - ~ -d t j , . , , )- p ( m O n ( m t , t ) -- d -- + J . . , ~ o [ p ( m ) n ( m , t ) ] dm.~ + 6 K e - 6 '

2 n (m . , d i n . d m ~- t ) n ( m . , t )- - d - - - - 2 . (ml , t )n ( m l , t ) I d tr " " ~ 0 ]+ / ~ [ 2 . ( m , t ) n ( m , t ) ] d m (B- l )

J , , , , ~ o O ~F r o m ( S a ) a n d ( 5 b ) w e o b t a i n

~ [ p ( m ) n ( m , t ) ] = p ( m ) { - O - ~ [ o ( m , F ) n ( m , t ) ] - ( d ( m ) + E ( t ) ) n ( m , t ) } (B-2 )O [ 2 . ( m , t ) n ( m , t ) ] ~. (m, t ) { O - ~ [ 9 ( m , F ) n ( m , t ) ]a t

- ( d ( m ) + E ( t) )n ( m , t ) } + n ( m , t ) O2.(m,at. _ _ _ ) (B -3 )F r o m ( 1 2 a ) , ( 1 2 b ) , ( B - I ) , ( B - 2 ) a n d ( B - 3 ) , w e o b t a in

e _ 6 t 0 [ p ( m ) n ( m , t ) ] - ~ I n ( m , t )2 . ( m , t ) ]= e - 6 ' p ( m ) [ - O - ~ [ g ( m , F ) n ( m , t ) ] - d ( m ) n ( m , t ) ]

- $ . ( m , t ) { - O - - ~ [ g ( m , F ) n ( m , t ) ] } + n ( m , t ) g ( m , F ) O ---'---m -- n ( m , t ) [ 2 F ( t ) w ( m , F ) - 2 . (m o , t ) b ( m ) f [ F ( t ) ] ] . (B-4 )

U s i n g t h e p r o d u c t r u l e t w i c e w e o b t a i n0 2 .2 .O- -~(9n) + g n o m = 0 - ~ ( 2 .9 n )

a2, , (m , t )

(B-5)a n d

p ( m ) [ _ O _ ~ g ( m , F ) n (m , t ) ] _ d ( m ) n ( m , t ) ] = ( _ O _ ~ p g n ) + n g-ff-'m dP p d n ) .(B-6 )


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290E q ua t i o n ( B - 4) a n d s iz e l i m i t c o n t r o l c o nd i t i on s (14a) a nd ( 14b ) y i e l d

= e -~ 'n (m , t ) g (m , F ) d m - p ( m ) d ( m )d t ~m l (O- - e -6 t2v( t )w(m, F) + e6'A,,(mo, ) b ( m ) f [ F ] l d m

f mu(t)- - 6 e - ~ tp ( m ) n (m , t ) d m + 6 K e - 6 t. (B-7)dm~{OH e n c e t h e c o n d i t io n , d S / d t = O , impl ies Eq . (18) .W i t h t h e d e f i n i t io n

@G ( m , t ) = - g ( m , 1 : )~ - ~ - p ( m ) d ( m ) - e - 6 t2 e ( t ) w( m , F )+ e 6tg( m o , t ) b ( m ) f [ F ~ - 6 p ( m )

d i f f e r e n t i a t i on o f ( B -7 ) y i e l d sd2 S e _ 6 t n ( m ~ , am ~- ~ f = t )G ( m ~ , t ) - - ~ - - n (m , , t )G ( m , , t ) ~ ]

[ ' m ~ ( O f m ~ ( O"- - e -6 t | n (m , t )G(m , t ) d m - 62K e -6 t - e -6~Jm~(t) dmz(t)x [ { - o ~ ~ q ( m , F ) n ( m , t ) ] - ( d( m ) + E ( t ) ) n (m . t ) } G ( m , t )

+ n ( m , t ) O G ? ' t ) . 1 den. (B-9)F r om ( 18 ) t he s e c ond t e r m i n ( B -

Optimal Harvest Policy

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