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Some stochastic models in animal resource management Reed, William John 1975

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SOME STOCHASTIC MODELS IN ANIMAL RESOURCE MANAGEMENT by WILLIAM JOHN REED B.Sc, University of London, 1968 M.Sc, M c G i l l University, 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of MATHEMATICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July, 1975 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o lumbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and stud y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 A b s t r a c t T h i s t h e s i s i s c o n c e r n e d w i t h m a t h e m a t i c a l models f o r t h e management o f r e n e w a b l e a n i m a l r e s o u r c e s . D i s c r e t e - t i m e Markov p r o c e s s e s a r e used t o model t h e dynamics o f p o p u l a t i o n s l i v i n g i n f l u c t u a t i n g e n v i r o n m e n t s , and c o n t r o l models a r e d e v e l o p e d on t h i s b a s i s . Of p a r t i c u l a r i n t e r e s t i s t h e way i n w h i c h t h e r e s u l t s o f t h i s s t o c h a s t i c a n a l y s i s compare w i t h known r e s u l t s o f t h e a n a l y s i s b a s e d on s i m i l a r d e t e r m i n i s t i c p o p u l a t i o n models. I n t h e p r i n c i p a l model used i n t h i s t h e s i s i t i s assumed t h a t s u c c e s s i v e a n n u a l l e v e l s o f p o p u l a t i o n f o r m a d i s c r e t e - t i m e Markov p r o c e s s } w i t h t r a n s i t i o n s g overned by t h e e q u a t i o n X , = Z f ( X ) n+1 n n where f i s a f u n c t i o n known i n t h e l i t e r a t u r e on d e t e r m i n i s t i c h a r v e s t models as t h e reproduction function ( o r stock-recruitment relationship),• and {Z^} i s a sequence o f i n d e p e n d e n t , i d e n t i c a l l y d i s t r i b u t e d random v a r i a b l e s w i t h u n i t mean. T h i s model g e n e r a l i z e s t h e w e l l - k n o w n d e t e r m i n i s t i c model Xn+1 = f ( x n ) ' t o t h e c a s e o f a f l u c t u a t i n g e n v i r o n m e n t . T h i s d e t e r m i n i s t i c p r o c e s s i s a s p e c i a l c a s e o f t h e s t o c h a s t i c model, and h o l d s i n e x p e c t a t i o n . Comparison o f r e s u l t s d e r i v e d f r o m t h e two models a r e made, t h u s r e v e a l i n g t h e adequacy o r o t h e r w i s e o f e x i s t i n g t h e o r y b a s e d on t h e d e t e r m i n i s t i c model. The s t e a d y - s t a t e o f t h e s t o c h a s t i c model i s d i s c u s s e d f o r v a r i o u s forms o f t h e f u n c t i o n f , and t h i s d i s c u s s i o n i s e x t e n d e d t o t h e c a s e when t h e p o p u l a t i o n i s s u b j e c t e d t o r e g u l a r h a r v e s t i n g , under a s t a t i o n a r y h a r v e s t p o l i c y . R a t e s o f h a r v e s t w h i c h would l e a d t o t h e e v e n t u a l e x t i n c t i o n o f the p o p u l a t i o n a r e i n v e s t i g a t e d , and i t i s shown how, i n some c a s e s , r a t e s o f h a r v e s t w h i c h may n o t appear c r i t i c a l i n an a n a l y s i s o f t h e d e t e r m i n i s t i c model, can be c r i t i c a l f o r a p o p u l a t i o n s a t i s f y i n g t he s t o c h a s t i c model. The y i e l d i n s t e a d y - s t a t e f o r v a r i o u s s t a t i o n a r y h a r v e s t p o l i c i e s i s d i s c u s s e d and com p a r i s o n s a r e made w i t h s t e a d y - s t a t e y i e l d e s t i m a t e s f r o m t h e e q u i v a l e n t d e t e r m i n i s t i c model. I t i s shown how t h e l o n g - r u n a v e r a g e y i e l d f r o m any s t a t i o n a r y p o l i c y cannot exceed t h e maximum s u s t a i n a b l e y i e l d as computed f r o m t h e d e t e r m i n i s t i c model. A dynamic economic c o n t r o l model w i t h t h e o b j e c t i v e o f m a x i m i z i n g t h e e x p e c t e d d i s c o u n t e d revenue t h a t can be e a r n e d f r o m t h e r e s o u r c e i s d e v e l o p e d . Under c e r t a i n c o n d i t i o n s , o p t i m a l p o l i c i e s a r e c h a r a c t e r i z e d q u a l i t a t i v e l y . When t h e r e i s a p o s i t i v e ' m o b i l i z a t i o n c o s t ' a s s o c i a t e d w i t h h a r v e s t i n g , i t i s shown t h a t an o p t i m a l p o l i c y <• i s o f t h e ( S , s ) t y p e . When t h e r e i s no m o b i l i z a t i o n c o s t a p o l i c y o f ' s t a b i l i z a t i o n - o f - e s c a p e m e n t ' i s o p t i m a l . I n t h e l a t t e r case c o m p a r i s o n s a r e made w i t h t h e o p t i m a l l e v e l o f escapement as d e t e r m i n e d f r o m t h e e q u i v a l e n t d e t e r m i n i s t i c model. On t h e b a s i s o f t h e d e t e r m i n i s t i c model i t i s shown how t h e i i i presence of a p o s i t i v e mobilization cost can lead to the optimality of a p o l i c y of pulse harvesting. Bio-economic conditions which determine the optimality of a p o l i c y of conservation or e x t i n c t i o n are discussed, and r e s u l t s s i m i l a r to the known r e s u l t s obtained from a deterministic analysis are obtained. It i s shown how the presence of a p o s i t i v e mobilization cost can lower the c r i t i c a l discount rate. Control models based on multi-dimensional dynamic models f or populations with age-structure are discussed, although very few d e f i n i t i v e r e s u l t s are obtained. Also two-sex models are discussed, and a control model f or the se x - s p e c i f i c harvesting of P a c i f i c Salmon i s developed. In the l i t e r a t u r e , t h e o r e t i c a l reproduction functions have been derived by modelling various stages of the l i f e - h i s t o r y of a species. In th i s thesis some stochastic models have been used for t h i s , and reproduction functions, based on expected values, have been obtained, which are q u a l i t a t i v e l y s i m i l a r to those derived from analagous deterministic models. A l l the r e s u l t s of t h i s thesis are derived a n a l y t i c a l l y and are not based on data analysis or simulation techniques. i v Contents Introduction and Summary Chapter 1. Discrete-Time Population Models. 1.1 The Reproduction Function 1.2 Stochastic Models f o r Survival and B i r t h Processes 1.3 A Stochastic Reproduction Model Chapter 2. The Population i n Equilibrium. 2.1 Steady-State Behaviour of the Stochastic • Population Model 2.2 Steady-State when there i s Regular Harvesting 2.3 The Steady-State Harvest Y i e l d Chapter 3. An Economic Optimization Model. 3.1 Introduction 3.2 The Model 3.3 An Optimum P o l i c y 3.4 More General Harvest Cost Functions,, Chapter 4. Some Results Concerning Optimal Economic P o l i c i e s . 4.1 Introduction 4.2 The Case of a Zero M o b i l i z a t i o n Cost 4.3 The Optimality of Pulse Harvesting f o r a Deterministic Population Model with P o s i t i v e M o b i l i z a t i o n Cost., 4.4 Economic Determinants of Survival Chapter 5. Multi-Dimensional Population Models. 5.1 Introduction 5.2 A Density-Dependent Deterministic Model 5.3 A Stochastic Density-Dependent Model 5.4 Two-Sex Models ACKNOWLEDGEMENTS I wou l d l i k e t o than k P r o f e s s o r C o l i n C l a r k f o r t h e g u i d a n c e he has g i v e n me i n w r i t i n g t h i s t h e s i s . I would a l s o l i k e t o thank Mrs. C a r o l Samson f o r d o i n g t h e t y p i n g . 1 I n t r o d u c t i o n and Summary T h i s t h e s i s w i l l g e n e r a l l y be c o n c e r n e d w i t h t h e management o f renewable a n i m a l r e s o u r c e s . W i t h much o f t h e w o r l d ' s human p o p u l a t i o n u n d e r n o u r i s h e d , and w i t h many o f t h e w o r l d ' s a n i m a l r e s o u r c e s s e r i o u s l y o v e r e x p l o i t e d , o r even i n danger of e x t i n c t i o n , t h e need f o r r a t i o n a l management i s o b v i o u s . Good management w i l l n o t o n l y e n s u r e t h e c o n s e r v a t i o n o f v a l u a b l e s p e c i e s , b u t i t w i l l have as an o b j e c t i v e an e x p l o i t a t i o n w h i c h i s o p t i m a l i n some s o c i o - e c o n o m i c s e n s e . I n o r d e r t o d e v e l o p sound management p o l i c i e s i t i s n e c e s s a r y f i r s t t o have some s t r u c t u r a l knowledge o f t h e p r o c e s s e s i n v o l v e d . T h i s can be done w i t h t h e h e l p o f m a t h e m a t i c a l models. There i s i n e x i s t e n c e a c o n s i d e r a b l e l i t e r a t u r e c o n c e r n i n g m a t h e m a t i c a l models o f a n i m a l h a r v e s t i n g . However t o d a t e a l m o s t a l l p u b l i s h e d work i n t h e f i e l d makes use o f d e t e r m i n i s t i c p o p u l a t i o n m o d els, and as such i g n o r e s many v a r i a b l e and u n p r e d i c t a b l e f a c t o r s w h i c h may p l a y an i m p o r t a n t r o l e i n t h e p o p u l a t i o n dynamics and c o n s e q u e n t l y i n t h e management o f t h e r e s o u r c e . F o r i n s t a n c e f l u c t u a t i o n s i n the w e a t h e r , and i n e c o l o g i c a l f a c t o r s s u c h as t h e a v a i l a b i l i t y o f and c o m p e t i t i o n f o r f o o d , t h e i n c i d e n c e o f p r e d a t o r s e t c . , a r e o f t e n u n p r e d i c t a b l e b u t may play, an i m p o r t a n t r o l e i n t h e dynamics o f an a n i m a l p o p u l a t i o n . A s t o c h a s t i c p o p u l a t i o n model can o f t e n i n c o r p o r a t e t h e s e f l u c t u a t i o n s i n t h e fo r m o f random v a r i a t i o n . 2 , I n t h i s t h e s i s we s h a l l i n v e s t i g a t e some s t o c h a s t i c c o n t r o l -t h e o r e t i c models o f a n i m a l h a r v e s t i n g . What w i l l be o f p a r t i c u l a r i n t e r e s t w i l l be the way i n w h i c h s t o c h a s t i c f a c t o r s a f f e c t t h e r e s u l t s o f t h e d e t e r m i n i s t i c t h e o r y , and i n what way p r a c t i c a l management p o l i c i e s can be improved i n t h e l i g h t o f s t o c h a s t i c m o d e l l i n g . Can s t o c h a s t i c models h e l p t o e x p l a i n phenomena i n c o m p l e t e l y u n d e r s t o o d w i t h e x i s t i n g d e t e r m i n i s t i c models? These w i l l be t h e main o b j e c t i v e s o f t h i s t h e s i s . The p o p u l a t i o n models h i t h e r t o used i n d i s c u s s i n g management problems can be c l a s s i f i e d as b e i n g e i t h e r d i s c r e t e - t i m e o r c o n t i n u o u s -t i m e models. D e t e r m i n i s t i c models o f t h e f o r m e r t y p e a r e bas e d on d i f f e r e n c e e q u a t i o n s whereas d e t e r m i n i s t i c models o f t h e l a t t e r t y p e a r e b a s e d on d i f f e r e n t i a l e q u a t i o n s . The two t y p e s o f model p l a y a complementary r o l e i n t h e s t u d y o f p o p u l a t i o n dynamics. From t h e c o n t r o l p o i n t o f v i e w , t h e c o n t i n u o u s - t i m e model i s perhaps more a p p r o p r i a t e when h a r v e s t i n g t a k e s p l a c e c o n t i n u o u s l y t h r o u g h o u t t h e y e a r , o r a t l e a s t d u r i n g a s i g n i f i c a n t l y l o n g h a r v e s t s e a s o n . Many d e m e r s a l f i s h e r i e s , and a l s o t h e whale f i s h e r i e s , w o u l d f a l l i n t o t h i s c a t e g o r y . On t h e o t h e r hand t h e d i s c r e t e - t i m e model i s p a r t i c u l a r l y a p p r o p r i a t e f o r t h e h a r v e s t i n g o f s p e c i e s , whose p o p u l a t i o n dynamics e x h i b i t a c y c l i c b e h a v i o u r , and w h i c h a r e h a r v e s t e d o n l y d u r i n g a s h o r t s e a s o n each c y c l e . Examples o f e x p l o i t a t i o n o f t h i s t y p e w o u l d be t h e salmon f i s h e r i e s and s e a l f i s h e r i e s . P o s s i b l y a l s o t h e e x p l o i t a t i o n o f range a n i m a l s , i n t h e w i l d , c o u l d b e s t be s t u d i e d w i t h a d i s c r e t e -t i m e model. 3 Early work In the f i e l d of animal resource management was c a r r i e d out l a r g e l y by f i s h e r i e s b i o l o g i s t s . The comprehensive book of Beverton-Holt (1957) discusses many approaches to the problems of f i s h e r i e s management and both continuous-time and discrete-time models are discussed. Ricker (1954, 1958(a)) developed a discrete-time model based on the "stock-recruitment" r e l a t i o n s h i p which i s assumed to f u n c t i o n a l l y r e l a t e recruitment i n a f i s h e r y to parent stock i n the preceding year or generation. Economists have also studied the e x p l o i t a t i o n of animal resources, e s p e c i a l l y f i s h e r i e s . Economic equilibrium analyses have been performed by Gordon (1954), Turvey (1964) and many others. Dynamic economic c o n t r o l - t h e o r e t i c analyses based on det e r m i n i s t i c , continuous-time population models have been performed by Plourde (1970, 1971), Quirk and Smith (1970), Clark (to appear), Clark and Munro (1975), and Clark, Edwards and Friedlaender (1973). Deterministic, discrete-time models have also been used i n a c o n t r o l - t h e o r e t i c approach to dynamic, economic analysis (see Clark 1971, 1972, 1973(b)). A l l of the above investigations have been based on deterministic population models. As we have already mentioned, t h i s i s l i k e l y to be an o v e r - s i m p l i f i c a t i o n of the real-world s i t u a t i o n . Fluctuating environmental factors are l i k e l y to play, an important part i n the dynamics of any population l i v i n g " i n the wild". Also factors " i n t e r n a l " to the population such as the success of mating and s u r v i v a l behaviour, the number of b i r t h s per female etc. are l i k e l y to vary i n 4 an u n p r e d i c t a b l e way. A s t o c h a s t i c model c o u l d b e t t e r a p p r o x i m a t e t h e dynamics o f a p o p u l a t i o n l i v i n g i n t h e w i l d , t h a n c o u l d a d e t e r m i n i s t i c model. However the g a i n i n a c c u r a c y i n u s i n g a s t o c h a s t i c p o p u l a t i o n model i s u s u a l l y o f f s e t by a l o s s i n s i m p l i c i t y . Sometimes a d e t e r m i n i s t i c model can a d e q u a t e l y d e s c r i b e t h e p o p u l a t i o n dynamics. The q u e s t i o n as t o when a d e t e r m i n i s t i c model g i v e s an adequate d e s c r i p t i o n o f t h e dynamics o f an a n i m a l p o p u l a t i o n l i v i n g i n a f l u c t u a t i n g e n v i r o n m e n t , and when i t does n o t , has been d i s c u s s e d , t h e o r e t i c a l l y , a t l e n g t h , by May (1973). May i s c o n c e r n e d p r i m a r i l y w i t h t h e s t a b i l i t y o f p o p u l a t i o n s l i v i n g i n a f l u c t u a t i n g e n v i r o n m e n t , u n d i s t u r b e d by s y s t e m a t i c h a r v e s t i n g by man. A q u e s t i o n w h i c h has n o t been a d d r e s s e d i s how does e n v i r o n m e n t a l f l u c t u a t i o n e f f e c t t h e t h e o r e t i c a l a n a l y s i s o f a n i m a l h a r v e s t i n g based on d e t e r m i n i s t i c p o p u l a t i o n models? I n o t h e r words how r o b u s t i s t h e d e t e r m i n i s t i c t h e o r y o f a n i m a l h a r v e s t i n g t o d e p a r t u r e s from t h e a s s u m p t i o n o f s t r i c t d e t e r m i n i s m i n t h e p o p u l a t i o n dynamics? When does a d e t e r m i n i s t i c h a r v e s t model a d e q u a t e l y d e s c r i b e t h e e x p l o i t a t i o n o f an a n i m a l r e s o u r c e i n a f l u c t u a t i n g e n v i r o n m e n t ? A r e o p t i m a l h a r v e s t p o l i c i e s i n a d e t e r m i n i s t i c model a l s o o p t i m a l i n a s t o c h a s t i c model w h i c h i n c o r p o r a t e s t h e e f f e c t s o f a f l u c t u a t i n g ' e n v i r o n m e n t ? These q u e s t i o n s a r e r a t h e r i m p o r t a n t s i n c e t h e y r e l a t e d i r e c t l y t o t h e v a l i d i t y o f t h e e x i s t i n g t h e o r y . I n o r d e r t h a t t h e s e q u e s t i o n s be r e s o l v e d - , a s t u d y o f s t o c h a s t i c c o n t r o l models f o r a n i m a l h a r v e s t i n g i s needed. 5 The e x i s t i n g l i t e r a t u r e c o n c e r n i n g a n i m a l h a r v e s t i n g i n a f l u c t u a t i n g e n v ironment i s s c a n t . There have been s i m u l a t i o n s t u d i e s c o n c e r n i n g t h e y i e l d s from v a r i o u s h a r v e s t p o l i c i e s f o r a p o p u l a t i o n l i v i n g i n a f l u c t u a t i n g e n v i r o n m e n t , by R i c k e r (1958(b)); L a r k i n and R i c k e r (1964); T a u t z , L a r k i n and R i c k e r (1969); and Radway A l l e n (1973). G a l t o and R i n a l d i ( t o appear) have i n v e s t i g a t e d some o f t h e c o n c l u s i o n s o f R i c k e r (1958(b)) a n a l y t i c a l l y . S t o c h a s t i c c o n t r o l - t h e o r e t i c a n a l y s e s o f t h e o p t i m a l e x p l o i t a t i o n o f a n i m a l r e s o u r c e s have been p e r f o r m e d by Mann (1970), J a q u e t t e (1972, 1974) and Reed*(1974). I n t h e p a p e r s o f Mann and J a q u e t t e d i s c r e t e - t i m e models a r e used and t h e p a r a m e t e r s o f t h e f i r s t - o r d e r p o p u l a t i o n growth e q u a t i o n s ( d i f f e r e n c e e q u a t i o n s ) a r e c o n s i d e r e d as random v a r i a b l e s . I n t h e two-sex model o f Mann t h e time-sequence o f p a r a m e t e r v e c t o r s forms a d i s c r e t e - t i m e Markov p r o c e s s , whereas i n the p a p e r s o f J a q u e t t e , w h i c h assume a homogeneous p o p u l a t i o n , i t i s c o n s i d e r e d as a sequence o f i n d e p e n d e n t random v a r i a b l e s . B o t h a u t h o r s use Dynamic Programming t o o b t a i n o p t i m a l p o l i c i e s o f t h e " f e e d b a c k " t y p e . W i t h a f e e d b a c k h a r v e s t p o l i c y , t h e s i z e o f t h e h a r v e s t i n any y e a r w i l l depend on t h e s i z e o f t h e p o p u l a t i o n i n t h a t y e a r and p o s s i b l y i n p r e v i o u s y e a r s . Thus w i t h a s t o c h a s t i c p o p u l a t i o n model and a f e e d b a c k p o l i c y t h e s i z e o f t h e h a r v e s t i n any y e a r w i l l be a random v a r i a b l e , and t h u s n o t c o m p l e t e l y p r e d i c t a b l e i n advance. I n d e t e r m i n i s t i c c o n t r o l problems an o p t i m a l p o l i c y i s _ T h i s p a p e r c o m p r i s e s some o f t h e r e s u l t s p r e s e n t e d i n C h a p t e r s 3 and 4 of t h i s t h e s i s . 6 generally of the "open-loop" type. . With t h i s type of con t r o l both the population l e v e l and the size of the harvest i n any year are exactly predictable i n advance. This i s an important difference between deterministic and stochastic c o n t r o l models, and there may be some ambiguity i n applying an open-loop p o l i c y , optimal f o r a deterministic system, to the control of a population l i v i n g i n a f l u c t u a t i n g environment and e x h i b i t i n g stochastic population dynamics'. For instance Ricker (1958(b)) suggested at l e a s t two feedback i n t e r p r e t a t i o n s of the open-loop p o l i c y which maximizes maximum sustainable y i e l d i n a deterministic model. The "natural" feedback i n t e r p r e t a t i o n of an open-loop p o l i c y f o r the case of a population l i v i n g i n a f l u c t u a t i n g environment should be f a i r l y obvious i n most cases. However a study of optimal p o l i c i e s for a stochastic c o n t r o l model should help c l a r i f y t h i s . In t h i s thesis we s h a l l study some stochastic control models for animal harvesting and hope to throw some l i g h t on the questions raised. We s h a l l deal e x c l u s i v e l y with discrete-time models. The population models which we s h a l l consider w i l l be discrete-time Markov processes. The main model which i s discussed i n 1-.-3 and Chapters 2, 3 and 4 i s a stochastic version of the one-dimensional "reproduction function", harvesting model of Clark (1971, 1972, 1973(b)). I t i s based on the "reproduction function" or "stock-recruitment" r e l a t i o n -ship much discussed by f i s h e r i e s b i o l o g i s t s and dating back to the paper of Ricker (1954). In Chapter 5 we develop some multi-dimensional 7 models f or age- and sex-structured populations and discuss some of the problems associated with such modelling. Chapter 1 concerns one-dimensional discrete-time stochastic population models and does not involve harvest models. In t h i s introduction we s h a l l give a f a i r l y comprehensive summary of the main r e s u l t s of the th e s i s . It i s intended to be comprehensible f o r the non-mathematician. A One-Dimensional, Discrete-Time, Stochastic Population Model In much of the l i t e r a t u r e on deterministic discrete-time population models the sizes of the population i n successive years are assumed to be related by the difference equation Xn+1 = f ( x n } ' where f i s a function known as the reproduction function (or a l t e r n a t i v e l y the recruitment function or stock-recruitment r e l a t i o n s h i p ) . This of course assumes that the population can be completely described by a sing l e continuous v a r i a b l e , and i s not age- or sex-s p e c i f i c . The var i a b l e x i s often taken to represent t o t a l bio-mass, or t o t a l number of animals i n the population. The fac t that t h i s model ignores many fa c t o r s , (including environmental and other random f l u c t u a t i o n s , the age and sex s p e c i f i c i t y of the population etc.) coupled with the s t a t i s t i c a l errors associated with estimating popu-l a t i o n abundance, has confounded most attempts at estimating reproduction functions. Indeed Larkin (1973) has remarked, "So great are some of 8 th e e n v i r o n m e n t a l e f f e c t s t h a t ' s t o c k - r e c r u i t p l o t s a r e o f t e n r e m i n i s c e n t o f random s c a t t e r d i a g r a m s . " We i n c o r p o r a t e t h e e f f e c t s o f e n v i r o n m e n t a l v a r i a t i o n s i n t o t h i s model by i n t r o d u c i n g a random m u l t i p l i e r . We c o n s i d e r t h e model n+± n n where now {X } i s a sequence o f random v a r i a b l e s ( r . v . ' s ) . We n assume t h a t {Z } forms a sequence o f i n d e p e n d e n t , i d e n t i c a l l y -n d i s t r i b u t e d r . v . ' s w i t h mean one, and t h a t Z i s i n d e p e n d e n t o f X ' n n Thus we have t h a t {X } i s a d i s c r e t e - t i m e Markov p r o c e s s w i t h n E ( X n + l l V x ) " f ( x ) i . e . t h e r e p r o d u c t i o n f u n c t i o n h o l d s i n e x p e c t a t i o n . We s h a l l c a l l f t h e e x p e c t e d r e p r o d u c t i o n f u n c t i o n . Of c o u r s e t h i s d o e s n ' t h e l p w i t h t h e p r o b l e m o f e s t i m a t i n g f , b u t i t may g i v e a more a c c u r a t e model o f t h e dynamics o f a p o p u l a t i o n , l i v i n g i n t h e w i l d . I n a d o p t i n g t h i s model we a r e assuming t h a t t h e r e i s no t r e n d i n t i m e o f t h e e n v i r o n m e n t a l f l u c t u a t i o n s , b u t r a t h e r t h a t s t o c h a s t i c a l l y t h e y a r e time-homogeneous. We d i s c u s s t h e q u e s t i o n o f t h e v a l i d i t y o f t h i s model i n 1 . 3 . There i s no immediate b i o l o g i c a l j u s t i f i c a t i o n f o r i n t r o d u c i n g t h e random e f f e c t s i n t h i s m u l t i p l i c a t i v e f o r m , a l t h o u g h i t seems r e a s o n a b l e t h a t t h e s t a n d a r d d e v i a t i o n o f X , n s h o u l d n+1 « i n c r e a s e w i t h X . I f t h e random e n v i r o n m e n t a l f l u c t u a t i o n s a f f e c t n 9 the population s i z e i n a proportional manner, independently of the population s i z e , then the m u l t i p l i c a t i v e model seems reasonable. Data for various species of P a c i f i c salmon seem to support t h i s model (see Shephard and Whitler (1958), Ricker (1958(b)). For convenience we assume that there i s a maximum population l e v e l , m , say. Also we assume that the random m u l t i p l i e r s have a d i s t r i b u t i o n with d i s t r i b u t i o n function (df) $ on an i n t e r v a l [a, bi] , where 0 £ a <_ 1 <_ b . We require that b = m/f (p) where p i s the point at which f attai n s i t s maximum on [0, m] . This i s i l l u s t r a t e d i n F i g 0-1, below. 0 p m F i g 0-rl We f i r s t investigate the steady-state of t h i s model i n 2.1. Unlike a deterministic population model t h i s stochastic model w i l l not reach a steady-state with the population si z e remaining constant at 10 some point of equilibrium. Rather, i n steady-state, the unconditional p r o b a b i l i t y d i s t r i b u t i o n of the population s i z e w i l l remain constant, from year to year i . e . the p r o b a b i l i t y d i s t r i b u t i o n of the population s i z e w i l l be stationary. Observations of the s i z e of a population with dynamics such as t h i s , when i t i s i n an undisturbed steady-state, w i l l be represented by a scatter of points whose d i s t r i b u t i o n w i l l r e f l e c t a stationary p r o b a b i l i t y d i s t r i b u t i o n of the model. Just as i n the deterministic reproduction function model, more than one steady-state i s possible i n general. In the deterministic model, stable equilibrium points are those solutions to x = f(x) for which | f ' ( x ) | < 1 . In our stochastic model stationary p r o b a b i l i t y d i s t r i b u t i o n s have d i s t r i b u t i o n functions p(t) which are solutions to P(t) = m <& ( j ^ r - ) dp(x) . (2) 0 W (This can e a s i l y be established from (1) by a con d i t i o n a l p r o b a b i l i t y argument). Rather than t r y to f i n d solutions to the i n t e g r a l equation (2), which w i l l depend very much on the nature of the functions $ and f , we look for conditions which imply the existence of a stationary d i s t r i b u t i o n and t r y to f i n d some of the properties of such a d i s t r i b u t i o n . We have not completely solved t h i s problem. We 11 look at some s p e c i a l cases. We assume that the d.f $ has a continuous density. We consider f i r s t l y the case f o r which af'(O) > 1 . This means that at low l e v e l s the population s i z e w i l l increase almost surely. In such a case there i s a maximum l e v e l r , such that with p r o b a b i l i t y one the population s i z e w i l l grow to exceed r and subsequently remain greater than r . (see F i g 0.2) For most simple forms of the function f the set [r, m] w i l l be a "communicating set", i n the sense that i t i s possible to go from any one population l e v e l i n that set to any other l e v e l i n that set, i n a f i n i t e number of years. In t h i s case we are able to show d i r e c t l y from a theorem of Doob (1953) that f o r any i n i t i a l p o s i t i v e population l e v e l , the  system tends to a unique l i m i t i n g steady-state with the stationary  d i s t r i b u t i o n having a p o s i t i v e density almost everywhere on [r, m]. (Thm. 2.1.1). 12 The f a c t that the l i m i t i n g stationary d i s t r i b u t i o n i s s t r i c t l y p o s i t i v e on [r, m] i s rather i n t e r e s t i n g . P r a c t i c a l l y i t means that i f the population were observed i n steady-state for long enough, i t would be seen to assume a l l distinguishable l e v e l s i n the i n t e r v a l [r, m] . Thus i n steady-state, a l l population l e v e l s between r and m can occur and indeed eventually, w i l l occur. Furthermore, i t i s easy to see from F i g 0.2 that the spread r to m of the steady-state d i s t r i b u t i o n depends on the spread a to b of the d i s t r i b u t i o n of the random m u l t i p l i e r s . For a small degree of environmental f l u c t u a t i o n thetrangev6fopopulata>on->- : j , ; l e v e l s i n steady-state w i l l be small and vice-versa. It i s important to note that t h i s r r e s u l t does not depend on the s p e c i f i c assumption of a m u l t i p l i c a t i v e population model of the form (1). The same r e s u l t holds for any Markovian model for which the stochastic kernel F(t;x) = p r { X n + 1 <. t|X Q = x} has a continuous density which i s non-zero only i n a region s i m i l a r to that shown i n F i g 0.2. A f i r s t - o r d e r discrete-time Markov model of t h i s type i s a natural generalization of the f i r s t order reproduction function model. I t seems that t h i s i s the f i r s t time that such a model has been used to explain the range of population l e v e l s occurring i n steady-state. Oster and May (1975) have shown how apparently random l e v e l s of population can occur i n a population s a t i s f y i n g a deterministic over-compensation reproduction function model. 13 If the population were ever disturbed from i t s steady-state, for instance by harvesting, i n such a way that the l e v e l dropped below r , the population i f subsequently l e f t undisturbed would return eventually to the same steady-state on [ r , m] . A population s a t i s f y i n g t h i s model would be very robust. In t h i s sense i t corresponds to a deterministic reproduction function model with f'(0) > 1 . Such a model has a stable equilibrium population l e v e l . What happens when af' (0) <_ 1 ? In t h i s case i t i s possible "at low population l e v e l s f or the population s i z e to decrease. We have not been able to completely solve t h i s case. It seems that there are three p o s s i b i l i t i e s : -(a) a steady-state with a stationary d i s t r i b u t i o n with s t r i c t l y -p o s i t i v e density on [0, m] i s the long-run outcome. (b) the process i s " n u l l recurrent" i n the sense used i n the theory of Markov Chains, and described below. (c) the population tends eventually to e x t i n c t i o n almost surely. I f case (a) were to r e s u l t , i n steady-state a l l values i n (0, m] would eventually occur. In steady-state the population could become a r b i t r a r i l y small. However i t could not stay small forever. It would eventually return to exceed any given l e v e l l e s s than m , with f i n i t e expected time of return. If case (b) were to occur, then for any i n i t i a l population l e v e l , the population might dwindle to a r b i t r a r i l y small sizes and stay i n the neighbourhood of zero for a r b i t r a r i l y long periods. However 14 the s i z e of the population would eventually grow back to exceed any l e v e l l e s s than m , but the expected time of return would be i n f i n i t e . For a l l p r a c t i c a l purposes the population would be e x t i n c t f o r a r b i t r a r i l y long periods. We are able to show that, eventual e x t i n c t i o n (case (c)) occurs when the expected reproduction function i s concave (compensatory), and i s such that E(X , Ix = x) = f(x) < x , f o r a l l x . (Thm 2.1.2) n+11 n — In the deterministic model for which f(x) <_ x , a l l x , e x t i n c t i o n also occurs. However i n that case the s i z e of the population decreases monotonically to zero, whereas i n the stochastic model the progress toward e x t i n c t i o n i s not n e c e s s a r i l y monotone. I t could be, i n t h i s case, that there are years, or even runs of years i n which the population s i z e increases, but e x t i n c t i o n i s nevertheless the eventual outcome. A model such as t h i s could be rather useful i n describing the dynamics of an over-exploited population. So f a r our r e s u l t s have only concerned the existence of steady-states and the spread of corresponding stationary d i s t r i b u t i o n s . How does a steady-state stationary d i s t r i b u t i o n compare q u a n t i t a t i v e l y with the equivalent equilibrium i n deterministic theory? We show i n 2.1 that i f a stationary d i s t r i b u t i o n e x i s t s and i f the expected reproduction function i s concave over the range of the stationary d i s t r i b u t i o n then the mean (expected) steady-state population l e v e l i s l e s s than the equilibrium l e v e l i n the equivalent deterministic model. Thus an 15 equilibrium l e v e l estimated from a deterministic population model might be greater than the average of steady-state l e v e l s observed i n a f l u c t u a t i n g environment. what i s known as c r i t i c a l depensation i n deterministic theory (see 1.1(c)). This i s characterized by an i n a b i l i t y of the population to sustain i t s s i z e at low population l e v e l s . B i o l o g i c a l l y t h i s could be explained by adaptations which can only function e f f i c i e n t l y at su i t a b l y high population l e v e l s . For instance v u l n e r a b i l i t y to predation might greatly increase at low population l e v e l s . Again there might be d i f f i c u l t y for animals to locate mates for breeding at low population l e v e l s . In 2.1 we consider a stochastic model which corresponds to The stochastic model shown i n F i g 0.3 exhibits t h i s property of c r i t i c a l depensation. bf (x) af(x) 0 c d r m F i g 0.3 We show i n 2.1 that the l e v e l c i s a c r i t i c a l l e v e l . I f 16 the population ever drops below c then a monotonic decrease to e x t i n c t i o n r e s u l t s . The l e v e l d i s a safety l e v e l . I f the population s i z e ever exceeds d then a steady-state with a stationary d i s t r i b u t i o n on [r, m] r e s u l t s . However for an i n i t i a l population i n the i n t e r v a l (c,d) e i t h e r outcome may r e s u l t , which one depending p r o b a b i l i s t i c a l l y on the s i z e of the i n i t i a l population. In some sense the i n t e r v a l (c,d) corresponds to the unstable equilibrium point i n a deterministic c r i t i c a l depensation model. If the population leaves (c,d) to enter [d, m] then steady-state on [r, m] ensues, whereas i f the population leaves (c,d) to enter (0,c] then e x t i n c t i o n ensues. The l e v e l d can also be thought of as a danger l e v e l . I f through harvesting say, the population were disturbed from steady-state on [r, m] i n such a way that the population s i z e dropped below d , then even i f subsequently l e f t undisturbed, e x t i n c t i o n of the population would become a p o s s i b i l i t y with a non-zero p r o b a b i l i t y of occurrence. With the population s i z e below the danger l e v e l d , a few unfavourable years could ensure the e x t i n c t i o n of the population. This could not happen at higher population l e v e l s . I f the population s i z e were to drop below d often enough then e x t i n c t i o n would eventually r e s u l t . The r e s u l t s on c r i t i c a l depensation do not depend on the assumption of a m u l t i p l i c a t i v e model of the form (1). It i s s u f f i c i e n t that the model be Markovian with the stochastic kernel having a density which i s non-zero only i n a region l i k e that shown i n F i g 0.3. The r e s u l t s are f a i r l y c l e a r i n t u i t i v e l y . Formal proofs can be e a s i l y derived, using the ergddic theorem of Doob (1953, p. 214). 17 S t e a d y - s t a t e when t h e r e i s r e g u l a r h a r v e s t i n g I n 2.2 we i n v e s t i g a t e t h e s t e a d y - s t a t e o f a p o p u l a t i o n s u b j e c t e d t o r e g u l a r h a r v e s t i n g . I n p a r t i c u l a r we a r e i n t e r e s t e d i n what r a t e o f h a r v e s t i n g can be s u s t a i n e d by a g i v e n p o p u l a t i o n . A l s o we a r e c o n c e r n e d w i t h how t h e r e s u l t s o f s t o c h a s t i c s t e a d y - s t a t e a n a l y s i s compare w i t h t h o s e o f d e t e r m i n i s t i c t h e o r y . We assume t h a t h a r v e s t i n g t a k e s p l a c e e v e r y y e a r ( o r c y c l e ) d u r i n g a s h o r t h a r v e s t s e a s o n . Such i s t h e case f o r many m i g r a t o r y f i s h p o p u l a t i o n s , i n c l u d i n g salmon. S e a l s a r e a l s o h a r v e s t e d i n t h i s way. We assume t h a t t h e h a r v e s t i n g p o l i c y i n o p e r a t i o n i s a s t a t i o n a r y p o l i c y i n t h e sense t h a t t h e h a r v e s t i n any y e a r depends o n l y on t h e s i z e o f t h e p o p u l a t i o n a t t h e b e g i n n i n g o f t h a t y e a r ' s h a r v e s t s e a s o n . We can c o n v e n i e n t l y denote a s t a t i o n a r y h a r v e s t p o l i c y by a f u n c t i o n h ( x ) , where h i s t h e s i z e o f a h a r v e s t t a k e n f r o m a p o p u l a t i o n o f s i z e x . We l e t u ( x ) = x - h ( x ) . We see t h a t u ( x ) r e p r e s e n t s t h e escapement from a h a r v e s t p o l i c y h when t h e i n i t i a l p o p u l a t i o n l e v e l i s x . S e v e r a l forms o f s t a t i o n a r y h a r v e s t p o l i c i e s w h i c h a r e commonly d i s c u s s e d a r e shown i n F i g 2.8. One form o f p a r t i c u l a r i n t e r e s t i s t h e " c o n s t a n t - e f f o r t " h a r v e s t , i n w h i c h i t i s assumed t h a t a f i x e d p r o p o r t i o n o f t h e c u r r e n t p o p u l a t i o n i s h a r v e s t e d each y e a r . I n t h i s c a s e h ( x ) = h*x and u ( x ) = u-x . T h i s might a p p r o x i m a t e l y r e p r e s e n t t h e case i n w h i c h a f i x e d s i z e h a r v e s t o p e r a t i o n i s m o b i l i z e d each y e a r (hence t h e name). F o r i n s t a n c e , i n a f i s h e r y , i f t h e same f l e e t were t o o p e r a t e f o r t h e same d u r a t i o n each y e a r , t h e n a p o l i c y 18 of the above type might approximately hold. In general the dynamics of a population s a t i s f y i n g our model under a stationary harvest p o l i c y h can be described by the equation X = Z f(u(X )) , n+1 n n where X^ represents the s i z e of the population immediately p r i o r to harvesting i n year n . A convenient graphical representation of t h i s process i s obtained by p l o t t i n g the escapement as x = u(y) on the reproduction-function diagram. In t h i s way for any X^ , measured on the ordinate, the possible range of values of X , ( = Z f(u(X ))) can be found, n+I n n again on the ordinate. An inspection of a diagram of t h i s type can reveal some very i n t e r e s t i n g r e s u l t s concerning the steady-state of the population under regular harvesting. The f i r s t case discussed i n 2.2 i s that of a compensatory  model (concave expected reproduction function) for which af'(O) > 1 (see F i g 0.4) x=uy Fig 0.4 19 For s i m p l i c i t y we summarize the r e s u l t s f or a constant-effort harvest p o l i c y . Similar r e s u l t s hold for other stationary p o l i c i e s . Undisturbed, the population w i l l be i n a steady-state d i s t r i b u t e d on [r, m] , (see F i g 0.4). If a constant e f f o r t harvest with proportional escapement u >, l / a f ' ( 0 ) i s applied the population w i l l at f i r s t be disturbed from i t s steady-state. However i t i s not d i f f i c u l t to show that i f such a harvest p o l i c y continues, the population w i l l reach a new steady-state d i s t r i b u t e d on [r^> m i J • Any rate of e x p l o i t a t i o n for which the proportional escapement i s  greater than l/af'(0) can be sustained by the population. The deterministic theory predicts that an e x p l o i t a t i o n f o r which the proportional escapement i s greater than l / f ' ( 0 ) can be sustained, whereas e x p l o i t a t i o n with proportional escapement les s than l / f ' ( 0 ) cannot be sustained. We have been unable to v e r i f y t h i s f or the stochastic model. The case for which the proportional escapement i s between l / f ' ( 0 ) and l/af'(0) corresponds to the undecided case when there i s no harvesting. I t could be that any of three outcomes re s u l t s - e x t i n c t i o n , n u l l recurrence, or a stationary d i s t r i b u t i o n . C e r t a i n l y in-.the • stochastic model an e x p l o i t a t i o n with  proportional escapement les s than l / f ' ( 0 ) cannot be sustained. If such a harvest p o l i c y were continually applied the population would eventually become ex t i n c t . However the progress towards e x t i n c t i o n would not n e c e s s a r i l y be monotonic, as the deterministic theory p r e d i c t s . The managers of an animal resource might be deceived into b e l i e v i n g that the population could sustain such a high l e v e l of e x p l o i t a t i o n on 20 t h e e v i d e n c e o f a few y e a r s i n w h i c h t h e s i z e o f t h e p o p u l a t i o n i n c r e a s e d . Y e a r s i n w h i c h t h e s i z e d e c r e a s e d c o u l d be d i s m i s s e d as a t y p i c a l l y 'bad' y e a r s . However w i t h s u c h a r a t e o f e x p l o i t a t i o n s u c h 'bad' y e a r s would n o t be a t a l l a t y p i c a l . Indeed s t a t i s t i c a l l y t h e y w o u l d dominate o v e r t h e 'good' y e a r s . I f t he p o p u l a t i o n were b e i n g h a r v e s t e d i n d e p e n d e n t l y o f o t h e r p o p u l a t i o n s , i t i s u n l i k e l y t h a t a c r i t i c a l r a t e o f e x p l o i t a t i o n w o u l d be c o n t i n u e d t o t h e p o i n t a t w h i c h t h e p o p u l a t i o n became e x t i n c t . When t h e p o p u l a t i o n s i z e became t o o l o w , t h e r e t u r n s w o u l d be so s m a l l as t o make h a r v e s t i n g u n e c o n o m i c a l and h a r v e s t i n g would l i k e l y c ease o r be r e d u c e d . The p o p u l a t i o n w o u l d be c a p a b l e o f r e g e n e r a t i n g i t s e l f t o i t s f o r m e r l e v e l s once h a r v e s t i n g had c e a s e d , a l t h o u g h t h i s m i g ht t a k e many y e a r s . On t h e o t h e r hand i f t h e p o p u l a t i o n were b e i n g h a r v e s t e d i n c o n j u n c t i o n w i t h a n o t h e r (more p r o d u c t i v e ) p o p u l a t i o n t h e n i t i s q u i t e p o s s i b l e t h a t a c r i t i c a l r a t e o f e x p l o i t a t i o n c o u l d be c o n t i n u e d t o t h e p o i n t a t w h i c h the p o p u l a t i o n became e x t i n c t . Such r e s u l t s based on d e t e r m i n i s t i c models a r e w e l l - k n o w n t o e c o l o g i s t s . F o r a c o m p e n s a t i o n model f o r w h i c h af'(0) < 1 , we can show t h a t an e x p l o i t a t i o n a l l o w i n g p r o p o r t i o n a l escapement l e s s t h a n l / f ' ( 0 ) cannot be s u s t a i n e d as t h e d e t e r m i n i s t i c t h e o r y p r e d i c t s . However we a r e u n a b l e t o d e t e r m i n e whether l o w e r r a t e s o f e x p l o i t a t i o n can be s u s t a i n e d as p r e d i c t e d by t h e d e t e r m i n i s t i c t h e o r y . A s i m i l a r a n a l y s i s i s p e r f o r m e d i n 2.2 f o r d e p e n s a t i o n models. ( D e p e n s a t i o n i s c h a r a c t e r i z e d by a r e p r o d u c t i o n f u n c t i o n , w h i c h , a t f i r s t i n c r e a s e s i n a convex way, and t h e n i s concave — see 1.1(b), ( c ) ) . 21 We d i s t i n g u i s h two cases. C r i t i c a l depensation i s when the population i s incapable of increasing i t s s i z e at low population l e v e l s . With normal depensation there i s a p o s i t i v e p r o b a b i l i t y that the population can increase i t s si z e at any given l e v e l . For a normal depensation model l i k e that shown i n F i g 0.5, a constant e f f o r t harvest allowing proportional escapement greater than u can be sustained by the population. Under such an e x p l o i t a t i o n the population w i l l reach a steady-state d i s t r i b u t e d on an i n t e r v a l above r . x=uy Fig 0.5 x-> The equivalent deterministic model predicts that harvests allowing proportional escapement les s than u^ cannot be sustained, while those f o r which proportional escapement i s greater than u^ can be sustained. This i s not true i n general for the stochastic model. For instance, i f as i n F i g 0.5, u^ < l/bf'(0) , then a proportional escapement near u^ w i l l lead to e x t i n c t i o n . In t h i s 2 2 c a s e a l i m i t a t i o n on h a r v e s t e f f o r t , d e s i g n e d as a c o n s e r v a t i o n  s a f e g u a r d and c a l c u l a t e d from a d e t e r m i n i s t i c p o p u l a t i o n model, might  be i n a d e q u a t e t o e n s u r e t h e c o n s e r v a t i o n o f t h e p o p u l a t i o n . Whether u i s t h e minimum p r o p o r t i o n a l escapement t h a t i s c o m p a t i b l e w i t h c o n s e r v a t i o n i s an open q u e s t i o n . C e r t a i n l y u c o n s t i t u t e s a t h r e s h o l d , i n t h e sense t h a t f o r p r o p o r t i o n a l escapements l e s s t h a n u , a r b i t r a r i l y l o w p o p u l a t i o n s i z e s can o c c u r , whereas i n s t e a d y - s t a t e , f o r a h a r v e s t a l l o w i n g p r o p o r t i o n a l escapement g r e a t e r t h a n u , t h e p o p u l a t i o n s i z e w i l l n e v e r be l e s s t h a n r . A g a i n i n t h e case o f c r i t i c a l d e p e n s a t i o n , r e s u l t s d i f f e r e n t f r o m t h e d e t e r m i n i s t i c t h e o r y a r i s e . x=u-,y x=uy c F i g 0.6 I n t h e s t o c h a s t i c model a c o n s t a n t - e f f o r t h a r v e s t a l l o w i n g a p r o p o r t i o n a l escapement g r e a t e r t h a n u can be s u s t a i n e d whereas one a l l o w i n g a p r o p o r t i o n a l escapement l e s s t h a n u cannot be s u s t a i n e d . The d e t e r m i n i s t i c t h e o r y p r e d i c t s t h e l e v e l o f p r o p o r t i o n a l escapement 23 t o be c r i t i c a l . A g a i n s t a n d a r d s o f c o n s e r v a t i o n based on a  d e t e r m i n i s t i c model wo u l d be i n a d e q u a t e . A model such as t h i s can e x p l a i n a sudden " c r a s h " i n an e x p l o i t e d p o p u l a t i o n . I f t h e p o p u l a t i o n were b e i n g r e g u l a r l y h a r v e s t e d w i t h a p r o p o r t i o n a l escapement s l i g h t l y g r e a t e r t h a n t h e c r i t i c a l l e v e l u , t h e n t h e p o p u l a t i o n w o u l d r e a c h a s t e a d y - s t a t e w i t h h i g h ( g r e a t e r t h a n r ) p o p u l a t i o n l e v e l s . However i f t h e h a r v e s t e f f o r t were i n c r e a s e d j u s t beyond the t h r e s h o l d l e v e l t h e n t h i s s t e a d y - s t a t e w o u l d be d i s t u r b e d and l o w e r p o p u l a t i o n l e v e l s would become i m m e d i a t e l y p o s s i b l e . I f t h i s i n c r e a s e d r a t e o f h a r v e s t i n g were t o c o n t i n u e e v e n t u a l l y t h e p o p u l a t i o n s i z e would drop b e l ow t h e c r i t i c a l l e v e l c , and f r o m t h e n on, r e g a r d l e s s o f whether h a r v e s t i n g were t o c o n t i n u e o r n o t t h e p o p u l a t i o n w o u l d m o n o t o n i c a l l y d e c r e a s e t o e x t i n c t i o n . Even f o r h a r v e s t - e f f o r t s o n l y s l i g h t l y i n e x c e s s o f t h e t h r e s h o l d s i z e , t h e c o l l a p s e o f t h e p o p u l a t i o n f r o m i t s s t e a d y s t a t e c o u l d be r e m a r k a b l y r a p i d , e s p e c i a l l y i f the e x p e c t e d r e p r o d u c t i o n f u n c t i o n i s s t e e p i n i t s m i d d l e p a r t . Indeed i n t h e d i a g r a m shown ( F i g 0.6), t h e c o l l a p s e from s t e a d y - s t a t e ( g r e a t e r t h a n r ) t o a l e v e l a t w h i c h t h e p o p u l a t i o n i s doomed (below c) c o u l d t a k e p l a c e i n o n l y two o r t h r e e y e a r s . I t m i g h t w e l l be t h e c a s e t h a t i t w o u l d n o t be r e a l i z e d t h a t  t h e c r i t i c a l l e v e l o f h a r v e s t i n g had been exceeded u n t i l i t were t o o  l a t e . T h i s c o u l d happen, e s p e c i a l l y , i f f o r one o r two 'good' y e a r s , t h e i n c r e a s e d r a t e o f h a r v e s t i n g were s u s t a i n e d . Any d e c r e a s e c o u l d be d i s m i s s e d as b e i n g t h e 'odd bad y e a r ' w i t h o u t i t b e i n g r e a l i z e d that the unfavourable environmental conditions would have a f a r l e s s damaging e f f e c t i f the rate of e x p l o i t a t i o n were lower. A s i m i l a r crash i s predicted by a deterministic depensation model (see Clark 1974). However as we have seen, i n that case, the threshold harvest rate i s greater than i t i s i n a f l u c t u a t i n g environment. Also i n the deterministic theory the collapse of the population i s monotonic, which i s not n e c e s s a r i l y the case i n the stochastic theory. The r e s u l t s above, for the greater part, do not depend on the m u l t i p l i c a t i v e form of the model. Rather they only depend on the Markov assumption and %Sv& tjhfei:minimumi awd£:maximum-- values of. reproduction for any given population ibevel. C r i t i c a l l e v e l s of e x p l o i t a t i o n and the range of steady-state population sizes seem to depend for a large part on minimum and maximum reproduction functions ( i . e . on min{X Ix =x} and max{X Ix =x}) rather than on the expected n+11 n n+11 n reproduction function. If standards of conservation need to be applied, they would be more r e l i a b l e i f based on the minimum and maximum reproduction functions rather than on a deterministic analysis of a deterministic reproduction function. Furthermore with reasonably large sets of data, estimation of maximum and minimum values of reproduction should be easier than estimation of a reproduction function. Harvest Y i e l d i n Steady-State Whereas section 2.2 i s concerned p r i m a r i l y with the e f f e c t of regular harvesting upon the population, section 2.3 discusses the 25 y i e l d i n steady-state from d i f f e r e n t harvest p o l i c i e s . What i s of sp e c i a l i n t e r e s t , i s how the steady-state y i e l d i n a f l u c t u a t i n g environment compares with that predicted by deterministic theory, and how such deterministic concepts as maximum sustainable y i e l d can be broadened to encompass the s i t u a t i o n of a f l u c t u a t i n g environment. In 2.3 we are concerned only with the actual s i z e of the harvest as measured i n the same units as the population i t s e l f , and not i n the y i e l d as an economic e n t i t y . (This we discuss i n Chapters 3 and 4). By the term y i e l d , then, we mean the number of units of population captured. For a deterministic model the y i e l d , i n equilibrium, from a given stationary harvest p o l i c y h i s h(x) , where x i s a stable s o l u t i o n to x = f°u(x) . This can be written as h(x) = x - u(x) = f(u(x)) - u(x) = g(u(x)) , where g(x) = f(x) - x . The deterministic maximum equilibrium y i e l d or, as i t has come to be known, the maximum sustainable y i e l d (m.s.y.) i s obtained when u(x): = x , where x i s the point at which g attai n s i t s maximum over [0, m] . Except i n the case when no y i e l d i s sustainable x w i l l be a point such that f'(x) = 1 . We see then that according to the deterministic theory the m.s.y. i s obtained by any p o l i c y allowing an equilibrium escapement of x . 26 In a stochastic model f or a f l u c t u a t i n g environment, the y i e l d , i n steady-state, w i l l of course vary from year to year. One way of comparing the steady-state y i e l d i n a f l u c t u a t i n g environment with the y i e l d i n a deterministic model i s to consider the expected value of the steady-state y i e l d i n the stochastic model. From the Strong Law of Large Numbers for Markov Process (Doob 1953 p. 220) we have that the expected steady-state y i e l d from any stationary p o l i c y i s the same as the long-run average y i e l d of that p o l i c y , and so we can i n t e r p r e t the expected steady-state y i e l d i n t h i s way. Ricker (1958(b)) and Larkin and Ricker (1964) used computer simulation experiments to compare with the m.s.y. the long-run average y i e l d i n a f l u c t u a t i n g environment of d i f f e r e n t stationary p o l i c i e s , which they considered might correspond to the deterministic p o l i c y giving m.s.y. In 2.3 we investigate t h i s comparison a n a l y t i c a l l y and show that, due to c e r t a i n errors a r i s i n g from a bias introduced into the simulation model, some of the conclusions of these authors need to be modified. The f i r s t main r e s u l t (Thm 2.3.1) i s that the expected  steady-state y i e l d , from any given stationary p o l i c y i n a f l u c t u a t i n g  environment, cannot exceed the maximum sustainable y i e l d , as computed  from a deterministic model. The m.s.y. then gives an upper bound to the long-run average y i e l d that can be derived from the population, at l e a s t using a stationary p o l i c y . The question of when t h i s upper bound i s attained i s an i n t e r e s t i n g one. We show that the expected steady-state y i e l d i s e q u a l t o t h e m.s.y. o n l y i f t h e l e v e l o f escapement x o f m.s.y. i s s u s t a i n a b l e i n t h e sense t h a t p r { X ,., > x l x = x} = 1 ( i . e . i f a f ( x ) > x) n+1 — n — and o n l y i f t h e s t a t i o n a r y p o l i c y i s e f f e c t i v e l y one o f s t a b i l i z a t i o n - o f - escapement a t t h e l e v e l x . i . e . i f o v e r t h e range o f t h e s t e a d y - s t a t e d i s t r i b u t i o n t h e h a r v e s t p o l i c y i s h ( x ) = \ Q_, X <^  X x-x , X > X T h i s means t h a t i n e v e r y y e a r t h e e x c e s s p o p u l a t i o n o v e r and above t h e l e v e l x i s h a r v e s t e d . T h i s r e s u l t c o n t r a d i c t s one of t h e c o n c l u s i o n s o f R i c k e r (1958(b)) and L a r k i n and R i c k e r (1964) whose s i m u l a t i o n s seemed t o i n d i c a t e t h a t a v e r a g e y i e l d s g r e a t e r t h a n t h e m.s.y. c o u l d be o b t a i n e d i n f l u c t u a t i n g e n v i r o n m e n t s , w i t h t h e s i z e o f the e x c e s s i n c r e a s i n g w i t h t h e l e v e l o f random v a r i a t i o n . I n 2.3 we i n d i c a t e t h e s o u r c e o f the e r r o r i n t h e s i m u l a t i o n s . I t i s o f t e n s t a t e d t h a t one o f t h e o b j e c t i v e s o f r e g u l a t i o n o f t h e e x p l o i t a t i o n o f a f i s h e r y i s the m a x i m i z a t i o n o f s u s t a i n a b l e y i e l d . As we have seen t h i s i s n o t a v i a b l e o b j e c t i v e i n a f l u c t u a t i n g e n v i r o n m e n t . G u l l a n d (1968 p. 4) i n a d i s c u s s i o n o f t h e c o n c e p t o f m.s.y. s a i d , " . . . . t h e term m.s.y. i s a po o r one f o r t h e y i e l d w h i c h w o u l d be t a k e n w i t h optimum e f f o r t , s i n c e i t may n o t be s u s t a i n a b l e 28 and w i l l be exceeded i n years when good year classes are present." However i t i s not d i f f i c u l t to broaden the objective of maximizing sustainable y i e l d to encompass a f l u c t u a t i n g environment. A natural extension would be the maximization of expected steady-state y i e l d , which, provided we r e s t r i c t our attention to stationary p o l i c i e s , i s the same as the maximization of long-run average y i e l d . There are objections however to such an objective. It may well be, for instance, that maximization of expected steady-state y i e l d i s brought about at the expense of a high degree of v a r i a b i l i t y i n y i e l d , (for instance f o r a stabilization-of-escapement p o l i c y there could w e l l be years when no harvest i s taken.) C l e a r l y a high degree of v a r i a b i l i t y i s undesireable both f o r the harvesting and processing industries and for the consumer. Indeed i t may w e l l be that men's l i v e l i h o o d s depend upon the annual harvest, and a complete shutdown i n harvesting i n a given year would be a d r a s t i c p r i c e to pay f o r an increased long-run average y i e l d . In formulating a v i a b l e objective f o r regulation, the question of v a r i a b i l i t y of y i e l d should perhaps be considered. However the r e l a t i v e values of average y i e l d and s t a b i l i t y are d i f f i c u l t to assess and are l i k e l y to be highly subjective. The formulation of an objective along these l i n e s i s e s s e n t i a l l y can" economic question. Here l i e s the main shortcoming of the m.s.y. objective. Maximum sustainable y i e l d (and i t s extension maximum long-run average y i e l d ) i s not an economic concept. This i s discussed further i n Chapter 3 (see also Clark (1973(a)) and Clark, Edwards and Friedlaender (1973)). 29 In Chapter 2 however we leave aside economic considerations and investigate the problem of f i n d i n g a stationary p o l i c y to maximize expected steady-state y i e l d . We have not solved the problem completely. From the r e s u l t of Thm 2.3.1 given above, we have that i f the l e v e l  of escapement x , which gives m.s.y. i n the deterministic model i s  sustainable, then a p o l i c y of stabilization-of-escapement at x  maximizes expected steady-state y i e l d i n a f l u c t u a t i n g environment. We see that i n t h i s case the maximal average-yield p o l i c y i n a f l u c t u a t i n g environment i s a d i r e c t extension of the m.s.y. p o l i c y i n the deterministic model. If the l e v e l x i s not sustainable the question of which p o l i c y maximizes expected steady-state y i e l d remains open. It seems possible that i t would be a p o l i c y of the stabilization-of-escapement type. In 3.3 we show that a stabilization-of-escapement p o l i c y maximizes expected discounted y i e l d . In the control theory of d i s c r e t e -time Markov processes, under c e r t a i n circumstances, a p o l i c y which maximizes expected discounted reward converges, as the discount rate approaches unity, to a p o l i c y which maximizes long-run average reward (see Ross (1968, 1970)). We are unable to e s t a b l i s h t h i s convergence for t h i s model. However i f i t were to hold we would have that a p o l i c y of s t a b i l i z a t i o n of escapement would maximize long-run average y i e l d , and hence expected steady-state y i e l d . The simulation studies of Ricker (1958(b)) indicated that of the three p o l i c i e s he considered (constant-effort, s t a b i l i z a t i o n - o f -escapement, p a r t i a l s t a b i l i z a t i o n of escapement) ;that of s t a b i l i z a t i o n -30 of-escapement gave the greatest average-yield (also the greatest v a r i a b i l i t y ) . This was confirmed by the more extensive simulations of Larkin and Ricker (1964). (We note that t h i s conclusion i s not i n v a l i d a t e d by the bias i n the model, since comparisons were made between d i f f e r e n t harvest p o l i c i e s f or the same population model)• In an unregulated f i s h e r y the i n t e n s i t y of f i s h i n g or f i s h i n g e f f o r t can be thought of as being proportional to the s i z e of the f i s h i n g f l e e t . A very simple and d i r e c t form of regulation of a f i s h e r y i s a l i m i t a t i o n on the s i z e of the f i s h i n g f l e e t . (This can be performed, f o r instance, by s e l l i n g a given number of f i s h i n g l i c e n c e s ) . With t h i s kind of regulation a question which n a t u r a l l y a r i s e s i s what i s the "optimum" s i z e of the f l e e t ? More generally we can ask, what i s the optimum f i s h i n g e f f o r t ? The answer to t h i s of course depends on the c r i t e r i o n of optimality. For the c r i t e r i o n of maximization of expected steady-state y i e l d used i n Chapter 2, the question mathematically can be expressed as what i s the best fixed rate of harvest to maximize the expected steady-state y i e l d over a l l constant-effort harvest p o l i c i e s ? In a deterministic model, since equilibrium y i e l d i s maximized by an escapement x , the constant e f f o r t harvest with proportional escapement x/f(x) maximizes equilibrium y i e l d . Ricker (1958, p. 1004) i n the summary of h i s paper makes the claim that "the best constant rate of e x p l o i t a t i o n when stocks f l u c t u a t e (over the range examined) i s the same, or very close to the best rate when there i s no environmental v a r i a t i o n . Ricker does not report any simulation 31 experiments i n which t h i s question was investigated and no arguments appear to be given i n support of t h i s claim. It i s not at a l l c l e a r that the rate of harvest which gives m.s.y. i n the deterministic model maximizes the expected steady-state y i e l d over a l l constant-effort harvest p o l i c i e s i n a f l u c t u a t i n g environment. In 2.3 we show that^for an expected reproduction function of b ' the form f(x) = ax , b < 1 , indeed the optimum rate i n a f l u c t u a t i n g environment i s the same as predicted by a deterministic model, but t h i s r e s u l t depends heavily on the s p e c i a l form of the function f . For more general expected reproduction functions the question as to which f i x e d rate of harvest gives maximum expected steady-state y i e l d , remains open. Another subject of inquiry i n 2.3 i s a comparison of .the expected steady-state y i e l d of a given stationary harvest p o l i c y , i n a f l u c t u a t i n g environment, with the equilibrium y i e l d of that p o l i c y as predicted i n a deterministic model. Following Galto & R i n a l d i (to appear) we c a l l the l a t t e r the nominal y i e l d of the p o l i c y . Such comparisons were c a r r i e d out by Ricker (1958(b)) and Larkin and Ricker (1964) but again t h e i r conclusions are i n v a l i d . We are not able to obtain, i n general, a comparison of the expected steady-state y i e l d of a p o l i c y with i t s nominal value, for a l l stationary p o l i c i e s . However i n 2.3 we have some r e s u l t s f o r some rather important s p e c i a l cases. We show that i n a compensatory model (concave expected  reproduction function), for any constant-effort harvest p o l i c y the 32 expected steady-state y i e l d (or long-run average y i e l d ) i n a  f l u c t u a t i n g environment i s s t r i c t l y l e s s than the nominal y i e l d of  the p o l i c y . (Thm. 2.3.2 Cor. 1) Thus a y i e l d estimate f o r a constant-effort harvest p o l i c y based on a deterministic compensation model w i l l be i n excess of the long-run average y i e l d r e a l i z e d i n a stochastic model f or a f l u c t u a t i n g environment. No such general r e s u l t holds f o r stabilization-of-escapement p o l i c i e s . We show (Thm 2.3.2 Cor. 2) that the expected steady-state y i e l d of a stabilization-of-escapement p o l i c y can be eit h e r l e s s than, equal to or greater than the nominal y i e l d of the p o l i c y . If the l e v e l  S at which escapement i s s t a b i l i z e d i s sustainable then the expected  steady-state y i e l d of the p o l i c y i s the same as the nominal y i e l d . I f S i s not sustainable and i s no greater than x , the escapement l e v e l that gives m.s.y. i n the deterministic theory, then  the expected steady-state y i e l d i s les s than the nominal y i e l d of the  p o l i c y . If S i s not sustainable and i s greater than x , then the  expected steady-state y i e l d can be greater than or l e s s than the nominal  y i e l d , which case depending on the stationary d i s t r i b u t i o n of the  population. An example i s given i n which the expected steady-state y i e l d exceeds the nominal y i e l d of the p o l i c y . Theorem 2.3.2 considers somewhat more general stationary harvest p o l i c i e s . P o l i c i e s for which the escapement function u(x) i s continuous non-increasing and concave and for which the deterministic escapement l e v e l i s no greater than x , are considered. (This could include p o l i c i e s such as modified-stabilization-of-escapement and in c r e a s i n g - e f f o r t harvest (Fig 2.8(b), ( e ) ) ) . For such harvest p o l i c i e s , under the assumption of a compensatory population model, the expected steady-state y i e l d i s no greater than the nominal y i e l d . Equality occurs only f o r p o l i c i e s which are e f f e c t i v e l y of the stabilization-of-escapement type, with the s t a b i l i z e d l e v e l of escapement being sustainable. From the r e s u l t s of 2.3 w,e see that estimates of long-run average y i e l d which are based on a deterministic population model are l i k e l y to be inadequate for a population l i v i n g i n a f l u c t u a t i n g environment. In many cases the deterministic y i e l d estimates w i l l be too high, i n some cases too low. S i m i l a r l y p o l i c i e s designed to maximize long-run average y i e l d and derived from a deterministic model are possibly sub-optimal for a population l i v i n g i n a f l u c t u a t i n g environment. An Economic Optimization Model In Chapter 3 we develop a dynamic optimization model f or the ex p l o i t a t i o n of a population l i v i n g i n a f l u c t u a t i n g environment. At the outset of any optimization problem one i s faced with the c r i t i c a l question as to what constitutes an optimum. Mathematically t h i s involves defining an objective (or c r i t e r i o n ) function and a set of admissible p o l i c i e s over which the objective function i s to be maximized. 34 Early work i n the f i e l d of animal resource management, ca r r i e d out l a r g e l y by f i s h e r i e s b i o l o g i s t s , sought to maximize the sustainable y i e l d of the population. A s i m i l a r objective for a population l i v i n g i n a f l u c t u a t i n g environment would be the maximization of expected steady-state y i e l d . Provided we were dealing only with stationary p o l i c i e s t h i s would be the same as maximizing long-run average y i e l d . However neither of these objectives includes economic considerations and as such both ignore many factors which are important i n the e x p l o i t a t i o n of animal resources. The importance of economic considerations i n the optimum regulation of f i s h e r i e s has long been r e a l i z e d . The paper of Gulland and Robinson (1973) surveys some of the more important economic aspects of f i s h e r i e s management. With much of the world's human population starving or severely undernourished i t could be argued that maximization of y i e l d as food, and not maximization of some economic objective should be the prime objective of mankind. However even assuming that maximization of global food production i s a reasonable objective, i t does not ne c e s s a r i l y follow that maximization of production from one resource w i l l n e c e s s a r i l y enhance mankind's progress towards t h i s goal. The resources - human, te c h n i c a l , c a p i t a l etc. - necessary to bring about an increase i n y i e l d from one p a r t i c u l a r resource could possibly be u t i l i z e d more e f f e c t i v e l y i n producing food i n another area. We can regard the economic revenue (or rent) derived from a given resource as a measure of the net production of commodity for mankind from that 35 p a r t i c u l a r r e s o u r c e . I n m a x i m i z i n g t h i s revenue we a r e f i n d i n g a most e f f i c i e n t u t i l i z a t i o n o f t h e r e s o u r c e , and th u s c o n t r i b u t i n g t o t h e g l o b a l m a x i m i z a t i o n o f p r o d u c t i o n o f commodity. Economic c o n c e p t s e q u i v a l e n t t o t h e d e t e r m i n i s t i c m a x i m i z a t i o n o f s u s t a i n a b l e y i e l d and t h e s t o c h a s t i c m a x i m i z a t i o n o f e x p e c t e d s t e a d y - s t a t e y i e l d w o u l d be m a x i m i z a t i o n o f s u s t a i n a b l e economic revenue and m a x i m i z a t i o n o f e x p e c t e d s t e a d y - s t a t e economic r e v e n u e , r e s p e c t i v e l y . These c o n c e p t s however a r e e q u i l i b r i u m c o n c e p t s . They t e l l us where we ought t o be, but n o t how t o get t h e r e f r o m where we a r e now. A l s o o b j e c t i v e s such as t h e s e i g n o r e an i m p o r t a n t a t t r i b u t e o f human b e h a v i o u r , namely t h e p r e f e r e n c e f o r consumption o f commodity now, r a t h e r t h a n a t some f u t u r e d a t e . T h i s phenomenon o f t i m e -p r e f e r e n c e can be i n c o r p o r a t e d i n t o an o p t i m i z a t i o n o b j e c t i v e by means o f a t i m e - p r e f e r e n c e ( o r d i s c o u n t ) f a c t o r , w h i c h d i s c o u n t s f u t u r e revenues t o a p r e s e n t v a l u e a t some base c a l e n d a r d a t e . I t c o u l d be argued t h a t a p o l i c y w h i c h i s optimum f o r mankind w i l l a l l o w f u t u r e g e n e r a t i o n s a s h a r e i n t h e r e s o u r c e a t l e a s t e q u a l t o t h a t o f t h e p r e s e n t g e n e r a t i o n . T h i s i s perhaps an e t h i c a l and p o l i t i c a l q u e s t i o n . A t p r e s e n t most government d e c i s i o n s i n v o l v e t i m e - p r e f e r e n c e however i m p l i c i t l y , and we o f f e r t h i s as a j u s t i f i c a t i o n f o r i t s i n c l u s i o n i n o u r model. C e r t a i n l y i f we a r e t o r e g a r d our o p t i m i z a t i o n model as b e i n g d e s c r i p t i v e , i n t h a t i t h e l p s i n t h e u n d e r s t a n d i n g o f t h e e x p l o i t a t i o n o f r e s o u r c e s , t h e n t i m e - d i s c o u n t i n g s h o u l d be i n c l u d e d , s i n c e i t p l a y s an i m p o r t a n t p a r t i n d e t e r m i n i n g the b e h a v i o u r o f r e s o u r c e 36 e x p l o i t e r s . I t s i n c l u s i o n i n a normative model i s open to debate. The relevance of time-discounting i n resource management problems has been discussed by S. V. C i r i a c y Wantrup (1952), A. Scott (1965), and others. For a deterministic model, an economic objective which includes time-discounting i s the maximization of the t o t a l discounted revenue that can be earned from the resource (see Clark 1973(b)). For any given i n i t i a l population l e v e l , and any given p o l i c y the t o t a l discounted revenue which w i l l be gained from the resource, can i n p r i n c i p l e be determined. Clark (1971, 1973(b)) has c a l l e d t h i s the present-value of the resource for a given i n i t i a l population l e v e l and a given p o l i c y . He found p o l i c i e s to maximize t h i s present-value for a l l i n i t i a l population l e v e l s . An analagous objective i n a stochastic model for a population l i v i n g i n a f l u c t u a t i n g environment would be the maximization of the expected t o t a l discounted revenue. Again for any given i n i t i a l population l e v e l , and any given p o l i c y , i t i s reasonable to speak of the expected t o t a l discounted revenue as the present-value of the resource f o r that p o l i c y and that i n i t i a l population l e v e l . The aim then of an optimization procedure would be to f i n d a p o l i c y which maximizes the present-value for a l l i n i t i a l population l e v e l s . This i s the objective we puruse i n Chapter 3. A dynamic optimization objective which does not include time-discounting, would be the maximization of the long-run expected average revenue earned from the resource ( i . e . the l i m i t as n -*•"•<*> of 37 the expected average revenue earned over n periods). If we were to consider only stationary p o l i c i e s then t h i s would be the same as the maximization of long-run average economic revenue which we have already seen i s the same as the maximization of expected steady-state revenue. However i n Markov control models of t h i s sort, maximization of expected average revenue i s not always brought about by a stationary p o l i c y (see Ross (1970) for some counter-examples). In Chapter 3 we seek to maximize the expected t o t a l discounted revenue (or present-value) that can be earned from the resource. We assume that:-(a) every unit of population harvested has a fi x e d s e l l i n g p r i c e , p . We can think of p as being the value to mankind of one unit of population captured. (b) there i s a fixed m o b i l i z a t i o n cost K , incurred every time a harvest i s undertaken (but only then). Such a lump cost i s assumed to cover such expenses as the transportation of men and equipment to the harvest grounds, the purchase of non re-usable equipment, the h i r i n g of men etc. (c) a marginal harvest cost c(x) which i s the unit cost of harvesting when the population i s at the l e v e l x . Roughly c(x) w i l l depend on the time required to capture one unit of population when the population i s at l e v e l x . It i s reasonable to assume that t h i s time would increase (or at le a s t n'o't decrease) as x decreased. We assume that c(x) i s non-increasing i n x . We l e t x^ be the zer o - p r o f i t l e v e l , i . e . we l e t x n be such that c ( x n ) = p . If 38 c(x) < p f o r a l l x we set x^ = 0 . Harvesting below x^ y i e l d s no p r o f i t . (d) there i s a discount (or time-preference) f a c t o r , a < 1 , assumed constant which discounts revenues at some future time to a present-value at some base calendar date. Thus a revenue r earned i n n time-period n i s assumed to have a present-value a I l r n a t t n e beginning of time-period 1 . The time discount factor a i s related to a time discount rate r by 1 a - 1+r In the case of a p r i v a t e l y owned resource i n a competitive environment, r can be thought of as the i n t e r e s t rate on money, or (more pr e c i s e l y ) the opportunity cost of c a p i t a l . Under these assumptions the net revenue earned i n period n , from a harvest of size h^ > 0 , given that the population l e v e l p r i o r to harvesting was X i s n p h n -which can be rewritten as •X n c ( t ) d t - UK , X -h n n R(X ) - R(X -h ) - i K 5(h ) , where n n n n R(x) = px -x c ( t ) d t 0 and 6(x) = \ 0 , x = 0 1. , x > 0 This formulation also includes the case when no harvest i s undertaken (h =0) and no p r o f i t i s earned. Discounted to i t s value at the n beginning of year 1 , the revenue i n year n from a harvest h^ i s a n{R(X ) - R(X -h ) - B6(h )} . n n n n For a harvest sequence h = {h^} the expected discounted revenue that w i l l be earned from the resource i s 00 E{ I a n[R(X ) - R(X -h ) - K 6(h )]} . . n n n n n=l This i s the objective function we seek to maximize. We seek a p o l i c y which w i l l maximize t h i s f or a l l i n i t i a l population l e v e l s ( i . e . f o r a l l values of ) . In order to completet'the d e f i n i t i o n of the maximization problem, we need to define what constitutes an admissible p o l i c y . We suppose that a harvest can be undertaken every year during a short harvest season p r i o r to the breeding season. If the population l e v e l immediately p r i o r to harvesting i s X then we require that the n harvest s i z e h s a t i s f y n 0 < h < X , n = 1, 2, — n — n This i s the only r e s t r i c t i o n we s h a l l place on a harvest p o l i c y . Any 40 p o l i c y which s a t i s f i e s t h i s condition w i l l c onstitute an admissible p o l i c y . We seek to maximize the expected t o t a l discounted revenue that can be earned from the resource over a l l admissible p o l i c i e s . We are assuming then that the sequence h = {h } constitutes the c o n t r o l v a r i a b l e of our optimization problem, and thus that the resource manager can exactly c o n t r o l the s i z e of the harvest i n any year. We also assume that i n making his decision as to what s i z e harvest should occur i n any given year, the resource manager has complete knowledge of the current population l e v e l . This l a t t e r assumption i s c l e a r l y an o v e r - s i m p l i f i c a t i o n for most harvested populations l i v i n g i n the wild. Usually, at best, information concerning animal abundance w i l l be subject to considerable s t a t i s t i c a l e rror. However without t h i s assumption of perfect knowledge, the mathematical co n t r o l problem becomes one of control with "noisy" data and mathematically t h i s greatly complicates the problem. Bearing i n mind the fac t that both of the above assumptions concerning the harvest are hardly l i k e l y to be met completely i n any real-world s i t u a t i o n , the optimal p o l i c i e s derived from our analysis should perhaps be regarded as id e a l s - the best that could be done under i d e a l circumstances. Of course i n developing the optimization model we have had to make many other s i m p l i f y i n g assumptions about the economic environment, about the population dynamics etc. The normative value of the r e s u l t s of the optimization should be assessed i n the l i g h t of these underlying assumptions. Apart from the i n c l u s i o n of a mob i l i z a t i o n cost, the economic 41 optimization model developed here p a r a l l e l s that of Clark (1973(b)) developed f o r a deterministic population model. The s p e c i a l case when the m o b i l i z a t i o n cost K = 0 i s a d i r e c t stochastic g e n e r a l i z a t i o n of the model of Clark. The deterministic optimization model i s a s p e c i a l case of the stochastic optimization model. Although we have considered some economic factors i n our model others have been ignored. They include v a r i a t i o n i n the demand and s e l l i n g p r i c e with the s i z e of the harvest, and the costs associated with excessive v a r i a b i l i t y i n the y i e l d . In seeking to maximize the expected t o t a l discounted revenue that can be earned from the resource we are seeking a p o l i c y which i s economically optimal for the management of the resource as a whole. Thus we are considering the case where the resource i s e f f e c t i v e l y c o n t r o l l e d by one operator, or si n g l e management agency. For an uncontrolled common property resource a d i f f e r e n t s i t u a t i o n a r i s e s - one i n which competitive e x p l o i t a t i o n tends to r e s u l t i n an equilibrium population at the l e v e l , x^ , of z e r o - p r o f i t (see Gordon (1954), Hardin (1968)). Clark (1973(b)) has shown how t h i s can be considered as the maximization of present-value with a zero discount f a c t o r ( i n f i n i t e discount r a t e ) . The purpose of the optimization analysis of Chapter 3 i s twofold. F i r s t l y the r e s u l t s can be regarded as normative i n that they o f f e r suggestions as to what i s the optimum u t i l i z a t i o n of the resource. Here i t i s of p a r t i c u l a r i n t e r e s t to see how the optimal p o l i c i e s f o r a population l i v i n g i n a f l u c t u a t i n g environment compare with those for 42 a population whose dynamics are deterministic. Secondly to some extent they can be regarded as d e s c r i p t i v e . I f i t i s assumed that a given resource i s e f f e c t i v e l y c o n t r o l l e d by one or a small group of operators and that they are acting optimally for the given objective then the optimization analysis can help i n understanding the process of e x p l o i t a t i o n and help i n understanding the way i n which economic and b i o l o g i c a l factors i n t e r a c t i n the determination of an optimal p o l i c y . In t h i s way we can hope to gain i n s i g h t into the causes of c e r t a i n bio-economic phenomena, such as the e x t i n c t i o n of animal species. In 3.2 and 3.3 the optimization problem i s formulated mathematically i n terms of Dynamic Programming, and a connection i s established between t h i s problem and a c e r t a i n problem i n stochastic inventory theory. Under c e r t a i n conditions on the expected reproduction  function f and the marginal harvest cost function c , i t i s shown  that there i s an optimal p o l i c y , which, to use the language of  inventory theory, i s of the (S,s) type, i . e . there are two numbers S and s with S < s , such that a harvest i s made i n any year i f and only i f the population l e v e l i n that year exceeds s , i n which case a harvest down to S i s made (Thm. 3.3.1). In the case of a zero set-up cost an optimal p o l i c y i s a  s t a b i l i z a t i o n of escapement p o l i c y (Thm. 3.3.2). Similar r e s u l t s have been obtained by Jaquette (1972, 1974) under rather d i f f e r e n t sets of assumptions, and a d i f f e r e n t population model. 43 The reason why an (S,s) p o l i c y a r i s e s i s f a i r l y e a s i l y understood i n t u i t i v e l y . We would l i k e to s t a b i l i z e the escapement at some l e v e l E , say. However because of the presence of the m o b i l i z a t i o n cost, some harvests down to E , notably the small ones w i l l y i e l d a negative economic return as well as depleting the population. Only large harvests are worth undertaking and an optimal p o l i c y i s to wait u n t i l the population reaches a higher l e v e l s , and then reduce i t to a l e v e l S lower than (or possibly equal to) E . These r e s u l t s (Thms. 3.3.1, 3.3.2) characterize the optimal p o l i c i e s q u a l i t a t i v e l y . We have not i n general been able to determine the l e v e l s S, s e x p l i c i t l y , nor the optimal l e v e l at which escapement i s s t a b i l i z e d when there i s no m o b i l i z a t i o n cost, (though see 4.2 for some s p e c i a l cases). Numerical techniques have been developed for the determination of optimal (S,s) p o l i c i e s i n inventory theory (e.g. Veinott and Wagner (1965)), and i t should be possible to adapt these computing techniques to the model here. These r e s u l t s hold for values of the discount f a c t o r a , 0 <_ a < 1 . In the l i m i t i n g case a = 1 , the expected t o t a l discounted revenue w i l l not n e c e s s a r i l y be f i n i t e . Thus the analysis has no meaning i n t h i s case. However Ross (1968, 1970) has shown that under c e r t a i n circumstances, as a -* 1 a p o l i c y optimal f o r maximizing expected discounted revenue, converges to a p o l i c y optimal for maximizing long-run expected average revenue. We have not been able to formally e s t a b l i s h t h i s convergence here, but i t i s worth keeping i n mind that an (S,s) p o l i c y may well maximize long-run 44 expected average economic revenue when there i s a m o b i l i z a t i o n cost, and a stabilization-of-escapement p o l i c y do likewise when there i s no m o b i l i z a t i o n cost. As we' have already seen maximization of long-run expected average economic revenue would be a very reasonable objective i f time-discounting were to be excluded. The r e s u l t of the optimality of a stabilization-of-escapement p o l i c y for the case when there i s no m o b i l i z a t i o n cost i s analagous to a r e s u l t of Clark (1971, 1973(b)) based on a s i m i l a r optimization model for a deterministic population. Clark shows that t o t a l discounted revenue i s maximized by s t a b i l i z i n g the population at a given l e v e l . When the dynamics of the population .are deterministic such a complete s t a b i l i z a t i o n i s possible. The same s i z e of harvest r e s u l t s every year. The main assumption concerning the population model, i n the proof of Theorems 3.3.1, 3.3.2 i s that the expected reproduction function be concave, and further i n the case of a p o s i t i v e m o b i l i z a t i o n cost that i t be increasing. Thus we are r e s t r i c t e d to compensation models for population growth. The proofs of Theorems 3.3.1, 3.3.2 assume a constant marginal harvest cost c(x) = c , but some extensions to t h i s are considered i n 3.4. In p a r t i c u l a r i t i s shown that an (S,s) p o l i c y and a stabilization-of-escapement p o l i c y are optimal i n the cases with and without a mo b i l i z a t i o n cost r e s p e c t i v e l y , when the marginal harvest cost i s of the form c(x) = c/x , provided that the z e r o - p r o f i t l e v e l x n i s sustainable and that the reproduction function s a t i s f i e s a 45 weak a d d i t i o n a l hypothesis. Other authors have considered t h i s a rather an important s p e c i a l case since i t approximates the s i t u a t i o n i n which the marginal cost i s proportional to the time taken to harvest a unit of population at a given l e v e l of population (see Clark (1973(b)). These conditions on the population model and the marginal harvest cost are s u f f i c i e n t for the optimality of (S,s) or s t a b i l i z a t i o n -of-escapement p o l i c i e s . It i s not at a l l clear whether they are necessary. The question as to what form an optimal p o l i c y might take i f an (S,s) p o l i c y ( i n the case of a p o s i t i v e m o b i l i z a t i o n cost) were not optimal i s discussed i n 3.4. Mathematically the problem of " c o n t r o l l i n g " a population with the objective of maximizing some output from the population, i s s i m i l a r to c e r t a i n problems i n inventory theory as has been pointed out by Jaquette (1972). In stochastic inventory models the outputs are usually considered as random variables and the inputs are con t r o l l e d with the objective of minimizing c e r t a i n losses. In the economic optimization problem f o r a population l i v i n g i n a f l u c t u a t i n g environment the inputs (growth) to the population are stochastic and the outputs are con t r o l l e d with the objective of maximizing c e r t a i n economic returns. The s i m i l a r i t y between the economic optimization model of Chapter 3, and a stochastic inventory model of Scarf (1960) has been exploited i n the proof of the theorems of Chapter 3. Another way of looking at the problem of optimal economic c o n t r o l of a population i s to regard i t as a problem i n c a p i t a l theory. This has been pointed out by Clark and Munro (1975). The population can be regarded as c a p i t a l stock, capable of growth, and the harvests can be regarded as consumption of c a p i t a l . The usual objective considered i n c a p i t a l theory i s the maximization of t o t a l net discounted present value. There i s a considerable l i t e r a t u r e on deterministic models for optimal c a p i t a l accumulation, and the relevance of t h i s work to animal resource problems has been discussed by Clark and Munro (1975) and Clark (to appear). Much less a ttention has been given to the optimal accumulation problem i n an uncertain environment. In the rel a t e d theory of economic growth, Levhari and Srinivasan (1969) and Brock and Mirman (1972) have discussed the existence of optimal p o l i c i e s and the convergence of the c a p i t a l stock under optimal consumption to a stationary p r o b a b i l i t y d i s t r i b u t i o n . Some r e s u l t s concerning the optimal economic p o l i c y i n a f l u c t u a t i n g  environment when there i s no m o b i l i z a t i o n cost In the case of a zero m o b i l i z a t i o n cost the economic optimization model of Chapter 3 d i r e c t l y extends the optimization model of Clark (1971, 1973(b)) to the case of a population l i v i n g i n a f l u c t u a t i n g environment. I t i s an i n t e r e s t i n g question as to how the optimal p o l i c i e s of the two models compare. This i s investigated i n 4.2. Of p a r t i c u l a r i n t e r e s t i s the question as to when a p o l i c y optimal i n the deterministic model i s optimal for a population l i v i n g 47 i n a f l u c t u a t i n g environment. Clark (1973(b)) has shown that an optimal p o l i c y i n the deterministic model i s the "open-loop" p o l i c y i n which the population escapement l e v e l i s s t a b i l i z e d at some l e v e l x , say. In t h i s model the optimal harvest i n every year w i l l be the constant f ( x ) - x . The most obvious i n t e r p r e t a t i o n of t h i s p o l i c y to the s i t u a t i o n of a population l i v i n g i n a f l u c t u a t i n g environment, would be to regard i t as a "feedback" p o l i c y of stabilization-of-escapement at the l e v e l x , i n which case the excess above x ( i f such exists) i s harvested each year. Of course for t h i s type of p o l i c y the s i z e of the harvest would vary from year to year, and there might well be years i n which no harvest were undertaken. As we have already seen a feedback p o l i c y of stabilization-of-escapement i s indeed optimal for the stochastic model f o r a population l i v i n g i n a f l u c t u a t i n g environment. However the optimal l e v e l , S , at which escapement i s s t a b i l i z e d may w e l l be d i f f e r e n t from the l e v e l x optimal i n the deterministic model. In t h i s case i f the optimal deterministic p o l i c y were employed for a population l i v i n g i n a f l u c t u a t i n g environment the e x p l o i t a t i o n would be sub-optimal. An important question then i s how does the optimal stochastic l e v e l S compare with the optimal deterministic l e v e l x ? 48 Clark (1973(b)) has shown that the optimal deterministic l e v e l of escapement x i s the point at which k(x) = oQ(f(x)) - Q(x) attain s i t s maximum on [x^, m] , where Q(x) = R(x) - R ( x Q ) • We r e c a l l that R(x) represents the revenue obtained by immediately harvesting the population to e x t i n c t i o n when the current l e v e l i s x . Since x^ i s the l e v e l of z e r o - p r o f i t i t follows that Q(x) i s the maximum immediate revenue that can be obtained from the resource. The function k(x) represents the annual increment i n the discounted value of immediate value of the resource, when the current population l e v e l i s x . where <S i s the d.f of the random m u l t i p l i e r s Z . The function £(x) represents the expected annual increment i n the discounted value of the immediate value of the resource when the current population l e v e l i s x . We denote by a the point at which &(x) attai n s i t s maximum on [ XQ> m l • In 4.2 we show that i f the l e v e l a i s sustainable i n the sense that pr{X ,.. > a I X = a] = 1 , then a i s the optimal l e v e l n+1 — ' n at which escapement i s s t a b i l i z e d i n the stochastic model f or a population l i v i n g i n a f l u c t u a t i n g environment. I f a i s not In 4.2 for the stochastic model we consider £(x) = a Q(tf(x)) d*(t) - Q(x) n 49 sustainable then we can show that the optimal l e v e l S i s greater or equal to a . In the important s p e c i a l cases for which c(x) = constant or * c(x) = c/x we can show that a = x . Thus we have that i f the  marginal harvest cost c(x) i s constant or of the form c/x , then the optimal l e v e l of s t a b i l i z e d escapement i n the stochastic model fo r a f l u c t u a t i n g environment i s greater than or equal to the optimal  l e v e l of s t a b i l i z e d escapement i n the equivalent deterministic model. Furthermore, i n t h i s case the deterministic optimal l e v e l  of escapement i s optimal for a population l i v i n g i n a f l u c t u a t i n g  environment provided that t h i s l e v e l i s sustainable. (Thm. 4.2.2 Cors. 1, 2) For other forms of the marginal cost function c(x) , we show that the optimal stochastic l e v e l of escapement S can be e i t h e r * greater than or le s s than the optimal deterministic l e v e l x which case depending on the concavity or convexity of the function xc(x) (Thm. 4.2.2). From these r e s u l t s we see that optimal economic p o l i c i e s derived from a deterministic population model may well be sub-optimal for a population l i v i n g i n a f l u c t u a t i n g environment. Pulse Harvesting as an Optimal Economic P o l i c y In some of the f i s h e r i e s of the world, e s p e c i a l l y those which are exploited by large modern f i s h i n g f l e e t s capable of t r a v e l l i n g long distances, a phenomenon, which has been termed "pulse-harvesting" 50 has been observed. In t h i s type of harvesting, rather than taking a f a i r l y constant sized catch every year, the f l e e t w i l l heavily f i s h the resource i n one year, and then wait a number of years, during which time the population can regenerate, before returning to repeat the operation. Usually i n such a strategy more than one f i s h e r y i s involved. The f l e e t w i l l t r a v e l i n successive years to each of a number of widely scattered f i s h e r i e s , heavily e x p l o i t i n g each i n turn. However i n 4.3 we are able to show that such a p o l i c y of pulse-harvesting can be economically optimal even for the e x p l o i t a t i o n of a s i n g l e population, provided that the m o b i l i z a t i o n cost i s s u i t a b l y high. We have shown i n 3.3 that an optimal economic p o l i c y when there i s a m o b i l i z a t i o n cost i s a p o l i c y of the (S,s) type. Such a p o l i c y may involve years i n which no harvest i s undertaken (when the population l e v e l i s below s ). Although i n general the time to regenerate from the l e v e l S to the l e v e l s w i l l be a random va r i a b l e , a p o l i c y of pulse harvesting could well be a r r i v e d at, based on, say, the mean of t h i s regeneration time. Roughly we would expect the distance between S and s and hence the expected regeneration time to increase with the m o b i l i z a t i o n cost. In 4.3 we consider a deterministic model which o f f e r s a p l a u s i b l e explanation for pulse harvesting. We note that the (S,s) p o l i c y of 3.3 i s optimal for a deterministic population reproduction function model with a m o b i l i z a t i o n cost, since t h i s i s j u s t a s p e c i a l case of the model of 3.2 i n which the d i s t r i b u t i o n of the random 51 m u l t i p l i e r s Z has a l l i t s mass concentrated at the point x = 1 . n I f i t takes more than one year for the (deterministic) population to regenerate from the l e v e l S to the l e v e l s ( i . e . i f f(S) < s) then the optimal p o l i c y w i l l be a pulse harvest, with the. period of the pulse being equal to the time i t takes for the population to regenerate from S to s . I n t u i t i v e l y we would expect that the period of the optimal pulse harvest would increase as the m o b i l i z a t i o n cost increased, since the greater the m o b i l i z a t i o n cost, the l e s s often one would want to incur i t . The main r e s u l t (Thm. 4.3.1) of 4.3 establishes t h i s and in d i c a t e s the way i n which the s i z e of the m o b i l i z a t i o n cost determines the period of the optimal pulse harvest. We show that there i s an increasing sequence of numbers {k^} with kg = 0 such that the period of an optimal harvest i s of s i z e n i f the m o b i l i z a t i o n cost K l i e s i n the i n t e r v a l [k ,, k ] . n-1 n I f K = k^ then both an n-period and an n+l-period harvest are optimal. The numbers k^ can be computed e x p l i c i t l y . I n t u i t i v e l y also, we would expect that the l e v e l down to which the population should be harvested, i n an optimal p o l i c y , would decrease as the s i z e of the period of the optimal pulse harvest increased with the m o b i l i z a t i o n cost. In other words, the l e s s frequently the resource i s harvested, the heavier we would expect the e x p l o i t a t i o n to be i n those years i n which a harvest occurs. We are able to show i n 4.3 that t h i s i s the case when the marginal harvest cost i s constant. 52 We see then that when there i s a large m o b i l i z a t i o n cost, a p o l i c y of pulse-harvesting i s compatible with the objective of maximizing discounted p r o f i t , even f or a sing l e resource i n i s o l a t i o n . For a large f i s h i n g f l e e t that has to t r a v e l a long distance to the f i s h i n g grounds, the mobili z a t i o n cost would indeed be high. Of course t h i s model for a single resource ignores many economic f a c t o r s , such as the costs associated with keeping the harvest equipment and men i d l e during the years i n which no harvest i s undertaken. However i f i t i s possible to mobilize the same operation to harvest a number of d i f f e r e n t widely scattered resources, t h i s consideration w i l l be of les s importance, since i n t h i s case i t w i l l be possible to pulse harvest each of the resources successively. For a large modern f i s h i n g f l e e t capable of t r a v e l l i n g large distances to exp l o i t a number of widely scattered resources, a p o l i c y of pulse harvesting w i l l be compatible with the objective of maximizing discounted p r o f i t , since the cost of mobili z a t i o n of the harvest e f f o r t w i l l be an important part of the t o t a l costs of the whole harvesting process. P r o f i t Maximization and the E x t i n c t i o n of Animal Species In the past the world has seen the e x t i n c t i o n or near e x t i n c t i o n of a number of animal species e.g. passenger pigeon, b u f f a l o , blue whale. This apparently senseless destruction i s usually a t t r i b u t e d to the common-property nature of these resources and to the accompanying phenomenon which economists c a l l the " d i s s i p a t i o n of economic rent" -53 (see Gordon (1954), Hardin (1968)). Clark (1973(a), (b)) however has shown, using a deterministic population model, that even i f the resource i s owned or c o n t r o l l e d by a si n g l e agency i t may s t i l l be economically optimal to exterminate the species. This argument i s based on the phenomenon of time-discounting, and Clark shows how, with the adoption of a s u f f i c i e n t l y high time-preference rate, i t can be economically optimal for the resource e x p l o i t e r to harvest the population down to a l e v e l of e x t i n c t i o n . He suggests that t h i s i s one of the main reasons behind the near extermination of the blue whale, and the continuing over-e x p l o i t a t i o n of other whale stocks, and suggests that absolute standards of conservation based on b i o l o g i c a l rather than economic factors are necessary to ensure the s u r v i v a l of some animal species. Clark's analysis was based on a deterministic population model, and on an economic optimization model which did not include a m o b i l i z a t i o n cost. It i s of i n t e r e s t to see i f the conclusions of Clark are v a l i d f o r a model of a population l i v i n g i n a f l u c t u a t i n g environment and to see i f the presence of a m o b i l i z a t i o n cost e f f e c t s these conclusions. We pursue these questions i n 4.4./" In 4.4 we consider a population being exploited optimally with regard to the objective of maximizing expected discounted economic revenue (or p r o f i t ) , as defined i n 3.2. We assume that conditions s u f f i c i e n t f o r the optimality of an (S,s) p o l i c y hold and look for economic conditions which imply that either e x t i n c t i o n i s optimal (S=0) or conservation i s optimal (S>0) . 54 As i n the deterministic theory, (Clark (1973(b))), harvesting below the z e r o - p r o f i t l e v e l , x^ , not only y i e l d s a negative unit return but also reduces ( i n p r o b a b i l i t y ) the l e v e l of the population i n the following year, and thus cannot be the action of an optimal p o l i c y . Thus regardless of other economic and b i o l o g i c a l factors we have that extermination of the population i n any harvest  can only be optimal i f i t i s p r o f i t a b l e to harvest the l a s t surviving  animals. This r e s u l t holds even for an i n f i n i t e time-preference rate, which Clark (1973 (b)) has shown corresponds to the case of the e x p l o i t a t i o n of a common-property resource. For marginal cost functions such as c(x) = c/x or c(x) = c/x , the zero p r o f i t l e v e l i s p o s i t i v e since the cost of harvesting becomes unboundedly high f o r low enough population l e v e l s . In t h i s case a p o l i c y of e x t i n c t i o n could never be optimal. Assuming that x^ = 0 , what other conditions are necessary for the optimality of an e x t i n c t i o n policy? In 4.4 we show that a large value of the time-preference rate coupled with a high m o b i l i z a t i o n cost can lead to the optimality of e x t i n c t i o n . These r e s u l t s are contained i n Thms. 4.4.2 and 4.4.4 and are conveniently summarized i n the following diagram. The diagram i s for the s i t u a t i o n i n which i t i s p r o f i t a b l e to harvest the l a s t surviving animals (x Q=0) . The e f f e c t i v e time-preference rate , r , and the m o b i l i z a t i o n cost, K ., can be represented j o i n t l y , by a point (r,K) i n the plane. Under given A I No e x p l o i t a t i o n i s p r o f i t a b l e i n th i s region Fi g . 0.7 assumptions on the marginal cost function c(x) (see 3.3) the nature of an optimal p o l i c y for maximizing expected discounted p r o f i t i s determined by which of the three regions as indicated i n the diagram, the point (r,K) f a l l s i n . We can think of the curve f as representing c r i t i c a l values for the time-preference rate f o r a given m o b i l i z a t i o n cost (r=r(K)) or v i c e versa (K=K(r)) . We have not been able to express these equations f o r V i n simple form. At the end points, R(m) represents the maximum p r o f i t that can be earned from the resource i n any sing l e harvest, and f'(0) - 1 represents the maximum annual growth rate i n the population (or i t s immediate harvest value). A l t e r n a t i v e l y we can think of f'(0) - 1 as representing the maximum of the marginal expected annual growth i n the population as the population increases by unit amount (or i n the immediate harvest value of the population as that value increases by unit amount.) (see 4.4) I I f the mob i l i z a t i o n cost exceeds the maximum revenue that can be earned from the population i n any one year (K>R(m)) then c l e a r l y i t i s never p r o f i t a b l e to harvest. II If i t i s p r o f i t a b l e to harvest the l a s t surviving animals and i f  the time-preference rate i s too high for the given m o b i l i z a t i o n cost  (r>r(K)) , or a l t e r n a t i v e l y i f the mobil i z a t i o n cost i s too high f o r the  given time-preference rate (K>K(r)) , then maximization of expected  discounted p r o f i t i s brought about by harvesting the population to  e x t i n c t i o n i n the f i r s t year of harvesting. The exact c a l c u l a t i o n of the c r i t i c a l l e v e l s K(r) , r(K) i s rather complicated. However we have shown i n Thm. 4.4.3, under c e r t a i n conditions on the marginal cost function c(x) (see 3.3), that regardless of the set-up cost, e x t i n c t i o n w i l l be optimal i f r > f'(0) - 1 (see F'i'g. 0.7). In t h i s case we could say that the resource i s sub-marginal i n expectation since the marginal expected annual growth i n the immediate value of the population i s le s s than the marginal annual growth i n wealth invested at the i n t e r e s t rate r . In t h i s case wealth invested at the rate r w i l l grow f a s t e r than the immediate value of the population. We have then that i f i t i s p r o f i t a b l e to harvest the l a s t surviving animals and i f the resource  i s sub-marginal i n expectation ( i . e . r > f'(0) - 1) then regardless  of the m o b i l i z a t i o n cost, maximization of expected p r o f i t i s brought  about by harvesting the population to e x t i n c t i o n i n the f i r s t year of  harvesting. This r e s u l t holds equally f or the deterministic model which i s a s p e c i a l case of the stochastic model. 57 II I If the time-preference rate and the m o b i l i z a t i o n cost are  s u f f i c i e n t l y low (f<r(K) or K<K(f)) then maximization of expected  discounted p r o f i t i s brought about by a p o l i c y of conservation-stabi- l i z a t i o n of escapement at a p o s i t i v e l e v e l . What exactly constitutes s u f f i c i e n t l y low l e v e l s f o r r and K ? As we have already mentioned the exact c a l c u l a t i o n of the c r i t i c a l l e v e l s i s complicated. However i n Thm. 4.4.3 and i n the remarks following Thm. 4.4.4 we have determined "safety" l e v e l s for the time-preference rate and m o b i l i z a t i o n cost, represented by the curve A i n F i g . 0.7. r rb I f K < K = sup xe[0,m] R(tf(x))d$(t) - (l+r)R(x) then conservation i s optimal, and s i m i l a r l y i f rb R ( t f ( x ) ) d * ( t ) - K r < r = sup \ [0,m] R(x) - 1 then conservation i s optimal. The f i r s t of these r e s u l t s indicates a possible way that e x t i n c t i o n could be avoided. If the m o b i l i z a t i o n cost were to be reduced to a l e v e l below , then a p o l i c y of e x t i n c t i o n would be economically sub-optimal. However such a reduction i n m o b i l i z a t i o n cost would be of no a v a i l i f the resource were sub-marginal i n expectation ( i n t h i s ease i s negative). In the s p e c i a l case of a zero m o b i l i z a t i o n cost the r e s u l t s II and III reduce to give necessary and s u f f i c i e n t conditions f o r the 58 optimality of a p o l i c y of conservation (or e x t i n c t i o n ) . I f i t i s  p r o f i t a b l e to harvest the l a s t surviving animals and i f there i s no  m o b i l i z a t i o n cost involved i n harvesting, then maximization of  expected p r o f i t i s brought about by harvesting the population to  e x t i n c t i o n ( i n the f i r s t year of harvesting) i f the resource i s  sub-marginal i n expectation ( f < f ' ( 0 ) - l ) , and by a p o l i c y of  conservation (stabilization-of-escapement at a p o s i t i v e l e v e l ) i f the  resource i s not sub-marginal i n expectation (r>f'(0)-l) . Again t h i s r e s u l t holds equally for the deterministic population model. How do these r e s u l t s compare with those of Clark (1973(b)), derived from a deterministic population model and what i s t h e i r s i g n i f i c a n c e ? Clark shows that with the adoption of excessive time-preference rates, a p o l i c y of maximization of discounted p r o f i t w i l l lead to the e x t i n c t i o n of the population, provided that i t i s p r o f i t a b l e to harvest the l a s t surviving animals. He suggests that t h i s phenomenon of discounting future revenues excessively may i n large part be responsible for the near-extinction of the blue whale and for the serious over e x p l o i t a t i o n of other whale stocks. His analysis does not include a m o b i l i z a t i o n cost. Broadly our r e s u l t s are i n agreement with these conclusions. If we assume that the objective of maximising expected discounted p r o f i t s reasonably r e f l e c t s the aims of the resource e x p l o i t e r s , then our r e s u l t s i n d i c a t e that the adoption of too high a time-preference rate w i l l lead to the e x t i n c t i o n of a population, l i v i n g i n a f l u c t u a t i n g environment (as described by our model). When there i s no m o b i l i z a t i o n cost the c r i t i c a l value of the time-preference rate i s the same regardless of whether a de t e r m i n i s t i c or stochastic population model i s used. The presence of a m o b i l i z a t i o n cost does however e f f e c t the c r i t i c a l value of the time-preference r a t e , the l a t t e r decreasing as the former increases. I n t u i t i v e l y we would expect that a combination of a high m o b i l i z a t i o n cost and a high time-preference rate would lead to a p o l i c y of e x t i n c t i o n . A resource e x p l o i t e r would wish to incur a high m o b i l i z a t i o n cost as seldom as p o s s i b l e , and i n the extreme case where future returns mean very l i t t l e to him, he would choose to incur i t only once, harvesting the population to e x t i n c t i o n i n that one year. We stress that t h i s argument does not apply to a high i n i t i a l cost, but to m o b i l i z a t i o n costs, which have to be paid each time a harvest i s undertaken. The presence of high m o b i l i z a t i o n costs may be a feature of the whaling industry, which, coupled with high time-preference rates adopted by whalers, i s leading to the e x t i n c t i o n of various whale species. Indeed the whaling industry has used the argument of high m o b i l i z a t i o n costs to j u s t i f y t h e i r demands f o r high quotas. Using data from the International Commission on Whaling (1964), Clark (1973(a)) estimated f'(0) f o r the blue whale population to be about 1.1. For the blue whale then, (provided that the marginal harvest cost function were to s a t i s f y the conditions of 3.3), even i f there were no mob i l i z a t i o n costs involved, the adoption of a time-preference rate higher than 10%, would lead the whalers to maximize t h e i r p r o f i t by hunting the blue whale population to e x t i n c t i o n . With a mobil i z a t i o n cost, time-preference rates even l e s s than 10% could lead to e x t i n c t i o n . The r e s u l t s of 4.4 are based upon a compensatory (concave-expected reproduction function) model. In such a model, with a stabilization-of-escapement p o l i c y , the only way e x t i n c t i o n can come about i s as a planned p o l i c y (see 2.2), and i n 4.4 we have looked for economic conditions which w i l l make such a p o l i c y optimal. On the other hand f o r a population whose dynamics are not of t h i s form (notably i f they s a t i s f y a depensation model) e x t i n c t i o n could e a s i l y come about u n i n t e n t i o n a l l y as the r e s u l t of too heavy an e x p l o i t a t i o n (see 2.2). Age-Structure Models The model discussed i n Chapters 2, 3 and 4 assumes that the population can be described by a s i n g l e continuous v a r i a b l e . In so doing the model ignores such aspects of the population as the age-and s e x - d i s t r i b u t i o n s , which play a part i n determining the dynamics of the population. Provided that the age- and s e x - d i s t r i b u t i o n s remain f a i r l y constant, t h i s may not be too serious an omission. However i t may be possible to u t i l i z e the age-structure or the sex-structure of the population to obtain a greater y i e l d or economic rent from i t . This 61 i s discussed i n Chapter 5. In 5.2 we formulate an age-structure model, based on the L e s l i e (1945) model. At f i r s t we formulate a deterministic model, and then i n 5.3 we formulate a stochastic model f o r a population l i v i n g i n a f l u c t u a t i n g environment. We discuss a number of co n t r o l models based on t h i s population model, but obtain optimal solutions only i n a very l i m i t e d s p e c i a l case. The model of L e s l i e (1945) assumes that the population can be divided into k age-classes, (for instance:- animals le s s than one year old, one year olds, two year olds, k-1 year o l d s ) . The vector x^ represents the d i s t r i b u t i o n of the population by age cla s s i n year n . I t i s assumed that the dynamics of the population s a t i s f y the equation = L x ~n where L i s the so - c a l l e d L e s l i e matrix. L = where the c o e f f i c i e n t s I. represent the proportions of survivors 62 from one age-class to the next, and the c o e f f i c i e n t s f represent the f e c u n d i t i e s , or average numbers of young born per member of each age c l a s s i n any year. The main drawback of t h i s model i s that i t i s l i n e a r , and as such does not allow for any i n t e r n a l regulation of the population s i z e . L e s l i e shows that the population grows exponentially (or decays), with a rate depending on the dominant c h a r a c t e r i s t i c root X of the matrix L . He also shows that the d i s t r i b u t i o n (by age) of the popu-l a t i o n tends to a l i m i t i n g form determined by the r i g h t c h a r a c t e r i s t i c vector k of L corresponding to the dominant root X (see 5.1). Both L e s l i e (1948) and Emlen (1973) have considered extensions of t h i s model where the growth rate i s dependent upon the current population s i z e ( t o t a l number of animals) i n a way s i m i l a r to that i n the one dimensional l o g i s t i c model. In 5.2 we develop a s l i g h t l y more general density-dependent growth model. We assume that = 1 T ? n + l 1<+ 3' -x ~n ~n where 3' i s a row-vector of p o s i t i v e elements r e f l e c t i n g the demands of i n d i v i d u a l s of each age-class upon the a v a i l a b l e resources. In t h i s model the proportions of animals which survive from one age-class to the next depends on the current population l e v e l and are represented by l^/l + § ' * x n (i =l»--',k) . In some ways i t i s a multi-dimensional version of'the Beverton-Holt one-dimensional reproduction function model (see 1.1) More generally we consider the model ~n+± ~n ~n where g(x) i s some function representing the e f f e c t on s u r v i v a l proportions, of a population of given s i z e x . For the former model we show that undisturbed the population tends to an equilibrium at v _ (A-l)k i f A > 1 (where A and k are the dominant c h a r a c t e r i s t i c root and r. vector of L ) and to e x t i n c t i o n i f A <_ 1 . (Thm 5.2.1) For the more general model we show that i f an equilibrium f o r the population e x i s t s , i t w i l l have the d i s t r i b u t i o n k , and w i l l be at q:k where q, i s a s o l u t i o n to A g(q;k) = 1 On the basis of t h i s population model we formulate a number of optimization problems for maximizing sustainable y i e l d and sustainable economic rent ( p r o f i t ) . F i r s t l y we consider the case when each age-class can be harvested independently. We formulate the optimization problem as a non-linear program but do not solve i t . 64 Secondly we consider the case where the resource e x p l o i t e r controls only one v a r i a b l e (the harvest e f f o r t say) which i s assumed to have d i f f e r e n t e f f e c t s on each age-class. This includes as a s p e c i a l case the s i t u a t i o n i n a f i s h e r y of a mesh-size l i m i t a t i o n , where a l l f i s h under a c e r t a i n age (size) can escape harvesting. Again we formulate the optimization problem as a non-linear program but do not solve i t . The only case that we are able to solve a n a l y t i c a l l y i s that case when the harvest e f f o r t acts uniformly on each age c l a s s . We assume that f o r a given harvest e f f o r t chosen by the c o n t r o l l e r , the same proportion of animals from each age-class w i l l be captured. This proportion depends on the harvest e f f o r t . In th i s case we show how the problem of maximizing sustainable y i e l d or sustainable economic rent reduces to a one-dimensional problem. In 5.3 we look at a stochastic version of the population model developed i n 5.2. We consider the entries i n the L e s l i e matrix as random v a r i a b l e s . In th i s case we have X = g(X ) Z X , ~n+l ° ~n n ~n where Z_ i s a matrix of random variables 65 with E(Z ) = L . We assume that {Z } i s an i . i . d . sequence of n n random matrices with Z independent of X , but we do not require n ~n the random va r i a b l e s that constitute the entries i n a given Z^ to be independent. Thus our model allows random f l u c t u a t i o n s i n the environment to e f f e c t the various age-classes i n a correlated way, but assumes that the random environmental f l u c t u a t i o n s are uncorrelated i n time (Markov model). In the harvesting model we consider only the case where the resource manager controls only a single v a r i a b l e (harvest e f f o r t ) which captures the same proportion of animals from each age-class. In t h i s case under a concavity assumption on the function g , we are able to show by Dynamic Programming that an optimal p o l i c y to maximize expected discounted y i e l d i s of the stabilization-of-escapement type, where the l e v e l at which escapement i s s t a b i l i z e d depends on the a g e - d i s t r i b u t i o n of the population (Thm 5.3.1). This i s i l l u s t r a t e d i n F i g . 0.8, for a population with two age-classes. x 1 , no. i n f i r s t age-class Fig 0.8 . 6 6 The proof i s given for the case of maximizing expected discounted y i e l d i n biomass (animals of d i f f e r e n t age-classes are i n general assumed to be of d i f f e r e n t sizes) and does not appear to early generalize to the case of maximizing expected discounted economic rent ( p r o f i t ) . The r e s u l t s of Chapter 5 concerning age-structured models appear to be of very l i t t l e p r a c t i c a l value. Two-Sex Models Besides ignoring the age-structure of a harvested population, the model of Chapters 2, 3 and 4 ignores the sex-structure of the population also. C l e a r l y t h i s o v e r - s i m p l i f i e s the dynamics of any r e a l population. However from the d e s c r i p t i v e point of view t h i s s i m p l i f i c a t i o n may not be too serious, i f the males and females of the species are harvested i n proportions approximately the same as t h e i r proportions i n the population as a whole. On the other hand i f the harvesting technology i s such that i t exerts a greater pressure of m o r t ality on one sex or the other, (e.g. as with the harvest of sperm whales) then a model which does not discriminate sexes may be d e s c r i p t i v e l y inadequate. When we look at an optimization model from the normative >point of view, we are faced with the question as to whether the optimal y i e l d or economic revenue can be increased by u t i l i z i n g the sex-structure of the population. However before considering the implementation of a harvesting p o l i c y which would upset the r a t i o of the sexes one should 67 be aware of possible dangers of such a p o l i c y , both to the population i t s e l f and to the environment. This i s b r i e f l y discussed i n 5.4. In that section also we look at a hypothetical model for s e x - s p e c i f i c harvesting of the various species of P a c i f i c Salmon based on the experimental observations of Mathisen (1962). In h i s experiments with Alaska Red Salmon, Mathisen found that there was a " b i o l o g i c a l surplus of males" with regard to the a c t i v i t y of breeding, and that one male could adequately service up to f i f t e e n females. He suggests that t h i s surplus could be harvested without impairing the reproductive capacity of the population, but says, " I t remains for future research to devise ways and means by which such a surplus can be harvested." In 5.4 we develop a model where we assume that t h i s can be achieved. The model i s s p e c i f i c a l l y for a si n g l e race of f i s h , with spawning beds of l i m i t e d geographical extent, and spawning season of l i m i t e d temporal extent. I t i s assumed that t h i s race can be i d e n t i f i e d p r i o r to i t s entry into freshwater, and that i t i s possible to harvest the males and females independently. The model assumes that one male i s capable of s e r v i c i n g a given (deterministic) number, S , of females, and w i l l do so provided that there are u n f e r t i l i z e d females a v a i l a b l e . Thus from a breeding population of X^ females and X^ males, the number of females la y i n g f e r t i l e eggs w i l l be min(X^, S X^) . I t i s assumed that the average numbers of female eggs and male eggs l a i d f e r t i l e by a f e r t i l i z e d female are f ^ and f ^ . In general f ^ and f ^ can be random var i a b l e s but for s i m p l i c i t y i n 5.4 we assume that they are deterministic 68 q u a n t i t i e s . I t i s assumed that some of these eggs hatch, survive a sojourn i n fresh water, migrate to the ocean, survive a sojourn i n the ocean and eventually return to the coastal waters ready f o r migration to t h e i r spawning grounds. I t i s assumed that the length of t h i s l i f e - c y c l e i s the same for a l l f i s h (e.g. 4 years), and that over t h i s time period the proportion of survivors of each sex, i s a random va r i a b l e depending on the s i z e of the population. Thus from B = f e r t i l i z e d female and male eggs, the numbers of female and male survivors who return for breeding i s assumed to be q(B) Z B , 2 where q i s a function R -> [0, 1] , and Z i s a diagonal matrix with random variables Z^ , Z^ d i s t r i b u t e d on the i n t e r v a l [0, 1] i n the diagonal p o s i t i o n s . The random v a r i a b l e s Z^ , Z^ represent the proportions of female and male survivors under d i f f e r e n t f l u c t u a t i n g environmental conditions, when there i s no pressure of competition,.and the 'f.un'cto-ion sqrvrepresentsiitheaeffecltfao-fs competition., on these s u r v i v a l c o e f f i c i e n t s . iWeaas'sume ithatitqcr.satisfies a c e r t a i n concavity condition. At t h i s stage, p r i o r to t h e i r entry into fresh water the adult f i s h can be harvested. We assume that the harvester has knowledge of the abundance of each sex, and can harvest each sex separately. Of those f i s h which survive the harvest a random proportion 6 of males and females survive the journey up the r i v e r s to t h e i r spawning grounds. If X = ~n Y n l 0 x n | denotes the population, by sex, subject to 69 harvest i n a given season, then i n the absence of harvesting the population x n +-^ o n e l i f e - c y c l e hence w i l l be Xn+1 = °n m ± n { X 0 ' S X l } q ( 6 n m l n { X 0 ' S Z J where f = i s the vector of fe c u n d i t i e s . We use Dynamic Programming to f i n d a p o l i c y which w i l l maximize the expected discounted y i e l d from t h i s model. We show that an optimal p o l i c y s t a b i l i z e s (where possible) the number of f e r t i l i z e d females, and harvests any excess of e i t h e r sex, not contributing to t h i s number. Graphically t h i s can be represented as i n F i g . 0.9. number" t h a r v e s t o n l y m a l e s * - , down to l i n e of males In t h i s region harvest both males and females to P . utt ****************** B In t h i s region harvest only females down to l i n e . F i g 0.9 X Q - number of females Of course t h i s model makes many assumptions concerning the 70 harvest operation and the information a v a i l a b l e to the harvester etc. which are c l e a r l y , at present, u n r e a l i s t i c . At best t h i s model gives . a t h e o r e t i c a l idea of the best p o l i c y under i d e a l circumstances. The r e s u l t v e r i f i e s the statement of Mathisen that greater y i e l d s can be obtained by s t a b i l i z i n g the number of f e r t i l e females, than by s t a b i l i z i n g the number of f i s h who escape the harvest. Again i n t h i s model we have considered only the problem of maximizing discounted y i e l d (males and females are assumed i n general to have d i f f e r e n t s i z e s ) , and not the problem of maximizing economic return. The introduction of a cost function dependent on the current population s i z e complicates the problem. If i t i s assumed that the harvester has complete knowledge of the population s i z e (by sex), but cannot implement a sex s p e c i f i c harvest, but rather can only c o n t r o l a harvest e f f o r t which w i l l remove the same proportion from each sex, then a p o l i c y , optimal f o r maximizing expected discounted y i e l d , i s a stabilization-of-escapement p o l i c y , i n which the s i z e of the escapement depends on the sex-d i s t r i b u t i o n of the population (see F i g . 0.8). Stochastic Models f or B i r t h and Survival Processes In Chapter 1 we discuss the reproduction function f o r deterministic discrete-time population models, and look at some stochastic models f or the underlying processes. Beverton and Holt (1957) derived a r e l a t i o n s h i p of the form 71 E = rfbP • (1> r e l a t i n g the number of new r e c r u i t s , R , to a f i s h e r y , from a parent stock of s i z e P . They assumed that the number of eggs l a i d , E , was proportional to the s i z e of the parent stock, and that the time from when the eggs were l a i d u n t i l the time of recruitment, could be divided into a number of l i f e - s t a g e s . In each l i f e - s t a g e they supposed that the s u r v i v a l of young-fish was p a r t i a l l y density-dependent, and s a t i s f i e d a d i f f e r e n t i a l equation of the form I f ~ < ^ 2 N > t £ [ V T r + 1 ] ( 2 ) where N(T^) represents the number of f i s h a l i v e at the s t a r t of the r1"*1 l i f e - s t a g e , and N(T^ +^) the number a l i v e at the end of that r r l i f e - s t a g e , and and % represent the c o e f f i c i e n t s of density-independent and density-dependent mortality, over that l i f e - s t a g e . Solving these equations they show that the number of r e c r u i t s i s r e l a t e d to the egg-production by a function of the form (1) above, and hence i s also related to the parent population by a function of s i m i l a r form. Such a function has been c a l l e d the Beverton-Holt stock recruitment r e l a t i o n s h i p . I t i s concave and non-increasing. Such a stock-recruitment r e l a t i o n s h i p has been termed compensatory. Another feature of the Beverton-Holt model i s that as P -* 0 0 , R approaches an asymptote at a/b . Thus there i s an upper l i m i t to the t o t a l 72 recruitment, regardless of the s i z e of the parent stock. If some of the parent stock are assumed to survive from one year to the next, and over that year the change i n the number of survivors i s assumed to s a t i s f y a d i f f e r e n t i a l equation of the form (2), then the t o t a l number of adults a l i v e i n one year (new r e c r u i t s plus surviving adults) i s re l a t e d to the parent stock the year before by a reproduction function, which i s not, i n general of the form (1), but i s however compensatory and asymptotic. Clark (1974 and to appear) considered a gene r a l i z a t i o n of the s u r v i v a l process (2) for any given l i f e - s t a g e . He assumed that over any l i f e - s t a g e the s i z e of the population was changing i n accordance with the d i f f e r e n t i a l equation dN (3) ^ = - <KN) , where § i s a p o s i t i v e convex function. Under t h i s assumption he showed that the number of survivors of any l i f e - s t a g e i s r e l a t e d to the i n i t i a l number for that l i f e - s t a g e by a concave, increasing function. Successive composition of such functions shows that the number of new r e c r u i t s i s related to the egg-production, and hence to the parent stock by a concave, increasing function. I f the parent population i s assumed to undergo a s i m i l a r s u r v i v a l process, then there w i l l be a reproduction function r e l a t i n g population sizes i n . successive years, of the compensatory type. In 1.3 we consider a stochastic version'of the generalized Beverton-Holt s u r v i v a l process (3). It i s based on the well-known 73 stochastic process known as a Birth-Death Process. Rather than assume, as the d i f f e r e n t i a l equation model does, that the s i z e of the population i s a continuous v a r i a b l e , we assume that i t i s a d i s c r e t e random v a r i a b l e , with i t s value N(t) at time t assuming an integer value. We assume that the p r o b a b i l i t y of a death i n any small time-interval during the l i f e - s t a g e i s density dependent, and that more than one death cannot occur simultaneously. P r e c i s e l y we suppose that pr{one death i n ( t , t+A] I N(t) = n} = X A + o(A) n and pr{more than one death i n ( t , t+A] | N(t) = n} = o(A) , where -*-s a sequence of non-negative c o e f f i c i e n t s representing the expected number of m o r t a l i t i e s per unit time f o r various population l e v e l s . We can see how t h i s model compares with the deterministic generalized B - H model by considering E{N(t+A) I N(t) = n } = n - A A + o(A) . n For the deterministic generalized B - H model we have {N(t+A) | N(t) = n} = n - <j>(n)A + o(A) . We see then that the sequence corresponds to the function <j>(n) i n the deterministic model. Both give time-rates for m o r t a l i t y for various population l e v e l s . We show i n Theorem 1.2.1, that i f the sequence LA } i s convex, then the expected number of survivors of a given l i f e - s t a g e i s r e l a t e d to the i n i t i a l population s i z e by a concave increasing function. Thus the s u r v i v a l process i s , i n expectation, compensatory. This generalizes the r e s u l t of Clark (1974 and to appear). Further we show that i f the sequence ^ s concave for small values of n , and then convex, the expected s u r v i v a l function exhibits depensation (Cor. 1). This again generalizes a r e s u l t which holds for the deterministic model. We can compare the stochastic and deterministic models by considering the sequence {A } as the values of the function cj> n assumes at the points n = 0, 1, 2,... . In t h i s way, we show (Cor. 2) that for a convex <j> , compensation i s more severe i n expectation i n the stochastic model than i n the deterministic model, and that f o r a function <j> which i s i n i t i a l l y concave, depensation i s l e s s severe i n expectation i n the stochastic model than i n the deterministic model. We also consider i n 1.3 the e f f e c t of stochastic b i r t h s . We assume that the number of b i r t h s per parent i s a Poisson random va r i a b l e with mean F , i . e . -F_r e F pr{r b i r t h s from a given parent} = ^ — , r = 0, 1, 2, .... If the s u r v i v a l process from b i r t h to recruitment i s compensatory i n expectation (as i s the stochastic model above, when {A^} i s convex) then the expected number of r e c r u i t s produced by 75 P parents i s a concave increasing function of P . The proof of t h i s r e s u l t depends heavily on the assumption of a Poisson d i s t r i b u t i o n for the number of b i r t h s . It i s not at a l l cle a r that i t would hold for more general d i s t r i b u t i o n s . The f a c t that c e r t a i n forms of the reproduction function, usually derived from deterministic models f or b i r t h and s u r v i v a l , can also be derived as expected reproduction functions from stochastic models, under s i m i l a r sets of assumptions, lends v a l i d i t y to t h e i r use, when these underlying assumptions are believed to hold. Chapter 1 Discrete-Time Population Models 1.1 The Reproduction Function In much of the l i t e r a t u r e on d e t e r m i n i s t i c , discrete-time population models, the sizes of the population i n successive years (or seasons) are assumed to be r e l a t e d by a difference equation, Xn+1 = f ( x n } * The function f i s known as a reproduction function. In f i s h e r i e s l i t e r a t u r e the stock-recruitment r e l a t i o n s h i p , or recruitment  function i s more often r e f e r r e d to. This r e l a t e s the recruitment (the amount of f i s h vulnerable to harvesting) i n one year with the escapement i n the previous year (the amount of f i s h which escapes the harvest.) The s i z e v a r i a b l e x , which i s usually allowed to assume r e a l values, i s often taken to represent the t o t a l biomass of the population, or again, to represent the t o t a l number of animals i n the populat ion. This l a t t e r measure i s approximately continuous f o r large population s i z e s . In adopting a model i n which successive values of the s i z e v a r i a b l e are f u n c t i o n a l l y r e l a t e d , several i m p l i c i t s i m p l i f i c a t i o n s concerning the population are assumed. F i r s t l y the age and sex d i s t r i b u t i o n s within the population are assumed to be of no importance, i n determining reproduction, growth and m o r t a l i t y . Secondly v a r i a b l e environmental factors such as the weather, the a v a i l a b i l i t y of food, the incidence of predators and competitors etc are ignored. 77 For a population i n a r e l a t i v e l y stable, undisturbed state, the age and sex d i s t r i b u t i o n of the population may we l l remain f a i r l y constant, and thus appear to play l i t t l e part i n determining successive population s i z e s . However for a population which i s subjected to a systematic harvest pressure, t h i s may not be the case. The technology of the harvest may bring about d i f f e r e n t pressures of mo r t a l i t y on d i f f e r e n t age or sex groups. For instance a harvest which sel e c t s f o r s i z e may cause an increased pressure on adult males, say. Variable environmental factors may or may not play an important part i n determining the reproductive behaviour of the population, depending on the population and i t s environment. That there i s great annual v a r i a t i o n i n population s i z e for many species i s evident from data records (see f o r instance Cushing and Har r i s , 1973). One l i k e l y source of t h i s v a r i a t i o n i s v a r i a b i l i t y i n the environment. Grossly over-simple as t h i s deterministic difference equation model may appear, i t has been the basis of a considerable l i t e r a t u r e concerning harvest p o l i c i e s i n f i s h e r i e s and other areas. Much work has been done i n t r y i n g to f i n d a t h e o r e t i c a l basis f o r the reproduction function of various species, e s p e c i a l l y f i s h (see Symposium on Fis h Stocks and Recruitment (1973)). Mathematical models have been formulated f o r the various stages of the l i f e - h i s t o r y of a species, and attempts have been made at fi n d i n g parametric forms f o r the reproduction function. When one considers the number and degree of s i m p l i f y i n g assumptions usually necessary to a r r i v e at a trac t a b l e mathematical model i t i s hardly s u r p r i s i n g that these attempts have 78 met with, at best, a l i m i t e d success. S i m i l a r l y attempts to estimate reproduction functions from empirical data have often not been successful, even when a s p e c i f i c parametric form of the function has been assumed. For instance Beverton and Holt (1957) could f i n d no d i s c e r n i b l e r e l a t i o n s h i p between parent stock and recruitment f o r the North Sea p l a i c e f i s h e r y . In deriving t h e o r e t i c a l reproduction functions i t i s usually assumed that they r e s u l t from two, not ne c e s s a r i l y independent, processes. The f i r s t i s the b i r t h , growth and mortality of young animals, and the second i s the growth and mo r t a l i t y of parent stock. We represent t h i s i n F i g . 1.1. Fi g . 1.1 We s h a l l r e f e r to, as a s u r v i v a l r e l a t i o n s h i p , the r e l a t i o n s h i p between the number of survivors of a given process, and the number of animals present at the s t a r t of the process. In the above, there i s a s u r v i v a l r e l a t i o n s h i p f o r both parents and young for the time period a f t e r the b i r t h . 79 We s h a l l now ou t l i n e some of the simpler forms of the reproduction function found i n the l i t e r a t u r e . C l a s s i f i c a t i o n and terminology seem to vary with author. We follow here the scheme adopted by Clark (to appear). (a) Normal Compensation. f ( x ) , »2 x, popn. i n year n Fi g . 1.2 This type of reproduction function i s characterized by a concave non-decreasing curve. Successive population l e v e l s monotonically converge, regardless of the i n i t i a l population to the stable equilibrium l e v e l x e The most common form of pure-compensation model found i n the f i s h e r i e s l i t e r a t u r e i s that known as the Beverton-Holt r e l a t i o n s h i p . This has form 80 (1) f(x) = a X 1 + bx ' where a and b are constants. This form was derived by Beverton and Holt (1957,' p. 48ff) as a s u r v i v a l r e l a t i o n s h i p r e l a t i n g t o t a l egg production with the number of new r e c r u i t s to the population. They assume that the time period between egg l a y i n g and recruitment may be divided i n t o a number of separate i n t e r v a l s ( l i f e - s t a g e s of the young f i s h ) and that during each of these i n t e r v a l s the mor t a l i t y rate consists of a density dependent and a density independent part. More e x p l i c i t l y they•assume that during the r l i f e - s t a g e of the young f i s h , the number of survivors N(t) s a t i s f i e s the d i f f e r e n t i a l equation /o\ 1 dN . r , r.T. (2) K dT = " ( y l + M 2 N ) ' r— r where y^ and are constants. Solving t h i s , i t can be shown that the number of survivors th at the s t a r t of the (r+1) l i f e - s t a g e i s re l a t e d to the number of ttl survivors at the s t a r t of the r l i f e - s t a g e by the r e l a t i o n a N r r N " r+1 1 + b N r r where a , b are constants, r r Combining these r e l a t i o n s for a l l l i f e - s t a g e s , i t can be shown that the number of new r e c r u i t s R , i s rela t e d to the egg production E by the r e l a t i o n s h i p 5 81 More generally Beverton and Holt observed that t h i s r e l a t i o n holds even r r when the mortality rates y^ and vary i n time. I f we assume that each adult f i s h lays p eggs, and then dies, we have from (3) a reproduction function t W 1 + bp x ' which i s of the form (2). Only i n s p e c i a l cases do adults die (or leave the fishery) a f t e r spawning (e.g. P a c i f i c Salmon). I f we allow some s u r v i v a l process for the adult f i s h a f t e r spawning, even i f i t i s of the same form as (2), then i n general the reproduction function i s not of the Beverton Holt type. However i f the two s u r v i v a l processes are independent and both of the type (2), then the reproduction function which r e s u l t s w i l l be the sum of two functions of type (1) with d i f f e r e n t c o e f f i c i e n t s , and as such w i l l be a concave, non-decreasing function — i . e . a purely compensatory reproduction function. It i s often assumed, for convenience, that the reproduction function has the Beverton-Holt form (2), even though adults may survive from one year to the next. It i s worth noting that a population which s a t i s f i e s the l o g i s t i c growth law, we l l known i n ecology, i f observed at discre t e time points, s a t i s f i e s the Beverton Holt r e l a t i o n s h i p . The l o g i s t i c law assumes a continuous growth i n population, with population s i z e 82 x ( t ) at time t s a t i s f y i n g — = rx(K-x) . If we denote x(n) by x^ , then i n t e g r a t i n g the above over [n, n+1] gives Kr e x n Xn+1 Kr , 1 + ' X K n which i s of the Beverton-Holt form (1) A feature of the Beverton-Holt model i s that f(x) -*• a/b as x -* » . We see that there i s an upper l i m i t to the number of animals returning i n any given year. For large values of a , f approaches the asymptote very r a p i d l y , (see F i g . 1.3). This could describe a population which possesses a strong s e l f - r e g u l a t i o n mechanism. x F i g . 1.3 83 With a reproduction function l i k e t h i s , recruitment would appear independent of parent stock si z e over a wide range of values for the l a t t e r quantity. Clark (to appear and 1974) considers a s u r v i v a l r e l a t i o n s h i p more general than the form (2) of Beverton and Holt. If f o r each l i f e - s t a g e , the number of survivors N(t) s a t i s f i e s a d i f f e r e n t i a l equation (4) f = - * r ( N ) , where <j> i s a convex function of N , then the r e l a t i o n s h i p between the number of survivors at the end of a l l the l i f e - s t a g e s considered, and the i n i t i a l number of animals i s a concave increasing function i . e . the s u r v i v a l r e l a t i o n s h i p i s purely compensatory. We s h a l l consider a stochastic generalization of t h i s i n the next section. Another convenient form of a pure-compensation reproduction function often assumed i s f(x) = ax° , where a and b are constants, b < 1 . Cushing (1971) has f i t t e d such a curve to stock recruitment data for a number of species of f i s h . A feature of t h i s model i s that f'(0) = «° . Thus i f we assume t h i s model, we nec e s s a r i l y assume that the population i s capable of an a r b i t r a r i l y high growth rate. 84 (b) Depensation x F i g . 1.4 A depensation curve i s characterized by a convex part for small x , followed by a concave part for larger x . A population for which the per parent increase i n recruitment increases for small l e v e l s of parent population and then decreases for larger l e v e l s , would have a depensation reproduction curve. This might be the case i f the population were subject to predation, and could f i n d greater safety i n numbers up to a c e r t a i n point. In fact i f any s u r v i v a l function were les s e f f i c i e n t l y performed at low population l e v e l s , a depensation curve might r e s u l t . Another example might be where the animals have d i f f i c u l t y i n l o c a t i n g a mate at low population l e v e l s . (c) C r i t i c a l Depensation This i s a s p e c i a l case of the depensation model, where the The upper point of i n t e r s e c t i o n x i s a stable equilibrium point whereas the lower point of i n t e r s e c t i o n x^ i s an unstable equilibrium point. The l e v e l x^ i s a c r i t i c a l l e v e l . I f the V population drops below x i t w i l l decrease monotonically to e x t i n c t i o n . e Otherwise i t converges monotonically to the equilibrium l e v e l x g . (d) Overcompensation This i s characterized by a dome-shaped curve. F i g . 1.6 x e 86 This model assumes that the per parent change i n recruitment a c t u a l l y decreases for large population l e v e l s . I f i n an "overcrowded" environment, competition for a v a i l a b l e resource, including breeding t e r r i t o r y , resulted i n a lessened reproductive c a p a b i l i t y , such a curve might r e s u l t . It might be an appropriate model for populations which "crash" when the l e v e l becomes too high. If f ' (x ) > - 1 the population w i l l approach the e equilibrium x e monotonically, or by way of damped o s c i l l a t i o n s , whereas i f f ' ( x e ) < - 1 equilibrium at a sing l e point w i l l not occur but rather o s c i l l a t i o n s around x w i l l r e s u l t (Clark (to appear) and e Oster & May (1975)). A common form of the overcompensation model i s due to Ricker (1954, 1958(a)) and i s given by -bx f(x) = axe , a, b constants. Many other t h e o r e t i c a l reproduction functions have been derived. Various stages of the l i f e - h i s t o r y can be modelled by d i f f e r e n t s u r v i v a l r e l a t i o n s h i p s of any of the above type. A composition of these functions would give the o v e r a l l s u r v i v a l r e l a t i o n s h i p . Independent s u r v i v a l of young and parents can be modelled by considering the sum of the separate s u r v i v a l r e l a t i o n s h i p s . • There i s no l i m i t to the complexity of such models. In the following section we look at some stochastic models for various stages of the l i f e - h i s t o r y . 87 1.2 Stochastic Models f or Survival and B i r t h Processes In the l a s t section we outlined the Beverton-Holt d i f f e r e n t i a l equation model for s u r v i v a l over a given l i f e - s t a g e . Most l i k e l y i n r e a l i t y , the factors e f f e c t i n g such a s u r v i v a l process, w i l l vary from year to year. The s u r v i v a l process would perhaps be better modelled by a stochastic process. This we do i n t h i s section and also look at the e f f e c t s of a stochastic b i r t h - r a t e . We s h a l l be concerned p r i m a r i l y with a sing l e stage of the l i f e - h i s t o r y . We s h a l l suppose that E i s the number of animals a l i v e at the beginning of t h i s period, and that R i s the number of survivors at the end of the period. We s h a l l denote the time period of the l i f e - s t a g e by the i n t e r v a l [0, T] . (a) A Simple Binomial Model Consider the Beverton-Holt model i n which m o r t a l i t y i s s t r i c t l y density dependent i . e . = 0 i n equation (2). In t h i s case the s u r v i v a l r e l a t i o n s h i p i s R = i - T b i • S ( E ) s a ? -The proportion of survivors i s R = 1 E 1 + bE ' Let us suppose that t h i s model holds i n expectation, and that the p r o b a b i l i t y of s u r v i v a l of an i n d i v i d u a l i s l/(l+bE) , and that 88 the s u r v i v a l of i n d i v i d u a l s are s t o c h a s t i c a l l y independent events. We then have that R i s a random va r i a b l e depending on the parameter E . The d i s t r i b u t i o n of R i s binomial. Its mean i s E ( R ) = T T b E " S ( E ) and i t s variance V< R> = E T T b E • < 1 - I T ^ ) = ^ 7 = M S ( E ) ] 2 For large values of E we can approximate t h i s binomial d i s t r i b u t i o n by a normal d i s t r i b u t i o n with mean S(E) and variance 2 b[S(E)] . In t h i s case we have R = Z S(E) , where Z i s a normal v a r i a t e with mean 1 and variance b . In t h i s case then, where the p r o b a b i l i t y of s u r v i v a l decreases as 1/(1+bE) , with the i n i t i a l density, E , the number of survivors i s a random multiple of the expected survival! function R. S< E) ti ^ ^^^ / . 1 • •• ' F i g . 1.7 89 In F i g . 1.7, R can l i e anywhere between 0 and E , with an approximately normal d i s t r i b u t i o n as shown. In l a t e r chapters we s h a l l f i n d i t very convenient to introduce a stochastic element into the reproduction function i n t h i s m u l t i p l i c a t i v e form (see 1.3). Unfortunately i f we t r y combining, by composition, successive l i f e - s t a g e s , such a convenient form as the above no longer holds. However i t might not be unreasonable to assume that the p r o b a b i l i t y of s u r v i v a l over the combined stages be 1/1+bE . Cer t a i n l y , for density dependent s u r v i v a l i t must be a decreasing function of E , at l e a s t for large E . (b) A "Birth-Death Process" model for the generalized Beverton-Holt  s u r v i v a l process. As we mentioned i n 1.2, Clark (to appear) has shown that i f over the l i f e - s t a g e [0, T] , the number of survivors N(t) s a t i s f i e s (1) § = - < K N ) where <f> i s a p o s i t i v e convex function, then the s u r v i v a l r e l a t i o n s h i p between N(T) = R and N(0) = E , i s a concave increasing function. We s h a l l c a l l t h i s the generalized Beverton-Holt s u r v i v a l process. In t h i s section we s h a l l develop a stochastic "Birth-Death Process" model (or more s t r i c t l y a "Pure-Death Process" model) which w i l l be analagous to t h i s . A r e s u l t s i m i l a r to Clark's holds when we consider the expected number of survivors as a function of the i n i t i a l number. A "Pure Death" process i s a continuous time Markov Chain with integer state space, i n which t r a n s i t i o n s from any state are possible only to that state or to the state represented by the integer one l e s s than that representing the present state. P r e c i s e l y , i f we l e t N(t) be a random v a r i a b l e denoting the number of survivors from an i n i t i a l population E , a l i v e at time t , we suppose that pr{one death i n ( t , t + A ] IN(t) = n} = X A + 5 ( A ) n and pr{more than one death i n ( t , t + A ] |N(t) = n} = o ( A ) , where /C n^} 1 S some non-negative sequence representing the expected number of m o r t a l i t i e s per unit time, for various population l e v e l s . We can see how t h i s model compares with the deterministic generalized Beverton-Holt model by considering E{N(t+ A ) I N(t) = n} = (n-l)A A + n(l-A A ) + Q ( A ) n n = n - X A + 9 ( A ) . n For the deterministic generalized B-H model we have {N(t+ A ) | N(t) = n} = n - <j>(n)A + 9(A) . We see that the sequence corresponds to the function <j>(n) . Both give a time rate f o r mortality, f or each population l e v e l . We s h a l l denote by Mg(t) > t n e expected number of survivors at time t from an i n i t i a l population E at time 0 . We s h a l l l e t P E > n ( t ) = p r { N ( t ) = n|N(0) = E} We can w r i t e E > L ( t ) = E{N(t)|N(0) = E} = I n-p ( t ) . n=0 We s h a l l be c o n c e r n e d w i t h t h e e x p e c t e d number o f s u r v i v o r s a t tim e T , i . e . w i t h E(R) = M^T) , and i n p a r t i c u l a r i n how t h i s r e l a t e s t o E . We need t h e f o l l o w i n g w e l l known r e s u l t s . Lemma 1.2.1 (The Forward E q u a t i o n ) The f u n c t i o n s p ( t ) s a t i s f y : E ,n V n ( t ) • Xn+1 P E > n f i « ^ V g E ? B S X t ) - ' for n = 0, 1,...,E where we define X = 0 , and p ( t ) = 0 , for U E ,m m > E . Proof. By conditioning on the population size at time t we have E pr{N(t+A) = n|N(0) = E} = £ pr{N(t+A) = n|N(t) = r} p (t) r=0 ' r • W P E s n + l ( t ) + ^ V ^ E . n ^ and therefore P E , ' n ( t ) - A n + 1 ^E,Jl^ V f i=?ES ( t ) ' ^ e ' d ' Lemma 1.2.2 (The Backward E q u a t i o n ) The f u n c t i o n s p„ s a t i s f y E,n J 92 P E , ' n ( t ) " " XE P E , n ( t ) + \ P E - l , n ( t ) > f ° r n = P . 1 — * Proof. By conditioning on the population s i z e at time A , we get E pr{N(t+A) = n|N(0) = E} = I pr{N(t+A) = n|N(A) = r} • r=0 pr{N(A) = r|N(0) = E} = P E - l , n ( t ) V + P E . n ^ ^ E ^ + ft(A) P E , ' n ( t ) " " X E P E , n ( t ) + A E P E - l , n ( t ) ' q - e ' d -We are now ready to prove a r e s u l t corresponding to Clark's r e s u l t f o r the generalized B-H s u r v i v a l model. Theorem 1 . 2 l l For a convex sequence {^n^ t n e expected number of survivors, E(R) , i s a concave non-decreasing function of i n i t i a l population E . In other words, i f the instantaneous time rate of expected mortality i s a convex function of the population s i z e then the expected s u r v i v a l r e l a t i o n s h i p i s purely compensatory. Proof. From the Backward Equation we have V ( t ) W f c ) + \ *E-lM We w i l l write V f o r the difference operator. We have then that v ' ( t ) VM^t) = M ^ c t ) - V t ) = - - ^ L — E+l E+l Now (t+A) = I I r pr{N(t+A) = r|N(t) = n} p (t) * 1 n=0 r=0 E + 1 ' n E+l = I [(n-l)X A + n(l-X A)] p (t) G n n r r r j .. n E+l = I [n - A A] p (t) n=0 n E + 1 , n " M E + l ( t ) " E a N ( t ) l N ( 0 ) " E + 1 } A I t follows that (t) = - E^ A N (- t) |N(0) = E + 1} and hence that E+l A I — 'E+l n=0 XE+1 -, l Since p E + 1 n ( t ) ^ 0 , for a l l n, E we have VM^(t) >_ 0 . It follows that E(R) = Mg^Ty^is^inon-decreasing i n E , which proves part of the theorem. For the concavity, we consider E+2 A E+l A v \ ( t ) = I j - 5 - P f + 2 n ( t ) - I ^ P e + 1 (t) ^ n=0 \+2 E + 2 ' n n=0 XE+1 E + 1 ' n E+2 A A = n=0 [ W PE+2,n ( t ) " A ^ P E + l , n ( t ) ] • ( s l n C e PE+l,E+2< 94 E+2 A X p_.. (t) Z [ ^ PE+2 n ( t ) " j T - ( PE+2 n ( t ) + t ' " )3 n=0 AE+2 E + 2 ' n AE+1 E + 2 ' n AE+2 (from Backward Equation) E+2 X X X X V r n /^ .\ n .. , n n+1 , . n=0 AE+2 E + 2 ' n • XE+1 E + 2 > n ' ^ AE+l AE+2 E + 2 > n + 1 X 2 + X X PE+2 n ( t ) ] AE+l AE+2 E + 2 ' n (from the Forward Equation) By r e l a b e l l i n g indices we have that E+2 E+2 2 V n + l PE+2,n+l ( t ) = I X n - l X n PE+2,n ( t ) n=0 n=0 ' and hence we have that 2 1 E + 2 V M (t) = I (X - A - X + X )X p w , 0 (t) E A . A _ u E+l E+2 n-1 n n E+2,n E+l E+2 n=0 E+2 7 A — I Xn ( V XE+l " V V l ) p E + 2 , n ( t > AE+l AE+2 n=0 Now i f {A } i s convex, we have VA .. < VA^,. , and hence n n-1 — E+l V 2M E(t) <_ 0 . Thus ^g(t) i - s concave i n E f o r a l l t . I t follows that E(R) = ^ ( T ) i s a concave function of E . This completes the proof. C l e a r l y i f the expected s u r v i v a l r e l a t i o n s h i p to ex h i b i t 95 depensation, the sequence {^n) must be non-convex over some range, The following c o r o l l a r y gives a condition f or depensation to occur. Coro l l a r y 1 I f f o r n < n„ , the sequence {A } i s concave then the — 0 n expected s u r v i v a l function exhibits depensation. Proof. We have from the proof of the theorem, that E+2 1E+l"E+2 n=0 2 x If E < n_ - 1 , then VA . > VA,,,, for n < E + 1 , and hence for — 0 n-1 — E+l — ? E <_ n Q - 1 we have V MgCt) >_ 0 , and so we see that E(R) = M^(T) i s a convex function of E f o r E <_ n^ - 1 . q.e.d. Clark (to appear) has proved a s i m i l a r r e s u l t f o r the generalized Beverton-Holt deterministic model. We now ask how the two models compare quantitatively? The next r e s u l t shows that compensation when i t occurs i s more severe i n the stochastic case than i n the equivalent deterministic case, and depensation when i t occurs i s l e s s severe i n the stochastic case than i n the deterministic case. To make precise comparisons, we s h a l l denote the deterministic v a r i a b l e by X(t) . The generalized B-H model i s then g = -KX) , X(0) = E . 96 We l e t D(E) be the number of survivors from the i n i t i a l population E , for t h i s model i . e . D(E) = X(T) . For the stochastic model we l e t N(t) denote the random v a r i a b l e for the number of survivors at time t from the i n i t i a l population N(0) = E . We l e t S(E) be the expected number of survivors at time T from the i n i t i a l population E i . e . S(E) = E(N(T)|N(0) = E) . To compare the two models we suppose that the sequence {X^} of values f o r the stochastic model i s the sequence of values of the function <j)(X) at X = 0, 1, 2,... . Corollary 2 If the function c|>(X) i s s t r i c t l y convex then S(E) < D(E) i . e . compensation i s more severe i n the stochastic model, and i f <J>(X) i s s t r i c t l y concave on [0, n Q] then S(E) > D(E) f o r E <_ n Q i . e . depensation i s less severe i n the stochastic model. Fi g . 1.8 97 Proof. Consider f i r s t l y the case tj>(X) s t r i c t l y convex. We have from the proof of the theorem that Mg'Ct) = - E(<KN(t)) |N(0) = E) Now <j> i s convex so we have by Jensens' slrlneqjaal'ity;. that E(c(.(N(t)) |N(0) = E) > <f>(E(N(t) |N(0) = E)) = K ^ C t ) ) . It follows that Mg'O:) < - (|)(ME(t)) But the equation of the deterministic model i s X'(t) = -<KX) . At any given l e v e l the process i s decaying f a s t e r than X . Hence, since MgCO) = X(0) , we conclude that MgCt) < X(t) , t > 0 , which fo r t = T gives the r e s u l t . Consider now when c)>(X) i s s t r i c t l y concave on [0, n^] . In t h i s case for E <_ n^ , we have by Jensen's Inequality that Mg'OO > - <|)(ME(t)) . But X' (t) = -<|>(X) , and so i n th i s case X(t) i s decaying f a s t e r than M ^ t ) . We conclude that, MgCt) > X(t) , E <_ n Q , which f o r t = T gives the r e s u l t . Both the forward and backward d i f f e r e n t i a l equations (lemmas 1 and 2) can be solved e x p l i c i t l y . For instance consider the backward equation. P E ; n ( t ) • - V E , n ( t ) + V E - l , n ( t ) ' 0 < n < E Let n (s) = r<*> - S t e p (t)dt be the Laplace Transform of ] 0 ' n The transformed backward equations are Y n ( s ) " - T F n E - l , n ( s ) > ' E+s ' and by induction we get yr / \ _ E E - l n+1 . . Now p n n ( t ) = pr{N(t) = n|N(0) = n} = pr{no deaths i n [0, t]|N(0) = n} n = e and hence 1 n ( s ) = , . n.n A J s n+s It follows that n ( s ) = ^ \ * ( s ) , n r=n A where <t>^(s) = i s the Laplace Transform of the exponential r f s density function 99 -X t (t) = X_ e r Inverting (2) we get (t) = — $ Jf $ & ' • • 4t $ Ct) :,nv J X n ~ n+1* ^ *E V ; P E , where f it g(t) denotes the convolution f(t-5)g(?)d? 0 From t h i s we can obtain e x p l i c i t expressions f o r the moments of the number of survivors N(t) . For instance using (2) we get that the Laplace Transform y E ( s ) of the expected number of survivors ^ ( t ) > at time t i s y E ( s ) = fm _ s t E E e I n P F n ( t ) d t = I n n (s) 0 n=0 E ' n n=0 E ' n E E = I -r n Ms> n=0 n E r n E X X , , X Xtj n=l n n+s n+l+s. E-.-f E E E = 1 ~ I ( n x -X ^ X +s ' b y P a r t i a l f r a c t i o n s n=l n j = l t=n t j j fs expansion Changing the order of summation, we get E X. i E X j = l j-!-s n=l n t=n t j ? 1 = I b,(E) -± • i j X. +s j= l J+S (3) 100 j E A where b. (E) = X . I j± H ( — ^ - ) . J J n=l A n t=n V X j Hence i n v e r t i n g (3) we get that E -X.t MgCt) = £ b.(E) e 3 3=1 3 Similar expressions can be obtained for higher moments. 2 We consider the s p e c i a l case A = kj , where k i s a constant. This i s a stochastic equivalent of the Beverton-Holt model i n which mortality i s purely density dependent. (Beverton & Holt 1957, p. 4 8 f f ) . This i s the model we considered i n (a) of t h i s section. In t h i s case closed form expressions f o r the c o e f f i c i e n t s b..(E) can be found. By some algebra, we can show M E ) = r ' (E+j)!(E-j)! = 2 i <f?i>'< 2£> Expressions such as t h i s are much too complicated f o r any p r a c t i c a l use i n population modelling. However the r e s u l t s of Theorem 1.2.1 and i t s c o r o l l a r i e s are useful i n showing that q u a l i t a t i v e l y the expected s u r v i v a l r e l a t i o n s h i p of a stochastic version of the generalized Beverton-Holt model i s no d i f f e r e n t from the corresponding deterministic version. I t would be of i n t e r e s t to know some second order properties of t h i s stochastic model. How, f o r instance, does the variance of the survivors change with the i n i t i a l population l e v e l ? We have not been able to determine t h i s . 101 (c) The E f f e c t of Stochastic Births In the models (a) and (b) of t h i s section we have considered the number of survivors R as a random v a r i a b l e depending on the parameter E , the i n i t i a l population. In the case of a s u r v i v a l process for young animals E represents the number of newborn animals or the number of eggs l a i d , and R represents the number of o f f s p r i n g that survive to become adults. In r e a l l i f e , the numbers of eggs l a i d from parent populations of the .same fi x e d s i z e are l i k e l y to vary from year to year. We can model t h i s by assuming that E i s a random v a r i a b l e depending on a parameter P , the si z e of the parent population. We are interested i n how the random v a r i a b l e R , the number of o f f s p r i n g who survive to become adults depends on the s i z e , P , of the parent population. Let us suppose that the number of b i r t h s f or each parent i s a Poisson random v a r i a b l e with mean F . The t o t a l number of b i r t h s for P parents i s then a Poisson random v a r i a b l e with mean FP . We have TO pr{E = r} = e _ F P , r = 0, 1, 2, (4) Another r a t i o n a l e f or the above d i s t r i b u t i o n of E , for c e r t a i n species, might be the following. Suppose each parent can produce at most one o f f s p r i n g t h i s being done with a p r o b a b i l i t y F (assumed no greater than one i n t h i s case). The t o t a l number of b i r t h s from P parents i s then a binomial v a r i a t e , with mean FP and variance 102 PF(l-F) . For large P t h i s can be approximated by a Poisson v a r i a t e with mean FP . Eith e r way, the Poisson d i s t r i b u t i o n seems a reasonable model, provided that b i r t h s are not density dependent as would be the case f o r t e r r i t o r i a l animals. We s h a l l be interested i n the expected number of o f f s p r i n g , who survive to become adults, from P parents. Let us denote t h i s by R(P) . We have R(P) = E ( R|P parents) = E { E ( R | E b i r t h s ) | P parents} where the expectation operator outside the curly brackets i s over the number of b i r t h s E . We s h a l l denote the expected number of survivors from E b i r t h s by S(E) as before. i . e . E ( R | E b i r t h s ) = S(E) . It follows then that R(P) = I S(E) e " F P Igi* E=0 *" D i f f e r e n t i a t i n g w.r.t. P we have ; l ( P ) . -F I S(E) + e - F P J S(E) I2|£i E-0 E ! E-0 E ! oo , E -FP r (FP) r = F e I - ^ i [S(E+1) - S(E)] by r e l a b e l l i n g i n d i c e s . E=0 h-If S(E) i s increasing i n E , we have R ' ' ' ( P ) > 0 , and hence R(P) 103 i s increasing i n P . D i f f e r e n t i a t i n g again, we get R"(P) = F e " F P I - i ^ i [VS(E+1) - VS(E)] E=0 *" . f f i t I mf ,2 S ( E ) . E=0 If S(E) i s concave i n E , then R"(P) <_ 0 and R(P) i s a concave function of P . We have thus proved the following r e s u l t . For a b i r t h process described by the Poisson random va r i a b l e (4), and a s u r v i v a l process with an expected s u r v i v a l r e l a t i o n s h i p which i s purely compensatory (concave, non-decreasing), the expected number of surviving o f f s p r i n g i s a compensatory function of the parent population s i z e . Thus f o r the models (a) and (b) of t h i s section with Poisson b i r t h s the r e l a t i o n s h i p between the expected number of surviving o f f s p r i n g and the number of parents i s compensatory. The above proof depends heavily on the assumption of a Poisson d i s t r i b u t i o n for the number of b i r t h s and o f f e r s no i n d i c a t i o n that the r e s u l t w i l l hold f o r more general b i r t h d i s t r i b u t i o n s . The models discussed i n t h i s section are i n t e r e s t i n g p r i m a r i l y f o r t h e i r q u a l i t a t i v e r e s u l t s . The exact d i s t r i b u t i o n s are too clumsy f or handling i n co n t r o l models, besides which, the basic assumptions made are much too sweeping f o r any r e a l use i n a more general theory. Instead i n the sequel we s h a l l use a rather l e s s s p e c i f i c stochastic version of the general reproduction function r e l a t i o n s h i p discussed i n 1.1. This i s described i n the next section. 104 1.3 A Stochastic Reproduction Model As was pointed out i n 1.1 the assumption of a f u n c t i o n a l r e l a t i o n s h i p Xn+1 = f ( x n } between successive population l e v e l s , contains many i m p l i c i t assumptions concerning the population. F i r s t l y i t assumes that the population i s homogeneous with respect to b i r t h growth and mortality p o t e n t i a l s i . e . that i t i s not age or s e x - s p e c i f i c . This may not be an unreasonable model for a population with a r e l a t i v e l y stable age and sex d i s t r i b u t i o n , or f o r a population which consists of only one year c l a s s . (Most species of P a c i f i c Salmon would approximately s a t i s f y ' t h i s l a t t e r condition. In t h i s case time points would be four years apart. Most of the f i s h born i n one year, return to spawn four years l a t e r and die a f t e r spawning). A second i m p l i c i t assumption i s that b i r t h , growth and mortality depend only on the s i z e of the population and not on any other v a r i a b l e s . C l e a r l y v a r i a b l e external factors such as the weather, the a v a i l a b i l i t y of food, the incidence of predators etc. play an important part i n the dynamics of the population. In a more complex eco-system model these factors would a l l be included. Also " i n t e r n a l " factors may vary. For instance the number of eggs l a i d or young born by an i n d i v i d u a l mother may vary, as may the success of mating behaviour, the success of s u r v i v a l behaviour etc. 105 Regardless of whether these v a r i a b l e factors are ultimately due to "chance" or not, we can conveniently incorporate them i n a population model as random fac t o r s . If the f l u c t u a t i o n s from year to year of these "external" and " i n t e r n a l " v a r i a b l e s are uncorrelated i n time we can reasonably model the population process as a f i r s t order Markov process i n d i s c r e t e time. The e f f e c t s of age and sex s p e c i f i c i t y i n the population are more d i f f i c u l t to incorporate i n a one dimensional model. For a given population the age and sex d i s t r i b u t i o n could be regarded as st o c h a s t i c , since unknown, and so the e f f e c t on reproduction could also be regarded as stochastic. However i n t h i s case the stochastic e f f e c t s w i l l be seriallyccorr-eiat-edt6Hg . The f i r s t - o r d e r Markov model we s h a l l develop w i l l make no attempt to incorporate t h i s aspect of population dynamics and we s h a l l assume that the population i s neither age nor sex s p e c i f i c . We s h a l l discuss some models for age and sex s p e c i f i c populations i n Chapter 5. The model we s h a l l assume i s that the population l e v e l i n year n i s a random va r i a b l e X^ , with the sequence { x n} forming a discrete-time Markov process. We s h a l l suppose that the one-step t r a n s i t i o n s of the process are given by the equation X = Z f ( X ) (1) n+1 n n where ^ i s a sequence of independent i d e n t i c a l l y d i s t r i b u t e d random variables ( i . i . d . r.v.s.) with Z independent of X and f i s a n n reproduction function of the type discussed i n 1.1. We s h a l l assume 106 that the expected value of the random variables Z i s one and then we n have E(X n + 1|X n = x) = f(x) . (2) In other words the reproduction function model of 1.1 holds in expectation in our stochastic model. We shall find i t convenient to assume that there is an upper limit, m say, to the size of the population, and we shall therefore assume that the state-space of the process { x n^ 1 S t n e interval [0, m] . To do this we need assume that the random variables Z are n distributed on an interval [a, b] , with 0<_a<_b<a>, and we let m = bf(p) , where p is the point at which f attains i t s supremum. This i s perhaps best understood with the help of a picture. m bf (x) = max(X _,, Ix n+11 n f(x) = E(X n + 1|X n >'x af (x) = min(X ,., X n+1 n Fig. 1.9 The i l l u s t r a t i o n shows the case of an over-compensatory expected reproduction function, but a s i m i l a r p i c t ure can be drawn for other types of expected reproduction function. If we assume that the random variables Z have d i s t r i b u t i o n n function $ on [a, b] , then the one-step t r a n s i t i o n p r o b a b i l i t i e s of the Markov process {X } are n F(t|x) = pr{X < t | x = x} = $(77-7) , x > 0 1 n+1 — 1 n f(x) and |0 t < 0 F(t|0) = pr{X < t l x = 0) = < n+l — n 1 1 t > 0 In the l i t e r a t u r e on Markov Processes the function F(t|x) i s known as a stochastic kernel. Can the use of a model such as t h i s be j u s t i f i e d ? There are two ways we can j u s t i f y the v a l i d i t y of a mathematical model. F i r s t l y there i s the empirical approach. We can t r y to show that the model " f i t s " with observed data for c e r t a i n animal populations. Secondly there i s the deductive approach i n which we try to derive the model from more basic general p r i n c i p l e s . This i s what Beverton and Holt t r i e d to do for t h e i r s u r v i v a l process model, (see 1 .1(a)). However at present i n the science of population ecology there are very few general p r i n c i p l e s from which more complex mathematical models can be developed. For instance when Beverton and Holt followed a re d u c t i o n i s t approach to deduce t h e i r s u r v i v a l r e l a t i o n s h i p (see 1.1(a)) 108 the assumptions they needed to make about the "lower-order" model were no more s e l f - e v i d e n t l y true than the deduced model i t s e l f . The d i f f e r e n t i a l equation model 1.1.2 that they assumed to hold over the s u r v i v a l period i s no more obviously true or r e a d i l y v e r i f i a b l e than i s the s u r v i v a l function 1.1.3 which they derived from i t . Although we s h a l l deduce the stochastic model given here from a continuous time model, we make no claim for the v a l i d i t y of the l a t t e r . Indeed i t i s f a i r l y obvious that the assumptions of t h i s "lower order" model can only at best be approximately true, and also that by changing the model s l i g h t l y , a d i f f e r e n t deduction w i l l r e s u l t . However i t may be i l l u m i n a t i n g to r e i n t e r p r e t the assumptions of our stochastic model. We s h a l l also give some empirical j u s t i f i c a t i o n for i t . Let us f i r s t l y consider the most simple, continuous time density independent, deterministic exponential growth model 1 dx , , - — = b - d = r , x dt where b i s the b i r t h - r a t e , d the death rate, and r the net growth rate. It i s f a i r l y easy to see that x(t) = x ( 0 ) e r t . A discrete-time model derived from t h i s i s the geometric model kr x = e x , n+1 n where k i s a constant r e f l e c t i n g the length of the i n t e r v a l between 109 successive time points. As i t stands t h i s model i s density-independent and describes geometric growth or decay. We can include a density-dependent f a c t o r by assuming that the instantaneous growth rate over the time i n t e r v a l depends on the i n i t i a l l e v e l x . In t h i s case we would have n kr(x ) ( 3 ) Xn+1 = X n " 6 = f ( x n } S a y ' Of course i n most cases i t i s i n v a l i d to assume that the instantaneous growth rate over the time i n t e r v a l between successive time points depends only on the i n i t i a l l e v e l x^ . This i s the assumption Ricker (1954, 1958) made for h i s overcompensation model of 1.1(d). However even i f we assume a more general model /i \ 1 dx , . (4) x dt" = P ( X ) ' i n which the instantaneous growth rate depends on the current population l e v e l , a r e s u l t l i k e (3) holds. In t h i s case r ( x n ) represents some sort of average instantaneous growth rate. I t i s the exponential rate at which the population would have to grow over the t h n year, to go from x to x ,, . n n+1 To incorporate a random element i n t h i s process we suppose that t h i s average instantaneous growth rate i s subject to random shocks from year to year, i . e . rather than suppose that the average instantaneous growth over the u"^ time period i s r(x ) we suppose n that i t i s r(x ) + e m where {e } i s a sequence of i . i . d . random n n n 110 v a r i a b l e s . In t h i s case we have that the population i n year n i s a random v a r i a b l e X , and that {X } i s a Markov process with n n kr(X ) + e (5) X . = X e n n = Z f(X ) n+1 n n n ke where Z^ = e , and f i s as defined i n (3). This i s j u s t the model (1). However we note that to a r r i v e at t h i s form we have needed to assume that the random shocks occur i n the average instantaneous growth rate. It would perhaps be more r e a l i s t i c to assume that random shocks occur i n the actual instantaneous growth rate. In t h i s case we would replace (4) by 1 dx / \ . - — = p(x) + e n , where e i s constant over the n time period. In t h i s case the n form (5) would not r e s u l t . A yet more r e a l i s t i c model would perhaps be 1 dx / \ . / . - N 1 x d l = P ( X ) + £ n ( t ) ' where e n ( t ) i - s a white-noise process (Wiener Process) over the n*"*1 time period. Again i n t h i s case the form (5) would not i n general r e s u l t . We see that the p a r t i c u l a r form (5) i s the consequence of very s p e c i a l assumptions concerning the continuous time system. Such a deduction o f f e r s very l i t t l e j u s t i f i c a t i o n f o r our d i s c r e t e time stochastic model. I l l When we look f o r an empirical j u s t i f i c a t i o n f o r our stochastic model we note that there are two main assumptions. The f i r s t i s that the random e f f e c t s are uncorrelated i n time. It seems that f o r some populations the va r i a b l e factors may well be correlated e.g. the s i z e of a predator population. However other e f f e c t s may we l l be uncorrelated i n time. Examples of th i s might be the e f f e c t s of weather and the a v a i l a b i l i t y of food. We s h a l l suppose that we are concerned with a population for which the va r i a b l e factors are uncorrelated i n time. A second main assumption of our model i s that the random e f f e c t s of external factors e f f e c t the population growth i n a way that i s p r o p o r t i o n a l l y independent of the population s i z e . Again f o r some va r i a b l e environmental e f f e c t s t h i s may be a reasonable assumption, but not so for others. A test f o r t h i s could be performed by p l o t t i n g l og ^ n +-^ against log . Under the hypothesis of our model, deviations from the trend curve should be i . i . d . r.v.s. for we have log X n + 1 = g(log x n ) + ? n , where g(x) = logof°exp(x) , and £ n = log . (We note that i f f i s of the form ax^ , then g w i l l be l i n e a r ) . Such a p l o t was c a r r i e d out by Shepard and Withler (1958) for Skeena River Sockeye Salmon, for which over f i f t y years of data 112 are a v a i l a b l e . They performed no precise s t a t i s t i c a l test but concluded from t h e i r p l o t that v a r i a t i o n s from the trend l i n e did not vary with population s i z e . They conclude (p. 1021) "This pattern of v a r i a t i o n probably r e s u l t s from the influence of randomly f l u c t u a t i n g environmental conditions which a f f e c t the s u r v i v a l of the sockeye independently of t h e i r abundance. Such factors would a f f e c t both large and small populations i n the same proportional manner. Examples of such factors might include adverse temperatures, or scouring of spawning grounds by floods, both of which would probably destroy the same proportion of deposited eggs whether the spawning were sparse or dense." It seems then that our model i s reasonable at l e a s t f o r some populations. A model of t h i s type has been used by Ricker (1958) and Larkin and Ricker (1964) f o r computer simulation studies. In t h e i r models the reproduction function f was modified by " m u l t i p l i e r s obtained by random s e l e c t i o n from a table of factors whose frequencies are d i s t r i b u t e d as i n a normal frequency d i s t r i b u t i o n . The p o s i t i v e (favourable) factors are applied as m u l t i p l i e r s and the negative ones as d i v i s o r s to the mean reproduction function. This procedure produces an asymmetrical d i s t r i b u t i o n of progeny s i z e s , which seems to accord f a i r l y w e l l with what i s usually observed i n nature among such species as pink or chum salmon" (Ricker 1958, p. 994). Galto and R i n a l d i (to appear) have discussed a n a l y t i c a l l y some of the questions r a i s e d by Ricker and they use a model of the above type. In the next chapter we s h a l l i nvestigate the long-run behaviour of a population governed by t h i s model and look at some of the e f f e c t s that harvesting has upon i t . 113 Chapter 2 The Population i n Equilibrium-. 2.1 Steady-State Behaviour of the Stochastic Population Model In t h i s section we s h a l l mainly be concerned with the long-run, steady-state behaviour of a population governed by the stochastic model of 1.3, X , = Z f (X ) , n+1 n n where i s a sequence of independent i d e n t i c a l l y d i s t r i b u t e d random variables with unit mean and d i s t r i b u t i o n function $ on [a, b] . Many of the r e s u l t s w i l l hold for populations governed by more general Markov processes {X } . n However before we go into the steady-state of the stochastic models i t w i l l be worth looking at the deterministic model of 1.1 x ,, = f ( x ) . n+1 v n The dynamics of a population governed by t h i s model can be determined by successive compositions of the reproduction function f . n-f o l d composition X n = f ( x n - l > - f ° f < V 2 > " = f ° O f ( x 0 > Graphically t h i s can be represented by "bouncing" a l i n e back and f o r t h between the reproduction curve and the 45° transf e r l i n e . 114 F i g . 2.1 In the case of c r i t i c a l depensation (see 1.1) i l l u s t r a t e d i n F i g . 2.1, i t i s easy to see that the population w i l l tend to equilibrium e i t h e r at or 0 , depending on whether the i n i t i a l population l e v e l i s greater than or l e s s than the l e v e l x u In general for deterministic reproduction function models, stable equilibrium points are those solutions to x = f(x) for which | f ' ( x ) | < 1 (see Clark (to appear)). The population w i l l tend to one or another of these points, which depending on the i n i t i a l population l e v e l . For a model with increasing reproduction function the other solutions correspond to unstable equilibrium points, l i k e x^ i n F i g . 2.1. For a reproduction function e x h i b i t i n g over-compensation (dome-shaped), a s o l u t i o n for which f'(x) < -1 i s also unstable and indicates that stable or unstable c y c l i n g w i l l occur (see Clark (op. c i t . ) ) . We now turn our attention to the stochastic model and ask what w i l l be the behaviour of the population when i t has reached a "steady-state" condition, having been undisturbed for many generations? C l e a r l y i n t h i s case due to the random flu c t u a t i o n s each year we cannot have the population remaining at a fixed equilibrium l e v e l . However we can have the p r o b a b i l i t y d i s t r i b u t i o n of the population remaining f i x e d . This i s known as s t a t i o n a r i t y and i s an analagous concept i n stochastic theory to that of equilibrium i n deterministic theory. When i n steady-state the successive values of the population would be represented by a scatter of points. The d i s t r i b u t i o n of t h i s s c a t t e r would r e f l e c t a stationary p r o b a b i l i t y d i s t r i b u t i o n of the population process. population converges to a stationary d i s t r i b u t i o n . Such i s the concern of the ergodic theorems i n the theory of Markov processes. We would l i k e to show that the p r o b a b i l i t y d i s t r i b u t i o n of the For a given i n i t i a l population l e v e l X n = x 0 ' l e t P n ( t ) = p r ( X n < t) By conditioning on X we have n rm (t) .= pr(X n+1 < t X = x)d p (x) — 1 n n 116 i . e . p n + 1 ( t ) = m F(t|x) d p (x) (1) 0 n where F(t|x) = $( J~, . ) i s the stochastic kernel of the process i (xj {X } as discussed i n 1.3. n Equation (1) determines the dynamics of the stochastic model since we know that P Q ^ ) n a s a x x l t s mass at x^ . It i s analagous to the successive compositions f o r the deterministic model. We wish to show that the sequence of d i s t r i b u t i o n functions {p n(t)} converges to a proper p r o b a b i l i t y d i s t r i b u t i o n p(t) say. If t h i s happens, following the customary terminology we say that the process i s ergodic. In t h i s case we would have from (1) that rm P(t) = F(t|x)d p(x) (2) 0 In other words once the p r o b a b i l i t y d i s t r i b u t i o n of the population i s at p(t) i t remains at p(t) i n subsequent generations. As indicated before t h i s i s c a l l e d a stationary d i s t r i b u t i o n . Any s o l u t i o n to (2) which i s a p r o b a b i l i t y d i s t r i b u t i o n i s a stationary d i s t r i b u t i o n f o r the population. The p r o b a b i l i t y d i s t r i b u t i o n of the population could tend to any one of these stationary d i s t r i b u t i o n s which depending (in p r o b a b i l i t y ) on the i n i t i a l population l e v e l . I t i s also possible that (2) has no so l u t i o n which i s a f i n i t e - v a l u e d measure, but does have a s o l u t i o n which i s not f i n i t e - v a l u e d . In t h i s case the p r o b a b i l i t y d i s t r i b u t i o n of the population would not converge to a proper p r o b a b i l i t y d i s t r i b u t i o n . However some phys i c a l i n t e r p r e t a t i o n could be given i n t h i s case (see H a r r i s , 1956). I t corresponds to what i s known as null-recurrence i n the theory of Markov Chains. Equation (2) determines the possible steady-states of the stochastic model i n the same way that the equation x = f(x) determines the steady-states of the deterministic model. Rather than t r y to f i n d solutions to (2) we s h a l l look f o r conditions which imply the erg o d i c i t y of the population process i n some s p e c i a l cases, and t r y to f i n d some of the properties of the stationary d i s t r i b u t i o n s . We s h a l l assume that the d.f. $ i s s t r i c t l y p o s i t i v e on [a, b] i n the sense that there i s a p o s i t i v e p r o b a b i l i t y associated with each Borel set of p o s i t i v e measure. We s h a l l assume also that $ has a continuous density <j) , say, on [a, b] . From the assumption of s t r i c t p o s i t i v i t y , $ w i l l be p o s i t i v e on [a, b] (except possibly on a set of measure zero). Then for x > 0 the stochastic kernel F(t|x) = $(J~. .) also has a density k(t|x) , say, continuous £ (x) i n t and k(t|x) - ^ ( ^ The density k i s p o s i t i v e (except possibly on a set of measure zero) for af(x) <_ t < _ b f ( x ) , and i s zero outside t h i s region. Furthermore i t i s continuous i n x on (0, m] because f and $ are continuous. By induction we can see that f o r any i n i t i a l population X = x the d i s t r i b u t i o n of the population i n year n w i l l have a 118 density. C a l l i t II . Then n n 1(t) = k ( t | x Q ) n (t) = n m II 1(x)k(t|x)dx ^ n-1 (3) 0 Following the usual terminology for Markov processes we s h a l l c a l l a set T of states (population l e v e l s ) transient i f lim pr(X eT) = 0 n n-*°° It i s almost sure that the process (population) w i l l f i n a l l y leave a transient set eventually. C l e a r l y a l i m i t i n g stationary p r o b a b i l i t y d i s t r i b u t i o n can have no mass on a transient set. ^Haf(x) bf(x) 0 r m 0 Fi g . 2.2 We consider the case f o r which af'(0) > 1 . In t h i s case there w i l l be a transient set (0,r) . In F i g . 2.2(a), which i s for ei t h e r a 119 compensation or depensation model provided that f i s increasing, i f the i n i t i a l population i s i n the set (0,r) i t w i l l , i n every subsequent generation almost surely increase u n t i l i t exceeds r . Once i t has entered the set [r, m] i t w i l l stay there with p r o b a b i l i t y one. Thus the set (0,r) i s transient. Furthermore the set (r,m) i s a communicating set i n the sense that i t i s possible to go from any one l e v e l i n that set - to any subset having p o s i t i v e measure i n a f i n i t e number of years. In F i g . 2.2(b) which i s f o r an over-compensation model the l e v e l r i s determined by the l e s s e r of af(m) and the x-coordinate of the point of i n t e r s e c t i o n of y = x and y = af(x) . The set (0,r) i s a transient set, for i f the population originates i n (0,r) i t w i l l eventually enter [r, m] and subsequently stay there. Again the set (r,m) i s communicating. We s h a l l now prove the e r g o d i c i t y of processes l i k e those depicted i n F i g . 2.2, for which, for low enough population l e v e l s , the population w i l l annually increase i n s i z e almost surely. We s h a l l use the ergodic theorem of Doob (1953, p. 214). As we have already seen the set (0,r) i s transient, and with p r o b a b i l i t y one the process w i l l eventually enter the set [r, m] and subsequently stay there. We " s h a l l then only consider the process on [r, m] and assume an a r b i t r a r y i n i t i a l p r o b a b i l i t y d i s t r i b u t i o n on [r, m] . The density kernel k(t|x) i s continuous on the closed square [r, m] x [r, m] 120 and hence i s bounded there. This s a t i s f i e s the conditions of Doob's ergodic theorem. We have then that the p r o b a b i l i t y d i s t r i b u t i o n of the population tends to a l i m i t i n g stationary d i s t r i b u t i o n on [r, m] . Furthermore t h i s stationary d i s t r i b u t i o n i s independent of the i n i t i a l d i s t r i b u t i o n on [r, m] , i s s t r i c t l y p o s i t i v e on [r, m] and has a density II(t) , say, s a t i s f y i n g rm n(t) = k(t|x) n(x) dx . (4) r We state t h i s as a theorem The exact condition given by Doob i s that the stochastic kernel s a t i s f i e s Doeblin's hypothesis (p. 192). If we l e t F be the f i e l d of Borel sets i n [r, m] then Doeblin's hypothesis can be stated as:-there ex i s t s a f i n i t e - v a l u e d measure <)> of sets A e F with <j)([r, m]) > 0 , and an integer v >_ 1 , and an e > 0 such that pr(X^ e A | x Q = £) <_ 1 - e , i f <|>(A) <_ e for every ? e [r, m] When as we have assumed there i s a density kernel k(t|x) continuous on [r, m] x [r, m] Doeblin's hypothesis holds, f o r i f $ i s Lebesgue measure and v = 1 p r ( X 1 e A|X Q = O = k(t|£)dt < K (KA) A where K = max k(t|x) , and i f we l e t e = -^j-r , we have f o r r i K + l x, t e [r, m] <j)(A) <_ e , that p r ^ e A | X Q = O = 1 - e , which i s Doeblin's hypothesis. We note that t h i s argument does not hold i f we consider the state-space [0, m] , for the density kernel i s not bounded over [0, m] x [0, m] . 121 Theorem 2.1.1 Consider models l i k e those i n F i g . 2.2(a), (b) for which af'(0) > 1 ( i . e . for which i t i s sure that the population w i l l increase annually at low enough population l e v e l s ) , and for which the random m u l t i p l i e r s Z have a continuous density function, p o s i t i v e on [a, b] n (except possibly on a set of measure zero). For any i n i t i a l population e (0, m] the p r o b a b i l i t y d i s t r i b u t i o n of the population i n year n tends to a stationary d i s t r i b u t i o n , s t r i c t l y p o s i t i v e on [r, m] and having a density function s a t i s f y i n g (4). This stationary d i s t r i b u t i o n i s independent of the i n i t i a l population l e v e l . The fact that the stationary d i s t r i b u t i o n on [r, m] i s s t r i c t l y p o s i t i v e i s rather i n t e r e s t i n g . I t means that i n steady-state any population s i z e between r and m may occur. Roughly speaking, i f a population s a t i s f y i n g t h i s model had reached steady-state and were observed for long enough i n t h i s condition v i r t u a l l y a l l values i n the i n t e r v a l [r, m] would eventually occur. It i s easy to see from F i g . 2.2 that the range r to m of the stationary d i s t r i b u t i o n depends on the range a to b of the d i s t r i b u t i o n of the random m u l t i p l i e r s . I f the range i n environmental conditions i s large then the range of possible sizes of the population i n steady-state w i l l be large, and low population l e v e l s can occur. The proof of Thm 2.1.1 does not depend on the assumption that the population s a t i s f i e s a m u l t i p l i c a t i v e model of the form of 1.3. 122 I t w i l l hold for any Markov model for which the stochastic kernel has a density continuous i n both v a r i a b l e s and p o s i t i v e on a set l i k e those shown i n F i g . 2.2. We now consider some other cases. Let us suppose that af'(O) < 1 . In t h i s case i t i s possible at low population l e v e l s f o r the population si z e to decrease i . e . af(x) < x < bf(x) (see F i g . 2.3) bf(x) af(x) 0 F i g . 2.3 In t h i s case we cannot use the ergodic theorem of Doob, since Doeblin's hypothesis i s not s a t i s f i e d because of the fac t that the density kernel k ( t ' x ) = f W i s unbounded i n the neighbourhood of (0,0) . (This also prevents the use of the ergodic theorem of F e l l e r (1966, p. 265), the kernel not s a t i s f y i n g F e l l e r ' s r e g u l a r i t y condition). We have not been able to determine the asymptotic properties of the process i n t h i s case. I t 123 seems that there are three p o s s i b i l i t i e s . (I) There i s the p o s s i b i l i t y that the process i s ergodic and that there i s a l i m i t i n g stationary d i s t r i b u t i o n , s t r i c t l y p o s i t i v e on [0, m] . In t h i s case a r b i t r a r i l y small l e v e l s of the population could occur i n steady-state. foHowever iKew^opulationucouldrnotirstay small forever. It woula eventually f-et-urnm|o exceed anyilevel'P^iLess 1 triaii cm) rewiith. a f i n i t e expected time of return. (II) A second p o s s i b i l i t y i s that with p r o b a b i l i t y one the population eventually becomes ex t i n c t ( i . e . X^ -> 0 , w. p. 1 (n-*30)) . We are able to show that t h i s occurs i f the expected reproduction function i s concave and i s such that f(x) < x , x > 0 i . e . f o r a compensation model, for which i n expectation, the population cannot sustain i t s si z e at any l e v e l . C e r t a i n l y i n a deterministic model for which f(x) < x , the population tends to e x t i n c t i o n . For the stochastic model we have E(X ^ Ix = x) = f(x) < x , x > 0 n+1' n — — The process {-X } i s a non-positive sub-martingale (Doob (1953, p. 294)), n f o r the con d i t i o n a l expectations s a t i s f y E ( - x n + i l - V -*r•••.-*,> >-~\ Thus from the convergence theorem for sub-martingales (Doob, p. 324) we have that with p r o b a b i l i t y one li m - X = -X say, e x i s t s . n oo n-*» 124 Thus the d i s t r i b u t i o n functions P ^ t ) converge to a l i m i t p(t) and from (1) we have that p(t) i s stationary s a t i s f y i n g rm p(t) = F(t|x)dp(x) 0 (N.B. p can be a degenerate d i s t r i b u t i o n function). Suppose the process has reached i t s asymptotic state, then X and X both have d i s t r i b u t i o n s p . Thus n n+1 K E ( V - E ( X n + l ) - E ( Zn f ( V > = E ( f ( V ) since and X n are independent, and since E ( z n ) = 1 • B u t by Jensen s Inequality , since f i s concave we have E(f(X )) < f(E(X )) n — n A Jensen's Inequality which i s used here and frequently i n the sequel, states that f or any s t r i c t l y concave function <j> and random v a r i a b l e X , E(4>(X)) 1 4>(E(X)) , with equality only i f X has a degenerate d i s t r i b u t i o n . The proof follows from the f a c t that <KX) " <f>(E(X)) - (X-E(X))' <j)'(E(X)) < 0 X * E(X) Taking expectations gives the r e s u l t . I f <J> i s n o n - s t r i c t l y concave equality can occur for non-degenerate d i s t r i b u t i o n s of X . 125 and thus that E(X ) < f(E(X )) . n — n But also we have that f(E(X )) < E(X ) since we have assumed that n — n f (x) <^  x , a l l x , and so we have that E(X n) = f ( E ( X n ) ) . It follows that E(X ) = 0 , and since X i s non-negative n n that X^ = 0 with p r o b a b i l i t y one. In t h i s case then the population approaches e x t i n c t i o n with p r o b a b i l i t y one. We state t h i s r e s u l t as a theorem. Theorem 2.1.2 For the stochastic population model X = Z f(X ) for which r n n n the expected reproduction function f(x) i s concave (compensatory) and assumes a value l e s s than x for each p o s i t i v e x , the population tends to e x t i n c t i o n with p r o b a b i l i t y one. i . e . i f E(X , J x = x) = f(x) < x , for x > 0 then X 0 w.p. 1 . n+11 n n Unlike the analagous deterministic model, the convergence to e x t i n c t i o n i n t h i s case i s not n e c e s s a r i l y monotonic. I t could be that there are years or even runs of years i n which the s i z e of the population increases, but nevertheless e x t i n c t i o n i s the eventual outcome. Although such a model i s hardly l i k e l y to describe a population i n i t s natural state, i t could well describe a population subject to regular harvests and i n t h i s respect i s rather important (see 2.2). 126 We note that the same outcome of almost sure eventual e x t i n c t i o n occurs i f the d i s t r i b u t i o n of the m u l t i p l i e r s Z has a n p o s i t i v e mass at zero i . e . i f i n any year there i s a p o s i t i v e p r o b a b i l i t y of the population become ex t i n c t . ( I l l ) A t h i r d p o s s i b i l i t y i n the case of F i g . 2.3 i s that the set [0, m] i s a " n u l l - r e c u r r e n t " set, which roughly means that, although the process o r i g i n a t i n g i n some small set E w i l l eventually return to E , almost surely, the expected time of return may be i n f i n i t e . An example of a n u l l - r e c u r r e n t Markov chain with denumerable state space i s a random walk on the integers, where at each step an increase or decrease can occur, each with p r o b a b i l i t y one-half. In t h i s case the random walk can take a r b i t r a r i l y long journeys out towards i n f i n i t y or negative i n f i n i t y and the expected return times are i n f i n i t e . In t h i s case i n the l i m i t the unit p r o b a b i l i t y mass i s spread evenly over the integers, a l l integers being "equally probable" and there i s no stationary d i s t r i b u t i o n . In the population model the population might dwindle to a r b i t r a r i l y small sizes and stay i n the neighbourhood of zero for a r b i t r a r i l y long periods. I t would however eventually grow back to exceed any l e v e l (less than m) , but unlike the case ( I ) , the expected time of return would be i n f i n i t e . For a l l p r a c t i c a l purposes the population would be e x t i n c t f or a r b i t r a r i l y long periods. If t h i s n u l l recurrence were to occur there would be no stationary p r o b a b i l i t y d i s t r i b u t i o n p s a t i s f y i n g (1), but there might well be an invariant measure p s a t i s f y i n g (1) with the p-measure of the whole space [0, m] being i n f i n i t e (see Ha r r i s 1956). It would be of considerable i n t e r e s t to f i n d the conditions determining which of the above three outcomes occurs. When we consider the steady-state of a population subjected to regular harvests, we would l i k e to know conditions which imply the e x t i n c t i o n or conservation of the population (see 2.2). It seems that the density <j> and the slope f'(0) w i l l be important i n determining which outcome occurs. We s h a l l now consider a stoc h a s t i c model corresponding to the c r i t i c a l depensation model of the deterministic theory (see 1.1(c)). A c h a r a c t e r i s t i c of the det e r m i n i s t i c c r i t i c a l depensation model i s that at c e r t a i n low l e v e l s the population i s unable to sustain i t s s i z e . I f the population ever drops below a c e r t a i n c r i t i c a l l e v e l then i n subsequent generations i t decreases i n s i z e and eventually becomes ex t i n c t . The stoch a s t i c models i n F i g . 2.4 and 2.5 both have t h i s property, f o r i n both cases i f X^ i s les s than c , then with p r o b a b i l i t y one X ,.. i s l e s s than X n+1 n In F i g . 2.4 for any i n i t i a l population l e v e l i n [c, m] , there i s a p o s i t i v e p r o b a b i l i t y that the population l e v e l w i l l drop below c i n a given f i n i t e number of years. It follows then that with p r o b a b i l i t y one the population w i l l eventually drop below c and subsequently decrease to zero. The set (0, m] i s transient. For any i n i t i a l population l e v e l , e x t i n c t i o n of the population i s the ultimate r e s u l t . 128 y 0 c m O c d r m F i g . 2.4 F i g . 2.5 This model i s of l i t t l e use i n describing a population i n i t s natural state, but could be rather u s e f u l i n describing a population subjected to regular e x p l o i t a t i o n (see 2.2). In the model depicted i n F i g . 2.5, i f the i n i t i a l population i s i n the set [d, r] then i n subsequent generations i t w i l l grow u n t i l i t exceeds the l e v e l r . Once i t has entered the set [r, m] i t w i l l stay there forever. Thus under su i t a b l e r e g u l a r i t y conditions on the density kernel, for any i n i t i a l population l e v e l i n [d, m] , the population tends to a steady-state with stationary d i s t r i b u t i o n on the set [r, m] . This can be proved l i k e Thm 2.1.1. If the i n i t i a l population i s i n the set (0, c] then with p r o b a b i l i t y one i t decreases monotonically to zero. If the i n i t i a l population'is i n the set (c,d) then e i t h e r a steady-state on [r, m] or e x t i n c t i o n can r e s u l t , which case depending on whether the process leaves (c,d) to enter [d, r ] or to enter [0, c] . This w i l l depend i n p r o b a b i l i t y on the i n i t i a l l e v e l of the population. In t h i s case the p r o b a b i l i t y d i s t r i b u t i o n of the population tends to a stationary d i s t r i b u t i o n which i s a convex combination of the stationary d i s t r i b u t i o n on [r, m] and the d i s t r i b u t i o n with a l l i t s mass at zero. This i s a s o l u t i o n to (2). The l e v e l c i s a c r i t i c a l l e v e l f o r the population. If i t ever drops below c then e x t i n c t i o n w i l l r e s u l t . The l e v e l d can be thought of as a danger l e v e l . I f the population were ever disturbed from a steady-state on [r, m] , for instance by harvesting, i n such a way that i t dropped below the l e v e l d L} then even i f harvesting were to stop, e x t i n c t i o n would become a p o s s i b i l i t y with p o s i t i v e p r o b a b i l i t y of occurrence. With the population below the danger l e v e l d a few unfavourable years could bring about a s i t u a t i o n i n which the e x t i n c t i o n of the population would be i n e v i t a b l e . This could not happen at higher population l e v e l s . I t could happen that the population could be reduced below the l e v e l d many times and each time regenerate i t s e l f , but i f t h i s were to happen often enough then with p r o b a b i l i t y one a combination of unfavourable years s u f f i c i e n t to bring about the e x t i n c t i o n of the population would r e s u l t . In some sense the i n t e r v a l (c,d) corresponds to the unstable equilibrium point i n the deterministic c r i t i c a l depensation model (see F i g . 2.1). If the population leaves (c,d) from the upper end then steady-state on [r, m] r e s u l t s , whereas i f i t leaves (c,d) from the lower end then steady-state at e x t i n c t i o n r e s u l t s . Again these r e s u l t s on c r i t i c a l depensation models do not depend on the assumption of a stochastic model of m u l t i p l i c a t i v e form 130 as i n 1.3. They w i l l hold f o r any Markov model with a continuous non-zero density kernel i n and only i n the regions shown i n Figs. 2.4, 2.5. So f a r the r e s u l t s of t h i s section have only concerned the existence of steady-states and the range of corresponding stationary d i s t r i b u t i o n s . How does a steady-state stationary d i s t r i b u t i o n f o r the stochastic model we are considering compare q u a n t i t a t i v e l y with the equivalent equilibrium i n the deterministic theory? Let us consider only concave expected reproduction functions (compensation models). In t h i s case i n the deterministic model there w i l l be only one equilibrium point, x say, where x i s the so l u t i o n .to f(x) = x . Let us suppose that x i s a stable equilibrium point. Suppose i n the stochastic model that a steady-state with a stationary d i s t r i b u t i o n on some i n t e r v a l e x i s t s . Suppose that X and hence X ,, have the n n+1 stationary d i s t r i b u t i o n . Then X = E(X . • ) • = E ( Z f ( X j ) = E ( f ( X ) ) n+1 n n n since and X^ are independent, and since E ( z n ) = 1 • Now by Jensen's Inequality E ( f ( X n ) ) < f ( E ( X n ) ) = f(X) . Hence we have that X < f(X) . I t follows from the concavity of f that X < x . Thus we have shown that f o r a stochastic compensation model of the form of 1.3 (concave f) , i f the population tends to a steady-state, the mean of the l i m i t i n g stationary d i s t r i b u t i o n i s l e s s than equilibrium l e v e l i n the corresponding deterministic model. In steady-state, the average population l e v e l f o r a population l i v i n g i n a f l u c t u a t i n g environment would be l e s s than the equilibrium l e v e l computed from a deterministic model. In the next section we discuss the steady-state of a population l i v i n g i n a f l u c t u a t i n g environment (as described by our model) when there i s regular harvesting. 2.2 Steady-State When There i s Regular Harvesting In t h i s section we s h a l l consider the steady-state of the stochastic population process when there i s regular harvesting. In p a r t i c u l a r we s h a l l be interested i n what rates of harvesting can be sustained by a given population and what rates lead to the e x t i n c t i o n of the population. It w i l l be of i n t e r e s t to see how these r e s u l t s compare with those derived from equivalent deterministic models. We s h a l l assume that any harvesting which takes place does so during a short harvest season each year (or cycle) thereby ensuring that natural growth and mortality i n the population are unimportant during t h i s period. Such i s approximately the case for many migratory f i s h populations including the pink, coho and sockeye salmon of the P a c i f i c which have a four (or f i v e ) year l i f e - c y c l e and which are 132 harvested, mainly during a short period p r i o r to t h e i r entry into fresh water for spawning. S i m i l a r l y most exploited species of se a l are harvested only during a short period each year when they "haul out" onto t h e i r breeding grounds. In t h i s section we s h a l l consider only stationary harvest  p o l i c i e s i . e . we s h a l l assume that the decision whether to harvest or not, and i f so the s i z e of the harvest i n any year (or cycle) n , say, depends only on the size of the population , i n that year and not on the h i s t o r y of the population and i t s harvests. We can conveniently describe such a p o l i c y by a function h : [0, m] -»- [0, m] with h(x) <_ x where h(x) represents the s i z e of the harvest (measured i n the same units as the population i t s e l f ) from a population of s i z e x . We l e t u(x) = x - h(x) . Clea r l y u(x) represents the escapement from a population of s i z e x when the harvest p o l i c y h i s operative. If X^ i s the population s i z e i n year n immediately p r i o r to harvesting, then for a stationary p o l i c y h , the y i e l d i s n ( ^ n ) and the escapement i s u ( ^ n ) • The population i n year n + 1 p r i o r to harvesting i s Z f(u(X )) , where Z i s a random m u l t i p l i e r . This 6 n n n can be represented by the following diagram. 133 y i e l d 1 X target population h ( x n ) HARVEST natural mortality u(X n) escapement y i e l d | h ( X n + 1 ) REPRODUCTION NATURAL MORT-ALITY AND GROWTH X n + 1 = Z n £ < " < X n » •> target population reproduction and growth F i g . 2.6 The population process {X } represents the l e v e l s of n population immediately p r i o r to harvest i n each season. The dynamics of t h i s process are determined by equation 2.1.1 only now the expected reproduction function, f , i s replaced by the composition fou(x) . A convenient diagrammatic representation of the dynamics of a deterministic population subject to a regular stationary harvest p o l i c y has been used by K. Radway A l l e n (1973). I t i s an extension of the method used i n F i g . 2.1. The reproduction function f(x) i s plo t t e d on the ordinate against x on the abscissa i n the usual way, whilst the escapement function i s plotted as u(y) on the abscissa against y on the ordinate (see F i g . 2.7). For an i n i t i a l population l e v e l x^ measured on the ordinate the escapement u(x^) i s found on the abscissa. For t h i s parent population the resultant target population = f(u(x^)) i n the following year, i s found again on the ordinate, and so on. By x=u(y) 134 T X r s y x 3 y=f(x) X 2 x 1 F i g . 2.7 successively "bouncing" a l i n e back and f o r t h between the reproduction function and the escapement function the target populations i n subsequent years can be determined. I t i s easy to see as before that i f f and u are non-decreasing functions, then the population w i l l converge to a stable point of equilibrium where the escapement curve crosses the reproduction function from below. depicted g r a p h i c a l l y i n F i g . 2.8. The escapement i s p l o t t e d as u(y) against y . proportion of the target population i s removed each harvest season. In t h i s case h(x) = hx and u(x) = ux . Such a harvest p o l i c y could a r i s e , f or instance i n a f i s h e r y , where a f i x e d s i z e f i s h i n g f l e e t applies a constant " f i s h i n g e f f o r t " throughout the harvest season. The f i s h i n g e f f o r t i s a measure of the i n t e n s i t y of f i s h i n g . It could Some commonly discussed stationary harvested p o l i c i e s are F i g . 2.8(a) i l l u s t r a t e s the s i t u a t i o n i n which a f i x e d F i g . 2'.8 Modified s t a b i l i z a t i o n of escapement s — An "S, s p o l i c y " 136 f o r instance be represented by the number of nets i n the water at any time. I f we assume that throughout the harvest season the rate of capture i s proportional to the si z e of the f i s h i n g e f f o r t , and that natural growth and mortality are unimportant during t h i s time, then the number N(t) of survivors at time t from an i n i t i a l population s i z e x s a t i s f i e s | f = -F , N(0) = x , where F i s the rate of capture. At the end of the harvest season we have that the number of survivors (or escapement) N(T) i s - F T ' N(T) = x e or u(x) = ux , where u i s a constant depending on the e f f o r t applied. For t h i s reason a p o l i c y i n which a f i x e d proportion of the population i s captured each year has been c a l l e d a constant e f f o r t  harvest p o l i c y (see Ricker (1958 (b))). F i g . 2.8 (b) i l l u s t r a t e s the s i t u a t i o n where the rate of e x p l o i t a t i o n h(x)/x increases with the s i z e of the target population. F i g . 2.8 (c) i l l u s t r a t e s what Ricker (1958 (b)) has c a l l e d a stabilization-of-escapement p o l i c y . An attempt i s made to s t a b i l i z e the escapement at some l e v e l S , say. A harvest i s made i f and only i f the target population exceeds S , i n which case the excess over and above S i s harvested. We s h a l l show that such a p o l i c y i s optimal for c e r t a i n c r i t e r i a (see 2.3 and Chapter 3). For population l e v e l s much i n excess of S , harvesting a l l of the excess might require a very high l e v e l of harvest e f f o r t . I f th i s were to happen only infrequently i t would not be worth maintaining a harvest operation large enough to handle these rare large harvests. The p o l i c y i l l u s t r a t e d i n F i g . 2.8 (d) represents an attempt at stabilization-of-escapement with an upper l i m i t to the s i z e of the e f f o r t than can be mobilized. Ricker (1958 (b)) has c a l l e d t h i s p o l i c y p a r t i a l stabilization-of-escapement. In a randomly f l u c t u a t i n g environment s t a b i l i z a t i o n - o f - e s c a p e -ment may mean that i n some years no harvest may be undertaken. I f men's l i v e l i h o o d s were to depend upon the harvest such a p o l i c y would be d r a s t i c . F i g . 2.8 (e) shows a modified form of s t a b i l i z a t i o n - o f -escapement i n which some harvest i s allowed at low population l e v e l s . Radway A l l e n (1973) considers such a p o l i c y . A combination of (d) and (e) i s possible. F i g . 2.8 (f) shows what we s h a l l c a l l an (S,s) p o l i c y following the usage i n inventory theory. No harvest i s permitted unless the population i s i n excess of s i n which case the excess above S i s removed. We s h a l l show that t h i s i s an optimal p o l i c y i n c e r t a i n s i t u a t i o n s (see Chapter 3). We have seen how the dynamics and steady-states of a population with deterministic dynamics can be studied with the use of a diagram 138 l i k e F i g . 2.7. A s i m i l a r diagram f o r a stochastic population model can y i e l d some very i n t e r e s t i n g r e s u l t s concerning the steady-states of a population i n a f l u c t u a t i n g environment when a stationary harvest p o l i c y h i s operative. The dynamics of such a population are described by P n + l ( t ) = rm F(t|x) d p (x) 0 where F(t|x) = *( t / f o U ( x ) ) and the stationary d i s t r i b u t i o n s are given by rm P(t) = F(t|x) d p(x) 0 Let us f i r s t consider a compensation model i n which at su i t a b l y low l e v e l s the population w i l l , with p r o b a b i l i t y one, increase i t s s i z e ( i . e . af'(O) > 1) (see F i g . 2.9). Let us suppose that there i s a stationary harvest p o l i c y h(x) i n operation. For s i m p l i c i t y we w i l l assume that t h i s p o l i c y i s of the constant-effort type. Similar analysis w i l l be possible f or more general forms of harvest p o l i c y . For any given i n i t i a l target population X^ = x^ , say, measured on the ordinate the range of the target population i n the following year, X^ = Z^f(ux^) , can again be found on the ordinate, i n a s i m i l a r way as i n F i g . 2.7. If there were no harvesting the population would s e t t l e down to a steady-state d i s t r i b u t e d on [r, m] (Thm 2.1.1). I f now a x=uy constant e f f o r t harvest with proportional escapement u > l/af ' ( 0 ) , as i n F i g . 2.9 were applied, the population would be disturbed from i t s steady-state. I f t h i s harvest p o l i c y were to continue the population would eventually s e t t l e down to a new steady-state with stationary d i s t r i b u t i o n on [ r ^ , m\p as i n F i g . 2.9. (This can be proved i n the same way as Theorem 2.1.1). We see then that any rate of e x p l o i t a t i o n  i n which the proportional escapement i s greater than 1/af'(0) can  be sustained by the population. If a harvest e f f o r t f o r which the proportional escapement were les s than l / f ' ( 0 ) were to be continually applied then e x t i n c t i o n of the population would eventually r e s u l t . This follows from Theorem 2.1.2 since fou(x) < x for x > 0 , i n t h i s case. Any rate of  e x p l o i t a t i o n with proportional escapement l e s s than l / f ' ( 0 ) cannot  be sustained. This i s also predicted by the deterministic model. The deterministic model also predicts that harvesting with proportional 140 escapement greater than l / f ' ( 0 ) • can be sustained. We are unable to determine whether t h i s i s v a l i d for the stochastic model. I t corresponds to the case i n 2.1 with three possible outcomes - e x t i n c t i o n , n u l l recurrence or a stationary d i s t r i b u t i o n . I t could be that conser-vation standards based on a deterministic model would be inadequate for a population l i v i n g i n a f l u c t u a t i n g environment. As we have seen i f the harvest e f f o r t were so high that the proportional escapement were less than l / f ' ( 0 ) then e x t i n c t i o n would eventually r e s u l t . However i n the stochastic model, unlike the deterministic model, the approach toward e x t i n c t i o n would not n e c e s s a r i l y be monotonic. The managers of a resource could be deceived into b e l i e v i n g that the population could support such a high rate of e x p l o i t a t i o n on the evidence of some years i n which the population increased. Other years i n which the population decreased could be written o f f as unlucky "bad" years. However with such a high rate of e x p l o i t a t i o n these "bad" years would not be at a l l a t y p i c a l . Indeed s t a t i s t i c a l l y they would dominate over the "good" years i n which the population increased. I f harvested independently of other populations i t i s u n l i k e l y that a population s a t i s f y i n g a compensation model such as t h i s could ever be harvested to e x t i n c t i o n . When population l e v e l s became too low harvesting would l i k e l y cease. L e f t to i t s e l f the population would be capable of eventually regenerating i t s e l f to i t s former l e v e l s , although t h i s might take many years. However i f the population were being harvested i n conjunction with another (more productive) population 141 then i t i s quite possible that a c r i t i c a l rate of e x p l o i t a t i o n would be continued to the point at which the population became e x t i n c t . Such r e s u l t s based on deterministic models are well-known to e c o l o g i s t s . Similar r e s u l t s to the above hold for an over-compensation model, f o r which af'(O) > 1 . We now turn our attention to depensation models. We consider f i r s t l y a model for normal depensation (1.1(b)), i n which there i s a p o s i t i v e p r o b a b i l i t y of the population s i z e increasing at any l e v e l . Figs. 2.10 (a) and (b) both represent models for normal depensa-t i o n . In both cases, undisturbed, the population would be i n steady-state d i s t r i b u t e d on the i n t e r v a l [r, m] . F i g . 2.3 also represents a model for normal depensation. However we are unsure whether t h i s i s a v i a b l e model for an undisturbed population since we do not know whether steady-states other than 142 For the models of F i g . 2.10 (a) and (b) undisturbed the population w i l l reach a steady-state with stationary d i s t r i b u t i o n on [r, m] . A l l l e v e l s of population between r and m w i l l occur. With a constant-effort harvest allowing a proportional escapement greater than u as shown i n both diagrams, a new steady-state w i l l come about with a stationary d i s t r i b u t i o n on an i n t e r v a l greater than r . No population s i z e l e s s than r w i l l occur. (This can be proved l i k e Thm 2.1.1). E x p l o i t a t i o n with a proportional escapement greater than u can be sustained by the population. The deterministic model for normal depensation predicts that e x p l o i t a t i o n with a proportional escapement greater than u^ can be sustained while e x p l o i t a t i o n with proportional escapement les s than u^ cannot be sustained (see F i g . 2.10). This i s not true i n general i n the stochastic model. I f as i n both diagrams u^ < l/bf'(0) , then a s i t u a t i o n l i k e that shown i n F i g . 2.4 ar i s e s and an e x p l o i t a t i o n with proportional escapement only s l i g h t l y greater than u^ w i l l lead to ex t i n c t i o n . In t h i s case a l i m i t a t i o n on harvest e f f o r t , designed as  a conservation safeguard and derived from a deterministic population  model would be inadequate to ensure the conservation of the population. Whether u i s the minimum proportional escapement that i s compatible with conservation i s an open question. The case of a stationary constant-effort harvest allowing a proportional escapement between 1/bf'(0) and u corresponds to the unsolved case of F i g . 2.3. Cer t a i n l y u constitutes a threshold. For e x p l o i t a t i o n with proportional escapement greater than u , the population l e v e l w i l l 143 always be i n excess of r . For an e x p l o i t a t i o n with proportional escapement les s than u a r b i t r a r i l y low population l e v e l s can occur. (This corresponds to a switch from the s i t u a t i o n of F i g . 2.2 (a) to that of F i g . 2.3). F i n a l l y we look at the e f f e c t s of harvesting upon a population e x h i b i t i n g c r i t i c a l depensation as shown i n F i g . 2.5. See F i g . 2.11. x=u y x=uy m , i bf (x) f(x) af(x) r F i g . 2.11 Undisturbed the population would be i n steady-state with l e v e l s i n the i n t e r v a l [r, m] occurring. Harvesting with a proportional escapement greater than u (see F i g . 2.11) could be sustained by the population. For continued harvests of t h i s nature the population would reach a steady4state with population l e v e l s greater than r occurring (the stationary d i s t r i b u t i o n would be on an i n t e r v a l greater than r) . However hafvesting-rM-lowing a^ffSpof ^ ionai-Ses&apemgrit'less^than u ' could not be sustained by the population. (This i s the s i t u a t i o n of F i g . 2.4). 144 The deterministic theory predicts that harvesting with a proportional escapement greater than u^ can be sustained whereas that with a proportional escapement les s than u^ cannot be sustained. This i s not true for the stochastic model f o r a population l i v i n g i n a f l u c t u a t i n g environment. The c r i t i c a l l e v e l of proportional escapement i s d i f f e r e n t i n the case of a stochastic model and i n the case of a deterministic model. A conservation safeguard based on a  l i m i t a t i o n of harvest e f f o r t and developed from a deterministic  population model could be inadequate to protect a population l i v i n g i n  a f l u c t u a t i n g environment. A model such as t h i s can explain a sudden "crash" i n an exploited population. If the population were being r e g u l a r l y harvested with a harvest e f f o r t allowing proportional escapement s l i g h t l y greater than u , then the population would reach a steady-state with high (greater than r) population l e v e l s . However i f the harvest e f f o r t were increased j u s t beyond the threshold value (proportional escapement u) then the steady-state would be disturbed and population l e v e l s below r would become immediately possible. If t h i s increased rate of harvesting were to continue the population s i z e would eventually drop below the c r i t i c a l l e v e l c , and from then on regardless of whether harvesting were to continue or not, the population would monotonically decrease i n s i z e to eventual e x t i n c t i o n . Even for h a r v e s t - e f f o r t s only s l i g h t l y i n excess of the threshold s i z e the collapse of the population from i t s steady-state with high population l e v e l s (greater than r) to a l e v e l at which the population i s doomed (below c) could take place i n a remarkably short 145 time, e s p e c i a l l y i f the expected reproduction function i s steep i n i t s middle part. I t might w e l l be the case that i t would not be r e a l i z e d that the c r i t i c a l l e v e l of harvesting had been exceeded u n t i l i t were too  l a t e to save the population from destruction. This could happen, e s p e c i a l l y , i f for one or two "good" years the increased rate of harvesting were sustained. Any decreases could be dismissed as exceptional "unlucky" years, without i t being r e a l i z e d that with such a high rate of e x p l o i t a t i o n , the "unlucky" years could s t a t i s t i c a l l y dominate over the "lucky" ones. A s i m i l a r crash i s predicted by a deterministic depensation model (Clark 1974). However as we have seen i n that case the threshold harvest rate i s greater than i t i s i n the stochastic model. Also i n the deterministic theory the collapse of the population i s monotonic which i s not n e c e s s a r i l y the case i n the stochastic theory. Apart from those r e s u l t s proved using the Martingale Convergence Theorem and Jensen's Inequality (Thm 2.1.2 and i t s consequences) the r e s u l t s of 2.1 and 2.2 are a l l proved using the Ergodic Theorem of Doob (1953, p. 214). We have not formally established the transience and recurrence of c e r t a i n sets but the i n t u i t i v e arguments are f a i r l y convincing and formal proofs could be developed. When we have used Doob 1s theorem we have assumed that the d i s t r i b u t i o n of the random m u l t i p l i e r s Z has a continuous, non-zero n (except possibly on a set of measure zero) density on [a, b] . However i n order to use the theorem to prove the e r g o d i c i t y of the population 146 process i t i s not necessary to assume that the population model i s of a m u l t i p l i c a t i v e form. In order to use Doob's theorem i t i s s u f f i c i e n t that the population process be Markovian and have a stochastic kernel F(t|x) which has a density continuous i n both variables on the recurrent set [r, m] x [r, m] . The r e s u l t s then, w i l l hold for a non-multipli-cative Markovian population model, which has a continuous density kernel and for which the curves max{X , , |x = x} and min{X ,, |x = x} n+11 n n+11 n are e s s e n t i a l l y l i k e the curves bf(x) and af(x) shown i n the various diagrams. As we have seen, the predictions of the stochastic theory often d i f f e r from those of the deterministic theory. I f standards of conservation are to be applied for the protection of the population they would be more r e l i a b l e i f derived from a stochastic analysis based on the maximum and minimum reproduction functions ( i . e . max{X , , ] x = x} n+11 n and min{X ., X = x}) rather than on a deterministic analysis of the n+11 n J deterministic reproduction function. With large enough data sets estimation of maximum and minimum reproduction functions should be no more d i f f i c u l t than the estimation of a reproduction function. In the next section we investigate the y i e l d from stationary harvest p o l i c i e s when the population i s i n steady-state. 2.3 The Steady-State Harvest Y i e l d Consider the deterministic population model Xn+1 = f <V 147 where f i s any reproduction function. For a stationary harvest p o l i c y characterized by the function h(x) , the escapement u(x) i s equal to x - h(x) , and the population l e v e l s , p r i o r to harvest s a t i s f y x = fou(x ) . n+1 n In equilibrium, p r i o r to harvest the population l e v e l i s x , where x i s a stable s o l u t i o n to x = f°u(x) . The equilibrium y i e l d i s h(x) = x - u(x) = f°u(x) - u(x) = g(u(x)) where the function g i s defined as g(x) = f(x) - x . The maximum equilibrium y i e l d , or the maximum sustainable y i e l d (m.s.y.) as i t i s known i n the f i s h e r i e s l i t e r a t u r e , i s obtained when u(x) =. x where x i s a point at which g(x) attai n s i t s maximum. We see then that the m.s.y. i s obtained for any stationary p o l i c y , for which the escapement function x~= u(y) i n t e r s e c t s the reproduction function y = f(x) at the point (x, f(x)) (see F i g . 2.7). 148 The value of the m.s.y. i s g(x) = f(x) - x . We now turn our attention to the stochastic model X^ +^ = Z^f(X n) . Let us suppose for a given stationary p o l i c y h(x) , a stationary d i s t r i b u t i o n f o r the process e x i s t s , i . e . l e t us suppose that the d i s t r i b u t i o n of the target population, tends to an equilibrium. The y i e l d , when the system i s i n equilibrium, w i l l be a random v a r i a b l e . In order to compare y i e l d s i n the stochastic and deterministic models, we s h a l l consider the expected y i e l d of the stochastic model, i n equilibrium. I f X i s a r.v. with the l i m i t i n g stationary d i s t r i b u t i o n , the expected equilibrium y i e l d i s E(h(X)) = E(X) - E(u(X)) . (1) From the Strong Law of Large Numbers for Markov Process (Doob, p. 220) we have that the expected equilibrium y i e l d i s the same as the long run average y i e l d of the process. We s h a l l compare the expected equilibrium y i e l d from the stochastic model with the equilibrium (or sustainable) y i e l d from the deterministic model. F i r s t l y , since X has the stationary d i s t r i b u t i o n we have that X and Zf°u(x) have the same d i s t r i b u t i o n . Taking expectations, remembering that Z and X are independent we get that E(X) = E(f°u(X)) . It follows from (1) that the expected equilibrium y i e l d can be written 149 E(h(X)) = E ( f o U ( X ) ) - E(u(X)) ' = E(g(u(X) . Now i f ¥ i s the d i s t r i b u t i o n function of the stationary d i s t r i b u t i o n , we can write t h i s as E(h(X)) = g(u(x))d¥(x) ft where ft i s the range of the d i s t r i b u t i o n ¥ . Since g(x) <_ g(x) , i t follows that E(h(X)) < g(x) , with equality occurring only i f u(x) = x for a l l values of X i n ft . The condition u(X) = x f o r a l l X i n ft , implies a constant escapement x , for a l l equilibrium values of the target population. For t h i s to occur i t i s necessary that the l e v e l x be sustainable, i n the sense that pr{X > x X = x} = 1 , (for an n+1 — 1 n escapement of x would not be possible from a target population < x) and also that u(x) = x over the range [af(x), bf(x)] . Outside of t h i s range u can behave i n any way, provided that convergence to a value i n [af ( x ) , bf(x)] i s assured with p r o b a b i l i t y one. We s h a l l say that any p o l i c y as described above i s e f f e c t i v e l y one of s t a b i l i z a t i o n  of escapement at x , since i t i s of t h i s type over the range of values experienced i n equilibrium, although i t can include p o l i c i e s such as the p a r t i a l s t a b i l i z a t i o n of escapement (Fig. 2.8(d)), the modified s t a b i l i z a t i o n of escapement (Fig. 2.8(e)) and the (S,<s) p o l i c y ( Fig. 2.8(f)) 150 We can thus state the r e s u l t proved above as the following theorem. Theorem 2.3.1 If f i s any expected reproduction function, and h i s any stationary p o l i c y f o r which a l i m i t i n g stationary d i s t r i b u t i o n f o r the population e x i s t s , then the expected equilibrium y i e l d from the p o l i c y h i s no greater than the maximum sustainable y i e l d i n the equivalent deterministic model. In f a c t , the expected equilibrium y i e l d i s s t r i c t l y less than the m.s.y., unless the l e v e l x , which gives m.s.y. i s sustainable, and the p o l i c y h , considered i s e f f e c t i v e l y one of s t a b i l i z a t i o n of escapement at the l e v e l x . This theorem i s quite important. I t t e l l s us that we cannot u t i l i z e the chance fl u c t u a t i o n s i n the reproduction dynamics to increase average y i e l d over the long run. The m.s.y. as cal c u l a t e d from a deterministic model i s an upper bound f or the long-run average y i e l d that can be derived from the resource using a stationary p o l i c y . The maximum long-run average y i e l d from a population obeying the  stochastic reproduction model with a stationary harvest p o l i c y i s no  greater than the maximum sustainable y i e l d computed from the equivalent  deterministic model. We s h a l l now turn our att e n t i o n to the problem of comparing equilibrium y i e l d s i n the stochastic and deterministic models f o r a 151 given stationary p o l i c y h . We s h a l l r e f e r to the equilibrium y i e l d t f o r a given p o l i c y computed on the basis of a deterministic population model as the nominal y i e l d for that p o l i c y . From the e a r l i e r development we have that the nominal y i e l d of a stationary p o l i c y h i s h(x) , where x = fou(x) . We s h a l l be interes t e d i n how the stochastic expected equilibrium y i e l d (or long-run average y i e l d ) f o r a given p o l i c y compares with the nominal y i e l d of that p o l i c y . We s h a l l r e s t r i c t our attention to concave reproduction functions, thus eliminating depensation models. Also we s h a l l consider only stationary harvest p o l i c i e s h , f o r which the escapement function i s continuous, concave and non-decreasing. This implies that both the s i z e of the harvest and the harvest rate (proportional s i z e of harvest) do not decrease with the s i z e of the target population. We do not consider p o l i c i e s l i k e the non-concave p a r t i a l s t a b i l i z a t i o n of escapement (Fig. 2.8(d)) or the discontinuous (S,s) p o l i c y (Fig. 2 . 8 ( f ) ) . We s h a l l suppose that under the harvest p o l i c y h , there i s a l i m i t i n g stationary d i s t r i b u t i o n . From 2.2 we know that a s u f f i c i e n t condition f o r t h i s i s that h'(0) < 1 - 1/af'(0) . We need the following lemma. Lemma 1 If f i s a s t r i c t l y concave function over a given domain, and u i s a function concave and non-decreasing over the range of f , then the composition uof i s concave over the domain of f . Furthermore the concavity i s s t r i c t i f u i s not a constant function. Proof. We wish to show that for a l l ^ x 2 and a l l t e (0,1) , that uof(tx + ( l - t ) x 2 ) I t u o f ( X l ) + (l-t)u°f(x 2) We can write the l.h.s. of the above as u ( t f ( X l ) + ( l - t ) f ( x 2 ) ) + [ u ( f ( t x 1 + ( l - t ) x 2 ) - u ( t f ( X l ) + ( l - t ) f ( x 2 ) ) ] Now f ( t x + ( l - t ) x 2 ) > t f ( x 1 ) + ( l - t ) f ( x 2 ) from the concavity of f , and hence i t follows, from the monotonicity of u ,- that the expression i n square brackets i s >_ 0 for a l l x^ ^ x 2 , with equality only i f u i s a constant function. Now u ( t f ( x 1 ) + ( l - t ) f ( x 2 ) ) > t u ( f ( X l ) ) + ( l - t ) u ( f ( x 2 ) ) and hence we have that the l.h.s. of the f i r s t equation i s >_ the r.h.s., with equality only i f u i s a constant function. This proves the lemma. We are now ready to compare the expected equilibrium y i e l d for a p o l i c y h , with the nominal value of i t s y i e l d . We s h a l l make the r e s t r i c t i o n on h , that the deterministic equilibrium escapement i s notgreater than x the l e v e l at which m.s.y. i s obtained. 153 Theorem 2.3.2 Consider a s t r i c t l y concave expected reproduction function f , and a stationary harvest p o l i c y h , f o r which the escapement function u i s continuous concave and non-decreasing, and for which the equilibrium escapement i n the deterministic model i s no greater than x , the escapement that gives m.s.y. I f , f o r the stochastic model under the p o l i c y h there i s a stationary equilibrium population d i s t r i b u t i o n f o r the population, then the expected equilibrium y i e l d from the p o l i c y h i s no greater than i t s nominal value. i . e . i f u(x) <_x , then E(h(X)) <_ h(x) Furthermore the expected equilibrium y i e l d from a given p o l i c y as above i s equal to i t s nominal value only i f i t i s e f f e c t i v e l y a p o l i c y of s t a b i l i z a t i o n of escapement at a sustainable l e v e l . Proof. Suppose X i s a r.v. with the stationary d i s t r i b u t i o n . As i n Theorem 1, we have that the expected equilibrium y i e l d i s E(h(X)) = E(X-u(X)) = E(f(u(X)) - u(X)) = E(g(u(X)) , since X and Zfou(X) have the same d i s t r i b u t i o n . The nominal y i e l d or deterministic equilibrium y i e l d of h 154 i s given by h(x) = x - u(x) = f°u(x) - u(x) = g(u(x)) , where x = f°u(x) i s the equilibrium population l e v e l , p r i o r to harvest. Since X and Zf°u(X) have the same d i s t r i b u t i o n , we have that u(X) and u(Zf°u(X)) have the same d i s t r i b u t i o n . I t follows from Jensen's Inequality that E(u(X)) = E(u(Zfo U(X))) <_u(E(fo U(X))) with equality only i f u i s l i n e a r . Now we have again by Jensen's Inequality since f i s concave, that E(f°u(X)) <_ f (E(u(X)) with equality only i f u i s constant over the range ft of X . Since u i s non-decreasing we have, then, that E(u(X)) < uof(E(u(X))) with equality only i f u i s constant over ft . Now from the lemma we have that u°f i s concave. Also we have that u(x) = u(fou(x)) = u°f(u(x)) , and that u(0) = 0 = uof(O) . Hence we have that uof(x) i s ,;greaier.atnafi ' x Jtfi, B.0?M(x) ] , .and .jless.thahivx i f o r x > u(x) . It follows from above, since 155 E(u(X)) <_ uof (E(u(X)) that E(u(X)) <_ u(x) , with equality only i f u i s constant over ft . Now, again by Jensen's Inequality, from the s t r i c t concavity of g we have E(h(X)) = E(g°u(X)) £g(E(u(X)) with equality only i f u(X) i s constant over ft . We have assumed that u(x) j£ x , and hence since E(u(X)) <_ u(x) <_ x we have from the monotonicity of g over [0, x] that E(h(X)) < g(u(x)) = h(x) , with equality only i f u i s constant over ft . As i n Thm. 1, we have that equality occurs only for a p o l i c y which i s e f f e c t i v e l y one of s t a b i l i z a t i o n of escapement at a sustainable l e v e l S , say. This completes the proof of the theorem. The theorem i s true for such concave stationary harvest p o l i c i e s as the i n c r e a s i n g - e f f o r t harvest (Fig. 2.8(b)), the s t a b i l i z a t i o n -of-escapement p o l i c y (Fig. 2.8(e)) and the modified s t a b i l i z a t i o n - o f -escapement p o l i c y (Fig. 2.8(e)), provided that i n a l l cases we assume that the equilibrium deterministic escapement i s no greater than that which gives m.s.y. In the case of a constant-effort harvest t h i s l a s t assumption 156 i s not necessary. We have the following r e s u l t . C o r o l l a r y 1 For any concave expected reproduction function f and any constant-effort harvest p o l i c y h(x) = hx for which an equilibrium d i s t r i b u t i o n e x i s t s , the expected equilibrium y i e l d i n a f l u c t u a t i n g environment from t h i s p o l i c y i s les s than the nominal y i e l d of the p o l i c y . Proof. I f the process i s i n steady-state then X ... and X both J n+1 n have the same d i s t r i b u t i o n . Furthermore X ,., = Z f(uX ) and hence n+1 n n X = E(f(uX ) < f(uX) n by Jensen's Inequality, where X i s the mean of the equilibrium d i s t r i b u t i o n . Now x < f (ux) f o r x < x and x > f(ux) for x > x . It follows that X < x and hence that hX < hx or E(h(X )) < hx n which proves the r e s u l t . How necessary i n proving the theorem i s the assumption that the equilibrium deterministic escapement x of the p o l i c y be no les s than the l e v e l x which gives m.s.y. and what are the implications of t h i s assumption? We show f i r s t l y that i n general the theorem i s not true without t h i s assumption. We consider the case of a p o l i c y of s t a b i l i z a t i o n - o f -escapement at a l e v e l S > x . This p o l i c y has escapement function u(x) x , x <_ S S , x > S As before the expected equilibrium y i e l d i s E(h(X)) = E(X-u(X)) = E(gou(X)) and the nominal y i e l d or deterministic equilibrium y i e l d i s h(x) = fou(x) - x which i n t h i s case reduces to h(x) = f ( S ) - S ,' since u(x) = S . If the l e v e l S i s sustainable we have that X >_ S with p r o b a b i l i t y one, i t follows that u(X) = S w.p.l., and hence that E(h(X)) = f(§) - S , and so i n t h i s case the expected equilibrium y i e l d i s the same as the nominal y i e l d . However i f S i s not sustainable the d i s t r i b u t i o n of X includes values le s s than S . Let us suppose that X i s d i s t r i b u t e d on Q = [p, q] where p <_ S . 158 Consider f i r s t l y the case S' < p < S , where S' i s the s o l u t i o n , l e s s than x to g(x) = g(S) . Such a s o l u t i o n e x i s t s from the concavity of g (see F i g . 2.12). F i g . 2.12 In t h i s case for r e a l i z a t i o n s x of X i n the i n t e r v a l [p,S) we have g°u(x) = g(x) > g(S) = f(S) - S , and for r e a l i z a t i o n s x of X i n the i n t e r v a l [S, q] we have gou(x) = g(S) = f(S) - S . Hence int e g r a t i n g over [p, q] = Q, we have E(h(X)) = E(g°u(X)) > f(S) - S = h(x) . In t h i s case then the expected equilibrium y i e l d i s greater than the nominal y i e l d f o r t h i s p o l i c y . In the case p < S' , for some r e a l i z a t i o n s x of X we have gou(x) > f(S) - S , and f o r other r e a l i z a t i o n s g°u(x) £ f(S) - S . In t h i s case the expected equilibrium y i e l d could be either greater or les s than the nominal y i e l d , which case depending on f , S and the d i s t r i b u t i o n $ of the random m u l t i p l i e r s Z n We have shown that Thm. 2 i s not true i n general without the assumption that u(x) <_ x . For what p o l i c i e s i s the equilibrium 159 o deterministic escapement u(x) no greater than x ? F i r s t l y we know already that the sustainable y i e l d i s maximized by keeping the population at the l e v e l x . For a p o l i c y which maximizes deterministic sustainable economic y i e l d the escapement i s greater than x (see Clark 1971)• For a s u i t a b l y high time-preference rate, the p o l i c y which maximizes discounted, economic y i e l d allows an escapement les s than x (see Clark 1971, 1973 and Chapter 4). Thus we see from Thm. 2 that i f a p o l i c y i s chosen to maximize sustainable y i e l d , or discounted economic y i e l d (for s u i t a b l e high discount r a t e ) , the long-run average y i e l d r e s u l t i n g from a population model which i s i n fac t stochastic may well be les s than the nominal y i e l d predicted from a dete r m i n i s t i c model. They can only be the same i f the chosen escapement l e v e l i s sustainable. We note that the above r e s u l t s concerning s t a b i l i z a t i o n of escapement p o l i c i e s are true for non-concave reproduction functions. We summarize these r e s u l t s as a c o r o l l a r y . Corollary 2 Consider a model with an expected reproduction function f of the compensation, over-compensation or depensation type, and a p o l i c y of s t a b i l i z a t i o n of escapement at a l e v e l S . (a) I f the l e v e l S i s sustainable then the expected stochastic equilibrium y i e l d i s the same as the nominal y i e l d of the p o l i c y . (b) I f the l e v e l S i s not sustainable, and i s no greater than x , the l e v e l of m.s.y., then the expected equilibrium y i e l d i s 160 l e s s than the nominal y i e l d of the p o l i c y . (c) I f the l e v e l S i s not sustainable and i s greater than x , then expected equilibrium y i e l d can be greater or les s than nominal y i e l d of the p o l i c y . I f the stationary d i s t r i b u t i o n i s on [p> q] > where S' < p < S , with S' as defined above, then the expected equilibrium y i e l d i s greater than the nominal y i e l d of the p o l i c y . Ricker (1958) used computer simulation experiments to compare the average y i e l d when there are chance f l u c t u a t i o n s i n the reproduction dynamics, with the maximum sustainable y i e l d predicted by a deterministic model. He was faced with a choice of stationary p o l i c i e s to employ i n the stochastic model, to correspond to that allowing equilibrium escapement x i n the de t e r m i n i s t i c model. We have already remarked that the m.s.y. i s obtained i n the deterministic model f o r any escapement function u , for which x = u(y) and y = f(x) i n t e r s e c t at the point (x, f(x)) . Ricker considered three a l t e r n a t i v e s . F i r s t l y he considered the " f i x e d rate", or constant e f f o r t harvest p o l i c y when the same proportion x/f(x) of the target population i s removed each year (see F i g . 2.8(a)). Secondly he considers the p o l i c y of s t a b i l i z a t i o n of escapement at x (Fig. 2.8(c)) and t h i r d l y the p o l i c y of p a r t i a l s t a b i l i z a t i o n of escapement (at the l e v e l x ( F i g . 2.8(d)) . Ricker used numerical simulation over 24 generations to determine the average y i e l d from each of these p o l i c i e s . . His conclusions were that, f i r s t l y , for.the f i x e d rate of harvest the average y i e l d could be eit h e r greater or l e s s than the m.s.y., which, 161 depending on the nature of the reproduction function. Secondly, for a s t a b i l i z a t i o n of escapement p o l i c y the average y i e l d i s greater than the m.s.y., and that the difference increases with the degree of v a r i a b i l i t y introduced, being from 12% to 41% for the lower degree of v a r i a b i l i t y used and being from 26% to 81% for the higher degree of v a r i a b i l i t y . T h i r d l y for p a r t i a l s t a b i l i z a t i o n of escapement, average y i e l d s are higher than m.s.y. but not as high as with complete s t a b i l i z a t i o n . These simulations were repeated by Larkin and Ricker (1964) using runs of 200 generations and the r e s u l t s were confirmed. These r e s u l t s do not agree with Theorem 1 , above. In order to understand why t h i s i s so we need to look at the way that the random factor was incorporated i n the simulation model of Ricker. We s h a l l show that a systematicsbias i s introduced into the model, with the stochastic factor. The apparent increases i n average y i e l d are due to t h i s b i a s , and the conclusions of Ricker (1958) and Larkin and Ricker (1964 are therefore i n v a l i d . The stochastic model used i s described by Ricker (1958, p. 994) " E f f e c t s simulating randomly-occurring environmental v a r i a b i l i t y are now to be super-imposed on the equilibrium y i e l d s i t u a t i o n s . An obvious model for such e f f e c t s w i l l be m u l t i p l i e r s obtained by random s e l e c t i o n from a table of factors whose frequencies are d i s t r i b u t e d as a normal frequency d i s t r i b u t i o n (reference given). The p o s i t i v e favourable factors are applied as m u l t i p l i e r s and the negative ones as d i v i s o r s to the mean reproduction indicated by the curve i n question." 162 This i s more e x p l i c i t l y stated i n Larkin and Ricker (1964, p. 1) "Scaling of the (standard normal) deviates to provide a s p e c i f i e d range of extremes was accomplished as i n Ricker (1958) by multiplying the deviates by an appropriate f a c t o r . The absolute quantity was then augmented by one and the deviates used as m u l t i p l i e r s or d i v i s o r s depending on sign." ' In other words a model such as ours, X , = Z f (X ) , n+1 n n was used. The random m u l t i p l i e r s Z are determined by n Z = { n l + k Y n , y n > 0 1/d-kkYY), y n < 0 , where ^- n^ ^ s a sequence of random standard normal deviates and k i s a non-negative constant. The d i s t r i b u t i o n function of the m u l t i p l i e r s Z i s n * ( t ) = G ( ^ ) , t < l GC-^1) , t > 1 where G i s the d.f. of a standard normal v a r i a t e . The mean of the d i s t r i b u t i o n $ i s greater than one. This can be shown as follows E(Z n) t t-1 r l 0 kt t ,1-1/tx ~2 } d t I 163 2 where g(t) = e ^ i s the density of a standard normal / 2 T 1 _-%t: d i s t r i b u t i o n . By a change of v a r i a b l e we get c 0 0 rO E(Z n) = (l+kz)g(z)dz + 0 (l+kz)g(z)dz + 1 1 - kz 0 n ^2  (kz) 1 - kz g(z)dz g(z)dz 1 + J&£_ g(z)dz > ± 1 - kz J —00 Furthermore the amount by which E(Z ) exceeds one, increases with k . n Thus we see that the model used i n both simulations i s biased i n the sense that, f o r a given escapement the expected recruitment i n the stochastic model i s greater than the recruitment i n the deterministic model which they use for comparison, and furthermore the degree of bias increases with what Ricker c a l l s "the degree of v a r i a b i l i t y . " The apparent increases i n y i e l d discovered by Ricker and Larkin and Ricker are due to t h i s bias i n the reproduction r e l a t i o n s h i p , and do not therefore support the conclusions which these authors draw. Indeed we have shown i n Thm. 1 that no stationary p o l i c y can increase the long-run average y i e l d i n a stochastic model, above the maximum sustainable y i e l d as computed from the equivalent deterministic model. Tautz, Larkin and Ricker (1969) have simulated the e f f e c t s of long term c y c l i c environmental f l u c t u a t i o n s for a discrete-time pop-164 u l a t i o n model. They introduce a random factor i n a s i m i l a r way as i n the papers of Ricker and Larkin and Ricker, and hence t h e i r numerical r e s u l t s are s i m i l a r l y biased and some of t h e i r conclusions are again i n v a l i d . The Optimum Stationary P o l i c y f o r the Stochastic Model, and V a r i a b i l i t y  i n Y i e l d An i n t e r e s t i n g question which a r i s e s from the inv e s t i g a t i o n s of Ricker i s , which stationary harvest p o l i c y gives maximum expected equilibrium y i e l d , (or maximum long-run average y i e l d ) i n a stochastic model? We have already answered t h i s f o r a s p e c i a l case, i n Thm. 1, fo r we have from that theorem that i f the l e v e l x which gives m.s.y.  i n the deterministic model i s sustainable, then the p o l i c y of  s t a b i l i z a t i o n of escapement at x maximizes expected equilibrium  y i e l d . This r e s u l t holds f o r any expected reproduction function. I f x i s not sustainable the question remains unanswered. I t seems l i k e l y that a p o l i c y of s t a b i l i z a t i o n of escapement maximizes expected y i e l d , although we have not been able to show t h i s i n general, or to determine an optimal l e v e l of escapement. To show that a p o l i c y of s t a b i l i z a t i o n of escapement i s optimal ( i n the sense of maximizing expected y i e l d ) we could t r y to show that f o r any f i x e d l e v e l S , the p o l i c y of s t a b i l i z a t i o n of escapement at S gives a greater y i e l d than any other p o l i c y allowing a deterministic equilibrium escapement S ( i . e . show that s t a b i l i z a t i o n 165 of escapement at S. i s maximal among p o l i c i e s u(y) which i n t e r s e c t f(x) at (S,f(S)) . Galto and R i n a l d i (to appear) have shown that i f S i s a sustainable l e v e l , the p o l i c y of s t a b i l i z a t i o n of escapement at S gives a greater expected y i e l d than the constant e f f o r t harvest g characterized by u(x) = Y(s)"x ' They assume that f i s concave. The paper of Galto and R i n a l d i , which, apart from the simulation papers of Ricker (1958) and Larkin and Ricker (1964), appears to be the only published work on equilibrium y i e l d s i n f l u c t u a t i n g environments, addresses another i n t e r e s t i n g question raised by Ricker (1958). Although s t a b i l i z a t i o n of escapement at a l e v e l S may give greater long-run average y i e l d than, say, a p o l i c y of constant e f f o r t harvest allowing proportional escapement S/f(S) the " v a r i a b i l i t y " of the annual y i e l d s may well be greater for the former p o l i c y . Great v a r i a b i l i t y i n annual y i e l d i s undesireable from the economic point of view, both for the fisherman (or harvester), and the intermediate industries and for the consumer. In defining an optimality c r i t e r i o n t h i s f a c t should, perhaps, be considered. Increasing long-run average y i e l d may be an undesireable goal i f the v a r i a b i l i t y of the annual y i e l d s increases too d r a s t i c a l l y as a r e s u l t of such actions. Formulation of an optimality c r i t e r i o n along these l i n e s i s b a s i c a l l y an economic problem. The r e l a t i v e values of y i e l d and s t a b i l i t y are d i f f i c u l t to assess, and are l i k e l y to be highly subjective. We do not address the problem here. 166 Ricker (1958) i n his simulation experiments summarized the v a r i a b i l i t y of the catch by two s t a t i s t i c s — the range s t a t i s t i c and the s t a t i s t i c for the numer of years i n which no catch was taken. His re s u l t s showed that f o r a given reproduction model, the p o l i c y of s t a b i l i z a t i o n of escapement gave the greatest range for annual y i e l d s and also gave the highest number of years i n which no catch was taken. The p o l i c y of p a r t i a l s t a b i l i z a t i o n (Fig. 2.8(e)) although giving a lower average y i e l d also gave l e s s v a r i a b i l i t y as measured the above two s t a t i s t i c s . The constant e f f o r t harvest, although giving lowest y i e l d s also gave lowest v a r i a b i l i t y . (We note"that these conclusions of Ricker are not in v a l i d a t e d by the bias i n the model discussed e a r l i e r , since i n t h i s case comparisons are not made between d i f f e r e n t reproduction models). Galto and R i n a l d i define two measures of v a r i a b i l i t y . F i r s t l y they consider the variance of the equilibrium y i e l d , and secondly they consider the maximum p o s i t i v e and negative deviations from the nominal y i e l d . They compare only a p o l i c y of s t a b i l i z a t i o n of escapement at a sustainable l e v e l S , and the constant e f f o r t p o l i c y allowing a proportional escapement S/f(S) . Up to a point t h e i r r e s u l t s confirm those of Ricker. They show that for a s u i t a b l y high variance of the random m u l t i p l i e r s , the variance of the y i e l d from the s t a b i l i z a t i o n of escapement p o l i c y exceeds the variance of the y i e l d from the constant e f f o r t p o l i c y . They also show that the maximum p o s i t i v e and maximum negative deviations of the y i e l d from i t s nominal value are greater i n the case of s t a b i l i z a t i o n 167 of escapement provided that S < x . If S > x the maximum p o s i t i v e and negative deviations of the y i e l d from the nominal value are greater i n the case of constant e f f o r t . From t h i s i t can be concluded that the range of the d i s t r i b u t i o n of the y i e l d i s greater f o r s t a b i l i z a t i o n of escapement i f S < x , and greater f o r constant e f f o r t i f S > x . The Optimum Constant Rate of Harvest Another question of i n t e r e s t a r i s e s from Ricker's (1958) paper. This can be put as follows. I f a constant e f f o r t harvest i s assumed, what i s the "best" rate of harvest to employ? In a f i s h e r y the harvest rate can be reckoned to depend on the f i s h i n g e f f o r t . This l a t t e r quantity can be thought of as a measure of the i n t e n s i t y of f i s h i n g . In an unregulated f i s h e r y the f i s h i n g e f f o r t can be thought of as being proportional to the size of the f l e e t . Thus i f the only regulation i n the f i s h e r y i s through the s i z e of the f l e e t (by s e l l i n g l i c e n c e s e t c . ) , the problem of choosing the "best" sized f l e e t can be represented mathematically as a problem of choosing the "best" rate of harvest. Of course the c r i t e r i o n of optimality i s one that i s open to much discussion (see introduction). Here we s h a l l only consider the problem of maximizing long run average y i e l d , (or expected equilibrium y i e l d ) . Ricker (1958, p. 1004) i n the summary of h i s paper makes the claim that, "the best constant rate of e x p l o i t a t i o n when stocks fluctuate (over the range examined) i s the same, or very close to the 168 best rate when there i s no environmental v a r i a t i o n . " Ricker does not report any simulation experiments i n which t h i s question was investigated and no arguments appear to be given i n support of t h i s claim. I t i s not at a l l clear that optimum stochastic rate of e x p l o i t a t i o n i s the same as the optimum deterministic one. We have been able to show that these optimizing rates are i n f a c t the same for the s p e c i a l case of the reproduction function f(x) = Ax b b < 1 . In t h i s case i f a = min Z > 0 , there i s n c e r t a i n l y a set (0, r ] of sustainable l e v e l s , since f'(0) = 0 0 , and hence there i s a l i m i t i n g stationary d i s t r i b u t i o n . For a fixed proportional escapement, u = 1-h , we have . .. Au b X 1 n+1 n n ' ; b b X = 1? Au X and hence by induction n-1 .n-J-1 (XbJ) , n X n = O^ Z.b ) A J = ° u ^ 1 X 0 b . j=0 J U Since {X^} converges to the l i m i t i n g stationary d i s t r i b u t i o n we have that {E(Xn>} converges to the mean of t h i s l i m i t i n g d i s t r i b u t i o n X(u) , say - h 1 _ b n h 3 X<.u) = (Au ) l i m E( n Z. ) n-x» j=o ^ n b J n j Now E( II Z. ) = exp{ I £n(E(Z.) )} by independence, j=0 3 j=0 3 b j b j and f o r each j > 0 , E(Z^ ) < [E(Z )] = 1 by Jensen's Inequality, n b J • and hence £ £n(E(Z.) ) i s a serie s of negative terms, and thus j=0 2 169 converges e i t h e r to a negative f i n i t e l i m i t or to negative i n f i n i t y . n b j Thus E( II Z. ) converges e i t h e r to zero, or to a l i m i t l e s s than 3=0 3 one. The former case i s not possible since the stationary d i s t r i b u t i o n i s on an i n t e r v a l away from zero, and hence we have 1 . X(u) = c(Au ) where c i s a constant independent of u with 0 < c < 1 . The deterministic process converges to the equilibrium l e v e l x(u) = (Au") and so we have X(u) = cx(u) , and the steady-state y i e l d s are s i m i l a r l y r e l a t e d , E(h(X)) = (1-u) X(u) = c(l-u)x(u) = ch(x) . It follows that the same u i s optimal i n both cases, i . e . the same fix e d rate of harvest i s optimal i n both cases. This optimal rate i s e a s i l y computed f o r i n the deterministic model the maximum sustainable y i e l d r e s u l t s from an escapement x , where f'(x) = 1 f'(x) = A b x b _ 1 and so 1 1-b x = (AB) The optimal escapement proportion i s f(x) b A ( A b ) 1 _ b and the optimal rate of harvest i s h = 1 - b . We have not been able to f i n d the constant rate of e x p l o i t a t i o n that leads to maximum expected equilibrium y i e l d , f o r more general reproduction functions. In the following chapter we look at a dynamic optimization model. 171 Chapter 3  An Economic Optimization Model 3.1 Introduction In t h i s chapter and the next we s h a l l again be concerned with the stochastic population model of 1.3. In Chapter 2 we discussed some aspects of the steady-state y i e l d s from stationary harvest p o l i c i e s . We touched upon the question of which stationary p o l i c y w i l l give the greatest expected equilibrium y i e l d , or the greatest long-run average y i e l d . In t h i s chapter we s h a l l look at optimization problems from a dynamic point of view. We s h a l l employ the method of Dynamic Programming, one of the main tools of stochastic optimal c o n t r o l theory. B a s i c a l l y we s h a l l be concerned with f i n d i n g p o l i c i e s which are optimal i n some sense. At the outset of any optimal c o n t r o l theory problem one i s faced with the c r i t i c a l question as to what constitutes an optimal p o l i c y . Mathematically t h i s involves defining an objective function and a set of admissible p o l i c i e s . An optimal p o l i c y i s then defined as a p o l i c y which maximizes the objective function over the admissible set. The f i r s t problem then i s to define an objective function and an admissible set which reasonably r e f l e c t the objectives of the resource owner or manager, and the p o l i c y options open to him. The r e s u l t s of optimal control theory can be interpreted i n 172 two ways. F i r s t l y , one can regard them as normative i n the sense that they o f f e r suggestions as to what the "best" p o l i c y i s . Of course the v a l i d i t y of any r e s u l t s i n applied mathematics can only be as good as the v a l i d i t y of the underlying model. We have already recognized that the population model under consideration i s a great o v e r - s i m p l i f i c a t i o n of the real-world s i t u a t i o n . S i m i l a r l y any objective function and set of admissible p o l i c i e s which we can develop w i l l most l i k e l y greatly over-simplify the complex of objectives and a l t e r n a t i v e s that the resource manager has to face i n the r e a l world. The normative value of any r e s u l t s should be assessed i n t h i s l i g h t , bearing i n mind that a model most l i k e l y ignores many factors which are important i n the decision making process of a resource manager. A second i n t e r p r e t a t i o n of the r e s u l t s of optimal control theory i s i n a d e s c r i p t i v e sense. I f we assume that the resource manager i s acting "optimally" f o r some s p e c i f i e d objective, then the optimal control theory r e s u l t s describe the process of e x p l o i t a t i o n and help to give ins i g h t and understanding into what are the important factors of the process and i n what way they i n t e r a c t . Also on the basis of the d e s c r i p t i v e model we can make predictions as to the outcome of the process under optimal e x p l o i t a t i o n . We see then that our choice of objective function w i l l quite l i k e l y depend on whether we wish to i n t e r p r e t the r e s u l t s i n a de s c r i p t i v e or normative way. The two approaches represent a dichotomy between "what i s " and "what should be" the objective of mankind with respect to the management of animal resources. We s h a l l 173 follow an economic objective which has some j u s t i f i c a t i o n both from the normative and d e s c r i p t i v e points of view. Early work i n the f i e l d of animal resource management, c a r r i e d out mainly by f i s h e r i e s b i o l o g i s t s , sought to maximize the sustainable y i e l d of the resource. This was based on deterministic population models. A s i m i l a r concept for a resource i n a f l u c t u a t i n g environment would be the maximization of long-run average y i e l d . This was an objective of Ricker i n h i s simulation studies (Ricker 1958(b)). We saw i n 2.3 how the long-run average y i e l d f o r any stationary p o l i c y i s the same as the expected equilibrium y i e l d , provided of course an equilibrium state e x i s t s . Both the deterministic sustainable y i e l d and the expected equilibrium y i e l d of the stochastic model are measured i n the same units as the population i t s e l f e.g. i n numbers of animals, or biomass etc. As such the objectives of maximizing sustainable y i e l d , or expected equilibrium y i e l d ignore economic fact o r s . C e r t a i n l y such economic considerations as the costs of harvesting, the s e l l i n g p r i c e of and demand- for animals caught, play an important part i n determining the actions of a resource manager. From the d e s c r i p t i v e point of view then, the objective of maximizing y i e l d , or average y i e l d seems inadequate. A better objective here would appear to be the maximization of p r o f i t or economic rent. When we turn to the normative aspect of the problem we are faced with the question of what should be the objective of man i n h i s u t i l i z a t i o n of animal resources. This i s perhaps an e t h i c a l and p o l i t i c a l question and we s h a l l not seek to give a d e f i n i t i v e answer to i t . With much of the world's population starving or s e r i o u s l y undernourished, i t could be argued that maximization of y i e l d should be our prime objective. However, even assuming that maximization of the t o t a l food production i n the world i s a worthwhile goal, i t does not follow n e c e s s a r i l y that maximization of production from one resource w i l l n e c e s s a r i l y enhance our progress towards t h i s goal. The resources, human, c a p i t a l t e c h n i c a l etc., necessary to bring about an increase i n y i e l d from a given animal resource could possibly be u t i l i z e d more e f f e c t i v e l y i n producing food i n another area. By reducing the complex of payoffs and costs of any course of action to a si n g l e measure, such as a cash value or a u t i l i t y value, the science of Economics attempts to f i n d a way of - comparing the consequences of d i f f e r e n t p o l i c i e s . Limited as such an approach may be i t does o f f e r a way of f i n d i n g a most e f f i c i e n t use of resources. We can thus regard an economic objective as normative bearing i n mind that p r o f i t , (or perhaps a better term here i s economic rent) can be thought of as a measure of the net production of commodity for mankind. There are many possible ways of defining an economic objective function. It depends on which consequences of a given action we consider important enough to include, and what assumptions we make concerning these consequences and concerning the economic environment. Apart from the presence of a m o b i l i z a t i o n cost, our economic model i s the same as that of Clark (1973b), who uses a deterministic population model. One important consequence of any p o l i c y , which a r i s e s i n a 175 stochastic model f o r a f l u c t u a t i n g environment and not for a deterministic one i s the v a r i a t i o n i n the annual y i e l d . Too great a v a r i a t i o n i n y i e l d i s c l e a r l y economically undesireable both from the point of view of the production industry and from the point of view of the consumer. The simulation r e s u l t s of Ricker (1959b) and Ricker and Larkin (1964) indicate that among a l l the p o l i c i e s considered, that of stabilization-of-escapement, although giving the highest average y i e l d , also gave the greatest v a r i a t i o n i n y i e l d . It i s important to determine the balance between y i e l d and s t a b i l i t y . How much annual v a r i a t i o n can be tolerated i n return f o r increased average y i e l d ? This i s b a s i c a l l y an economic question and could t h e o r e t i c a l l y , be resolved by assigning u t i l i t y values to uncertain outcomes. An objective function that maximized u t i l i t y would incorporate the adverse e f f e c t s of i n s t a b i l i t y i n the optimization model. Interesting, though i t may be, we s h a l l not pursue t h i s approach, and we s h a l l ignore the e f f e c t s of v a r i a t i o n i n the formulation of the objective function. The economic model and the form of the objective function and admissible set which we s h a l l assume, are discussed i n the next section. In t h i s and the next chapter we s h a l l look f o r p o l i c i e s to maximize the objective function and investigate the way economic and b i o l o g i c a l factors i n t e r a c t i n the determination of optimal p o l i c i e s , and further how they e f f e c t the behaviour of the population under optimal e x p l o i t a t i o n . Clark <1971, 1972, 1973(b)) has discussed these questions for a discrete-time deterministic population model and also (1973(a), and 176 to appear) f o r continuous-time population models. 3.2 The Model We s h a l l assume the b i o l o g i c a l population model of 1.3. We s h a l l suppose that the population , at time point n i s a random va r i a b l e d i s t r i b u t e d on an i n t e r v a l [0, ml , with {X } forming a n discrete-time Markov process with X = Z f ( X ) , n+1 n n where {Z^} i s a sequence of i . i . d . r.v.s. with unit mean and with Z^ independent of X^ , and where f i s the expected reproduction function. We s h a l l assume throughout the chapter that f i s concave and d i f f e r e n t i a b l e . Thus we s h a l l r e s t r i c t our attention to compensation models. We s h a l l assume that i n any year j u s t p r i o r to the breeding season a harvest can be made. As i n 2.2 we s h a l l assume that the harvest period i s short to ensure that natural m o r t a l i t y i s unimportant during t h i s period. I f h^ i s the s i z e of the harvest i n year n (measured i n the same units as the population i t s e l f ) then c l e a r l y h^ must s a t i s f y the a d m i s s i b i l i t y condition 0 < h < X . (1) — n — n < ' We s h a l l c a l l any sequence h = (h } of admissible harvests a harvest  p o l i c y . This i s the sole c r i t e r i o n of a d m i s s i b i l i t y which we s h a l l 177 require. We s h a l l seek to maximize the objective function over a l l harvest p o l i c i e s . Thus we s h a l l include p o l i c i e s i n which harvests may depend on the h i s t o r y of the population and i t s harvests, and also p o l i c i e s which are randomized. We s h a l l assume that the resource manager knows exactly the current l e v e l of the population, and that he i s able to bring about a harvest of given s i z e exactly. For most resources the former of these assumptions w i l l r a r e l y be met. At best the manager w i l l have an estimate of the current s i z e . However i f we t r y to incorporate t h i s uncertainty i n the model we are faced with a non-linear stochastic control model with "noisy" data, a problem about which very l i t t l e i s known. The l a t t e r assumption above i s again u n r e a l i s t i c . At best the manager can only approximately a t t a i n the target for the harvest s i z e . However i f we assume that the harvest s i z e i s a random v a r i a b l e depending on a parameter which the manager can control (the t o t a l harvest e f f o r t , say) the problem becomes more complicated. For any harvest p o l i c y h , the dynamics of the exploited population s a t i s f y the equation X = Z f(X -h ) . (2) n+± n n n We s h a l l make the following economic assumptions:-(a) Every unit of population harvested can be sold f or a f i x e d s e l l i n g p r i c e p . This s i m p l i f y i n g assumption corresponds to what economists c a l l "completely e l a s t i c demand". (b) If any harvest i s undertaken i n any year a f i x e d m o b i l i z a t i o n cost, 178 K , i s incurred i n that year. Such .a lump cost i s assumed to cover the s e t t i n g up of the harvest operation. It would involve such expenses as transporting the harvest equipment and men to the harvest grounds, the purchase of non re-usable equipment, the h i r i n g of men etc. (c) There i s a marginal cost of harvesting c(x) , which i s the unit cost of harvesting when the population i s at the l e v e l x . Roughly, c(x) w i l l depend on the time taken to harvest one unit of population when the l e v e l i s x . We s h a l l assume that c i s a non—increasing function. (d) There i s a discount or time preference f a c t o r , a < 1 , assumed constant, which discounts revenues earned at some future time to a "present-value" at some base calendar date. Thus a revenue, r ^ , earned during period n i s assumed to have a present-value a n r n at the beginning of period 1. Time-preference c e r t a i n l y plays a ro l e i n determining the nature of resource e x p l o i t a t i o n and thus from a d e s c r i p t i v e point of view i t should be included i n the economic model, e s p e c i a l l y i n the case where the resource i s owned by a sing l e f i r m i n a competitive economic environment. I t could be argued from the normative point of view that when the resource i s c o n t r o l l e d by a government or i n t e r n a t i o n a l agency, time-discounting should not be included i n the model, and that future generations should be e n t i t l e d to a share i n the resource at l e a s t equal to that of the present one. This perhaps i s an e t h i c a l and p o l i t i c a l question. At present government decisions involve, however i m p l i c i t l y , a time-preference. Besides the mathematical convenience, we o f f e r t h i s as a j u s t i f i c a t i o n f o r the i n c l u s i o n of time-discounting i n the model. The relevance of time-discounting i n resource management problems has been discussed by S. V. C i r i a c y Wantrup (1952), A. Scott (1965) and others. The discount factor a i s rela t e d to a discount rate, r , by 1 1+r In the case of a p r i v a t e l y owned resource i n a competitive economy, r can be thought of as the i n t e r e s t rate on money, or (more pr e c i s e l y ) the opportunity cost of c a p i t a l . If the population l e v e l i n year n immediately p r i o r to harvesting i s X^ , and the harvest that year i s of s i z e h > 0 , then the net revenue earned i n year n i s p h n -X n c ( t ) d t - K , X -h n n and i f no harvest i s undertaken, h = 0 , and there i s no net revenue. n Thus we have that the revenue i n year n i s R(X ) - R(X -h ) - 6(h )K n n n n where R(x) = px - [ c ( t ) d t and 6(x) = < 0, x = 0 1, x > 0 180 Discounted to a value at the beginning of year 1 t h i s revenue i s a n{R(X ) - R(X -h ) - 6(h )K} n n n n For a harvest p o l i c y h = {h^} , and an i n i t i a l population l e v e l X^ = x the expected revenue earned over the subsequent N periods discounted to a value at the beginning of year 1 i s h N C*(x) = E{ I a n[R(X ) - R(X -h ) - 6(h )K] | X = x} (3) JN , n n n n 1 n=l We define C Q ( X ) - 0 f ° r A H p o l i c i e s h , for a l l x . S i m i l a r l y the expected discounted revenue over an i n f i n i t e time-horizon, f o r an i n i t i a l l e v e l x and harvest p o l i c y h , i s 00 C h(x) = E{ Y a n[R(X ) - R(X -h ) - 6(h )K] I X = x} (4) ^ v n n n n 1 1 n=l Since X^ i s bounded, the expected one-period revenue i s bounded. I f a < 1 , the expected t o t a l discounted revenue i s f i n i t e . The convergence of the expected t o t a l discounted revenue requires that a < 1 . If there i s no time-discounting i t i s not possible i n general to define a t o t a l expected revenue for a given p o l i c y . The expression C (x) can be thought of as the expected  present-value of the resource for the harvest p o l i c y h and an i n i t i a l population l e v e l x . This w i l l be our objective function. We seek to maximize C (x) over a l l admissible harvest p o l i c i e s s a t i s f y i n g (1), f o r a l l x e [0, m] . 181 There are other choices of an economic objective function open to us. We could f o r instance seek to maximize the minimum revenue earned i n any year. Or we could seek to maximize the expected average revenue. Both of these l a t t e r choices do not depend on a time discount-rate. The maximization of expected present-value i s a natural extension of the maximization of present-value i n the deterministic models of Clark (1971, 1972, 1973 (a), (b), ( c ) ) . As we have previously mentioned the only r e s t r i c t i o n we have placed on a harvest p o l i c y h i s that i t s a t i s f i e s (1). Thus i n general, when there i s harvesting, the process w i l l not nece s s a r i l y be Markovian, f o r h^ and thus m a v w e l l depend on X, ,...,X .. , as well as on X . It w i l l be a Markov process i f 1 n-1 n the p o l i c y h i s Markovian i n the sense that the harvest h n i n year n depends only on n and the population l e v e l X^ and not on the hi s t o r y of the process. A Markovian p o l i c y can be represented by a sequence of functions {h n(x)} , h^ : [0, m] -* [0, m] with 0 < h (x) < x , f o r a l l n and x . — n — The process tX } w i l l be a time-homogeneous Markov process i f the harvest h^ depends only on X^ and not on n , nor on the h i s t o r y . Such a harvest p o l i c y we have c a l l e d stationary (2.2). We have seen how a stationary p o l i c y can be represented by a function h : [0, m] [0, m] with 0 <_ h(x) <_ x . We look for harvest p o l i c i e s to maximize (3) and (4). 182 3.3 An Optimum P o l i c y We s h a l l c a l l a p o l i c y h ah optimal N-peribd p o l i c y i f ,h C^(x) att a i n s i t s supremum over a l l admissible p o l i c i e s at h , for a l l x e [0, m] . We denote C N ( X ) by C J J ( x ) * S i m i l a r l y we define an optimal i n f i n i t e - p e r i o d p o l i c y , and we denote C (x) by C(x) . N-period p o l i c y e x i s t s , there i s one which i s Markovian, and i f an optimal i n f i n i t e - p e r i o d p o l i c y e x i s t s there i s one which i s stationary. Thus we can r e s t r i c t our search for an optimal p o l i c y to those sub-classes of admissible p o l i c i e s . regarded as the current expected value of the resource, under optimal e x p l o i t a t i o n given that there are N time-periods remaining u n t i l the time-horizon, and that the current population l e v e l i s x . S i m i l a r l y C(x) represents the current expected value of the resource under optimal e x p l o i t a t i o n given that the current l e v e l i s x . We could think of C(x) as being a ' f a i r ' s e l l i n g p r i c e for the resource. Bearing t h i s i n mind i t i s f a i r l y easy to see that ^ ( x ) s a t i s f i e s the Bellman equation of Dynamic Programming. It has been shown by Blackwell (1965) that i f an optimal We can now give another i n t e r p r e t a t i o n to C (x) It can be a max{R(x) - R(x-h) - K6(h) + E[C„ (z f(x-h))]} 0<h<x N _ 1 n or a max{R(x) - R(z) - K<S(x-z) + 0<z<x C. 'N-l ( t f ( z ) ) d * ( t ) } 183 for N = 1., 2,. . . . and x' e [0, m] . (1) It can be shown that C^(X) i s t n e unique s o l u t i o n to (1) (see Kaufman and Cruon (1967)). S i m i l a r l y C(x) s a t i s f i e s C(x) = a max{R(x) - R(z) - K6(x-z) + 0<z<x C(tf ( z ) ) d 4 ( t ) } (2) a f o r x e [0, m] , and moreover C(x) i s the unique s o l u t i o n to (2). Furthermore i t can be shown that as N -> °° , C^(x) converges uniformly to C(x) (Kauffman and Cruon (1967)). For the f i n i t e time horizon problem i t can be seen from (1), that the Markovian p o l i c y which when there are N periods remaining harvests down to the l e v e l z which maximizes the r.h.s. of (1), w i l l be optimal. This p o l i c y can be expressed i n the following way:-when there are N periods remaining harvest i f and only i f there e x i s t s a z e [0, x) such that R(x) - R(z) - K - • C N _ 1 ( t f ( z ) ) d * ( t ) > a C N _ 1 ( t f ( x ) ) d $ ( t ) a and i f so harvest down the l e v e l which maximizes the l.h . s . of the above. L e t t i n g P N(x) = -R(x) + C N _ 1 ( t f ( x ) ) d $ ( t ) a we can rewrite t h i s as:-harvest when there are N periods remaining i f and only i f there e x i s t s z e [0, x) such that P N ( Z ) - K > p ^ ( x ) > a n a i f s o 184 harvest down to a l e v e l which maximizes P^ over [0, x) . (3) We note that such a maximum w i l l e x i s t i f P„T i s continuous. N A s i m i l a r condition was met by Scarf (1960) i n h i s study of a dynamic inventory model. The s i m i l a r i t i e s between inventory models and population c o n t r o l models was pointed out by Jacquette (1972). Scarf introduces the idea of a K-convex function. Here following h i s method we say a continuous function <|>(X) i s K-concave on an i n t e r v a l I i f f o r a l l x, y e I , with x < y , *(x) - <Ky) " ( x - y ) ^ ( y ) (4) where <f>+(y) i s the r i g h t - d e r i v a t i v e of (j) at y . S t r i c t K-concavity i s defined likewise by making the i n e q u a l i t y i n (4) s t r i c t . We s h a l l need the following r e s u l t about K-concave functions. Lemma 1. Suppose <j>(x) i s continuous and s t r i c t l y K-concave on [p, q] , and l e t M = sup e|>(x) , and S = i n f { t : <j>(t) = M, t e [p, q]} . Then [p» ql there e x i s t s at most one s , S <^  s <_ q such that <j>(s) = M - K , and further i f such an s e x i s t s <f>(x) < M - K for x e (s, q] . P S Proof. We prove the lemma f i r s t for the case <j> d i f f e r e n t i a b l e on [p. q] • Suppose c > S i s a c r i t i c a l point f o r <j> , <j>'(c) = 0 then since (j> i s s t r i c t l y K-concave, we have from (4) <j>(S) - <Kc) < K , or <Kc) > M - K . Thus a l l c r i t i c a l points to the r i g h t of S l i e above the l i n e y = M - K . The lemma follows. To prove the lemma for <j) not d i f f e r e n t i a b l e we f i r s t note that i f c i s l o c a l minimum, with c > S , then <j)^(c) >^  0 , and thus since from the s t r i c t K-concavity of <j> , <j>(S) _ <Kc) - ( S - c ) ^ ( c ) < K we have cj>(c) > M - K . Let X = {x : x > S , <|)(x) = M - K} . We s h a l l show that X contains at most one point. Suppose i t . i s non-empty. I t i s compact since <(> i s continuous, and thus contains i t s infimum. Let b = i n f X then <j>(b) = M - K and b > S . Cl e a r l y for a l l x e (S,b) , <j>(x) > M - K . We s h a l l show that i f x e (b, q] then <j)(x) < M - K . Suppose t h i s i s not so, and there i s an x^ such that x^ e (b, q] , <j>(x^ ) ^ _ M - K . Since (j> i s continuous i t follows that the image of [b,x|ip under <|> w i l l be a closed i n t e r v a l [y^ y^] say. Let x^ e [b, x^] be such that C K X Q ) = y-^  • I f X Q ^ b then X Q i s a l o c a l minimum, and hence from above M X Q ) > M - K = b , contradicting <K X Q) = y^ . If x^ = b then f o r x > b <(>(x) ^ _ M - K . But also we have f or x < b that <}>(x) > M - K , and thus b i s a l o c a l minimum, and thus <j>(b) > M - K , 186 which i s a contradiction. Hence we have that <J>(X) < M - K , f o r x > b , and thus that X contains only the point b . We conclude that either X i s empty or contains one point, b , with <f>(x) < M - K for x > b , which proves the lemma. If we can show that the function ^ N(X) i s continuous and s t r i c t l y K-concave on [0, m] , and we l e t be the S of the preceding lemma and l e t s^ be the s of the lemma i f i t exi s t s and be m i f not then we have that the condition (3) i s met i f and only i f / X > s„ , i n which case a harvest down to S„ i s made. C a l l t h i s N N condition (5). Such a p o l i c y i s known i n the l i t e r a t u r e of inventory theory as an (S,s) p o l i c y . In the next theorems we s h a l l show that under c e r t a i n conditions an (S,s) p o l i c y i s optimal. We need Lemma 2. (a) I f J i s an i n t e r v a l containing the i n t e r v a l I , and <j> i s ( s t r i c t l y ) K-concave on J , then <$> i s ( s t r i c t l y ) K concave on I . (b) ( S t r i c t ) 0-concavity of a function on an i n t e r v a l I , i s equivalent to ( s t r i c t ) ordinary concavity on I . (c) I f 0 <_ K < M , then the K-concavity of a function on I implies the s t r i c t M-concavity of that function on I . (d) I f <j> i s ( s t r i c t l y ) K-concave on I , and \Ji i s ( s t r i c t l y ) M-concave on I , then f o r a, 3 > 0 , the function ac|> + $ty i s ( s t r i c t l y ) (aK+3M)-concave on I . (e) I f f i s non-decreasing and concave on I and ip i s non-decreasing 187 and ( s t r i c t l y ) K-concave on the i n t e r v a l f ( I ) , then the composition ip°£ i s ( s t r i c t l y ) K-concave on I . (f) I f f i s concave on I , and \p i s non-decreasing and ( s t r i c t l y ) concave on f ( I ) then the composition i|j°f i s ( s t r i c t l y ) concave on I . (g) I f f i s non-decreasing and concave on I , 5> i s a p r o b a b i l i t y d i s t r i b u t i o n on [a, b] and ip i s a non-decreasing K-concave function on [a i n f f(x) , b sup f ( x ) ] then the S t i e l t j e s i n t e g r a l x e l X E I b ijj(tf (x) )d$(t) i s a K-concave function of x on I • a (h) If f i s concave on I , $ i s a p r o b a b i l i t y d i s t r i b u t i o n on [a, b] and ip i s a non-decreasing concave function on [a i n f f(x) , b sup f ( x ) ] then the S t i e l t j e s i n t e g r a l x e l x e l i s a concave function of x on I . i|»(tf (x))d*(t) a Proof. The proofs of (a), (b), (c) and (d) follow d i r e c t l y from the d e f i n i t i o n (4). To prove (e) we f i r s t note that f or non-decreasing f , the r i g h t d e r i v a t i v e Ol>°f)+(y) = *+(f ( y ) ) - f ; ( y ) .. Now for x < y , i|>of(x) - i|iof(y) - (x-y) 0|;of )_j_(y) 188 t*of(x) - ^of(y) - (f(x) - f(y) ) i^(f(y»] + [f(x) - f(y) - (x-y)f_^(y)] ^ ( f ( y ) ) From the ( s t r i c t ) K-concavity of ^ we have that the f i r s t part i s < K , f(x) being less than f(y) , and from the monotonicity of and the concavity of f we have that the second part i s no greater than zero. Hence ^°f i s s t r i c t l y K-concave on I . Part (f) i s the same as Lemma 2.3.1. To prove (g), we have rb n i K t f ( x ) ) d $ ( t ) = l i m I t ( t f ( x ) ) ! * ( t - | | ) : } ] $ ( t . ) ] J a n - K » j=o J " J 2 2 and from parts (d) and (e) we have that n I K t , f ( x ) ) [ $ ( t . , 1 ) - * ( t , ) ] j=0 J 3 J i s a s t r i c t l y K-concave function of x on I . Hence i n the l i m i t , f b <Ktf(x))d$(t) a i s a K-concave function of x on 1 . The proof of (h) follows i n the same way as above from (d) and ( f ) . We are now ready to prove the optimality of (S,s) p o l i c i e s . We s h a l l f i r s t l y assume that the marginal cost function c(x) i s a constant function c(x) = c . In the next section we s h a l l consider 189 other forms of the cost function. Theorems 1(a) and 1(b) are for the case of a p o s i t i v e m o b i l i z a t i o n cost K > 0 , and Theorems 2(a) and 2(b) are for the case when there i s no mobiliz a t i o n cost, K = 0 . Theorem 3.3.1(a) If the expected reproduction function f(x) i s concave and non-decreasing on [0, m] and the marginal cost function c(x) i s constant on [0, m] , and i f the mobili z a t i o n cost K i s p o s i t i v e , then the functions ^ ( x ) a r e continuous and s t r i c t l y K-concave on [0, m] f o r N = 1, 2, Hence f or any f i n i t e time-horizon problem there e x i s t s an optimal p o l i c y which i s Markovian and i s determined by a sequence of pair s ^ ^ u ' S n ^ w n e r e a harvest i s made when N periods remain i f and only i f the population l e v e l i n that year exceeds s^ , i n which case a harvest down to the l e v e l S„ i s made. N i . e . h (x) = < n i s an optimal p o l i c y . x - S N , x > s N n Proof. We proceed by induction on N and assume that i s continuous and (aK)-concave on [0, m] . Then from lemma 2 (g), (c) we have that C ( t f ( x ) ) d * ( t ) a i s (aK)-concave and thus s t r i c t l y K-concave. I t follows from lemma 2 (b), (d) since R(x) = (p-c)x that 190 P N + l ( x ) = " R ( x ) + C N ( t f ( x ) ) d $ ( t ) a i s continuous and s t r i c t l y K-concave on [0, m] . Thus there are numbers S,TI1 , s. T I 1 with 0 < S^.- < s„ < m N+1 N+1 — N+1 N — which determine the action of an optimal p o l i c y when there are N + 1 periods remaining using the usual (S,s) r u l e (5). Thus we have from (2) CN+1 ( X ) -a [ P N + 1 ( x ) +R(x)] , x < s N + 1 0 1 [ W W - K + R ( x ) ] ' » X > SN+1 (6) The function C„,... i s continuous because P A T 1 1 i s and because N+1 N+1 PN+1 ( SN+1 } = PN+1 ( SN+1 ) + K ' F u r t h e r m o r e C N + 1 ± S n o n " d e c r e a s i n g -We can show t h i s by showing i t non-decreasing on [0, sj^+-|_] a n d o n ( s N + 1 , m] . For 0 < X ]_ < ^ < s N + 1 CN+1 ( X2 ) " S + l ( x l ) = a [ C N ( t f ( x 2 ) ) - C N ( t f ( X ; L ) ) ] d $ ( t ) ^ 0 a because C„T and f are non-decreasing. For s„, n < x, < x„ < m N ° N+1 1 2 — C N + 1 ( x 2 ) - C N + 1 ( X ; L ) = a[R(x 2) - R(x 1)] = a(p-c) ( x ^ x . ^ > 0 We now show that CL,,.. i s (aK)-concave on [0, ml . We wish to show N+1 that the condition (4) with aK on the r.h.s. holds f o r f ° r a l l x^ < x^ . F i r s t l y f o r x^ < <_ s N + ^ we use lemma 2 (d) and equation (6) bearing i n mind that ^ s K-concave and R i s l i n e a r , to show that (4) with aK on the r.h.s. holds. Secondly for s ^ + ^ < x l < x2 ^ with aK on the r.h.s. holds since R i s l i n e a r (lemma 2(a), ( c ) ) . F i n a l l y for x^ <_ SN+I < x2 w e n a v e f r o m (6) CN+1(X1) " CN+1(X2) ~ ( x l " X 2 ) VKI/V + = a { P N + 1 ( X ; L ) - P N + 1 ( S N + 1 ) + K + R( X ] L) - R(x 2) - ( X l - x 2 ) R ' ( x 2 ) } < aK from the l i n e a r i t y of R and the fac t the P„.- attain s — J N+1 i t s maximum at S„,, . N+1 Thus we have shown that i s continuous non-decreasing and (aK)-concave. The induction hypothesis holds f o r C (x) = 0 and hence i t o holds f or a l l N >_ 0 . Consequently P^ i s continuous and s t r i c t l y K-concave f o r N = 1, 2,...., and hence there e x i s t p a i r s {(S^jS^)} which determine an optimal p o l i c y by the usual (S,s) rule (5). Q.E.D, We have mentioned already that C^(x) converges uniformly i n x to C(x) , (Kauffman & Cruon (1967). It i s easy to see that C^(x) increases with N . Thus we have that P^(x):- increases, uniformly i n N to a l i m i t P(x) = -R(x) + C(tf(x ) ) d $ ( t ) . We a have that P(x) i s K-concave, and hence there are numbers S, s from lemma 1, which determine an optimal stationary p o l i c y . It can be shown that S N -> S , and -> s as N -> °° . Thus we have the following theorem. 192 Theorem 3.3.1(b) Under the conditions of Theorem 1 (a), for the i n f i n i t e time-horizon problem, there e x i s t s an optimal stationary p o l i c y determined by a pa i r of numbers S = l i m S„ and s = lim s„, , for N N N-x» N-*» which a harvest i s made i n any year i f and only i f the population l e v e l i n that year exceeds s , i n which case a harvest down to the l e v e l S i s made. i . e . the stationary p o l i c y h(x) = |x - S , x > s L.' 0 , x <_ s i s an optimal p o l i c y . We now consider the case when there i s no mobili z a t i o n cost, K = 0 . In t h i s case from (2) the condition (3) becomes:-' harvest when there are N periods remaining i f and only i f there e x i s t s z e [0, x) such that P^(z) > P N ( X ) > and i f so harvest down to the z which maximizes P„ over [0, xD . (7) N If P„ i s continuous and concave, and S„ i s where P„ N N N attain s i t s maximum over [0, m] then (7) can be expressed:-harvest when there are N periods remaining i f and only i f the population l e v e l exceeds SXT , i n which case harvest down to the N l e v e l S„ . N We can prove the concavity of P^ by induction exactly as i n theorem 1(a). However i n t h i s case we do not need the condition that 193 f i s non-decreasing. We prove the concavity of from lemma 2(h) which does not require the monotonicity of d i r e c t l y without assuming the monotonicity of f , i n the following way. Let x 2 > , and suppose an optimal p o l i c y when there are N periods remaining and the population i s at l e v e l x^ , involves harvesting down to x^ . Then C N ( X ; L ) = a[R( X ; L) - R( X ; L)] + a C ^ O ^ ) . Now C N ( X 2 ) ! a [ R ( x 2 > " R ( x x ) ] + a C N - l ( x l ) ' = a[R(x 2) - R(x L)] + C N(x 1) The r e s t of the proof of the concavity of P^ ( i n t h i s case not s t r i c t concavity) follows i n the same way as the proof of Theorem l ( a ) v Thus we have the following theorem, Theorem 3.3.2(a) If the expected reproduction function f(x) i s concave but not n e c e s s a r i l y monotone on [0, m] , and the marginal cost function c(x) i s constant on [0, m] and i f there i s no mobili z a t i o n cost (K=0) , then an optimal Markovian p o l i c y i s determined by a sequence {S^} where there are N periods remaining, a harvest i s made i f and only i f the population l e v e l exceeds i n which case the excess i s harvested, i . e . the Markovian p o l i c y ' 194 h n(x) 0 > X 1 S N i s optimal. In the same way as before we can prove a theorem for the i n f i n i t e time-horizon problem when there i s no m o b i l i z a t i o n cost. Theorem 3.3.2(b) Under the conditions of Theorem 2(a), for the i n f i n i t e time-horizon problem, there i s an optimal stationary p o l i c y determined by a si n g l e number S = lim , where a harvest i s made i n any year i f and only i f the population l e v e l i n that year exceeds S , i n which case the excess i s harvested i . e . the stationary s t a b i l i z a t i o n - o f -escapement p o l i c y h(x) = x - S , x > S*5 0 , x <_ S i s optimal. The previous theorems characterize the optimal p o l i c i e s q u a l i t a t i v e l y . We have not, i n general, been able to determine* the l e v e l s S, s etc. Some r e s u l t s i n t h i s d i r e c t i o n are given i n 4.2. Numerical algorithms have been developed for the determination of optimal (S,s) p o l i c i e s for inventory problems (for example Velnott and Wagner (1965)). It should be possible to adapt these computing techniques to the model here. In the case of a zero 195 m o b i l i z a t i o n cost the numerical determination of the optimal escapement l e v e l S should be quite straightforward. Clark (1971) has proved the optimality of a s t a b i l i z a t i o n - o f -escapement p o l i c y when there i s no mobiliz a t i o n cost f o r a deterministic population model equivalent to that of Thm 2. The proof of Thm 2 does not place any r e s t r i c t i o n on the d i s t r i b u t i o n function $ , except that i t be concentrated on some f i n i t e i n t e r v a l , and so the theorem holds for the case of $ being a degenerate d i s t r i b u t i o n with a l l i t s mass at the point 1 i . e . for Clark's deterministic population model. The same i s true f o r the proof of Thm 1. I t follows that an (S,s) p o l i c y i s optimal f o r the deterministic model when there i s a p o s i t i v e m o b i l i z a t i o n cost. This (S,s) p o l i c y for the deterministic model i s i n a sense a discrete-time version of "chattering c o n t r o l " (see L. C. Young, 1969). The reason f o r i t i s roughly as follows. We would l i k e to s t a b i l i z e the escapement at some f i x e d l e v e l E , say, as i n the case when there i s no mobilization cost. S t a b i l i z a t i o n of the population at t h i s l e v e l would correspond to the "singular path" of continuous-time optimal control theory. However because of the presence of the mob i l i z a t i o n cost, some harvests down to E , notably the small ones w i l l y i e l d , a negative return as well as depleting the population. Only large harvests are worth undertaking and an optimal p o l i c y i s to wait u n t i l the population reaches a higher l e v e l s and then reduce the population to a l e v e l S lower than (or possibly equal to) E . We s h a l l discuss the e f f e c t of the mobiliz a t i o n cost on the optimal p o l i c y 196 for" the deterministic model i n more d e t a i l i n 4.3. In the next section we consider more general forms f or the marginal harvest cost function c(x) . 3.4 More General Harvest Cost Functions In the proof of Theorems 1 and 2 of the l a s t section we assumed that the marginal cost function c(x) was constant over the range [0, m] . In t h i s section we look for more general forms of c(x) , for which (S,s) and stabilization-of-escapement p o l i c i e s are optimal. We s h a l l assume that c(x) i s non-increasing. I t seems reasonable that the unit cost of harvesting should grow (6r at l e a s t not decline) as the population s i z e decreases. We define the zero p r o f i t l e v e l , X Q as the so l u t i o n to c(x) = p , i f one e x i s t s , and as zero otherwise. F i g . 3.2 197 C l e a r l y the marginal return on harvesting i s negative when the target population i s of size l e s s than x^ . It w i l l never be p r o f i t a b l e to harvest below ^ X Q , (except possibly i n the case of an over-compensation expected reproduction function, when X Q i s greater than p the point at which f attai n s i t s maximum — we s h a l l ignore t h i s u n l i k e l y case.) For a decreasing function c(x) , the function R(x) = px -x c ( t ) d t i s a convex function with a minimum at x^ , as i n F i g . 3.3. Fi g . 3.3 We seek conditions on the function c(x) for which Theorems 1 and 2 hold. We approach t h i s by two methods. In the f i r s t s method, which only holds i n the case of a p o s i t i v e m o b i l i z a t i o n cost (K > 0, Theorem 1), we s h a l l impose conditions on how convex R can 198 be, i n terms of K i . e . on how much c(x) can decrease. The second method which works for K =^  0 , w i l l impose conditions on c , f and X0 ' The f i r s t method i s given i n Extensions ( i ) and ( i i ) of Theorem 1 (a), (b). Extension ( i ) Theorem 3.3.1 (a)(b) If the marginal harvest cost c(x) i s v decreasing and s a t i s f i e s the condition rm (A) [c(t) - c(m)]dt < K ( l / a - l ) 0 and i f , as i n the conditions of Theorem 3.3.1, the expected reproduction function f(x) i s concave and non-decreasing then the r e s u l t s of Theorem 3.3.1 (a), (b) hold i . e . when there i s a p o s i t i v e m o b i l i z a t i o n cost, Markovian and stationary (S,s) p o l i c i e s are optimal both i n the cases of f i n i t e and i n f i n i t e time-horizons of Theorem 3.3.1 (a) and Theorem 3.3.1 (b) respectively. Before proving t h i s r e s u l t , we observe that the condition (A) i s a condition on how c(x) decreases over the range [0, m] . We require that the shaded area i n F i g . 3.4 be less than Kr , where r = 1/a - 1 i s the discount rate. A l t e r n a t i v e l y we can think of condition (A) as a condition on the concavity of R(x) , over [0, m] , for we have [c(t) - c(m)]dt < Kr 199 m x F i g . 3.4 R ( X l ) - R(x 2) - (X;L-x2)R_|_(x2) = x„ X , [c(t) - c ( x 2 ) ] d t m [c(t) - c(m)]dt < Kr , for a l l x^, x 2 e [0, m] Thus we see that condition (A) implies that R i s Kr-concave on [0, m] We now give the proof of extension ( i ) Proof. Let x From (A) we have m [c(t) - c(m)]dt a(T+K) < K We l e t k be a number such that 200 a(x+K) < k < K . We proceed by induction as i n the proof of Thm 3.3.1(a) only now we make the induction hypothesis that i s continuous, non-decreasing and s t r i c t l y k-concave on [0, m] . Since -R(x) i s concave i t follows as before that P N + 1 ( x ) = -R(x) + •m 0 C N(tf-(x))d$(t) i s k-concave and thus from Lemma 3.3.2(c) that i t i s s t r i c t l y K-concave on [0, m] . Thus there e x i s t s numbers S„T,.. , s > T 1 1 which determine the N+1 N+1 action of an optimal p o l i c y when N periods remain and CN+1 ( X ) " a [ P N + 1 ( x ) + R(x)] , x < s N + 1 " [ W W - K + R « ] • x > S N+1 The continuity of C ^ + T . f ° l l ° w s as before. It i s also monotone non-decreasing. For x^ < x^ s j j + - ^ rb [ C N ( t f ( x 2 ) - C N ( t f ( X ; L ) ) ] d $ ( t ) > 0 because C„T and f are non-decreasing. N & F ° r SN+1 K X l < X2 ' CN+1 ( X2 } " CN+1 ( X1 ) = R ( X 2 ) " R ( X 1 } Now R i s non-decreasing on [x^, m] , so to show the monotonicity of C.7I1 on ( s . T I 1 , m] we need only show that x_-< s„,_ . I t i s cl e a r N+1 N+1 J 0 — N+1 that no harvest w i l l ever be made to a l e v e l lower than x^ , since t h i s 201 not only costs money but decreases the population. Thus we have S„,, > x„ and hence s„... > x„ , and we conclude that C„,.. i s N+1 — 0 N+1 — 0 N+1 non-decreasing on [0, m] since i t i s non-decreasing on [0, s ^ + ^ ] and ( S ^ . t , m] and i s continuous. N+1 We wish to show the s t r i c t k-concavity of o n [0» m J • F i r s t l y f o r x^ < x^ < S N + ^ S + i ( x i ) - S f i ( x 2 ) " ( x r x 2 ) c N + i ( x 2 ) + " ^ W V ~ PN+1 ( X2> " ( x r X 2 ) P N + l ( x 2 ) K + + a { R ( X ; L ) - R ( x 2 ) - ( X ; L - X 2 ) R | ( X 2 ) } N o w pN + i i s s t r i c t l y K-concave. Also from (A) R i s x-concave because R( X ; L) - R(x 2) - (x 1 - x 2 ) R | ( x 2 ) = X2 [c(t) - c ( x 2 ) ] d t ' X l rm [c(t) - c(x )]dt = T 0 Hence i n the above we have that C N + ^ i s a(K+x)-concave and hence s t r i c t l y k-concave on [0, s^ +^) . F ° r SN+1 - X l < X2 C N + l ( x l ) " CN+1 ( X2 ) " ( x r X 2 ) C N + l ( x 2 ) = " ^ ^ i ^ ~ R ( x 2 ) " ( X 1 ~ X 2 ) R + ( X 2 ) } + <_ ax < k , 202 and hence 1 S s t r i c t l y k-concave on [ sjj+i' m ^ • F o r X l K SN+1 - X2 CN+1 ( X1 } " CN+1 ( X2 } " ( x r X 2 ) C N + l ( x 2 } + = C t { P N + l ( x l ) " PN+1 ( SN+1 ) + K } + a { R ( x l ) " R ( x 2 } " ( x l " X 2 ) R + ( x 2 ) } <_ a(K+x) < k , since P„,,, a t t a i n s i t s maximum at S„T... and R(x) i s x-concave. N+1 N+1 We conclude that ^ ^ 1 S s t r i c t l y k-concave onn [0, m] . The induction hypothesis holds since i t holds for C Q ( X ) = 0 • The optimality of Markovian (S,s) p o l i c i e s follows as i n the proof of Theorem 1(a). The proof of the extension to Theorem 1(b) follows from t h i s i n exactly the same way as Theorem 1(b) follows from 1(a). We have already established that an optimum p o l i c y would never involve a harvest to a l e v e l lower than x^ , since t h i s would not only reduce the l e v e l of the population but would y i e l d a negative return. Thus i t would seem that the behaviour of c(x) on [0, x^] i s i r r e l e v a n t , provided of course that c(x) >_ p on t h i s i n t e r v a l . We would l i k e to replace the condition (A) by a condition r e s t r i c t i n g the amount by which c(x) can decrease on [x^, m] . If we impose the extra r e s t r i c t i o n that the zero p r o f i t l e v e l X Q i s sustainable i n the sense of 2.1 then for x >_ x^ we can write the Bellman equation (3.3.1) 203 C N(x) = a max {R(x) - R(z) - K6(x-z) + ze[x Q,x] C N _ 1 ( t f ( z ) ) d $ ( t ) } If the current population l e v e l x i s le s s than X Q , c l e a r l y any optimal p o l i c y w i l l not allow a harvest before the population has increased at l e a s t to the l e v e l X Q . I f we can prove the s t r i c t K-concavity of P^ on [x^, m] the optimality of (S,s) p o l i c i e s w i l l follow as before. We do t h i s i n the following extension. Extension ( i i ) Theorem 3.3.1(a), (b) If the z e r o - p r o f i t l e v e l x^ i s sustainable i n the sense that p r { X ^ > _ X Q | X Q = X Q } = 1 , and i f the marginal cost function c(x) s a t i s f i e s (B) m [c(t) - c(m)]dt < K ( l / a - l ) , X0 and i f as i n Theorem 3.3.1, f i s concave and non-decreasing, then for the case of a p o s i t i v e m o b i l i z a t i o n cost Markovian and stationary (S,s) p o l i c i e s are optimal as i n the f i n i t e and i n f i n i t e time-horizon cases of Thm 3.3.1(a) and Thm 3.3.1(b) res p e c t i v e l y . Proof. If the i n i t i a l population l e v e l i s below x^ , any optimal p o l i c y w i l l involve waiting, without harvesting at l e a s t u n t i l the population l e v e l exceeds x^ . Once i t has exceeded x^ , since no optimal harvest i s ever made to a l e v e l lower than x^ , and since 204 i s sustainable, i t follows that the population l e v e l w i l l never again drop below . The optimality of an (S,s) p o l i c y w i l l follow i f we can prove the s t r i c t k-concavity of on txQ> m ] • This can be done i n exactly the same way as i n Extension ( i ) , using now condition (B), and the r e s u l t follows. The condition (B) puts a r e s t r i c t i o n on how c decreases on IXQ> m ] • The proof of the extension ( i i ) above requires that X Q be sustainable. I t seems that the r e s u l t should hold without t h i s l a t t e r assumption. We say t h i s because 'nocpp.6ima3>rhar-vest '.will involve harvesting to a l e v e l below x^ , and so the behaviour of c(x) on [0, X Q ] i s apparently i r r e l e v a n t , even though the population l e v e l may drop below x^ . However without the assumption of the s u s t a i n a b i l i t y of X Q we are unable to apply Lemma 3.3.2(g) to prove that P N + - L i s k-concave on [x^, m] given that i s k-concave on the same i n t e r v a l . We have not found a proof which does not involve the assumption of the s u s t a i n a b i l i t y of x^ . At t h i s point we might ask how r e a l i s t i c are the conditions (A) and (B). In so f a r as they allow for a decreasing marginal cost c(x) they are better than the o r i g i n a l condition of Theorem 1(a). C. Clark (1973) o f f e r s the following r a t i o n a l e for the marginal cost function c(x) . Suppose there are N animals randomly dispersed throughout the hunting ground and that there i s a f i x e d p r o b a b i l i t y 0 of l o c a t i n g and catching a given animal i n a given unit of time. Then the expected waiting time u n t i l the f i r s t capture i s 205 N 1/[1 - (1-6). ] which f o r small 9 i s approximately equal to. 1/0N . If we assume that the marginal harvest cost c(x) i s proportional to the expected waiting time to. catch one animal when the population l e v e l i s x + 1 we have C ( X ) = TUmJ ' where c i s the cost incurred i n operating f o r one unit of time. We see that c i s decreasing and that •m [c(t) - c(m)]dt = ! Un(m+1) - . J 0 oo n—1 The expected cost of catching a p a r t i c u l a r animal i s c £ 6(1-0) = c/0 . 1 Hence we see that condition (1) c/6[£n(m+l) - ~=rr] < K ( l / a - l ) can m+± be interpreted as an upper bound on the cost of catching a p a r t i c u l a r animal. This expected cost might not be too high i f the animals were randomly d i s t r i b u t e d throughout a l i m i t e d geographical area during the hunting season. This might well be the case for animals which are hunted on or close to t h e i r breeding grounds which cover a r e l a t i v e l y small area. Populations of P a c i f i c Salmon and of seals possibly s a t i s f y t h i s requirement. A s i m i l a r cost function a r i s e s when the animals form into f l o c k s or schools which are randomly dispersed. Condition (A) could then be interpreted as an upper bound on the cost of l o c a t i n g and catching a p a r t i c u l a r f l o c k . Condition (B) can be interpreted i n a s i m i l a r way. 206 The conditions (A) and (B) put r e s t r i c t i o n s on the way the marginal harvest cost decreases. Roughly speaking condition (A) w i l l not be s a t i s f i e d i f the marginal cost decreases a considerable amount over the i n t e r v a l [0, m] but decreases only slowly at f i r s t . I t could well be, i n such a case, that the functions (x) are not K-concave, and (S,s) p o l i c i e s not optimal. Condition (B) can be s i m i l a r l y interpreted f o r the i n t e r v a l [ X Q , m] . Later we show that (S,s) p o l i c i e s are optimal f o r c(x) = c/x , a <_ 1 , provided f s a t i s f i e s a weak a d d i t i o n a l hypothesis, and that the l e v e l X Q i s sustainable. This condition can be interpreted i n a way s i m i l a r to the above. It could well be that some condition of the above type, on the way the marginal harvest cost decreases i s necessary f o r the optimality of (S,s) p o l i c i e s . The form of the function P(x) determines the form of an optimal i n f i n i t e period p o l i c y . The K-concavity of P i s not a necessary condition for the optimality of an (S,s) p o l i c y , although i t i s a s u f f i c i e n t condition. If the function P assumed a non-K-concave form of the type shown i n F i g . 3.5, the following optimal p o l i c y involving two escapement l e v e l s would be optimal:-207 h(x) = Fi g . 3.5 0 , for x <_ s^ , x e [ s 2 , s.^ ] x - f o r x e (s^, s 2 ) x - S 2 for x > • x Such a form of the function P might a r i s e , for instance, f o r ce r t a i n forms of the cost function, which although decreasing a considerable amount over. [0, m] , decrease only slowly at f i r s t . Harvests to the l e v e l S^ might make use of high growth p o t e n t i a l i n the population, even though costs would be high, whereas harvests to the l e v e l S 2 might make use of cheap harvest costs, even though the growth p o t e n t i a l would be lower at t h i s l e v e l . For our second approach to generalizing the theorems of 3.3 for more general cost functions we s h a l l look at a d i f f e r e n t ' s i z e ' measure of the population, or i n terms of language of stochastic processes, we s h a l l look at a d i f f e r e n t state v a r i a b l e . Rather than use the phys i c a l siz e of the population as sta t e - v a r i a b l e , we s h a l l consider the harvest 208 value of the population i . e . the cash value of the population i f i t were to be harvested completely and sold immediately. We s h a l l make the assumption that the l e v e l , X Q , of zero p r o f i t , i s sustainable. Then as we mentioned before i f the i n i t i a l population l e v e l i s below X Q any optimal p o l i c y w i l l allow no harvest at l e a s t u n t i l the population l e v e l has grown to exceed X Q . Under an optimal p o l i c y the population w i l l never drop below x^ , once i t i s i n excess of x^ . In t h i s case we can write the Bellman equation, for x >_ X Q , as (1) C„(x) = a max {R(x) - R(z) - K6(x-z) + W r i I ze[x 0,x] > b C N _ 1 ( t f ( z ) ) d $ ( t ) } a If the population i s harvested when the current l e v e l i s x , the maximum immediate return i s R(x) - R(x Q) - K or 0 , depending on which i s greater. We s h a l l define Q(x) = R(x) - R(x Q) = x tc ( x Q ) - c ( t ) ] d t X0 Disregarding the K , we can think of Q(x) as the immediate harvest  value of the resource. Since R ( x ) i s increasing"on [x^, m] , Q(x) w i l l have the same property, and furthermore the image under Q of [X Q , m] i s [0, R(m) - R ( X Q ) ] . I t follows that Q has an inverse Q defined 209 on [0, R(m) - R(x Q)] . If we l e t y = Q(x) and w = Q(z) , we can write the Bellman equation (1) i n the following way:-C N(Q 1 ( y ) ) = a max {y - w - KS(y-w) + we[0,y] b -1 C N - l ( t f ( Q ( w))> d $Ct)} a L e t t i n g D„T denote the composition G:°Q ^ t h i s becomes N '"N D^(y) = a max {y - w -KKS(y-w) + we[0,y] b -1 D N - l ( Q ( t f ( Q ( w > ) ) d $ ( t ) > (2) a The function D N ( y ) represents the optimal value of the resource, N periods remaining, when the current immediate harvest value of the resource i s y . Equation (2) i s a necessary condition for the optimality of . I f , from t h i s equation, we can prove the optimality of an (S,s) p o l i c y , i n terms of the s t a t e - v a r i a b l e y , (the immediate harvest value), then, since Q i s monotone on [0, Q(m)] , i t would follow that an (S,s) p o l i c y , i n terms of the o r i g i n a l state v a r i a b l e (the population size) i s optimal. The equation (2) i s s i m i l a r to the Bellman equation 3.3.1 i n Theorem 3.3.1(a). The only difference i s i n the "reproduction model". Instead of having z ->- t f ( z ) , t e [a, b] as i n 3.3 we now have w + Q ( t f ( Q _ 1 ( w ) ) ) , t e. [a, b] . To follow the proof of Theorem 3.3.1Ca), we need only e s t a b l i s h that condition (€) below holds. 210 Condition (C):-for every r e a l i z a t i o n t e [a, b] , of the random m u l t i p l i e r the function Q(tf(Q _ 1(w))) i s concave and non-decreasing i n w , on the i n t e r v a l [0, Q(m)] . In the proof of Theorem 3.3.2 we needed only the concavity of t f ( x ) , t e [a, b] . Thus we can show that a s t a b i l i z a t i o n of escapement p o l i c y i s optimal i f the cost function s a t i s f i e s the following condition. Condition (D):-for every r e a l i z a t i o n t e [a, b] of the random m u l t i p l i e r the function Q(tf(Q _ 1(w))) i s concave i n w on the i n t e r v a l [0, Q(m)] . We thus have the following extensions to Thm. 3.3.1 and Thm. 3.3.2. Extension ( i i i ) Theorems 3.3.1(a), (b) If the zero p r o f i t l e v e l x^ i s sustainable and i f the marginal cost function c(x) s a t i s f i e s condition (C) then, f o r the case of a p o s i t i v e m o b i l i z a t i o n cost, Markovian and stationary (S,s) p o l i c i e s are optimal for the f i n i t e and i n f i n i t e time-horizon problems re s p e c t i v e l y , as i n Theorems 3.3.1(a) and 3.3.1(b). 211 Extension (iv) Theorems 3.3.2(a), (b) If the zero p r o f i t l e v e l x^ i s sustainable and i f the marginal cost function c(x) s a t i s f i e s the condition (D), then for the case of no m o b i l i z a t i o n cost (K=0) , Markovian and stationary stabilization-of-escapement p o l i c i e s are optimal f o r the f i n i t e and i n f i n i t e time-horizon problems r e s p e c t i v e l y as i n Theorems 3.3.2(a) and 3.3.2(b). We now look at an important s p e c i a l case for which the conditions (C) and (D) are s a t i s f i e d . Consider the cost function c(x) = c/x , c a constant. This i s a rather important form of the cost function.ofWensawtearlier i n t h i s section how a cost function of t h i s type arose when costs are proportional to the expected waiting time f o r a catch, ( a c t u a l l y we ended up with c(x) = c/x+1 , but the dif f e r e n c e i s unimportant). Another well-known way of j u s t i f y i n g such a cost function i s as follows. Suppose that the harvest period i s short, so that the e f f e c t s of natural mortality i n the population i n that period are n e g l i g i b l e . We denote by F(t) , the instantaneous harvest e f f o r t at a time t , and we s h a l l suppose that the s i z e of the population N(t) , t time-units a f t e r the s t a r t of harvesting, s a t i s f i e s ~ = -F(t) N(t) . (3) The instantaneous harvest e f f o r t represents the instantaneous i n t e n s i t y of harvesting. In a f i s h e r y , f o r instance, i t could represent 212 the t o t a l number of nets i n the water at time t . We s h a l l consider a harvest from an i n i t i a l l e v e l x to a l e v e l x-6 . Suppose i t takes T time units to accomplish t h i s . We s h a l l suppose that the cost of harvest i s proportional to the t o t a l  harvest e f f o r t , which can be represented by F( t ) d t , 0 and so the cost of harvest i s rT F ( t ) d t , 0 where c i s a constant. Now in t e g r a t i n g equation (3) we have that T rx-6 F( t ) d t = -0 dN N x + o(S) It follows that the cost of harvesting from the l e v e l x to the l e v e l x - 6 i s x 6 + o(6) , and hence that the marginal harvest cost i s c/x . We now show that t h i s marginal cost function s a t i s f i e s the conditions (C) and (D) provided c e r t a i n other weak hypotheses are s a t i s f i e d . The condition (C) i s s a t i s f i e d i f the der i v a t i v e -— Q(tf(Q "*"(w))) i s p o s i t i v e and monotone decreasing on [0, Q(m)] dw for each t e [a, b] . Now / Q ( t f ( Q - \ w ) ) ) - I'wfw)) • t f C Q ^ W ) d " Q ' ( Q _ 1 ( w ) ) Since Q i s increasing the above d e r i v a t i v e w i l l be p o s i t i v e i f f increasing. Furthermore, since Q ^ i s increasing, the d e r i v a t i v e w i l l be decreasing i f Q ' ( t f ( x ) ) , ( . Q'(x) W i s decreasing i n x on txQ> m l • Now Q(x) = X x o [c(x Q) - c ( t ) ] d t , and so for the s p e c i a l case c(x) = c/x , we have Q'(x) = - ^ x 0 and Q'(tf(x)) c re- . c Hr t f ( x ) c c x o " X Q. ( x ) t f ' ( x ) t f ' ( x ) 214 t f ( x ) - x x-x,. 0 xf'(x) f(x) Now on [X Q , m] , the function t f ( x ) - x 0 x - x 0 i s decreasing and p o s i t i v e , for every t , i f f i s increasing and i f X Q i s sustainable. This i s so because i t represents the slope of the ray PQ j o i n i n g ( X Q . X Q ) to (x,tf(x)) , (see F i g . 3.6). x„ 1— J- — F i g . 3.6 If we make the assumption that -r^r 2" f (x) we have that t f ( x ) i s non-increasing and p o s i t i v e , Q'(tf(x)) Q'(x) t f ' ( x ) i s p o s i t i v e and decreasing and hence that (C) holds. 215 • The condition that x f ' ( x ) / f ( x ) be non-increasing i s a t e c h n i c a l one. I t i s s a t i s f i e d by the Beverton-Holt r e l a t i o n s h i p f(x) = ax/l+bx and by the function f(x) = ax^ . It i s also s a t i s f i e d -bx by the Ricker r e l a t i o n s h i p f(x) = axe , and by the quadratic or 2 " l o g i s t i c " form f(x) = ax - bx Under the above hypotheses conditions (C) and (D) are both s a t i s f i e d . We thus have the following s p e c i a l case of Extensions ( i i i ) and ( i v ) . Extension (v) Consider the case of a marginal cost function of the form c/x f o r which the z e r o - p r o f i t l e v e l X Q i s sustainable. If the reproduction function f i s increasing and s a t i s f i e s the condition that x f ' ( x ) / f ( x ) be non-increasing, we have i n the case of a p o s i t i v e m o b i l i z a t i o n cost that Markovian and stationary (S,s) p o l i c i e s are optimal i n the f i n i t e time-horizon and i n f i n i t e time-horizon problems respectively. In the case of no m o b i l i z a t i o n cost (K=0) , Markovian and stationary stabilization-of-escapement p o l i c i e s are optimal for the f i n i t e and i n f i n i t e time-horizon problems respective. We now show that the r e s u l t of Extension (v) holds for a marginal cost function of the form c(x) = where a i s p o s i t i v e a constant l e s s than one. 1 216 In t h i s case we have Q' ( t f ( x ) ) f ' ( x ) Q'(x) _c a ( t f ( x ) ) 1 c a x„ X f'(x) ltf(x).L_ _ x 0 a a x - x Q [ t f f x l ] f , w We s h a l l show that both [ t f C x ) ] " - X^ a a x - X Q and [ x t f (x) ] f'(x) are decreasing, p o s i t i v e functions of x on [x^, m] under the hypotheses of Extension (v). For the former expression we make the transformation y = x with y^ = x^ and l e t 1/a a g(y) = [ t f ( y i / a ) ] . We can then write the expression as g(y) - y 0 If we can show that g(yg) i l yQ > a n ( i that g(y) i s concave and increasing, then as i n Extension (v) we w i l l have that the expression i s a decreasing p o s i t i v e function of y (Fig. 3.6). Since y i s an increasing function of x i t w i l l follow that 217 [ t f ( x ) ] - x Q a a. x - x Q i s a p o s i t i v e decreasing function of x . To show that g(y ) >_ y.Qc ;iss;easyr for g(y Q) = [tf ( x 0 ) ] a >_ x^ = y Q , since x^ i s sustainable. To show that g i s concave, increasing we consider a-1 g'(y) = a [ t f ( y )] t f (y ) 1/a y ' f ( y 1 / a ) y 1 / a f ( y 1 / a ) Now by hypothesis both ^ and ^ f ^ \ - X a r e decreasing p o s i t i v e x r (.xj functions of x , and since x i s an increasing function of y we have that f ( y 1 / 0 S . f ( y 1 / a ) y 1 / a 7 7 — and T-r 1/ot , 1/a. y f (y ) are decreasing p o s i t i v e functions of y . 1 / a)T I t follows that ( v ^ , — ' - l i s also a decreasing p o s i t i v e | W " ) T function of y and hence that g'(y) has t h i s property. We have thus established the concavity and monotonicity of g , and from t h i s we have that 218 [ t f ( X ) ] a - X Q a a x - x Q i s a p o s i t i v e decreasing function of x on [x^, m] . We now need only show that [ t f f x y ] a f , ( x ) i s a p o s i t i v e decreasing function of x on [ xg> m l • To do t h i s we rewrite the expression as _1 f (x)x . ^ ( x ) / - " a f(x) 1 x J From the hypotheses we have that f'(x)x , p f O O / " " —f(x) a n d [ — ] are p o s i t i v e decreasing functions of x , and hence t t f f x / ^ has t h i s property. We conclude that n i f ^ c i w r i / \ [tf ( x ) . ] a x j s " a Q' ( t f ( x ) ) f ' (x) ttr{y/;] ~ ;rQ -/ x , , Q'(x) :a a---- L t f ( x ) J * W x xra- x_a v x - 0 0 i s p o s i t i v e and decreasing i.e.e'that condition (C), (and hence also condition (D)) are s a t i s f i e d . We thus have the following r e s u l t . 219 Extension (vi) Theorems 3.3.1(a), (b), 3.3.2(a), (b) The r e s u l t s of Extension (v) hold when the marginal cost function takes the form instead of c(x) = — , a < 1 a x c(x) = f , provided the other hypotheses are s a t i s f i e d . A s i m i l a r r e s u l t holds f o r a l i n e a r marginal cost function c(x) = d - c/x , where d/c > m 220 Chapter 4 Some Results Concerning Optimal Economic P o l i c i e s 4.1 Introduction The theorems of Chapter 3 were concerned with the q u a l i t a t i v e nature of optimal p o l i c i e s for the given economic model. In t h i s chapter we s h a l l look further at these optimal p o l i c i e s and at the behaviour of the population under optimal e x p l o i t a t i o n . Section 4.2 w i l l be concerned e x c l u s i v e l y with the case of no m o b i l i z a t i o n cost (K=0) of the l a s t chapter. There i t was proved that under c e r t a i n conditions, a s t a b i l i z a t i o n of escapement p o l i c y was optimal. We s h a l l investigate the optimal l e v e l of s t a b i l i z a t i o n and compare i t with the l e v e l of s t a b i l i z a t i o n which i s optimal for the deterministic model. The q u a l i t a t i v e r e s u l t s of Chapter 3 made no assumption concerning the d i s t r i b u t i o n of the random m u l t i p l i e r s Z , save that n i t had unit mean and was on some non-negative closed i n t e r v a l . In p a r t i c u l a r they hold for a d i s t r i b u t i o n with a l l i t s mass at one, i . e . for the deterministic model. This has been investigated i n some d e t a i l by Clark (1971, 1972, lgySjCbyfe-v^PXeVie-rs the^ca^'£%<&:P«?A ii t iy e mobilization cost for a d i s c r e t e time deterministic model has not, as f a r as we know, been considered. In 4.3 we consider t h i s case, and show how the presence of a m o b i l i z a t i o n cost can bring about an optimal p o l i c y of "pulse harvesting", i . e . one where a harvest i s undertaken p e r i o d i c a l l y with there being one or more intervening seasons i n which no harvest takes place. We show how the s i z e of the m o b i l i z a t i o n cost 221 determines the period of the "pulse". In the past the world has seen the e x t i n c t i o n or near e x t i n c t i o n of several animal species, e.g. passenger pigeon, sea cow, buf f a l o , blue whale. Clark (1971, 1973'(b)-)• M s h i s U s i u d i e l of de'tefministic population models has shown, that, even i n the hands of a si n g l e owner, i t may be economically optimal to exterminate the population. In 4.4 we examine economic conditions which determine whether maximisation of expected present-value leads to e i t h e r e x t i n c t i o n of the population or conservation of i t . Throughout t h i s chapter we s h a l l be concerned with the same m u l t i p l i c a t i v e model X ,., = Z f(X ) as was used i n the previous two n+1 n n chapters. We s h a l l assume throughout that f i s concave - i . e . we have a compensation model. We s h a l l use the same economic model as i n the previous chapter. Any further assumptions w i l l be e x p l i c i t l y mentioned. 4.2 The Case of a Zero M o b i l i z a t i o n Cost In the l a s t chapter we showed, that under c e r t a i n conditions, an optimal p o l i c y , when there i s no mobilization cost, i s a s t a b i l i z a t i o n -of-escapement p o l i c y (Theorem 3.3.2(a), (b)). In th i s section we s h a l l derive some q u a l i t a t i v e r e s u l t s on the optimal l e v e l of s t a b i l i z e d escapement.. F i r s t l y we s h a l l obtain bounds f or the optimal escapement l e v e l , and then f i n d i t e x p l i c i t l y i n a c e r t a i n s p e c i a l case. Secondly we s h a l l compare the optimal l e v e l of escapement i n our stochastic model with the optimal l e v e l of escapement for the equivalent deterministic model of Clark (1971, 1973(b)). We s h a l l show that the optimal stochastic l e v e l can be greater than, l e s s than, or equal to the optimal deterministic l e v e l . In the two s p e c i a l cases of a constant marginal cost function, and a marginal cost function of the form c(x) = c/x , the optimal stochastic escapement l e v e l i s never lower than the optimal deterministic l e v e l . In both these cases the two l e v e l s are the same i f the optimal deterministic escapement l e v e l i s sustainable. As i n 3.3 and 3.4 we s h a l l assume that e i t h e r : -(a) the marginal cost function c(x) i s constant and f i s concave, but not n e c e s s a r i l y monotone, or (b) the condition (D) of 3.4 holds i . e . for every r e a l i z a t i o n t of- the random m u l t i p l i e r s , the function Q (tf(Q _ 1(w))) i s concave i n w on the i n t e r v a l [x^, m] . We showed that t h i s condition was s a t i s f i e d f or a cost function c(x) = c/x , a <_ 1 , provided we also assume that the z e r o - p r o f i t l e v e l x^ i s sustainable, and the reproduction function f i s concave, increasing and s a t i s f i e s the t e c h n i c a l condition that x f ' ( x ) / f ( x ) be non-increasing. We showed i n Theorem 3.3.2 and Extension (iv) that under conditions (a) or (b) an optimal p o l i c y for the f i n i t e time-horizon problem was the Markovian stabilization-of-escapement p o l i c y y N — 1} 2 j • • • • j x < S. N and an optimal p o l i c y f or the i n f i n i t e time-horizon problem was the stationary stabilization-of-escapement p o l i c y h(x) = x - S , x > S 0 , x <_ S We s h a l l be concerned with f i n d i n g the l e v e l S We define £(x) = a Q(tf( x ) ) d * ( t ) - Q(x) , where Q , we r e c a l l , was defined i n 3.4 as Q(x) = R(x) - R(x Q) = r x [c(x Q) - c ( t ) ] d t and represents the immediate harvest value of the resource. Suppose I a t t a i n s i t s maximum on [x^, m] at the point a The point c w i l l be rather s i g n i f i c a n t with respect to the determina-t i o n of the optimal escapement l e v e l S . We define p as the point at which f(x) at t a i n s i t s maximum on [0, m] . C l e a r l y a i s no greater than p for i f a > p we have £» ( a) = _Q' ( 0) + a Q'(tf(a)) t f ' ( a ) d $ ( t ) < 0 224 since Q i s s t r i c t l y increasing, and f < 0 on (p, m] . But t h i s cannot be so since £ attains i t s maximum at a . The f i r s t lemma establishes a and p as lower and upper bounds on the optimal l e v e l of s t a b i l i z e d escapement S . Lemma 1. Except i n the t r i v i a l case x^ >_ p , the optimal l e v e l of escapement S i s bounded as a < S < p Proof. We s h a l l follow notation of Theorem 3.3.2. F i r s t l y we prove the r e s u l t f o r c(x) = c y and then f o r when (D) i s s a t i s f i e d using Extension (iv).» From Theorem 3.3.2 we have P N + 1 ( x ) = -R(x) + C N ( t f ( x ) ) d $ ( t ) (1) and C N + 1 ( X ) " < a [ P N + l ( x ) + R ( X ) ] ' x < S N+1 ^ W W + R ( X ) ] ' x > V l (2) We prove f i r s t l y by induction that the functions PAT and C„T N N are d i f f e r e n t i a b l e on (0,m) . This i s true f o r C Q(x) = 0 . Let us suppose i t i s true for C N . Since R and f are d i f f e r e n t i a b l e we have from (1) that has the same property. 225 From (2) we have that C ^ i s d i f f e r e n t i a b l e on (0,S^ +^) and on (S^+^,m) . To complete the induction we need only show that C > T I 1 i s d i f f e r e n t i a b l e at S„,.. . This follows e a s i l y because N+1 N+1 C N + l ( S N + l + h ) ~ CN+1 ( SN+1 ) p V c , l x m + - = aR (S ) h->0 and i • C N + l ( S N + l + h ) " CN+1 ( SN+1 ) >,:"*" c . N , h " ^ ^mPw^ + C W ] since P„,. i s d i f f e r e n t i a b l e and at t a i n s i t s maximum at S„,.. . N+1 N+1 We now e s t a b l i s h the upper bound by contr a d i c t i o n . We s h a l l prove that <_ p for a l l N . Suppose S^ > p , and that x e (p,S^) . Then since i s non-decreasing f o r a l l N ( i n proof of Thm. 3.3.2), and since f'(x) <_ 0 we have 0 1 C N + 1 ' ( x ) < af'(x) b C N ' ( t f ( x ) ) t d $ ( t ) <_ 0 . a But C^'(x) = 0 only on [0, X Q ] , and since we have assumed p > x^ we have a contr a d i c t i o n . I t follows that S^ <_ p for a l l N , and hence that S = l i m S^ i s also <^  p . N - H » We prove the lower bound, likewise f o r a l l S„ . We note N that i CN'(X) " 1 a[R'(x) + P N (x)] , x < S N aR'(x) , x > S N 226 Now P^ (x) •> 0 on (°> S N) since P N i s concave and attai n s i t s maximum at . Hence for a l l x e (0,m) . C N'(x) >_ aR'(x) . Now f o r x e ( X Q , P ) P N + 1 ' ( x ) = -R'(x) + f'(x) >_ -R' (X) + af' (x) b C N ' ( t f ( x ) ) t d$(t) a R ( t f ( x ) ) d $ ( t ) , since f' (x) >_ 0 onrt (x^,p) , and so we have P N + 1 ' ( x ) >_a'(x) , on (x Q,p) . Now £ attai n s i t s maximum on [X Q , H I ] at a . Suppose f i r s t l y that a e (x„,m), . It follows that a < S„,, since P„,, i s concave and 0 — N+1 N+1 obtains i t s maximum at S„,, . Secondly i f a = x„ we have N+1 0 S„T,, > x- = a and t h i r d l y i f a = m we have that P>T, ' (m ) > I'(m ) > 0 N+1 — 0 N+1 — — and hence from the concavity of P„ T l, that S„., = m = a . J N+1 N+1 In a l l three cases we have that SN+1 ^ ° • This i s true f o r N = 0, 1,... , and we conclude that S = l i m S > a . N-x» For the more general case when the condition (D) i s s a t i s f i e d we consider the transformed v a r i a b l e y = Q(x) In terms of t h i s v a r i a b l e we define the functions D N(y) = VQfV) V y ) - p N ( Q " 1 y ) In terms of these functions, from Extension (iv) we have the equations D N(Qt f Q 1 y ) d $ ( t ) and D N + l ( y ) = a[y + \ + 1 ( y ) ] , a [ y + HN+I(W] v < T y - N+1 y > T. N+1 where T N +-^ i s the point at which the concave function a t t a i n s i t s maximum on [0, Q(m)] . (Note that f o r convenience brackets around the arguments of some functions have been omitted.) Following the same procedure as before we can show Q(°) 1 T N + 1 - Q ( p ) ' The l e v e l at which escapement i s s t a b i l i z e d i s S„,, = Q~^(T ., N+1 N+1 Hence we get that ° ± SN+1 ± P ' 228 and the r e s u l t follows. We next show that when the l e v e l a i s sustainable, stabilization-of-escapement at the l e v e l o i s an optimal p o l i c y . Theorem 4.2.1 If the l e v e l a i s sustainable then s t a b i l i z a t i o n of escapement at the l e v e l a i s an optimal p o l i c y , ( i n the sense of maximizing expected present value). Proof. We proceed by induction and assume = a . In the case (a), when c(x) = c , we have from Thm. 3.3.2, that f o r a < x < p , PN+1 ( X ) = " R ( x ) + = -R(x) + a C N ( t f ( x ) ) d $ ( t ) a [R(tf(x)) + P N(a)]d$(t) , a since from the s u s t a i n a b i l i t y of a we have that t f ( x ) >_ a f o r a l l t e [a, b] and a l l x e [a, p] , and thus that C N ( t f ( x ) ) = a[R(tf(x) + P N ( a ) ] It follows then that for a < x < p P N + l ( x ) = " Q ( x ) + 0 1 Q(tf(x))d$(t) + [aP N(a) - ( l - a ) R ( x Q ) ] = £(x) + constant Now from Lemma 1 we know that" P_TJ. at t a i n s i t s maximum on [x„, ml N+1 L 0 somewhere on [a, p] . Thus from the above we have that a t t a i n s i t s maximum on [x^, m] at the same point as that at which I a t t a i n s i t s maximum on [a, p] , i . e . at a . Hence S N +-^ = 0 • To complete the induction we show that = a . We have P 1(x) = -R(x) and P 2(x) = -R(x) + a R(tf(x))d$(t) = £(x) - (1-a) R(x Q) , and hence that P„ attains i t s maximum at a . Thus S„T = a f o r 2 N a l l N > 2 . I t follows that S = l i m S„T = a . N-**> For the more general case (b) when (D) i s s a t i s f i e d , we proceed i n a s i m i l a r way using now the equations i n terms of the transformed v a r i a b l e y = Q(x) . For the induction we assume that TN = Q ( V = Q ( a ) We then get for Q(a) <_ y <_ Q(p) V i ( y ) = -y+ D N(Qt f Q ^dHt) = -y + a [Q(tfQ -^y) + H N(Qa)]d$(t) since from the sustainability of a we have 230 t f (Q 1 y ) > a for y >_ a and hence Q(tfQ > Q(a) = T N We can write H N + l ( y ) = ^ Q l y ) + c o n s t a n t • As before we get that f L T I 1 attains i t s maximum at T „ f 1 where N+1 N+1 Q = & i . e . at = Q(°) • T° complete the induction we show T 2 = Q(o) . This follows from the fact that b -1 Q(tfQ y)d*(t) H 2(y) = -y + a = ^(Q _ 1y) Thus we have that TN - Q ( V = Q ( a ) for a l l N , and hence that T = Q(S) = Q(o) or S = a . This completes the proof of the theorem. We s h a l l now compare the optimal l e v e l s of escapement for the deterministic and stochastic models. The following r e s u l t gives the optimal l e v e l at which escapement i s s t a b i l i z e d i n the deterministic model and i s due to Clark (1973X-b)). Lemma 2. In the deterministic model the present-value of the resource i s maximized by a p o l i c y of s t a b i l i z a t i o n of escapement at the l e v e l <5 at which k(x) = aQ(f(x)) - Q(x) attai n s i t s maximum on [x^, m] . Proof We know that there i s a s t a b i l i z a t i o n of escapement p o l i c y which maximizes the present value f o r a l l i n i t i a l values of the population. Suppose the optimal l e v e l of s t a b i l i z a t i o n i s at x , and that the i n i t i a l population i s at l e v e l x^ >_ x . The p r o f i t i n the f i r s t year i s R(x^) - R(x) and i n subsequent years i t i s R(f(x)) - R(x) . Thus the t o t a l present value CO = a(R(x ) - R(x)) + I a n ( R ( f ( x ) ) - R(x)) n=2 = (aR(f(x)) - R(x)} + aR(x.) 1-a 1 = r ^ - k ( x ) + a R ( X l ) - (1-a) R(x_) 1-a 1 0 We seek the x which w i l l maximize t h i s over [ X Q , m] . We can assume x^ ^ x , since we know that the p o l i c y of escapement at x maximizes the present-value f o r a l l x^ . The present-value i s maximized at the l e v e l 6 at which k(x) a t t a i n s i t s maximum over [x^, m] . The next lemma compares the l e v e l s a and 6 at which £(x) and k(x) a t t a i n t h e i r respective maxima. Lemma 3 Let <5 and a be the points at which 232 £(x) = a. Q(tf(x))d$(t) - Q(x) and k(x) = aQ(f(x)) - Q(x) , a t t a i n t h e i r respective maxima on txQ> m ] • If the function xc(x) i s s t r i c t l y convex then a <_ 6 with equality possible only i f a = 6 = m or a = 6 = x^ . If xc(x) i s s t r i c t l y concave then a >_ 6 with equality possible only i f a = 6 = m or a = 6 = X Q . I f xc(x) i s l i n e a r then a = 6 . Proof Consider £'(x) - k-*(x) = a f ' ( x ) [ Q ' ( t f ( x ) ) t d * ( t ) - Q'(f(x))] a f ' ( x ) [ c ( f ( x ) ) - c ( t f ( x ) ) t d * ( t ) ] Now for x > 0 l e t the r.v. Y be defined as Y = Zf(x) , where Z i s a r.v. with df$ . We have that c ( t f ( x ) ) t d $ ( t ) = f(x) E{Y c(Y)} and from Jensen's Inequality we have that the r.h.s. i s e i t h e r r e s p e c t i v e l y greater than, l e s s than or equal to 1 E(Y) c(E(Y)) = c ( f ( x ) ) f(x) depending on whether the function xc(x) i s s t r i c t l y convex, s t r i c t l y concave, or l i n e a r . Thus since f'(x) > 0 on ( X Q , P ) we have that on (Xg,p):-(x) < k' (x) i f xc(x) s t r i c t l y convex £'(x) > k'(x) i f xc(x) s t r i c t l y concave and £'(x) = k'(x) i f xc(x) i s l i n e a r . Now under both the assumptions (a) and (b) given at the s t a r t of t h i s section, we have that Q ' ( t f ( x ) ) f ' ( x ) Q*(x) i s decreasing i n x for each t e [a, b] , (NB (a) i s a s p e c i a l case of (b)) and hence that b Q ' ( t f ( x ) ) f 1 ( x ) t d $ ( t ) a Q*(x) i s decreasing i n x . I t follows that £'(x) = a Q ' ( t f ( x ) ) f ' ( x ) t d$(t) - Q'(x) has at most one zero i.e.athat I i s unimodal. S i m i l a r l y k i s unimodal. Consider the case £'(x) < k'(x) on (x^,p) . F i r s t l y i f 6 e (x Q,p) we have that k'(6) = 0 , and hence that £'(6) < 0 . It 234 follows that i n t h i s case a < 6 . If <5 = X Q , we have, since £-k- i s decreasing, that f o r x > x Q , £(x) < £(XQ) + k(x) - k(x Q) < £(XQ) , since k attai n s i t s maximum at X Q . I t follows that £ also a t t a i n s i t s maximum at X Q i . e . a = S = X Q . If 6 = m , then since a e [ X Q , m] we have a <_ 6 = m . Thus i f xc(x) i s s t r i c t l y convex we have that a <_ 6 with equality only possible i f a = <5 = X Q or m . The lemma i s s i m i l a r l y proved f o r the case xc(x) s t r i c t l y concave. If xc(x) i s l i n e a r we have that &'(x) = m'(x) on ( X Q , P ) and hence that &(x) and k(x) d i f f e r only by a constant on th i s i n t e r v a l . It follows that they both a t t a i n t h e i r maxima on [ X Q , p] and hence on [ X Q , m] at the same point (NB we have already shown i n Lemma 1 that a <_ p , and we have from the same lemma that 6 <_ p also since the deterministic model i s obtained f o r a s p e c i a l form of the d.f. $ .) This completes the proof of the lemma. Our f i n a l theorem compares the optimal l e v e l s at which escapement i s s t a b i l i z e d i n the stochastic and deterministic models. Theorem 4.2.2 The optimal l e v e l of s t a b l i z e d escapement S for the in f i n i t e - t i m e - h o r i z o n stochastic model compares with that 6 i n the equivalent deterministic model i n the following way. 235 (i) I f xc(x) i s s t r i c t l y concave i n x then S >_ 6 with equality possible only i f S = 6 = X g O r S = 5 = m- . ( i i ) I f xc(x) i s l i n e a r then S >_ 8 with equality i f the l e v e l a i s sustainable. ( i i i ) I f xc(x) i s s t r i c t l y convex and the l e v e l a i s sustainable then S <_ 6 with equality possible i n t h i s case only i f S = 6 = X Q or S = <5 = m . Proof In case ( i ) we have from Lemma 3 that a >_ <5 . Also from lemma 1 we have S >_ a . The r e s u l t follows. The proof of ( i i ) follows from lemma 3 and Theorem 1. The proof of ( i i i ) follows from lemma 3 and Theorem 1. We now consider two s p e c i a l cases of the cost function. F i r s t l y when the marginal cost function c(x) i s constant we have* that xc(x) i s l i n e a r . Also we have that k(x) = (p-c) (af (x)-1) and hence 6 = q where q i s the unique s o l u t i o n to f'(x) = 1/a i f such e x i s t s , and equals 0 i f f'(x) < 1/a on [0, m] and equals m i f f'(x) > 1/a on [0, m] . From the above theorem we have Corol l a r y 1 For a constant marginal harvest cost, the optimal stochastic l e v e l of escapement S i s no l e s s than q the optimal deterministic l e v e l of escapement, with S = q i f q i s sustainable. 236 The second s p e c i a l case we consider i s c(x) = c/x . If we make the assumptions of Extension (v) of 3.4 we have that condition (D) i s s a t i s f i e d . In t h i s case xc(x) i s l i n e a r and from the above theorem we have Corol l a r y 2 For a marginal harvest cost function c(x) = c/x , under the conditions of Extension (v) of 3.4 we have that the optimal l e v e l , S , of s t a b i l i z e d escapement i n the stochastic model i s no l e s s than the optimal l e v e l of s t a b i l i z e d escapement i n the equivalent deterministic model. The two l e v e l s are equal i f 6 i s sustainable. For both the other cases, c(x) = — , a < 1 a • x and c(x) = d - cx for which we have established condition (D) i n 3.4, we have that xc(x) i s concave, i, I t follows from ( i ) of the theorem above that S >_ 6 f o r both these cases. We are l e d to ask the question whether S i s ever le s s than 6 ? We f i n d an example of t h i s below. 2 Consider c(x) = c/x , and Beverton-Holt reproduction 237 function f(x) = ax/(1+bx) . In t h i s case 2 /tax > 2 Q ' ( t f ( x ) ) f ' ( x ) _ ^l+bx ; x 0 . (1+bx) Q'(x) 2 2 • 2 2 ,2 x - x Q a t (1+bx) 2 .tax . _ 2 1 (-l+bxj x 0 2 - 2 2 at x - x Q The monotonicity of t h i s can be proved i n the same way as i n Extension (vi) of 3.4 by making the s u b s t i t u t i o n 2 . , v ,ta/y 2 1+bvV y = x , g(y) = (-provided we assume as before that X Q i s sustainable. Thus condition (D) i s s a t i s f i e d and from Extension (iv) a stabilization-of-escapement p o l i c y optimal. In t h i s case we have that xc(x) = c/x which i s convex. We have from ( i i ) of the theorem above that i f " a ( / i s sustainable S <_ 6 with equality only i f S = 6 = x^ or m . In t h i s way we can construct an example for which S < 6 . We have thus seen that the optimal stochastic escapement l e v e l can be e i t h e r greater than, l e s s than or equal to the equivalent optimal deterministic l e v e l , which case depending on the cost function and the reproduction model. If the f l u c t u a t i n g dynamics of the resource were more nearly approximated by the above stochastic model rather than by a det e r m i n i s t i c model, i t could well be, the case, i f a deterministic model were used to c a l c u l a t e escapement l e v e l s that the resource would be maintained at a l e v e l higher or lower than i t s true optimum l e v e l . 4.3 The Optimality of Pulse Harvesting f o r a Deterministic Population  Model with P o s i t i v e M o b i l i z a t i o n Cost. In Chapter 3 we have shown how an (S,s) p o l i c y i s optimal under c e r t a i n conditions f o r the given stochastic model. As a s p e c i a l case of t h i s we can consider $ a degenerate d i s t r i b u t i o n with a l l i t s mass at the point one. This i s simply the deterministic population model Xn+1 = f ( x n } If the other conditions of Theorem 3.3.1(b) or one of i t s extensions are s a t i s f i e d , we w i l l have that f o r a p o s i t i v e m o b i l i z a t i o n cost, an optimal i n f i n i t e - p e r i o d deterministic p o l i c y w i l l be a stationary (S,s) p o l i c y . It may well be that i t takes more than one period f o r the population to regenerate from S to s , i . e . f(S) may be les s than s . In t h i s case we w i l l have what has been c a l l e d a p o l i c y of "pulse harvesting", i . e . a p o l i c y where a harvest i s undertaken p e r i o d i c a l l y , but r e g u l a r l y with one or more intervening years i n which no harvest i s undertaken. This i s the actual case f or some f i s h e r i e s , and the following theory may i n part explain why t h i s i s 239 so, although i n most r e a l - l i f e s i t u a t i o n s more than one population i s involved, with the harvest operation moving from one to another i n successive years. The main r e s u l t of t h i s section shows how the period of the optimal pulse harvest depends on the mobil i z a t i o n cost K . We s h a l l assume that we are dealing with an i n f i n i t e time-horizon problem and that an (S,s) p o l i c y i s optimal, i . e . we s h a l l assume that e i t h e r the conditions of Theorem 3.3.1(b) or one of i t s extensions ( i ) , ( i i ) , ( i i i ) , (v) or (vi) are met. The population model i s given by the difference equation. For any population l e v e l x e [0, m] the present value C(x) i s continuous i n K , the mobil i z a t i o n cost. x n+1 Lemma 1 Proof. Let C (x) be the present-value of the resource, with mobil i z a t i o n cost K Suppose that f or a given K , N(K) i s the optimal harvest period. Then C v(x) = Q. N(K) (x) - 1-a' (K) K where Q^^^(x) represents the return that would have ensued from t h i s p o l i c y i f there were no mobilization cost, and p i s the year of the f i r s t harvest (p depends on x and K) . For a mobil i z a t i o n cost K + e , w x ) ^ w x ) - r^m (K+e> -1-a for the optimal p o l i c y i n t h i s case y i e l d s a return as l e a s t as good as that with period N(K) a P C K + £ ( x ) > C K(x) - — ^ • £ 1-a Now p >_ 1 , and N(K) >_ 1 , and p a r a < , N(K) - 1-a 1-a and so we have C K + £ ( x ) l C K ( x ) - ^ e C l e a r l y C K(x) i s decreasing i n K , and t h i s establishes the continuity of C (x) K We r e c a l l the notation R(x) = px - c ( t ) d t 0 ^n We s h a l l write f (x) for the n - f o l d composition f°f° °f(x) 241 The next lemma gives the optimal escapement f or a given period of harvest. Lemma 2 If the optimal period of harvest N(K) i s n then the optimal l e v e l of escapement when there i s a harvest i s q n , where maximizes cf> (x) = a n R ( f n ( x ) ) - R(x) n over [ x ^ m] . Proof Suppose that the optimal (S,s) p o l i c y has escapement l e v e l S<m, and has harvest period n. Then f or an i n i t i a l population l e v e l m we have that n+1 C(m) = a(R(m) - R(S) - K) + - (R(f n(S)) - R(S) - K) Since t h i s (S,s) p o l i c y i s optimal for a l l i n i t i a l population l e v e l s we have that S maximizes n+1 -aR(x) + — - ( R ( f n ( x ) ) - R(x)) 1-a i . e . i t maximizes <f>n(x) = a n R ( f n ( x ) ) - R(x) over [ x Q , m] . We s h a l l l a t e r prove some r e s u l t s concerning the l e v e l s q n for the s p e c i a l case of a constant harvest cost i . e . R(x) = (p-c)x . Lemma 3 The optimal harvest period N(K) i s non-decreasing i n K 242 and increases at most i n steps of s i z e one. Proof For a mobili z a t i o n cost K , and an N-period harvest using the escapement l e v e l , the return f o r an i n i t i a l l e v e l m i s N+1 a(R(m) - R(q N) - K) + 2 _ ( R ( f W ( q ) - R(q ) - K) 1-a i . e . aR*(m) + 1-a N = 1, 2,...i.e. N(K) maximizes N~ ^ N^%^ ~ K ^ ' N o w N ^ maximizes t h i s over a l l v v - K A N(K) - W W - K W " K 1 - a N Let 1-a N+1 1 - a N Then N(K) i s the smallest N f or which Aj^(K) i s non-positive. Now for 6 > 0 V K + 6 ) • A N ( K ) + 1-a N. N 1-a N+1 • VK> + a " ( N " a ) N+1 * W (1-a )(1-a X) Since the second term i s p o s i t i v e , AN(K+6) w i l l f i r s t change sign f o r N _> N(K) (see F i g . 4.1) i . e . N(K+6) >_ N(K) which proves the monotonicity of N(K) i n K . VK + 6> ^ N(K+<5) F i g . 4.1 A N ( K ) We can make AN(K+<5) a r b i t r a r i l y close to A M(K) by choosing <5 s u f f i c i e n t l y small. Thus we can ensure that N(K+6) <_ N(K) + 1 by choosing 6 s u f f i c i e n t l y small (see F i g . 4.1). This proves that N(K) increases v o n l y n i j i steps of s i z e one. 243 We have shown i n the l a s t lemma that the period of the optimal harvest increases i n steps as the mobiliz a t i o n cost increases. In the following theorem we determine where these steps occur. This theorem t e l l s us exactly how the si z e of the mobiliz a t i o n cost determines the period of the optimal pulse harvest. Theorem 4.3.1 If the mo b i l i z a t i o n cost K l i e s i n the i n t e r v a l [k ., k ] where k- = 0 and n-1 n 0 , n+1 i J- - 0 1 r n„n ,_n. „, N 1 kn'= [ a ( f <«n» " R ( q n ) ] l-an r n+l„ . ..n+1 . . . _,. - , " ^ I = a T [ a R ( f ^n+1^ " R ( q n + 1 ) ] then an n-period pulse harvest down to the l e v e l i s optimal. Proof We have from Lemma 3 that N(K) i s a non-decreasing step-function with steps of s i z e one. Let us suppose that i t has steps at 0 = k , k , k , etc. We have from Lemma 1 that the present value function C(x) i s continuous i n the parameter K . Thus at the point K = k^ both an optimal n-period and an optimal (n+l)-period harvest w i l l y i e l d 8 the same return (n>_l) . 244 For an i n i t i a l value x^ = m , • the optimal value of an n-period harvest i s -n+1 a(R(m) - R(q ) - K) + ( R ( f n ( q ) - R(q ) - K) n . n n n 1-a i . e . aR(m) + — — ( a n R ( f n ( q )) - R(q ) - K) , - otn n n 1-a'. and the optimal value of an (n+1)-period harvest i s ^ + 7^n+T ^ ^ ( ^ C W " R ( q n + 1 ) " K ) 1-a Equating these two for the case K = k we get n ( a n R ( f n ( q n ) ) - R(q n) - k Q) ( a n + 1 R ( f n + 1 ( q ^ ) ) - R ^ ) - k Q) , n . n+1 1 - a 1 - a From which we get 1 n+1 k = —tz r- [a R (f (q.)) - R(q )] n a(1-a) ^n T I 1-a11 r n+1 ..n+1. „. ^ a T [ a R ( f ( V l } ) " R ( q n + 1 ) ] which proves the r e s u l t . To conclude t h i s section we prove some r e s u l t s about the optimal l e v e l s of escapement q^ , f o r n-period pulse harvests, under the assumption of a constant marginal harvest cost. In t h i s case R(x) = (p-c)x = Rx «• • say. 245 We s h a l l ignore the case i n which the population i s harvested to e x t i n c t i o n at any time. This means we can assume q > 0 . We n have seen i n Lemma 2 how q maximizes n <j> (x) = a n R ( f n ( x ) ) - R(x) . n In t h i s case then q i s the so l u t i o n to n a There i s a unique s o l u t i o n to t h i s f o r f n ( x ) i s concave, being the composition of increasing concave functions. I f {f n(x)} < — ^ a for a l l x e [0, m] , q^ = 0 , but we are ignoring t h i s case. I f d n 1 T~ {f (x) } > — for a l l x e [0, ml , q = m , but t h i s means no dx n n a harvest i s made and so we can ignore t h i s p o s s i b i l i t y also. We f i r s t l y show that (1) q < q, for a l l n = 1, 2, n — 1 Proof We have 1 _ d f n — ~ T~ f (x) n dx a = f ( f n 1 ( q n ) ) . f ( f n 2 ( q n ) ) . f ' ( q n ) q n i [ f ( q n ) ] n since to s a t i s f y Theorem 3.1(b) we assume f concave and increasing, 2 n—1 and hence f , . . . , f also have t h i s property. It follows that n — a f'Cq-L) and hence that q n ± q l ' We now show that (2) qn ^  f ~ V l > Proof d .n, . dx" f ( X ) n q a n 1 a n-1 f*«l> df f n _ 1 ( x ) n-d .n-1 ' dx" f ( x ) f ( q n ) * n - l From (1) we have f ' ( q n ) rl f ' ( q 1 ) and hence we have dx" f ( x ) f ( q j n d n - l at .n-1 . , Now r i s concave non-decreasing, since f i s , and thus we have f <«n> ^ V l 247 Since f has an inverse we have We also have ( 3 ) F i n a l l y we show that the sequence {q n^ *-s non-increasing. (4) q < q i n — n-1 Proof 1 _ d f n — - T~ f ( x) n dx But also f (f 1 1 1 ( q n ) ) d .n-1 dx" f ( X ) *n (i) 1 1 1 n a n-1 a a f ( q i } ' d l f d n - l ( I D Now from previous r e s u l t (2) .n, f (q n) > f ( q x ) and hence ^n-1. x f (q n) > q x from the equality of the r.h.s. of ( i ) and ( i i ) d .n-1 dx" f ( X ) n *n-l 248 It follows from the concavity of f that q < q .. T I — n-1 We thus see that for a constant marginal harvest cost the escapement l e v e l decreases as the period of the harvest increases with the m o b i l i z a t i o n cost. * 4.4 Economic Determinants of Survival In t h i s section we s h a l l examine conditions which imply that maximization of expected discounted p r o f i t leads to either e x t i n c t i o n of the population or conservation of the population. C. Clark (1971, 1973) using a discrete-time deterministic model has shown that even i n the hands of a s i n g l e owner i t may be economically optimal to exterminate the population i f the time-preference rate i s s u f f i c i e n t l y high. In the case of an uncontrolled common property resource i t i s well known how competitive e x p l o i t a t i o n can lead to the extermination of the population (see Hardin (1968), Gordon (1954)). Clark (1973) shows how t h i s case can be considered as a s p e c i a l case of maximization of discounted p r o f i t (present value) with an i n f i n i t e discount rate (a=0) . In t h i s s ection we s h a l l prove a v e r s i o n of the 'Extinction Theorem' of Clark (1973) for a stochastic population model. We s h a l l assume, i n the case of a p o s i t i v e m o b i l i z a t i o n cost, 249 that the conditions of Theorem 3.3.1 or one of i t s extensions are met „ and thus an (S,s) p o l i c y i s optimal, and i n the case of a zero mobi l i z a t i o n cost that the conditions of Theorem 3.3.2 or one of i t s extensions are met and thus that a s t a b i l i z a t i o n of escapement p o l i c y i s optimal. E x t i n c t i o n of the population i s optimal when S = 0 , and conservation i s optimal when S > 0 . F i r s t l y we note that i f 0 < x^ <^  p , then harvesting below the z e r o - p r o f i t l e v e l x^ not only y i e l d s a negative return, but also reduces the l e v e l of the population i n the subsequent year and thus cannot be the action of an optimal p o l i c y . Thus regardless of other economic and b i o l o g i c a l factors we have that:-extermination of the population at any given harvest can only  be an optimal p o l i c y i f i t i s p r o f i t a b l e to harvest the l a s t surviving  animals of the population. This i s true even f or the case of an i n f i n i t e time-preference rate (a=0) and thus we have the obvious and well-known r e s u l t that d i s s i p a t i o n of economic rent can only lead to e x t i n c t i o n i f i t i s p r o f i t a b l e to harvest the l a s t surviving animals of the population. We note that f o r the decreasing marginal cost functions ct c/x , a <_ 1 discussed i n the extensions of 3.4, we have X Q > 0 and so for these cost functions maximization of expected p r o f i t never leads to e x t i n c t i o n of the population. We s h a l l now assume x r t = 0 and look further f o r conditions which imply S = 0 or S > 0 • In order to ensure that an (S,s) p o l i c y i s optimal f o r the i n f i n i t e - p e r i o d problem with p o s i t i v e m o b i l i z a t i o n cost, K we s h a l l assume that e i t h e r (a) f i s concave and increasing and c(x) = constant. (Thm. 3.3.1(b)) or (b) f i s concave and increasing and c(x) s a t i s f i e s condition (A) of 3.4 (Extension ( i ) ) or (c) condition (C) of 3.4 be met (Extension ( i i i ) ) . In the case of a zero m o b i l i z a t i o n cost we s h a l l assume that (d) f i s concave, but not necessa r i l y monotone, and c(x) = constant. (Thm. 3.3.2(b)) or (e) condition (D) of 3.4 be met. (Extension ( i v ) ) . Under ei t h e r of these hypotheses a s t a b i l i z a t i o n of escapement p o l i c y i s optimal. e i t h e r When X . = 0 we have that Q(x) = R(x) = [p - c ( t ) ] d t R e c a l l from 4.2 that £(x) = a Q(tf(x))d$(t) - Q(x) which i n t h i s case i s rb £(x) = a R(tf(x))d$(t) - R(x) 251 Our f i r s t theorem gives a necessary and s u f f i c i e n t condition for e x t i n c t i o n (or conservation) of the population to be an optimal p o l i c y . I t s usefulness w i l l become apparent l a t e r , when we use i t to derive other r e s u l t s . Theorem 4.4.1 In the case of a p o s i t i v e m o b i l i z a t i o n cost (K>0) suppose that one of conditions (a), (b), (c) above i s met and that consequently an (S,s) p o l i c y i s optimal. In the case of no mobilization cost suppose that condition (.;;e) or (d) above i s met, and that consequently a stabilization-of-escapement p o l i c y i s optimal. Let \Jj(x) = l i m i|>N(x) N-*» where the functions ^ ( x ) a r e defined r e c u r s i v e l y by:-^ ( x ) = -R(x) ^ N ( t f ( x ) ) d $ ( t ) - a K d<S>(t) t:^ N(tf(x))+K>0 t:iP N(tf(x))+K<0 Suppose that i t i s p r o f i t a b l e to harvest the l a s t surviving animals ( i . e . x Q = 0 ) > then i f (i) iKx) ±_ 0 for a l l x e. [0, m] , maximization of expected discounted p r o f i t leads to a p o l i c y of e x t i n c t i o n . ( i i ) there e x i s t s x e [0, m] for which if)(x ) > 0 , maximization of expected p r o f i t leads to a p o l i c y of conservation (S>0) . 252 Proof We f i r s t - prove the theorem for the case c(x) = constant ((a), (d) above) and f o r the case (b) above. We give an i n t e r p r e t a t i o n to *KX) and prove i t s existence. Consider a s i t u a t i o n i n which the only options open to the resource owner i n any year are to eit h e r harvest the population to A e x t i n c t i o n or not to harvest at a l l . Let C^ (x) represent the optimal expected discounted revenue that can be obtained i n t h i s s i t u a t i o n , when N periods remain and the current l e v e l i s x . Define * A C„ (x) = 0 . We have that C„ s a t i s f i e s 0 N C N (x) = a max{R(x) - K , C N_ 1 (tf(x))d$(t)} = a R(x) + a max{-K, -R(x) + C N_ X ( t f ( x ) ) d $ ( t ) } We define * N ( x ) = -R(x) + C N_ 1 ( t f ( x ) ) d * ( t ) I t then follows that CN ( X ) = a[R(x) - K] , it N(x) + K < 0 a[R(x) + <j,N(x)] , ^ ( x ) + K > 0 and hence that Tj) , (x) = -R(x) + N+1 C N ( t f ( x ) ) d * ( t ) 253 = £(x) + a * N ( t f ( x ) ) d * ( t ) - a K d$(t) * N(tf(x))+K>0 i|)N(tf(x))+K<0 which, since ^ ( x ) = -R(x) + b * C Q ( t f ( x ) ) d * ( t ) = -R(x) , a i s exactly the previous d e f i n i t i o n . From the general theory of Dynamic Programming (Blackwell 1965) * we have that the sequence of functions {C^ } converges uniformly to a function C (x) . I t follows that {xp } converges uniformly to a function ip(x) • The sequence {C^ (x)} i s monotone i n Nj . This follows from the fac t then when N + 1 periods remain and the l e v e l i s x , oneecan do at le a s t as well as when N periods remain and l e v e l i s x , simply by following the optimal p o l i c y of the l a t t e r case f o r the f i r s t N periods. From t h i s we have that the sequence (x)} i s monotone and ^ ( x ) — 'Kx) f ° r a ± x x • . We now prove ( i ) by induction and suppose that n " ' n S± = S 2 =?....= S N = 0 . I t follows that f o r n £ N , (x) = C„ (x) , because when there are no more than N periods remaining the only options open are no harvest or a harvest to e x t i n c t i o n . From t h i s f a c t we have that P N + l ( x ) = " R ( x ) + C N ( t f ( x ) ) d * ( t ) = -R(x) + a b * C N ( t f ( x ) ) d * ( t ) = * ( a 254 Since ^ N ( x ) . S * K X ) f ° r a H N i t follows that PN+1 ( X ) - *<*> • If i > ( x ) ±0 , \/x e [0, m] we have that P N + 1 ( x ) <_ 0 , V x e [0, m] and thus that i t attai n s i t s supremum on [0, m] at 0 i . e . that S N + ^ = 0 • T r i v i a l l y S^ = 0 , and thus the induction i s complete and we have that S^ = 0 f o r a l l N = 1, 2,... and thus that S = 0 i . e . a p o l i c y of e x t i n c t i o n i s optimal. We prove ( i i ) by contradiction and suppose S = 0 , i . e . that P(x) a t t a i n s i t s supremum on [0, m] at 0 . This implies that P(x) £ 0 , V x e [0, m] . Since ^ j j ( x ) » a n d hence P j ^ ( x ) > i - s monotone non-decreasing i n N i t follows that P N(x) £ 0 , N = 1, 2, and hence that S N = 0 , N = 1, 2, From t h i s we have, as i n ( i ) that p ^ ( x ) = ^ ( x ) » N = 1, 2,.... and hence that P(x) = i K x ) • The condition that there e x i s t s x f o r which ij>(x ) > 0 contradicts the statement P(x) <_ 0 , V x e [0, m] . We conclude that S > 0 . In the more general cases (c) and (d) above, when the proof of the optimality of (S,s) p o l i c i e s involves the transformed s t a t e -v a r i a b l e y = R(x), the proof proceeds i n terms of transformed var i a b l e s . We define ^ ( y ) = -y and 255 ^ N + l ^ = m ^ + 0 1 4> N(RtfR ^ d ^ t ) <f>N(RtfR 1y)+K>0 - a K d$(t) <J)N(RtfR 1y)+K<0 where m(y) = a R(tfR 1y)d$(t) - y The necessary and s u f f i c i e n t condition f o r e x t i n c t i o n to be optimal w i l l be that $\yj g 'Mm * (y) <0 , \/ y e [0, R(m) ] N-*» which i n terms of the population l e v e l x , i s that <)>(R(x)) £ 0 V x e [0, m] Now i t i s easy to show by induction that <j>(Rx) = ^(x) and so the r e s u l t of the theorem holds. QED The above theorem gives conditions which determine exactly whether a p o l i c y of conservation or e x t i n c t i o n i s optimal. The functions IJJ can i n p r i n c i p l e be computed r e c u r s i v e l y , and the function ^(x) can be computed, at l e a s t approximately. However the theorem gives no simple i n d i c a t i o n of the way economic and b i o l o g i c a l parameters influence the optimality of conservation or e x t i n c t i o n p o l i c i e s . From the p r a c t i c a l point of view these are the important questions. We s h a l l study t h i s i n the theorems that follow. F i r s t l y we consider the e f f e c t of time-preference. 256 I n t u i t i v e l y we would think that an excessively high discount rate would lead to a p o l i c y of e x t i n c t i o n . We prove t h i s to be the case i n the following theorem. I t i s stated i n terms of the discount rate r = 1 / a - 1 . Theorem 4.4.2 There e x i s t s a c r i t i c a l value r of the discount rate such that maximization of expected discounted p r o f i t leads to:-(i ) e x t i n c t i o n of the population i f i t i s p r o f i t a b l e to harvest the l a s t surviving animals and i f the discount rate, r >_ r ( i i ) conservation of the population i f either i t i s not p r o f i t a b l e to harvest the l a s t surviving animals or i f the discount rate r < r . Proof We show that <Kx) i s non-decreasing i n the parameter a f o r any fixed x . We follow a proof by induction and assume ^ ( x ) non-decreasing i n a , f o r any x . A I t follows from the d e f i n i t i o n s that (x) and hence i|^ +^(x) are non-decreasing i n a . T r i v i a l l y i j^(x) = -R(x) i s non-decreasing i n a , being independent of a . Thus the induction i s proved and ^ ( x ) > a n d hence ijj(x) , are hon-decreasing i n a . Now sup ip(x) >_ 0 since 4* (0) = 0 , and also sup ijj(x) [0,m] [0,m] i s non-decreasing i n the parameter a . Let a = sup { a : sup I|J(X) =0} 0<a<l [0,m] 257 Then for a > a , sup ip(x) > 0 and hence from ( i i ) of Theorem 4.4.1 [0,m] conservation i s optimal. For a <_ a , sup iKx) = 0 and hence from [0,m] (i) of Theorem 4.4.1, e x t i n c t i o n i s optimal. L e t t i n g r = 1/a - 1 , and using the f a c t that e x t i n c t i o n can only be optimal i f X Q = 0 we get the r e s u l t . QED For the case of a p o s i t i v e m o b i l i z a t i o n cost, K > 0 , c a l c u l a t i o n of the c r i t i c a l discount rate r seems d i f f i c u l t . However we can prove a weaker r e s u l t which gives 'safe' and 'danger' l e v e l s f o r the discount rate. We need f i r s t l y a lemma. Lemma 1 Assume x^ = 0 . (i) I f £(x) <_ 0 , V x e [0, m] then ^(x) <_ 0 , \/x e [0, m] . * * ( i i ) I f there e x i s t s x e [0, m] such that £(x ) > aK , then iKx*) > 0 . Proof We prove ( i ) by induction and suppose that ^ N ( x ) £ 0 , Vx e [0, m] . We have that a[R(x) - K] , ^ ( x ) + K < 0 a[R(x) - * N(x)] , i|iN(x) + K >_ 0 and hence that C N (x) < aR(x) It follows that 258 W(x) = "R(x) + C N ( t f ( x ) ) d $ ( t ) < -R(x) + a R(tf(x)d$(t) = &(x) Hence If £(x) <^  0 , V/x E [ 0 , m] we have that * N + 1<x) £ 0 , \/x e [0, m] To complete the induction we observe that ^ ( x ) = -R(x) <_ 0 , x e [0, m] It follows that ^ ( x ) £ 0 ,\/n , and hence that I|J(X) <_ 0 . The proof of ( i i ) follows from the defining recursion. ^ N + 1 ( x ) = £(x) + a ^ N ( t f ( x ) ) d $ ( t ) - a K d$(t) iJ; N(tf(x))+K>0 ^ N(tf(x))+K<0 > £(x) - a K d$(t) = £(x) - SR.. - * * Hence iji(x) 1 £(x) - aK. . I f £(x ) > aK then I|J(X ) > 0 . In the case where the proof of the optimality of an (S,s) p o l i c y involves using the transformed v a r i a b l e y = R(x) , we have that (i) m(y) £ 0 Vy £ [0, R(m) ] implies <|>(y) < 0 , V y e [0, R(m) ] which i s equivalent to £(x) <_0 ,Vx E [0, m] implies ijj(x) 5.0 , V x e [0, m] since m(Rx) = £(x) and <f>(Rx) = ^(y) ( i i ) i f there e x i s t s y e [0, R(m)] such that m(y ) > aK then <Ky ) > 0 which i s equivalent to:- i f there e x i s t s x e [0, m] such that £(x ) > aK then ijj(x ) > 0 . Thus the Lemma holds i n t h i s case. We are now ready to f i n d 'safe' and 'danger' l e v e l s f or the discount rate r . Theorem 4.4.3 Let r. sup [0,m] R ( t f ( x ) ) d * ( t ) R(x) and r r sup [0,m] R ( t f ( x ) ) d * ( t ) - K R(x) - 1 If the discount rate r i s greater than or equal to r ^ then maximization of expected p r o f i t leads to e x t i n c t i o n of the population, providing that i t i s p r o f i t a b l e to harvest the l a s t surviving animals. If r < r ^ , then maximization of expected p r o f i t leads to a p o l i c y of conservation. 260 Proof We use Lemma 1. If r >_ r ^ , then Vx e [0, m] , rb I > a — R(tf(x))d$ R(x) i / e . £(x) <_ 0 . I t follows from Lemma 1 that i > ( x ) <_ 0 , Vx and hence from Theorem 4.4.1 that e x t i n c t i o n i s an optimal p o l i c y , provided of course that x^ = 0 If r < r 2 , then there ex i s t s x e [0, m] such that R(tf(x ))d$(t) - K Roa i . e . such that £(x ) > aK . From Lemma 1 we have that IJJ(X ) > 0 , and hence from Theorem 4.4.1 that a conservation p o l i c y i s optimal. QED In the case for which the marginal cost function, c(x) , i s constant (cases (a) and (d) above) or i n the cases i n which conditions (C) or (D) of 3.4 are met — i . e . Q(tfQ "'"(w)) concave i n w for every t e [a, b] (cases (c) and (e) above) — we can calc u l a t e the l e v e l r ^ e x p l i c i t l y . Consider f i r s t l y the case c(x) = c . In t h i s case R(x) = (p-c)x and 1 + r = sup [0,m] tf(x)d$(t) x 261 sup I i [0,m] X f (0) since f i s concave. .-1. More generally i f Q(tf(Q (w)) i s concave i n w , we have rb 1 + r = sup [0,R(m)] R(tfR 1 y ) d $ ( t ) and since R(tfR \ y ) ) = Q(tfQ "*"(y) i s concave i n y we have 1 + r = l i m { y-K) R(tfR \)d^(t) dy J - l R(tfR y)d*(t)}| = f ( R 1 ( 0 ) ) ( R X) (0) R'(tfR 1 ( 0 ) ) t d $ ( t ) = f'(0) since (R X ) (0)R'(0) = 1 So i n both cases r ^ = f'(0) - 1 . We can think of r ^ = f'(0) - 1 as the maximum expected annual growth rate i n the population, since max -} = f'(0) - 1 , [0,m] X 262 since f i s concave. A l t e r n a t i v e l y we can think of r ^ as the maximum of the marginal expected growth i n the population as the population s i z e increases by unit amount. . S i m i l a r l y we can think of r ^ as both the maximum expected annual growth rate i n the immediate harvest value of the population, and as the maximum of the marginal expected growth i n the immediate harvest value as that value increases by unit amount, since max ye[0,R(m)] = f'(0) - 1 , since we are assuming that R(tfR ''"(y)) i s concave for a l l t e [a, b] . We see then that i f the e f f e c t i v e discount rate, r , i s greater than r ^ , then wealth invested at the i n t e r e s t rate r w i l l have an annual growth greater than the annual growth of the immediate harvest value of the population. S i m i l a r l y wealth thus invested w i l l have a'marginal growth greater than the marginal expected growth of the immediate harvest value of the population. In economic terms we could say that i f r >_ r ^ , then the resource i s sub-marginal i n expectation. The l e v e l s r ^ and r ^ of Theorem 4.4.3 give "danger" and "safe" l e v e l s f o r the discount rate. I f the discount rate i s high enough so that the resource i s sub-marginal i n expectation ( i . e . i f r exceeds the danger l e v e l r.,) then maximization of expected p r o f i t R(tfR 1 ( y ) ) d $ ( t ) - y } = li m { y-K) ,-1 R(tfR "(y)d$(t) - y •> a 263 w i l l lead to the e x t i n c t i o n of the population provided that i t i s p r o f i t a b l e to harvest the l a s t surviving animals. If the discount rate i s l e s s than the safe l e v e l then conservation i s optimal. In the case of an unregulated common-property resource we can think of the e f f e c t i v e discount rate as being i n f i n i t e (Clark 1973). In t h i s case then, except i n the case of a population model for which f' (o) = oo (such as f (x) = ax b) , the resource w i l l always be sub-marginal i n expectation. The sole determinant of conservation or e x t i n c t i o n w i l l be whether i t i s p r o f i t a b l e to harvest the l a s t surviving animals. We now look at the case of no mobil i z a t i o n cost (K=0) ; In th i s case we have from Lemma 1 and Theorem 4.4.1, that i f x_ = 0 , a 0 necessary and s u f f i c i e n t condition f or the optimality of an e x t i n c t i o n p o l i c y (S=0) i s that £(x) £ 0 ,Vx e [0, m] . The safe and danger l e v e l s r ^ and r ^ of Thm. 4.4.3 are equal and thus both equal to the c r i t i c a l l e v e l r of Thm. 4.4.2. We have then that r = r 2 = r x = f'(0) - 1 and the following c o r o l l a r y . C o r o l l a r y (Thm. 4.4.3) If there i s no m o b i l i z a t i o n cost, then maximization of expected discounted p r o f i t leads to e i t h e r ( i ) e x t i n c t i o n of the population i f I t i s p r o f i t a b l e to harvest the 264 l a s t surviving animals and i f the discount rate r >_ f'(0) - 1 ( i . e . i f the resource i s sub-marginal i n expectation). ( i i ) conservation of the population i f e i t h e r i t i s not p r o f i t a b l e to harvest the l a s t surviving animals or i f the discount rate i s les s than f' (0) - 1 . This i s a stochastic version of the 'extinction' theorem of Clark (1973). It gives a necessary and s u f f i c i e n t condition on the discount rate for the optimality of conservation (or e x t i n c t i o n ) . We now investigate the e f f e c t of the mobil i z a t i o n cost i n determining the optimality of conservation or e x t i n c t i o n . For s i m p l i c i t y we s h a l l ignore the case (b) above ( i . e . we s h a l l assume eith e r c(x) = constant (case (a)) or that Q(tfQ "*"(w)) i s concave increasing i n w , V t e [a, b] (case ( c ) ) ) . We s h a l l show how excessively high values of the mob i l i z a t i o n cost K can lead to the optimality of an e x t i n c t i o n p o l i c y . Theorem 4.4.4 For any value of the discount rate r , there i s a c r i t i c a l value K of the mo b i l i z a t i o n cost. This value depends on r and i s non-increasing i n r . (a) I f K < K , then maximization of expected p r o f i t leads to a p o l i c y of conservation. (b) I f K <_ K < R(m) then maximization of expected p r o f i t leads to a p o l i c y of e x t i n c t i o n provided of course that i t i s p r o f i t a b l e to harvest 265 the l a s t surviving animals. (c) If K >_ R(m) then i t i s never p r o f i t a b l e to harvest and optimal e x p l o i t a t i o n leaves the resource untouched. Proof We show f i r s t l y that for each x , ip(x) i s non-increasing i n K . We use induction and assume that ^ ( x ) non-increasing i n * K . I t follows that (x) and hence ^N +T/X) are non-increasing, i n K . Since ij^(x) = -R(x) i s independent of K the induction i s complete. We have that i|>(x) i s non-increasing i n K , and hence sup iKx) has the same property. [0,m] Let K = i n f {K : sup i|i(x) = 0 } [0,R(m)] [0,m] If K < K then sup iKx) > 0 since sup i|>(x) i s . [0,m] [0,m] monotone i n K , and hence i n t h i s case from Thm. 4 . 4 . 1 conservation i s optimal. If K <_K < R(m) , then sup i^(x) = 0 , and from Thm. 4 . 4 . 1 e x t i n c t i o n i s optimal, provided x^ = 0 . If K >_ R(m) , i t i s never p r o f i t a b l e to harvest, and hence optimal e x p l o i t a t i o n leaves the resource untouched. The fa c t that K i s monotone non-increasing i n r , follows from the fa c t that sup iKx) i s non-increasing i n r for [0,m] any f i x e d K . QED 266 This theorem i s of some importance since i t suggests a way of ensuring the existence of the population i n the face of a high i n t e r e s t or discount rate. F i r s t l y we note that i f r ^ f'(0) - 1 ( i . e . i f the resource i s sub-marginal i n expectation), then K = 0 for we have i n t h i s case that sup 'J'(x) = 0 for any K < R(m) (Thm. 4.4.3). [0,m] Thus i f the resource i s submarginal i n expectation and i f i t i s p r o f i t a b l e to harvest the l a s t surviving animals, then e x t i n c t i o n of the population i s optimal for a l l values of the m o b i l i z a t i o n cost for which i t i s ever p r o f i t a b l e to harvest ( i . e . i f K < R(m)) . However i f the resource i s not sub-marginal i n expectation ( i . e . i f r < f'(0) - 1) then K > 0 since a conservation p o l i c y i s optimal for K = 0 . Thus i n t h i s case, reducing the m o b i l i z a t i o n cost to a l e v e l l e s s than the c r i t i c a l value, K , f o r the given discount rate i n operation would ensure that conservation was an optimal p o l i c y . A grant or subsidy could be offered to the resource owner every time he undertakes a harvest i n order to reduce the m o b i l i z a t i o n cost to a l e v e l below the c r i t i c a l one for the current discount rate. However i f the e f f e c t i v e discount rate i s so high as to make the resource sub-marginal i n expectation then such a strategy could be of no use i n ensuring conservation. We have seen i n Theorems 4.4.2 and 4.4.4 how excessively high values of the discount rate and m o b i l i z a t i o n cost can lead to the optimality of a p o l i c y of e x t i n c t i o n . These r e s u l t s can be conveniently 267 summarized i n a diagram. We s h a l l assume x_ = 0 Fi g . 4.2 The e f f e c t i v e discount rate and mo b i l i z a t i o n cost can be represented by a point (r,K) i n the F i g . 4.2. If the point l i e s i n Region I i t i s optimal to leave the resource unexploited (K>R(m)) . If the point l i e s i n Region I I . apoyV-ttie'^ curve r a p o l i c y of A, A. e x t i n c t i o n i s optimal (K >_ K or r >_ r) , provided of course that X Q = 0 . If the point l i e s i n Region I I I below curve T a p o l i c y of A. A. conservation i s optimal (K < K or r < r) . From the diagram i t i s easy to see how, i n the case, 268 r < f ( 0 ) - 1 , conservation can be ensured by reducing the m o b i l i z a t i o n cost, and how, i n the case r >_ f'(0) - 1 , such a strategy i s of no a v a i l . We have not been able to c a l c u l a t e the c r i t i c a l l e v e l K exactly for any given discount rate r . However we can f i n d a lower bound for K which gives a 'safe' l e v e l for the m o b i l i z a t i o n cost. We have from Lemma 1 that i f aK > sup it(x) [0,m] then conservation i s an optimal p o l i c y . Thus i f we have that K < K^ where b R(tf(x))d$(t) - (l+r)R(x)} a K^ = sup { , xe[0,m] then conservation i s optimal. The graph of K^ against r i s shown as a broken l i n e on F i g . 4.2. I t i s the same as the graph of x^ against K of Theorem 4.4.2. In the case of a constant marginal harvest cost, c(x) = c , K± = (p-c)[f(q) - q(l+r)] , where f'(q) = 1 + r . We conclude 'this section with some remarks on the e f f e c t of the s e l l i n g p r i c e and the marginal harvest cost. F i r s t l y i n the case of no m o b i l i z a t i o n cost (K=0) , i f X Q = 0 , we have seen that the optimality of an e x t i n c t i o n or 269 conservation p o l i c y depends only on the discount rate and i s independent of the s e l l i n g p r i c e and marginal harvest cost. Thus we have that i f K = 0 and the resource i s sub-marginal. In expectation (r>f'(0)-l) , then there i s a c r i t i c a l l e v e l of the s e l l i n g p r i c e , equal to c(0) , such that i f p >_ c(0) then e x t i n c t i o n i s optimal, whereas i f p < c(0) , then conservation i s optimal. I f  the resource i s not sub-marginal i n expectation, then conservation i s  an optimal p o l i c y , regardless of the s e l l i n g p r i c e . When K > 0 and the resource i s sub-marginal i n expectation the same r e s u l t applies. When i t i s not sub-marginal i n expectation we can f i n d a 'safe' l e v e l f o r the s e l l i n g p r i c e p to ensure conservation. If p > i n f af(x)-x>0 aK. + a' G(,tf(x))d$(t) - G(x) af(x) - x say , where fx G(x) = c( t ) d t , then then there e x i s t s x such that £(x ) > aK and thus from Lemma 1 conservation i s ensured, as an optimal p o l i c y . A s i m i l a r r e s u l t holds i f aK a' G(tf(x))d$(t) - G(x) p < sup af(x)-x<0 af (x) - x P2 » say. Thus we have for a p o s i t i v e m o b i l i z a t i o n cost (K>0) , i f the 270 resource i s sub-marginal i n expectation, (r>f'(0)^l) , then conservation i s ensured as an optimal p o l i c y i f p < c(0) , whereas e x t i n c t i o n i s an optimal p o l i c y i f p >_ c(0) . If the resource i s not sub-marginal i n expectation (r<f'(0)-l) then conservation i s optimal i f eit h e r pg*fp|0)-or if-hep <--maxfe^'Q^onpx^. . i i either ~ P > Pj, - or i f P < i.axic(0;, p 2> „ When the resource i s sub-marginal i n expectation, for K >_ 0 , we have seen that the determinant of the type of optimal p o l i c y i s the way i n which the marginal harvest cost, at low population l e v e l s , compares with the s e l l i n g p r i c e p . When K = 0 , and the resource i s not sub-marginal i n  expectation, conservation i s optimal regardless of the marginal harvest  cost. When K > 0 and the resource not sub-marginal we f i n d a 'safe' l e v e l f o r the marginal harvest cost, assuming that i t i s independent of population l e v e l ( i . e . c(x) = c) . If oK C P sup {af(x)-x} °1 ' af(x)-x>0 say, then there e x i s t s x f o r which £(x ) > aK , and hence conservation i s optimal. Thus we have that f o r a p o s i t i v e m o b i l i z a t i o n cost, and a  constant marginal harvest cost, c , i f the resource i s not sub-marglhal i n expectation then conservation i s optimal i f c < c 1 . 271 I t i s i n t e r e s t i n g to note f o r K > 0 that when the resource i s not sub-marginal i n expectation, ensuring that e i t h e r the mobi l i z a t i o n cost, or the marginal harvest cost ( i f i t i s constant) are s u f f i c i e n t l y low, can ensure conservation as an optimal p o l i c y . However i f the resource i s sub-marginal i n expectation, reducing the marginal harvest cost to a su i t a b l y low l e v e l can bring about the optimality of an e x t i n c t i o n p o l i c y . The r e s u l t s i n t h i s section concern optimal p o l i c i e s i n the sense developed i n 3.1 and 3.2. I f we assume that the resource i s being exploited optimally i n t h i s sense (perhaps an approximately optimal e x p l o i t a t i o n would be ar r i v e d at empirically) then the r e s u l t s of t h i s section could be regarded as d e s c r i p t i v e . They would o f f e r economic explanations f o r the e x t i n c t i o n of animal species, and also suggest economic means by which such actions could be avoided. On the other hand i f we view the model i n a normative way then the r e s u l t s of t h i s section should perhaps be regarded as q u a l i f i e r s to the normative value of the optimal p o l i c i e s found i n Chapter 3. They indi c a t e economic conditions which imply that the extermination of the population i s an "optimal" p o l i c y . C l e a r l y i n such a case the maximization of expected discounted revenue i s too l i m i t e d an objective. On theoother hand, the r e s u l t s also i n d i c a t e when the objective of maximizing expected discounted revenue i s compatible with a p o l i c y of conservation. 272 Chapter 5  Multi-Dimensional Population Models 5.1 Introduction In the preceding chapters we have described the population by a sin g l e continuous v a r i a b l e , representing i t s s i z e . We suggested that t h i s measure of s i z e could be the number of animals i n the population or the t o t a l biomass of the population. As we mentioned i n Chapter 1, i t i s "a\ great s i m p l i f i c a t i o n to assume that successive values of t h i s s i z e measurement are f u n c t i o n a l l y r e l a t e d , even i f we assume random v a r i a t i o n s i n the parameters. We have ignored such aspects as the age and sex d i s t r i b u t i o n s within the population. C l e a r l y these play an important part i n determining population dynamics. When we tr y to develop more complex models to incorporate age and sex structure we do so with two aims i n mind. F i r s t l y can we more r e a l i s t i c a l l y describe the dynamics of a population by including these f a c t o r s , and secondly assuming the f i r s t aim i s accomplished, can we develop a con t r o l model which w i l l enable us to u t i l i z e the age or sex structure of the population to increase the y i e l d or economic rent that we can earn from i t ? There have been many papers concerned with the f i r s t aim. The basic continuous time models of demography assume an age-structure i n the population, (see K e y f i t z 1968). The d i s c r e t e -time model of L e s l i e (1945), which we s h a l l discuss l a t e r , i s for an age-structured population. Two-sex models have also been developed 273 (see f o r instance Goodman 1968). However many of these models were developed for human populations and are l i n e a r . As such they allow f o r no s e l f - r e g u l a t i o n by the population, and are inadequate f o r discussing the dynamics of animal populations l i v i n g " i n the wild". The second problem of u t i l i z i n g the age-structure of a population i n harvest strategy has att r a c t e d le s s attention. Beverton and Holt (1957) developed a model f o r a f i s h e r y which recognized d i f f e r e n t y i e l d s from d i f f e r e n t aged f i s h . This model has been used as the basis of an economic analysis by Clark, Edwards and Friedlaender (1973). A discrete-time c o n t r o l model has been studied by Beddington and Taylor (1973). In keeping with the rest of t h i s thesis we s h a l l look only at d i s c r e t e time models. We s h a l l develop an age-structure model which allows f o r s e l f - r e g u l a t i o n of the population through density-dependent mortality. In t h i s respect i t i s an improvement on the L e s l i e model. F i r s t l y i n 5.2 we look at a deterministic model, and then i n 5.3 look at a stochastic version. We discuss the equilibrium behaviour of the deterministic model i n 5.2. We also i n d i c a t e how i t can be used as a 4 model for competing populations. We use t h i s model as the basis of a control model, deterministic i n 5.2 and stochastic i n 5.3. However with respect to the aim mentioned e a r l i e r — c a n we u t i l i z e the age-structure to increase y i e l d or rent? — we have made l i t t l e progress. We discuss t h i s i n 5.2. Our analysis concerns a much more l i m i t e d problem. We assume that the resource manager controls only a sing l e v a r i a b l e — a ) 274 0 harvest e f f o r t which we assume acts uniformly on each age-class. We assume that the manager has information concerning the current s i z e and a g e - d i s t r i b u t i o n of the population and can u t i l i z e t h i s information i n determining the s i z e of the harvest e f f o r t he applies. We look at the deterministic problems of maximizing sustainable y i e l d and rent i n 5.2. In 5.3 we look at the dynamic problem, using a stochastic model, of maximizing expected t o t a l discounted y i e l d . Since i t i s probably d i f f i c u l t and c o s t l y to obtain accurate information concerning the s i z e , by age-class, of a population i n the wild, the p r a c t i c a l usefulness of t h i s analysis i s l i k e l y l i m i t e d . In 5.4 we discuss some of the problems associated with developing c o n t r o l models for a sex-structured population. We study a sex model for the harvesting of a salmon population, based on the experimental work of 0. Mathisen (1962). In the development of age-structure models we s h a l l take as s t a r t i n g point the discrete-time deterministic model of P. L e s l i e (1945). We o u t l i n e the basic L e s l i e theory i n t h i s introduction. It i s assumed that the population i s divided into k age classes. The vector represents the population d i s t r i b u t i o n i n year n , with x n i = 1 ,k denoting the number of animals i n the i 1 " * 1 year of t h e i r l i f e at the n t b time point. The L e s l i e model assumes that x ,, = L x , ~n+l ~n where L i s the so-called L e s l i e matrix. 275 Al*l ' V l L = l2 ' 13 where , i = 2,...,k , represents the proportion of members of the ( i - l ) t ' 1 age-class who survive to become members of the i t b " age-class the following year, f , i = l , . . . , k , i s the fecundity of the i t b age-class ( i . e . the average number of b i r t h s per year per animal i n the th i age-class), and i s the proportion of new b i r t h s i n the year [n, n+1) that survive to become members of the f i r s t age-class at time point n + 1 . I t i s assumed that the parameters f ^ - , , i = l , . . . , k i , do not depend on the given year (date) and do not depend on the value of x n . Thus the model i s temporally homogeneous and l i n e a r . The main r e s u l t s of the theory concern the l i m i t i n g behaviour of , and the stable a g e - d i s t r i b u t i o n . It can be shown that a s u f f i c i e n t condition for the convergence of the a g e - d i s t r i b u t i o n to a stable form i s that at l e a s t two consecutive f be non-zero (see for instance P o l l a r d (1966)). For most animal populations t h i s i s not an unreasonable assumption, although for a species l i k e those of P a c i f i c Salmon, which, for the 276 main part, breed only once i n t h e i r l i f e at a given age, i t i s an assumption which may not be s a t i s f i e d . A broader s u f f i c i e n t condition i s that the matrix L be p o s i t i v e regular, i n the sense, that there N e x i s t s an N such that the matrix L has a l l p o s i t i v e elements. Under t h i s assumption the matrix L has a p o s i t i v e dominant character-i s t i c root of m u l t i p l i c i t y one with corresponding r i g h t and l e f t c h a r a c t e r i s t i c vectors which consist only of p o s i t i v e elements. Furthermore t h i s i s the only c h a r a c t e r i s t i c root for which the elements of the corresponding c h a r a c t e r i s t i c vectors are a l l p o s i t i v e (Brauer 1962). The matrix L i s non-singular and has s p e c t r a l decomposition L - A 1 Z 1 + + \ \ » where A , j = 1,...,k are the c h a r a c t e r i s t i c roots of L , and Z. = k., h.' where k. and h.' are the normalized r i g h t and l e f t 3 ~3 ~3 ~3 ~3 c h a r a c t e r i s t i c column and row vectors corresponding to A (j=l,...,k) It follows that Z.Z. = 0 ( i ^ j ) and Z.Z. = Z. . We have then that i 3 i i i L n = A.V + + x f z . , 1 1 k k and hence that, as n -»• °° , 5 N - L ? 0 - X l V 0 + ° a i since A^ > |A | , 3=2, k . Thus we have that ?n = £&l*0>h + o ( X ? 277 The a g e - d i s t r i b u t i o n of the population tends to that of the r i g h t c h a r a c t e r i s t i c vector corresponding to the dominant c h a r a c t e r i s t i c root of L . This i s a stable equilibrium age d i s t r i b u t i o n . For large n the si z e of the population behaves l i k e A^ . If X^ > 1 the population grows geometrically ( i t "explodes"). If A < 1 the population decays to zero. Only i n the case A^ = 1 does the s i z e of the population tend to a non-zero, f i n i t e l i m i t . These basic r e s u l t s were obtained by L e s l i e (1945). Since then the model has been extended i n many ways. Williamson (1959) and Goodman (1968) modified the model to include both sexes, and Lefkovitch (1965) developed a modification for organisms grouped by l i f e - s t a g e s rather than age. C u l l and Vogt (1974) have shown that l i m i t cycles i n the population d i s t r i b u t i o n can occur when the matrix L i s not p o s i t i v e regular. The main drawback with t h i s model i s that the fecundity and s u r v i v a l c o e f f i c i e n t s f ^ , Z , are density independent, i . e . they do not depend on x . Because of t h i s there i s no i n t e r n a l regulation of the population s i z e and t h i s r e s u l t s i n exponential growth (or decay). • The model may describe f a i r l y accurately a population i n i t s early stages of growth i n a new habitat, when there are more than s u f f i c i e n t resources a v a i l a b l e , but i t does not allow for a regulation i n population growth imposed by the l i m i t s of a v a i l a b l e resources. I f the population were harvested i n such a way that i t s s i z e , r e l a t i v e to a v a i l a b l e resources, were kept small then the model might well present an adequate p i c t u r e . Beddington and Taylor (1973) have 278 used the L e s l i e model i n looking at the problem of maximizing sustainable proportional y i e l d from a population of given s i z e , by c o n t r o l l i n g the a g e - d i s t r i b u t i o n . They show that the maximum i s obtained by completely harvesting the oldest age-class (thereby e f f e c t i v e l y reducing the age of the population) and by p a r t i a l l y harvesting a second age-class. We s h a l l look at some ways of generalizing the L e s l i e model so that i t w i l l take into account density dependent and stochastic fa c t o r s . We s h a l l be concerned with f i n d i n g optimal harvest p o l i c i e s . Before discussing a stochastic model i n 5.3 we s h a l l look at a density-dependent deterministic model i n the following section. 5.2 A Density-Dependent Deterministic Model The model of L e s l i e i s l i n e a r , and we have seen that once the population has reached i t s stable age d i s t r i b u t i o n , the s i z e grows geometrically with parameter A^ . As a f i r s t step i n extending h i s model L e s l i e (1948) sought a modification for which the s i z e of the population would s a t i s f y the l o g i s t i c law, once the stable a g e - d i s t r i b u t i o n was reached. If we l e t N = e.'-x , where e' = (1,...,1) , the l o g i s t i c law i s n - ' ~n ~ v > > ' > & A.N N - I n n+1 A -1 1 + N K n A population s a t i s f y i n g the l o g i s t i c law has i n i t i a l l y a geometric growth with parameter A^ , known as the i n t r i n s i c growtl The growth rate decays as the population s i z e increases, and the 279 population eventually becomes stable at the si z e K . L e s l i e showed that for the age-structure model L x x -n+1 X -1 1 + - ± — N K n the population s i z e s a t i s f i e d the l o g i s t i c law, once the population had reached i t s stable age d i s t r i b u t i o n . However f o r a population with an i n i t i a l age d i s t r i b u t i o n d i f f e r e n t from the stable d i s t r i b u t i o n , the early growth of the population s i z e i s not l o g i s t i c . A s l i g h t extension of the l o g i s t i c model of L e s l i e i s L x < u " + ' 1 + g -x ~n where 3' i s a non-negative row vector r e f l e c t i n g the r e l a t i v e demands upon a v a i l a b l e resources, of i n d i v i d u a l s of the various age-classes. More generally we might consider.a model (2) x = g(x ) L x , ~n+l ~n ~n where g i s a function from R •+ [0, 1] and represents the e f f e c t of population density upon s u r v i v a l . We should expect that the function g(x) be "decreasing" i n some sense, at l e a s t f or "large" values of x . This would ensure that mortality increases with the s i z e of population, and thus compensates f o r high population d e n s i t i e s . Some other convenient forms of the function g(x) might be 280 g(;) = (l+B'-x)' g(x) = l+ce'-x) 1 n > 0 g ( ? ) = TZb where 0 < b < 1 and a <_ (min 3.)"'" ^  , to ensure g(x) <^  1 , or i , , -3-x g(x) = a e ~ . ~ In assuming a model of the form (2) we are assuming the density-dependent e f f e c t on mortality operates i n the same way i n each age c l a s s . There i s no immediate b i o l o g i c a l j u s t i f i c a t i o n f o r t h i s . Indeed i t seems l i k e l y that i n many populations the e f f e c t s of high population density would be f e l t more strongly among c e r t a i n age classes e.g. the very young or the very o l d . Also we are assuming that density has no e f f e c t on the fecundities of various age-classes. This i s c l e a r l y not the case f o r t e r r i t o r i a l animals, which do not breed without f i r s t e s t a b l i s h i n g a t e r r i t o r y . Limited as t h i s model may be i n b i o l o g i c a l a p p l i c a t i o n i t does incorporate some aspects of density dependent regulation which are not present i n the 'linear model of L e s l i e . In the remainder of t h i s section we s h a l l be concerned p r i m a r i l y with the model (1) which can be thought of as m u l t i -dimensional version of the one-dimensional model with Beverton-Holt reproduction function. ax f ( x ) = rr~bx" • We s h a l l i n d i c a t e how the r e s u l t s apply to other forms of the function g(x) . In section 5.3, where we develop a more general stochastic model, we s h a l l give a concavity condition on the function g which ensures that density-dependent e f f e c t acts i n a compensatory way. Let us then consider the deterministic model x_ , n = — • L x 1 + 6'-x n+1 , . _, ~n -n As before we s h a l l assume that.the matrix L i s p o s i t i v e regular and has s p e c t r a l decomposition L = A Z, + + A Z , 1 1 k k where Z. = k.-h' , k. and h'. being the normalized r i g h t and l e f t c h a r a c t e r i s t i c vectors of L corresponding to the c h a r a c t e r i s t i c root A. (j=l,...,k) . We have that L x o X l =  1 + § ' x o L ( W x - } 1+6. x 0 x„ = ~ 2 L ?o 1 + ^ 0 1 + g'x 0 + g' L x Q 282 By induction i t i s not d i f f i c u l t to show that L_^0 x = ~ n 1 + g'x 0 + §'L x Q + B'L n ; 0 Using the s p e c t r a l decomposition of L we can write t h i s as x = A i z i x o + + Xl\?0  i + §'( i. A ) ? + + §•(!. c l zi ) xo j = l J J j = l J J x i z i x o + • • • + Ak Zk X0 n-1 . n-1 . i + < I V ? ' z i x o + + «I ^)§'ZLx 1=0 i=0 k u = x i z i x o + + xk\ xo  A n - i A n - i 1 + ( ^ ) B ' Z l x 0 + + <JL_) B'Z kx 0 1 k Dividing by , assuming A^ > 1 i s the dominant c h a r a c t e r i s t i c root, we see that as n °° ( > r 1 ) z i x o + „. i , x = «. .|, + 0( ) e z i x o \ j X n = H h ^ + 0(-i) since = (h.Jx ^  . 283 We thus see that i f A^ > 1 , the population approaches equilibrium ( V 1 ) k i at —T -T—: • In the case A. < 1 i t i s easy to see that x -»--0 § 1 — J ~n We thus have the following r e s u l t . Theorem 5.2.1 For the deterministic age-structure model x ,, = L x ~n+l i • o» ~ n 1 + 3 x ~n where L i s the L e s l i e matrix, assumed p o s i t i v e regular, with dominant c h a r a c t e r i s t i c root A and corresponding r c h a r a c t e r i s t i c vector k , we have that as n -* °° , x^ approaches equilibrium at / , ^ _ i f A > 1 A. B and x^ approaches zero i f A <_ 1 . The l i m i t i n g stable age d i s t r i b u t i o n i s the same as that of the simple L e s l i e model and i s determined by the r i g h t c h a r a c t e r i s t i c vector corre-sponding to the dominant c h a r a c t e r i s t i c root of L . This i s exactly what we expect since the density-dependent factor e f f e c t s only the population s i z e and not i t s a g e - d i s t r i b u t i o n . For other forms of the function g(x) , we have that i f a l i m i t f o r x e x i s t s i t w i l l be a multiple of k, . This multiple ~n ~1 qk^; say can be determined from the equation 284 q ^ g C q ^ ) = q » since the population i s i n steady-state at i t s l i m i t i n g value. - 8 1 'X For example, i f g(x) = a e ~ - ~. , the l i m i t of x i s ~n qk^ where £nA^a q = We can use a model of t h i s form for a system of competing populations. F i r s t l y l e t us ignore a g e - d i s t r i b u t i o n i n the i n d i v i d u a l r x l \ populations and suppose that x = ( . \ represents the numbers of animals i n each of p populations. Consider the model x ,, = g(x )Mx , -n+1 ~n ~n where M i s a diagonal matrix, with diagonal entries m (j=l,...,p) representing the growth factors of the respective populations when there i s no competition e f f e c t s . The c h a r a c t e r i s t i c roots of M are m.,...,m , with 1 p corresponding r i g h t c h a r a c t e r i s t i c vectors e^,...,e , where e. i s th the p-vector with an entry 1 i n the j p o s i t i o n and zeros elsewhere. If one of the nu dominates the others,say m^, then the system tends to an equilibrium with a l l but t h i s one population becoming e x t i n c t . For the form g(x) = l/i(l+3fi"*x-). , the equilibrium of the m^  - 1 system i s at — e . P-, ~1 285 If two or more of the m. are equal and dominate the others, J say = = m^  , then the system tends to equilibrium with these r populations surviving and the others becoming e x t i n c t . The r e l a t i v e sizes of these populations are the same as the r e l a t i v e sizes of the i n i t i a l values of these r populations. The equilibrium of the population i s where the a_.'s are given by the equations m 1g(a 1e.+....a e ) = 1 1 1-1 r ~ r and 1 _2 2 r r where x 3 i s the i n i t i a l s i z e of the j t h population ( j = l , . . . , r ) . For a system of p competing age-structured populations we can l e t A M x = ~n n 2 x ~n where x? i s the vector f o r the ag e - d i s t r i b u t i o n of the j 1 " population. 286 We l e t L = where i s the L e s l i e matrix for the j 1 - 1 1 population, and consider the model Xn+1 = g ( ? n ) L ?n The c h a r a c t e r i s t i c roots of L are a l l the c h a r a c t e r i s t i c roots of L^ , , • • • , Lp , and the corresponding c h a r a c t e r i s t i c vectors are the c h a r a c t e r i s t i c vectors of the L (j=l,...,p) with zeros i n appropriate p o s i t i o n s . Thus i f the dominant c h a r a c t e r i s t i c root of one of the L. , J dominates the dominant roots of the other L^ , then the system tends to equilibrium with a l l but the one population becoming ex t i n c t . This population, which corresponds to the l a r g e s t of the dominant roots, has l i m i t i n g age structure given by the p r i n c i p a l r i g h t c h a r a c t e r i s t i c vector of i t s L e s l i e matrix. If more than one of the dominant c h a r a c t e r i s t i c roots of the p populations are equal, the system tends to equilibrium with a l l these corresponding populations represented, the others becoming e x t i n c t . The age d i s t r i b u t i o n of any given population i s determined by the p r i n c i p a l 1 287 r i g h t c h a r a c t e r i s t i c vector of i t s L e s l i e matrix, but the d i s t r i b u t i o n of r e l a t i v e sizes of the surviving populations depends upon t h e i r i n i t i a l values. The use of models of t h i s sort f or competing populations seems rather l i m i t e d , since, except i n the case of equal i n t r i n s i c growth rates, competitive exclusion r e s u l t s . Stable coexistence or stable l i m i t cycles are not possible i n t h i s kind of model. In the case of equal i n t r i n s i c growth rates, the r e s u l t i n g equilibrium i s n e u t r a l - when- the d i s t r i b u t i o n of the equilibrium populations i s disturbed i t neither returns to the old d i s t r i b u t i o n , nor grows away from i t , but simply assumes t h i s new form. We s h a l l now look at some control models for age-structured populations. As always we face the problem of formulating a c o n t r o l model whose mathematical structure i s reasonably simple, and which r e f l e c t s the control options of the resource manager. In an i d e a l s i t u a t i o n , i t would be possible, not only for the resource manager to determine exactly the current s i z e of each age-class, but also to be able to control each age-class independently. The f i r s t of these conditions i s seldom met. The s t a t i s t i c a l problem of estimating animal abundance i s one that i s usually subject to large error, e s p e c i a l l y for f i s h populations. Determination of the age of an i n d i v i d u a l f i s h (or animal i n general) usually, i n p r a c t i c e , involves a s t a t i s t i c a l procedure. The problem of estimating abundance by age-class i s then, one that involves at l e a s t two s t a t i s t i c a l procedures. The compounded error i s l i k e l y to be great. However i f we t r y to 288 incorporate t h i s "error i n observation" into a control model, the model becomes considerably more complex. We are faced with the problem of control with "noisy" data. The s o l u t i o n of t h i s type of problem involves a " f i l t e r i n g " process. This i s a d i f f i c u l t problem for a l i n e a r dynamic model. We know of no r e s u l t s for the c o n t r o l of a non-linear stochastic system with noisy observations. We s h a l l not attempt to solve such a problem. Whenever necessary we s h a l l assume that the resource manager has perfect knowledge of the state of the population. Such a model probably has l i t t l e p r a c t i c a l use. I t may have some use as an i n t e l l e c t u a l t o o l . The second problem we are faced with i s to develop a control model with control v a r i a b l e s that reasonably r e f l e c t the resource manager's control options. The i d e a l s i t u a t i o n , i n which each age-class can be c o n t r o l l e d independently may be met f o r some harvested populations. For instance, i t may not be unreasonable to assume that seals, which are hunted on land, can be harvested i n t h i s way. For f i s h and whale populations, age-selective harvests seem d i f f i c u l t to achieve. In p r i n c i p l e , for f i s h caught i n nets, c o n t r o l of the mesh s i z e of the nets could do t h i s , but i t seems that smaller f i s h which escape from the nets are often so badly injured by the process thaf they do not survive. We s h a l l f i r s t l y formulate (though not solve) a model for t h i s i d e a l s i t u a t i o n of age-selective harvests. Next we s h a l l look at a model i n which there i s only one c o n t r o l v a r i a b l e . The resource manager i s assumed to be able to control the harvest e f f o r t . Such an 289 e f f o r t i s assumed to act d i f f e r e n t l y on each age c l a s s . This w i l l include the case of mesh-size l i m i t a t i o n i n a f i s h e r y , by simply assuming that the e f f o r t has no e f f e c t on the younger age-classes. We formulate t h i s i s as a non-linear program, but do not solve i t . e f f o r t acting i d e n t i c a l l y on a l l age-classes - i . e . the same proportion i s harvested from each age-class. We solve e x p l i c i t l y the problem of maximizing the sustainable y i e l d and the sustainable economic y i e l d from such a harvest. In the following section we solve the dynamic problem of maximizing discounted y i e l d from such a harvest, using a stochastic population model. This includes the deterministic population model of t h i s section as a s p e c i a l case. F i n a l l y we look at the much simpler problem of the harvest Let us look f i r s t l y then at the s i t u a t i o n of complete age-s e l e c t i v i t y . Suppose i n year n immediately p r i o r to the breeding season a proportion h? of the j t l x age-class i s removed ( j = l , . . . , k) If we l e t n u. = 1 - (j=l , . . . , k) 3 and denote by U the diagonal matrix with e n t r i e s u? we see that n 3 the harvested population s a t i s f i e s the differ e n c e equation (3) = g(U x ) L U x n~n n ~n and that the harvest y i e l d i n year n- i s p'(I-U )x n ~n 290 where p' = (p ,...,p^) represents the y i e l d s per i n d i v i d u a l i n each age-class. If there i s a cost c(U) associated with a harvest characterized by U , then the t o t a l discounted economic y i e l d i s given by 00 (4) I a n{p'(I-U )x - c(U ')} . - ~ n ~n n n=l The dynamic optimization problem i s to maximize (4) subject to the "equation of motion" (3), and subject to the constraints 0 £ u^ _< 1 (j=l,...,n) . An analysis of t h i s problem may be possible using the Discrete Maximum P r i n c i p l e (see Halkin 1966). We do not address the problem here but turn instead to the problem of maximizing the sustainable economic y i e l d or rent. If the same harvest e f f o r t , characterized by the matrix U i s applied every year, the population w i l l s a t i s f y the difference equation ?n+l " g ( U X n ) L U ?n ' The y i e l d i n year n i s p'(I-U)x . The population w i l l tend to equilibrium with an ag e - d i s t r i b u t i o n characterized by the right c h a r a c t e r i s t i c vector of LU corresponding to the dominant c h a r a c t e r i s t i c root X of t h i s matrix. The s i z e of the equilibrium populat ion w i l l depend on the function g(Ux) In the s p e c i a l case of the model (1) of 5.2, we have from Theorem 5.2.1 that the harvested population s a t i s f i e s 291 L U x ~n Xn+1 =  1 + 3'Ux - ~ n and that the population tends to equilibrium at The equilibrium y i e l d i s XAy-Dp'd-U)^ r •u ^ If there i s a cost c(U) associated with a harvest e f f o r t characterized by U , the problem of maximizing sustainable economic y i e l d , or rent i s to maximize ( A - D p ' d - U ) ^ — y — = ^ - C ( u ) •r • u ^ u over the admissible set 0 <_ u^ <_ 1 , (j=l,...,k) . A n a l y t i c a l l y t h i s seems d i f f i c u l t to perform. Bearing i n mind that k^ i s the only c h a r a c t e r i s t i c root of the matrix LU with a l l p o s i t i v e elements (Brauer 1962) we can express the problem as:-Maximize the objective function (A-Dp'(I-U)k = - - c(U) 3' U k Subject to the constraints 292 (i) 0 <_ u_. <_ 1 , j = 1,. . . ,k J ( i i ) k > 0 , j = 1,...,k ( i i i ) L U k = Ak (iv) A > 1 where L, g, p' are known constants and c i s a known function. This i s a non-linear program with non-linear objective function and non-linear constraints. There may be a computer program capable of solving such a problem numerically. There are other ways of expressing the N.L. program. For instance, expressing k^ i n terms of A^ and U we get the following?,,. Maximize:-(A-l) [ p 1 J l 1 ( l - u 1 ) A k " 1 + p 2£ 1Jl 2u 1(l-u 2)A k" 2+... .+P k* r • . \ u r . . u k _ 1 ( l - u k ) ] B.. JL u_ A k - 1 + $ JL A 0 u . u . k _ 2 + . . .-<;•.-.'.«. .e ..«,. a . - . B . . . t . ' j l t ^ 1 c i ( 1 u . , u . .. .u. I l l 2 1 2 1 2 i c l K 1 u k 11 k 1 k subject to:- " c ( u 1 > - - - » u k ) (i) det(LU-AI) = 0 ( i i ) 0 <_ u_. <_ 1 , j = 1,. . . ,k ( i i i ) A > 1 Expanding the determinant of LU - Al , and l e t t i n g v^ = , v. = u.£. v. 1 (j=2,...,k) we can write t h i s i n the form:-Maximize the objective function:-293 ( X - l ) [ p 1 ( A 1 T V 1 ) X k 1 + P 2 ( V l " V 2 ) k 2 + + P k ( V k - l " \ ) ] 3, v. A k - 1 + 3 9v 9A k" 2 + -,......3, v. 1 1 11 .... -r x k k subject to: - c ( u r . . . u k ) (i) -A k + v 1 f 1 A k 1 + v „ f a k 2 +...v, .f. J + v,f, = 0 1 1 2 2 k-1 k-1 k k ( i i ) 0 < v± < H± 0 < v 2 < ^ 0 £ v k < ( i i i ) A > 1 Eithe r of these l a s t two forms may be solvable numerically on the computer. We have not investigated t h i s . From the p r a c t i c a l point of view the r e s u l t s of such numerical optimizations would be severely r e s t r i c t e d by the q u a l i t y of the data inputs. Estimates of the parameters A , , , 8^ would l i k e l y have large variances. We now leave t h i s problem i n i t s general form and look at the simpler s p e c i a l case i n which the resource manager has co n t r o l of only a s i n g l e v a r i a b l e - the harvest e f f o r t . We s h a l l suppose that i n general t h i s harvest e f f o r t has a d i f f e r e n t impact on each age-class.-Suppose that the harvest i n the n t b year takes place over a time i n t e r v a l [0, T] and that over t h i s time i n t e r v a l there i s a constant harvest e f f o r t F^ , say. We s h a l l suppose that f o r each age-class the percentage harvest mortality i s proportional to F^ . 294 This i s a f a i r l y common assumption i n the f i s h e r i e s l i t e r a t u r e (see Beverton & Holt 1957). Although conceptually easy to understand, the f i s h i n g e f f o r t may i n p r a c t i c e be d i f f i c u l t to compute. It could for instance represent the number of nets of a f i s h i n g f l e e t that are i n the water throughout the harvest period. If at time t e [0, T] ttl the s i z e of the j age-class ( j - l , . . . , k ) i s y_. (t) , we have that dy.(t)-dt j ' where a^ i s a constant, (j=l,...,k) = - ( F a J y,(t) , (5) For the harvest i n the year n , we have that y.(0) = x n and y.(T) = u n x n (j=l,...,k) J J j 3 3 Solving (5) with the i n i t i a l condition we get -F t a. y.(t) = fe n ] 3 , (j=l,...,k) and hence we get that or where -F T a. n r n i J u. = [e ] a. n J u. = u 3 n -F T n u = e n If the cost of harvesting i s proportional to the harvest e f f o r t F , we have that the cost of the harvest i n year n i s n 295 - c in u , n c a constant. We s h a l l look at the problem of maximizing equilibrium (sustainable) economic y i e l d over a l l possible harvest e f f o r t l e v e l s . From the second of the N.L. programs above we get that t h i s maximum i s brought about by the p o l i c y u which solves the following N.L. program. Maximize:-(A-l) [p A (1-u "V1*"1 + P 2A 1A 2u 1 ( l - u 2)A k~ 2+...+p A ..A i T 1 k _ 1 ( l - u k ) ] -- a n t _ i a i " ^ " a 9 v_9 a " i~ '~" * *"'"av ^J,j.u A + B 2 A 1 £ 2 u 1 Z X +.... + P ^ ! " - ^ + c £n(u) subject to the constraints 1 a i 1 1 a i " ^ " a o 1 9 a i " ^ " " * * *^~ai ( i ) -X +2,.^ .^  -""X + ^ 1 ^ 2 f 2 u X . ..A kf f cu ' = 0 ( i i ) 0 £ u <_ 1 ( i i i ) X > 1 where £^ , f , a_. , B.. , p^ (j=l,...,k) and c are known constants. In p r i n c i p l e at l e a s t i t i s possible to solve t h i s program by the method of Lagrange M u l t i p l i e r s . This model can incorporate the s i t u a t i o n of a mesh s i z e l i m i t a t i o n . If we l e t a. = 0 , j < M , and a. = a j >_ M , then the th harvest e f f o r t acts equally on the j and older age-classes, while leaving untouched the younger age-classes. I f we solve the r e s u l t i n g N.L. program f o r each M = l , . . . , k , and then choose the value of M 296 which maximizes the optimal value of the objective function, we can determine the optimal s i z e of the mesh. This can probably be done numerically (the objective function and the constraints involve only r a t i o n a l functions, apart from the cost function). A n a l y t i c a l l y i t seems very d i f f i c u l t . We note that i n t h i s , and i n a l l the sustainable y i e l d problems so f a r discussed, observation of the abundance of each age-class i s not necessary, and the procedure does not therefore s u f f e r from the p r a c t i c a l l i m i t a t i o n s derived from imperfect observation of the population as discussed e a r l i e r . This i s not the case for a dynamic model, where the harvest e f f o r t w i l l depend on the current population l e v e l s . However estimation of the parameters of the model would l i m i t the p r a c t i c a l usefulness of the equilibrium r e s u l t s above. A l l of the procedures for maximizing equilibrium y i e l d carry over to more general forms of the density-dependence function g(x) , -8' 'x without great d i f f i c u l t y . For instance when g(x) = ae * ~ , we simply replace (A-l) by Jln(Aa) i n the objective function. We now r e s t r i c t our attention even further, and consider the s p e c i a l case i n which the harvest e f f o r t acts equally on each age-class. We s h a l l suppose that a proportion h i s harvested from each age-class every year. If we l e t u = 1 - h , we have that the dynamics of a population with density dependent model (1) are determined by the equation u L x x _ ~n ~n+l .. , „, 1 + u 8 x ~ ~n 297 This reaches equilibrium at . (X - l ) k u ~u u B' -k ~u where X and k are the dominant c h a r a c t e r i s t i c root and vector of u ~u the matrix uL . Cl e a r l y X =J.UX' and k = k where X and k are 3 u ~u ^ the dominant c h a r a c t e r i s t i c root and vector of L . The equilibrium y i e l d i s (1-u)(Xu-1) S'' k The maximum equilibrium economic y i e l d (or sustainable  economic rent) i s determined by maximizing v , , (1-u)(Xu-1) g ' . „ Y(u) = — + c in u over the range 1/A < u <_ 1 . (N.B. We require uA > 1 i n order that the population reach a non-zero equilibrium). We can solve t h i s problem by elementary c a l c u l u s . We have 1 P'' k Y'(u) = (-X + ~ ) + £ u §''k This decreases on (0,°°) . Also Y'(0 +) = « and li m Y'(u) < 0 Hence Y'(u) has exactly one zero i n (O,00) . I t follows that t h i s zero i s the global maximum of Y(u) over (O,00) . Let u be t h i s point. We have that 2 9 8 c + V c 2 + 4 K 2 u = -2 A K p' -k where K = 3' -k We are concerned with the maximum of Y(u) over ( 1 / A , 1 ] . We have that Y(l/A) = c £n Y < 0 A and Y ( l ) = 0 . If u >_ 1 then Y attai n s i t s maximum over ( 1 / A , 1 ] at 1 and the maximum value of Y i s 0 . If u < 1 then Y attai n s i t s maximum over ( 1 / A , 1 ] at u and the maximum value of Y i s p o s i t i v e (see F i g . 5 . 1 ) . u < 1 F i g . 5 . 1 The former case holds i f c + V c 2 + 4AK 2 2AK -i . e . i f A < 1 + £ : - K or A £ 1 + c -p' -k and the l a t t e r case i f 3' -k A > 1 + c - — -p'-k Thus we see that i f the i n t r i n s i c growth rate A i s s u f f i c i e n t l y large (greater than 1 + c 3'k/p'-k) i t i s possible to derive a sustainable economic rent from the resource, whereas i f A does not meet t h i s condition no rent can be sustained ( max Y(u) = u e ( l / ^ l ] Although t h i s necessary and s u f f i c i e n t condition f o r a sustainable rent appears above as a condition of the i n t r i n s i c growth rate i t can be given an economic i n t e r p r e t a t i o n . When the population has the stable a g e - d i s t r i b u t i o n , k , and i s at a l e v e l tk say, the marginal cost of harvesting, i n such a way as to not disturb the a g e - d i s t r i b u t i o n i s c/t , and the marginal s e l l i n g p r i c e i s g'k . We can write the necessary and s u f f i c i e n t condition for a sustainable rent as A — 1 i . e . that the marginal cost at the l e v e l , t ^ • k be l e s s than the marginal s e l l i n g p r i c e . But — ~ • k i s simply the undisturbed g * k equilibrium of the population, and so we see that the necessary and s u f f i c i e n t condition for a sustainable rent i s that there e x i s t s a l e v e l at which the population can be sustained for which the marginal s e l l i n g p r i c e exceeds the marginal harvest cost. We note that the above equilibrium analysis i s i n essence the same as the one-dimensional analysis of Clark (1971, 1973(b)). Once the population has reached i t s equilibrium age d i s t r i b u t i o n , harvesting and reproduction do not disturb i t from i t . The s i z e of the population i n successive years w i l l change according to a difference equation. I f the population a f t e r harvest i n year n i s tk , p r i o r to harvest i n year n + 1 i t w i l l be L-tk 1 + 3''(tk) or At 1 + t-3'k The maximum sustainable rent can be obtained from an analysis of the one-dimensional deterministic reproduction function . . Ax f(x) = 1 + (§'-k)x When we r e s t r i c t the harvest e f f o r t to act i d e n t i c a l l y on a l l age 301 classes, we reduce the problem to a one-dimensional form (at le a s t when the ag e - d i s t r i b u t i o n has reached stable equilibrium). In doing so we s i m p l i f y the analysis, but lose some of the features of an age-structure model which may be important from the management point of view. For other forms of the density-dependence function g(x) a s i m i l a r analysis can be performed. The maximum sustainable rent i s found by s o l v i n g : -Maximize: Y(u,q) = (l-u)q p'-k + c In u subject to:-(i) g(uq k) = (11) - < u <_ 1 This can be solved by the method of Lagrange M u l t i p l i e r s . In many cases however, l i k e the one already discussed, the equation ( i ) can be solved to give q e x p l i c i t l y i n terms of u . The maximization of the rent Y i s then a d i r e c t problem of maximization using elementary calculus. -8' *x For instance i f g(x) = a e ~ ~ the so l u t i o n to ( i ) i s £n(u Aa) . , . . . ^ = U M , ^ , and the maximization problem i s to maximize 1 _ P'^k Y(u) = £n(u Aa) z J-^ z- + c Jin u U 8'-k 302 over 1/A < u < 1 . If a model of the above type were employed i n a r e a l problem where the harvest e f f o r t acts equally on each age-class, i t would, of course, be unnecessary to estimate the c o e f f i c i e n t s of the L e s l i e matrix L , and the density-dependence function g(x) (at lea s t i f one was concerned only with equilibrium conditions.) One would only need to estimate the reproduction function r e l a t i n g successive sizes of populations with the stable d i s t r i b u t i o n . What t h i s model t e l l s us i s that, i f the e f f e c t of population density on mo r t a l i t y i s the same i n each age class ( i n the sense previously defined) and i f the e f f e c t of harvest e f f o r t on mortality has the same property, then, at lea s t as f a r as sustainable y i e l d or rent are concerned, one need only look at the one-dimensional problem i n which the sizes of successive populations with the stable age d i s t r i b u t i o n are re l a t e d by a reproduction function. uniform over a l l age-classes the above conclusion does not hold. In th i s case we would have a population model of the form I f the e f f e c t of population density on mortality i s not L x ~n 303 Let us suppose that t h i s model has a stable equilibrium. Under d i f f e r e n t harvest e f f o r t s , each having a uniform e f f e c t on mortality i n the various age-classes, the system would tend to d i f f e r e n t e q u i l i b r i a . In general these equilibrium populations w i l l have d i f f e r e n t a g e - d i s t r i b u t i o n and s i z e . In the previous case i n which population density had a uniform e f f e c t of mortality the e q u i l i b r i a reached, for d i f f e r e n t harvest e f f o r t s , a l l had the same i d i s t r i b u t i o n , but d i f f e r e n t s i z e s . Thus we see that, since the equilibrium age-distributions under d i f f e r e n t harvest e f f o r t s d i f f e r , the problem cannot be regarded as a one-dimensional one as previously. The e f f e c t of harvest e f f o r t on age-structure i s important i n t h i s case. Again i f we are concerned with a dynamic model, and not j u s t the equilibrium behaviour, the conclusion does not hold. We s h a l l discuss t h i s i n the context of a stochastic model i n the next section. In t h i s case i t turns out that the optimal harvest e f f o r t depends on both the ag e - d i s t r i b u t i o n and the s i z e of the current population. F i n a l l y i f we allow the harvest from each age-class to be co n t r o l l e d independently then the age-structure of the model w i l l be important and no reduction to one-dimensional form as above w i l l be possib l e . Indeed the age-structure of the model w i l l be important i n determining the optimal harvest. At le a s t as f a r as y i e l d i s concerned, the maximum sustainable y i e l d with t h i s kind of con t r o l i s at l e a s t as large as with the more r e s t r i c t i v e kind of con t r o l above. The i n t e r e s t i n g fl©esit-ign iSn n2 w e muGhtgreater willdigebgjVvIn _," other words, how much can the sustainable y i e l d be increased by 304 u t i l i z i n g the age-structure of the population? As we have indicated e a r l i e r we have made very l i t t l e progress with t h i s problem. If we t r y to develop an economic model i n which the harvest from each age-class can be c o n t r o l l e d independently, we are faced with the problem of f i n d i n g a r e a l i s t i c form for the cost function c(U) . It seems u n l i k e l y that the costs of catching i n d i v i d u a l s from each age-class w i l l be completely independent. The form of a r e a l i s t i c function c(U) w i l l surely vary with the nature of the animals being hunted. 5.3 A Stochastic, Density-Dependent Model In t h i s section we investigate a c o n t r o l model based on a stochastic version of the density-dependent age-class model of 5.2. J. P o l l a r d (1966) has studied a stochastic version of the o r i g i n a l density-independent L e s l i e model. Rather than assume that there i s a f i x e d proportion of survivors from the ( j - l ) S t to j t b age-class, he assumes that the p r o b a b i l i t y of such a s u r v i v a l i s £j (j=l,...,k) and that the s u r v i v a l s of i n d i v i d u a l s are s t o c h a s t i c a l l y independent events. He also assumes that the number of o f f s p r i n g born th by a member of the j age-class i s a random v a r i a b l e , F , with known d i s t r i b u t i o n . I f E ( F j ) = f j » t n e fecundity c o e f f i c i e n t of the L e s l i e model, then the L e s l i e model holds i n expectation. This model i s a s p e c i a l case of a multi-type branching process (Galton-Watson process) (see Harris 1963), and P o l l a r d points out that his main r e s u l t s can be derived from the theorems of the theory of such 305 processes. He obtains recurrence r e l a t i o n s f o r the f i r s t and second moments of the population vector X , and shows that i f the L e s l i e ~n matrix L has dominant c h a r a c t e r i s t i c root X > 1 then, as n -> °° , X > W k with p r o b a b i l i t y one, where W i s a (scalar) random A n v a r i a b l e and k i s the r. c h a r a c t e r i s t i c vector corresponding to X . In other words almost surely the age d i s t r i b u t i o n of the population converges to the same l i m i t i n g form as i n the deterministic L e s l i e model. Like the o r i g i n a l L e s l i e model, t h i s model ignores density-dependent factors i n the population growth. Although i t may o f f e r a reasonable d e s c r i p t i o n of the growth of a population when there are more than s u f f i c i e n t resources a v a i l a b l e , i t does not allow f o r i n t e r n a l regulation of the population. As a control model i t seems of l i m i t e d value unless one adopts the approach of Beddington and Taylor (1973) and assumes that, by harvesting, the population i s kept at such a low l e v e l that the environment can support the maximal growth of V which i t i s capable. One way of extending Pollard's model to include density-dependent factors would be to suppose that the s u r v i v a l p r o b a b i l i t i e s depend i n some way on the current population X^ . However i n doing t h i s the model loses i t s c h a r a c t e r i s t i c of being a branching process and i t seems d i f f i c u l t to analyse. 5 Instead we s h a l l adopt another approach and suppose that the s u r v i v a l c o e f f i c i e n t s Z z of the L e s l i e model are random va r i a b l e s with 306 d i s t r i b u t i o n s depending on the current population, and that the fecundity c o e f f i c i e n t s are random va r i a b l e s . More p r e c i s e l y we s h a l l work from the deterministic model (2) of 5.2 and suppose that {X } i s a Markov process with t r a n s i t i o n s ~n (1) X = g(X ) Z X ~n+l ° ~n n ~n where g i s a function R -> [0, 1] and {Z n} i s an i . i . d . sequence of matrices of random v a r i a b l e s , representing (fluctuating) s u r v i v a l and fecundity c o e f f i c i e n t s . z n z n 11 ' 12 ' z = n ,n The function g r e f l e c t s the e f f e c t of population density on s u r v i v a l proportions as i n 5.2. The random v a r i a b l e g(X )Z?. ~n l j represents the proportion of surviving o f f s p r i n g of parents of the ttl j age class when the population i s X. (j=l,...,k) , and the ~n random v a r i a b l e g(X^)Z^ represents the proportion of survivors from the ( j - l ) S t to j t b age-class when the population i s X (j=2,...,k) ~n If we assume that E(Z ) = L , the L e s l i e matrix, then the n model of 5.2 holds i n expectation. We s h a l l make no assumption on the independence of the component random va r i a b l e s of the matrix Z , but n 307 we s h a l l assume that the Z ^ are independent i n time, n = 1, 2, Thus our model allows random e f f e c t s i n the environment to e f f e c t the various age-classes i n a correlated way, but assumes that random environmental e f f e c t s are uncorrelated i n time. We s h a l l not study the asymptotic nature of t h i s model short of noting that there w i l l be a stable steady state p r o b a b i l i t y d i s t r i b u t i o n (over a subset of R ) for the age-structured population i f Doeblin's Hypothesis i s s a t i s f i e d (see Chapter 2). This w i l l c e r t a i n l y ' be the case i f the d i s t r i b u t i o n s of the random variables i n the matrix Z have den s i t i e s and i f there i s a set T d R "around" n the point zero, which i s transient i n the sense that the process leaves T at some f i n i t e time, with p r o b a b i l i t y one, and never returns to i t (cf 2.1). We s h a l l now discuss a co n t r o l model based, on t h i s stochastic population model. In 5.2 we looked at various c o n t r o l models for a deterministic version of our present model (NB. The deterministic model of 5.2 i s a s p e c i a l case of the stochastic model, obtained by l e t t i n g a l l the mass of the random va r i a b l e components of the matrix Z^ be concentrated at sin g l e points.) For the deterministic model we found i t easier to look at the system i n equilibrium. Only i n the very l i m i t e d s p e c i a l case of a sin g l e harvest e f f o r t , acting uniformly on each age-class, were we able to obtain an e x p l i c i t s o l u t i o n . For a stochastic model new problems present themselves i n an equilibrium a n a l y s i s . The deterministic equilibrium l e v e l i s easy to f i n d , but t h i s i s not so for the stochastic case. The stochastic equilibrium 308 p r o b a b i l i t y d i s t r i b u t i o n i s given by a complicated i n t e g r a l equation which we are unable to solve (cf 2.1). However when we look at a dynamic control model the method of Dynamic Programming i s a v a i l a b l e f o r both the deterministic and stochastic models. As for the one-dimensional model (Chapt. 3), when dealing with the stochastic case, we s h a l l concentrate on a dynamic control model and seek a q u a l i t a t i v e c h a r a c t e r i z a t i o n of an optimal p o l i c y . Before proceeding i t i s perhaps as well to c l a r i f y the re l a t i o n s h i p s which e x i s t between the various types of dynamic and equilibrium optimization c r i t e r i a . F i r s t l y , i f a steady-state e x i s t s for the stochastic modelj we can t a l k of the expected equilibrium y i e l d (or rent) f o r any stationary p o l i c y . We can seek a stationary p o l i c y to maximize expected equilibrium y i e l d . This corresponds to the deterministic optimization c r i t e r i o n of maximizing sustainable y i e l d . In the l i t e r a t u r e f o r dynamic optimization over an i n f i n i t e time-horizon two optimality c r i t e r i a are usually discussed (see Ross 1970). F i r s t l y there i s the c r i t e r i o n of maximizing the t o t a l  expected discounted y i e l d (or re n t ) . Here we assume a discount factor a < 1 . This i s the c r i t e r i o n adopted i n Chapter 3. Secondly there i s the c r i t e r i o n of maximizing long-run expected average y i e l d , (or rent ) . This i s defined as the l i m i t , as n -* °° , of the expected average y i e l d over n periods. How are these three optimality c r i t e r i a related? Suppose we assume that f o r a given stationary p o l i c y a steady-state e x i s t s . 309 Then we have from the strong law of large numbers f o r Markov processes (Doob 1953, p. 220), that under suitable r e g u l a r i t y conditions, the long-run expected average y i e l d for t h i s stationary p o l i c y i s the same as the expected equilibrium y i e l d for t h i s p o l i c y . Thus under these conditions maximizing expected equilibrium y i e l d i s the same as maximizing long-run expected average y i e l d over a l l stationary p o l i c i e s . We need t h i s l a s t condition, "over a l l stationary p o l i c i e s " , since i n some cases long-run expected average y i e l d i s maximized by a non-stationary p o l i c y (for some examples see Ross 1970). However i t can also be shown, under some conditions, that a stationary p o l i c y which maximizes t o t a l expected discounted y i e l d converges, as the discount rate a approaches unity, to a p o l i c y which maximizes long-run expected average y i e l d . Thus i n t h i s case we would have that a stationary p o l i c y which maximizes t o t a l expected discounted y i e l d would converge, as a -> 1 , to a stationary p o l i c y to maximize expected equilibrium y i e l d . Ross (1968) has proved t h i s convergence under a set of s u f f i c i e n t conditions, for a Markov process with non-denumerable state-space. Unfortunately these conditions are too r e s t r i c t i v e f o r the model we have developed. However i t i s worth keeping i n mind the idea that a stationary p o l i c y which maximizes t o t a l expected discounted y i e l d may converge to one to maximize expected equilibrium y i e l d . I f t h i s were the case the two optimal p o l i c i e s would be q u a l i t a t i v e l y the same. For the dynamic model developed i n t h i s section we s h a l l adopt the c r i t e r i o n of maximizing the t o t a l expected discounted y i e l d . We 310 s h a l l r e s t r i c t ourselves to the l i m i t e d case i n which the resource manager has c o n t r o l only over a s i n g l e - v a r i a b l e , the harvest e f f o r t , which we s h a l l assume acts uniformly on each age-class i . e . we assume that the same proportion i s harvested from each age-class. This i s the case we were able to solve for the equilibrium deterministic model i n the l a s t section. For the dynamic model, then, we represent an admissible  p o l i c y by a sequence u = {u^} , where 0 <_ u n <_ 1 , a l l n . The', number u^ represents the proportional escapement from each age-class i n year n , and i s r e l a t e d negative-exponentially to the harvest e f f o r t i n that year (see 5.2). Under such a c o n t r o l , the dynamics of the population are given by the equation (2) X = u g(u X )Z X ~n+l n n ~n n ~n We s h a l l suppose as before that the y i e l d from an animal of the j t b age-class i s p_. (j=l,...,k) . The y i e l d i n year n i s then (1-u )p'*X where p' i s the row vector of the p.'s . The expected n - ~n v j t o t a l discounted y i e l d f o r a p o l i c y u and an i n i t i a l population X^ = x i s then oo C U(x) = E{ I a I 1(l-u n)p'-X n|X ; L = x} (3) n=l We seek then to maximize (3) subject to the dynamic equation (2) for a l l i n i t i a l values x . We note here that t h i s model i s not r e a l l y an economic model, although i t includes time discounting. We 311 have not included any harvest cost function. The analysis which follows does not generalize d i r e c t l y to the case of a cost which i s a function of u other than a l i n e a r function. To solve the optimization problem above by Dynamic Programming we define f o r any admissible u = {u^} N cJjCx) = E{ I a I 1 ( l - u n ) p ' . X n | x i = x} n=l for N >_ 1 , and cJJCx) = 0 , for a l l x , u . The function ^J(x) represents as a function of the i n i t i a l population the expected discounted y i e l d under a p o l i c y u , over N time-periods with the resource becoming valueless at the end of the time horizon. We l e t C(x) = sup{C U(x) : u admissible} and = S U P ^ N ^ X ^ : U admissible} . We have from the general r e s u l t s of discounted Dynamic Programming (Kauffman & Cruon, 1967) that C^(x) converges to C(x) uniformly i n x and that there i s an optimal p o l i c y which i s Markovian and hence that the Bellman equation i s s a t i s f i e d : -(4) C. (x) = a max {(l-u)p'-x + E[C (ug(ux)Z x)]} N+1 _ . ~ N ~ n 0<u<l when the expectation i s taken over the random variables i n the matrix Z 312 If we write F(x) = g(x)*x and denote the j o i n t d i s t r i b u t i o n of the random variables of the matrix by $(Z) we can write (4) as C N + l ( x ) = a m a x { ( 1 _ U ) 5 ' ' X + 0<u<l C N(ZF(ux))d$(Z)} (5) As i n chapter 3 we use the equation (5) as the basis f o r our analysis. In the one-dimensional analysis of 3.3 we assumed that the reproduction function f(x) was concave. We make a s i m i l a r assumption here concerning the function F(x) . We s h a l l express i t i n terms of the function g(x) which r e f l e c t s the e f f e c t of density-dependence. We s h a l l suppose that g(x) i s r a d i a l l y concave and non-decreasing i n the sense that t g(tx) i s concave and non-decreasing i n t , f o r a l l x . What th i s means i s that we require the density-dependence to operate i n a purely compensatory way, i n the sense that as the population i s magnified i n any given year, keeping the ag e - d i s t r i b u t i o n unchanged, the expected number of survivors i n each age-class the following year, grows, but at a decreasing rate of increase. i . e . E ( X n + 1 | x = tx) i s non-decreasing and concave i n t , for a l l x . This condition c e r t a i n l y holds for the forms 313 g(x) = and g(x) = r-r- , b < 1 . 1 + 8 '-x (§'-x) Under t h i s assumption we are able to show that the following r a d i a l s t a b i l i z a t i o n of escapement p o l i c y i s optimal:-for any given population a g e - d i s t r i b u t i o n there i s an optimal escapement s i z e . I f the population si z e exceeds t h i s c r i t i c a l l e v e l the excess i s harvested (leaving the d i s t r i b u t i o n unchanged) whereas i f the population s i z e i s l e s s than the c r i t i c a l value no harvest i s made. This p o l i c y f or the case of two-age classes i s shown i n the following 2-dimensional p i c t u r e . no. i n f i r s t age-class F i g . 5.2 In the one-dimensional problem the p o l i c y which maximizes expected discounted y i e l d i s a s t a b i l i z a t i o n of escapement p o l i c y . In 314 the multi-dimensional problem the p o l i c y which maximizes expected discounted y i e l d also s t a b i l i z e s escapement, but the l e v e l at which the escapement i s s t a b i l i z e d depends on the ag e - d i s t r i b u t i o n . To prove the optimality of the above p o l i c y we f i r s t need a lemma. Lemma 1 If g(x) i s r a d i a l l y concave and non-decreasing then the function F(x) = g(x)x s a t i s f i e s the following:-(a) f o r any a^, >_ 0 , and any x F ^ x ) - F<a 2x) - — (a 2x) " x ^ - a . ^ = kx where k i s a scalar <_ 0 , depending on x , and where the d e r i v a t i v e 8F (a 2x) represents the matrix of p a r t i a l derivatives of F w.r.t. x O K at a 2x . (b) Vg(x)'x + g(x) >_ 0 , for a l l x , where Vg(x) i s the row-vector of p a r t i a l derivatives of g . The condition (a) i s a r a d i a l concavity condition f or F(x) , and (b) simply says that the magnitude of g(x)x i s r a d i a l l y increasing. Proof 3F —& = g(x)I + x • Vg(x) , 315 where the product x> g(x) i s the matrix product of a (kxl) and (lxk) matrices and so 9F F( a ; Lx) - F ( a 2 x ) - ( a 2 x ) - x ( a 2 - a ; L ) [a1g,(ia1x) - a 2g(a 2x) - (.a^a^) . ( g ( a 2 x ) + a r V g ( a 2 ; ) ) ]x = ta 1g(a 1x) - a 2g(a 2x) - ( a ^ a . ^ - ^ ag(ax)} a=a 2 = kx say. From the r a d i a l concavity of g we have that k <^  0 , and t h i s proves (a). » To prove (b) we note that tg(tx) = g(tx) + t Vg(tx) . Since tg(tx) i s non-decreasing i n t , putting t = 1 gives the r e s u l t . We are now ready to prove the optimality of the r a d i a l s t a b i l i z a t i o n of escapement p o l i c y . Theorem 5.3.1 Suppose that the function g(x) i s such that tg(tx) i s concave and non-decreasing i n t ( i . e . suppose the density-dependence operates i n a purely compensatory way). The following p o l i c y maximizes expected discounted y i e l d over a l l p o l i c i e s for which a sin g l e harvest e f f o r t acts uniformly on a l l age-classes:-\ 316 for any given population a g e - d i s t r i b u t i o n there e x i s t s a c r i t i c a l population s i z e . I f the s i z e exceeds t h i s c r i t i c a l value the excess i s harvested (keeping the a g e - d i s t r i b u t i o n unchanged.) If the s i z e i s l e s s than the c r i t i c a l value no harvest i s made. Proof We follow a proof by induction, and make the induction hypothesis that (A) VC N(x)'x >_ 0 , f or a l l x (B) C N ( a l X ) - C N(a 2x) - V C N ( a 2 x ) - x ^ - a ^ £ 0 for a l l a± <_ a 2 . ( i . e . we assume that i s r a d i a l l y concave and non-decreasing). Since C Q(x) = 0 , the.induction hypothesis i s t r i v i a l l y s a t i s f i e d for N = 0 . Bellman's equation i s C (x) = a max {p'x(l-u) + N ~ 0<u<l ~ C N(ZF(ux))d$(Z)} Let ~N+1, x Q x (u) = C N(ZF(ux))d$(Z) - E'-x u We show f i r s t l y that Q^^ 1 1^ *"S c o n c a v e ^ n u • ^ e have that N+1, . N+1. . , . d nN+l. Q x ( U1> " Q x ( U 2 } " ( V Vdu" Q x ( U ) u=u„ C N(ZF( U ; Lx))d$(Z) - C N(ZF(u 2x)d$(Z) 317 ^ U1~ U2^ 8F V C N(ZF(u 2x)-Z- ^ (u 2x)-x d$(Z) {C N(ZF( U ; Lx) - C N(ZE(u 2x)) - 7C N(ZF(u 2x))[ZF(u 1x) - ZF(u 2x) ] }d$(Z) + 9F V C N ( Z F ( u 2 x ) ) { Z [ F ( U l x ) - F(u 2x) - ( U j - i ^ ) — (u 2x)x] }d$(Z) From the induction hypothesis (B) we have that the f i r s t of the above i n t e g r a l s i s £. 0 , and from Lemma 1(a), the expression i n the square brackets i n the second i n t e g r a l can be written kx , where k depends on x and k < 0 . Thus we have that the l.h.s. of the above i s k VC (ZF.(u 2x))Z x d$(Z) u 2g(u 2x) k VC N(ZF(u 2x))-ZF(u 2x)d$(Z) <_ 0 , since k <_ 0 and VC N(ZF(u 2x) • ZF(u 2x) >_ 0 from the induction hypothesis (A). Thus we have proved that Q^+"'"(u) i s a concave function of u . To f i n d the optimal proportional escapement when the population l e v e l i s x and N + 1- periods remain, we need f i n d the u N+1 N+1 which maximizes Q over [0, 1] . Suppose Q attai n s i t s x x ~ ~ N+1 maximum over [0, °°) at u(x) , then from the concavity of Q i t follows that an optimal p o l i c y w i l l be to harvest i f and only i f u(x) < 1 , i n which case the harvest w i l l allow a proportional escapement u(x) . In other words an optimal p o l i c y w i l l be to harvest i f and only i f ||x|| >_ ||xu(x)|| i n which case a harvest down to x u(x) i s made. Now we show that x u(x) depends only on the d i s t r i b u t i o n of the vector x (or i n geometric terms on i t s di r e c t i o n ) and not on i t s magnitude. We note that N+l u = N+l ax a x for a l l u, a ^  0 . From t h i s we have that f o r any u >_ 0 .N+l. . 1SI+1 .u. ^N+l r , ~N+1. Q (u) = Q (7) £ 0. (u(ax)) = Q (au(ax)) X aX a. ciX X *" N+l and hence Q i s maximized over [0, °°) at u = au(ax) . It X ~ follows then that u(x) = au(ax) , and thus that x u(x) = ax u(ax) .. Thus we see that x u(x) depends only on the d i r e c t i o n or d i s t r i b u t i o n of x and not on i t s magnitude. We have then that f o r any age - d i s t r i b u t i o n or d i r e c t i o n s p e c i f i e d by a unit vector e , there "is a c r i t i c a l l e v e l S^"1""*" = u(e) . From the concavity of Q (u) we have that i f a vector x = xe i s such X *** N+l that x > S g , then an optimal p o l i c y when N + l periods remain N+l and the population i s at x i s to harvest down to S e . If e N+l ~ x <_ S , then the optimal p o l i c y i s not to harvest. Thus we have 319 C N(ZF(x)d$(Z) , i f | |x| | •< S N+1 x C N(ZF(S x»x)d$(Z) + ap'(x-S--x), i f ||x|| > S N+1 where x denotes the unit vector i n the d i r e c t i o n x . We now show that the induction hypotheses (A) & (B) hold for N + 1 . F i r s t l y we show (A) holds. I f x i s such that I I x I I < S ~+"^ , we have x V CN+1 ( X )' X = 0 1 8F V C N(ZF(x))Z ^ (x)-x d$(Z) V C N(ZF(x))Z[g(x)I + x Vg(x)]-x d$(Z) - a V C N(ZF(x))-Zxd$(Z)g(x) fg(x) + Vg(x)-x] g(?) = a V C N(ZF(x))-ZF(x)d$(Z) g(x) + Vg(x)-x > 0 since from (A) V C (ZF(jp) 'ZF(x) >_ 0 for a l l Z , and since from Lemma 1(b) the expression i s square brackets i s > 0 . 320 If x i s such that x > S N+l x V C N + 1 ( x ) - x = a p'x >_ 0 . Thus we have that (A) holds for N + l For (B) we consider, for any a^ <_ a^ , the quantity E = CN+l ( al K ) " CN+l ( a2 x ) " V CN+l ( a2 x )' x ( ar a2 ) TJT I I I I ..N+l . I I I I JN+± l i L a, I IxI I > S- and a | |x| | > S-_ , we have that N+lx= E = ata^g'-x - p'-x S^ + 1] - a f a ^ ' - x - p'-x S^ + 1] - a(a 1~a 2)p'-x = 0 I I I I I I I I nN+1 If a 1| |x| | , a 2 | |x[ I <_ we have E = a 8F [ C N ( Z ~ ( a l x ) ) " C N ( Z ? ( a 2 x ) ) " V C N ( Z F ( a 2 x ) ) Z 9x" ( a 2 x ) x ( a 2 - a ; L ) ] d * ( Z ) = a [ C N ( Z ? ( a l x ) ) " C N ( Z ~ ( a 2 ? ) ) ~ V C N ( Z F ( a 2 x ) ) Z ( ? ( a l x ) " F ( a 2 x ) ) ] d * ( Z ) + a 3F V C N ( Z F ( a 2 x ) ) Z [ ~ ( a l x ) " F ( a 2 x ) " 3 7 ( a 2 x ) x ( a r a 2 ) ] d $ ( Z ) Now from the induction hypothesis (B) we have that the expression i n the square bracket i n the f i r s t i n t e g r a l i s £ 0 , and from Lemma 1(a) we have that the expression i n the square bracket i n the second i n t e g r a l can be written i n the form kx , where k depends on x and k <_ 0 Thus we have that 321 E < a V C N ( Z F ( a 2 x ) ) - Z F ( a 2 ? ) d*(Z) a 2 g ( a 2 x ) J From the induction hypothesis (A) i t follows that E <_ 0 . If a. I I x l I < S? +^ and a„| I x l I > S?+^" we have 1 1 1 - 1 1 — x . 2 M ~ M x E = a z-C N(ZF(a 1x))d$(Z) - a C N ( Z F ( S x + 1 x ) ) d $ ( Z ) , , -N+1 . - a D'(a 2x-S^ *x) - ( a ^ - a ^ a P/*? C N ( Z F ( a l X ) ) d $ ( Z ) - aa l E'-x - [a C N ( Z F ( S ^ + 1 x))d$(Z) - a S ^ + 1 p'-x] x r-N+l. . -N+1.0N+1 ., | | a[Q x ( a ^ - Q x ( S x /| |x| |)] N+1 But we have that Q (u) atta i n s i t s maximum over [0, 00) at X S^"*""*"/1 |xj | , and hence E < 0 . Thus i n a l l cases, a < a , we have, X ~ X . z. E <_ 0 . The induction hypothesis (B) holds f o r N + 1 . We have C Q(x) = 0 , and hence (A) and (B) hold for N = 0 . Thus we have, by the induction, that (A) and (B) hold f o r a l l N . It follows that f o r a l l l ' N an optimal p o l i c y i s the r a d i a l s t a b i l i z a t i o n N of escapement p o l i c y characterized by the quantities S x From the uniform convergence of ^ ( x ) to C(x) we have that Q x(u) converges uniformly i n u to 322 Q x(u) - C(ZF(ux)d$(Z) - g'-x u It follows that Q x ( u ) 1 S concave, and since C(x) s a t i s f i e s the Bellman equation, C(x) = ap'-x + a max {Q (u)} , ~ ~ oi^l 1 x we have that an optimal p o l i c y of the i n f i n i t e time-horizon i s the r a d i a l s t a b i l i z a t i o n of escapement p o l i c y characterized by the quantities S*- , the points at which the functions Q~(u) a t t a i n t h e i r maxima over X X [0, °°) , for a l l non-negative unit vectors x . This completes the proof of the theorem. This theorem applies to the dete r m i n i s t i c model of 5.2. I t i s a s p e c i a l case of the above. We have thus solved the deterministic case both for the c r i t e r i o n of maximizing equilibrium y i e l d and for the c r i t e r i o n of maximizing discounted y i e l d . Bearing i n mind the e a r l i e r remarks about the r e l a t i o n s h i p s between various optimality c r i t e r i a , we see that i f the maximum expected discounted y i e l d c r i t e r i o n does indeed converge (as a -> 1) to the maximum expected equilibrium y i e l d c r i t e r i o n , then the p o l i c y which maximizes expected equilibrium y i e l d i s q u a l i t a t i v e l y a s t a b i l i z a t i o n of escapement p o l i c y of the type above (indicated i n F i g . 5.2). We stress the r e s t r i c t i o n s we have placed on an admissible p o l i c y . By considering only c o n t r o l p o l i c i e s i n which the same proportion i s harvested from each age-class, we have e f f e c t i v e l y reduced the dimensionality of the problem. However i n doing so we have 323 begged an important question. That i s the question, "How much better can we do by u t i l i z i n g the age-structure of the population i n our harvest p o l i c y ? " In the above optimization we have u t i l i z e d only the information concerning the age-structure. We have not exploited the dynamics of an age-structured population. As we mentioned e a r l i e r i n 5.2, i n p r a c t i c a l s i t u a t i o n s i t i s probably very d i f f i c u l t to obtain accurate data on the current age-structure of a hunted population. This very much l i m i t s the p r a c t i c a l usefulness of the r e s u l t above. In t h i s section we have only considered the problem of maximizing discounted y i e l d and not economic p r o f i t , or rent, as i n the l a s t section. This i s b a s i c a l l y for a mathematical reason. For a cost function c(u) other than a l i n e a r function, the above theorem does not e a s i l y generalize. I t i s the same type of problem as we faced i n Chapter 3. There we were able to generalize the main theorem of 3.3 for non-linear cost functions, i n 3.4. It seems l i k e l y that Theorem 5.3.1 i s true for some non-linear cost functions. We would be p a r t i c u l a r l y interested i n proving i t for the case c(u) = -An u , which represents the case of cost being proportional to harvest e f f o r t . F i n a l l y we observe we have made no assumptions of the random matrices , save t h e i r temporal independence. Our model then i s not r e s t r i c t e d to an age-structure model for a si n g l e population. As i n 5.2, i t could represent a number of competing populations, each possibly with an age-structure. It could also represent a two-sex model for a si n g l e population i n the way indicated i n the following section. 324 5.4 Two-Sex Models In a l l the models discussed so far i n t h i s and the previous chapters we have not considered the sex composition of a population. C l e a r l y our models si m p l i f y the dynamics of any r e a l population. The question we face i s whether t h i s s i m p l i f i c a t i o n i s a serious shortcoming, or not? Descriptive models f or the dynamics of two-sex populations have been developed (see for instance Goodman (1953, 1968)). I f we are concerned only with the s i z e of a population l i v i n g i n a "natural s t a t e " we probably gain l i t t l e by considering a sex-distributed population, at l e a s t for a large population. This i s because fl u c t u a t i o n s i n proportions of male and female b i r t h s and deaths when averaged over large numbers are l i k e l y to be small and to have l i t t l e impact on population growth. For a small population t h i s may not be the case. Again when we look at a population which i s subject to systematic harvesting a single-sex model may be inadequate for a descr i p t i o n of the population dynamics. For instance i t may well be that the harvesting procedure se l e c t s for larger animals, which may well be predomiantely from one sex. The harvest of sperm whales i s an example of t h i s . In t h i s case the dynamics predicted by a one-sex model are l i k e l y to be considerably i n error. When we turn to a control model we are faced with a d i f f e r e n t question. Here we are p r i m a r i l y i n t e r e s t e d i n whether the resource manager can u t i l i z e the sex structure of the population dynamics i n 325 order to increase h i s y i e l d or economic rent. This i s c e r t a i n l y achieved, i n many cases, f o r a population which i s husbanded rather than hunted. For instance a rancher w i l l u t i l i z e male and female c a t t l e i n d i f f e r e n t ways. The 'husbandman, however, controls not only the harvests from h i s population, but also to some extent i t s environment. This i s not the case f o r the hunter. The hunted population l i v e s " i n the wil d " and the hunter or resource manager controls only the harvests from the population. In developing a sex model for the harvesting of such a population i t i s important to know the behavioural r o l e s of the two sexes with respect to the growth and s u r v i v a l of the population. For instance we might be concerned with an animal population i n which one male can p o t e n t i a l l y service a large number of females (e.g. sperm whales). I t might seems at f i r s t sight that one could sustain a high y i e l d from t h i s population without impairing i t s reproductive capacity by harvesting only males, and leaving s u f f i c i e n t male escapement to service the females or better sustaining the females at some optimum l e v e l of production and allowing a male escapement only s u f f i c i e n t to service these females. Mathisen (1962) has suggested such a p o l i c y for Alaska Red Salmon, based on h i s observations of spawning behaviour . However i f t h i s were the case, one would face the question as to why.the species had evolved i n such a way as to allow the presence of t h i s "surplus" of males unnecessary for f e r t i l i z a t i o n . If we assume some sort of p r i n c i p l e of e f f i c i e n c y i n evolution, i t would 326 seem that these apparently "surplus" males perform some function. It might be that they are there as a safeguard against a d r a s t i c increase i n male death rate, i n order that a l l female eggs be f e r t i l i z e d i n a l l circumstances. Again i t might be a matter of natural s e l e c t i o n - by competition f o r a v a i l a b l e females only the f i t t e s t males would reproduce. Another p o s s i b i l i t y i s that the males perform some function e s s e n t i a l f o r the s u r v i v a l of the population. In the case of a mammalian species t h i s might be, for instance, protecting the females and infants from predation during gestation and l a c t a t i o n . Another p o s s i b i l i t y i s that males gather food for the whole population during the reproduction period. If the surplus males are harvested possibly these functions might not be performed. The e f f e c t s on the population might not be f e l t for many years, as would be the case with genetic s e l e c t i o n . Again the e f f e c t s might only be f e l t i n "bad" years as would be the case of a "surplus" of males forming a safety margin. On the other hand the ef f e c t s would be immediate i f these apparently "surplus" males perform some e s s e n t i a l s u r v i v a l function. Unlike the husbandman the hunter cannot perform these functions f o r the population. i We see then, that i n order to b u i l d an adequate sex-structured model for the harvesting of a population' i t i s important that the f u l l behavioural roles of the sexes be known. Such a model i s l i k e l y to be s p e c i f i c to a given population. The development of such models i s a problem more for the b i o l o g i s t than f o r the applied mathematician. An example of inadequate model i s that of Mann (1970). He 327 assumes that the numbers of male and female deaths per season are random multiples of the respective male and female population s i z e s . He also assumes that the numbers of male and female b i r t h s are random multiples of the t o t a l population s i z e . This model, although stochastic, i s l i n e a r , and l i k e the L e s l i e model, does not allow for any i n t e r n a l regulation of the population. This i s one weakness of the model as a basis for a harvest model. Mann perhaps p a r t l y overcomes t h i s objection by assuming a density cost, which could impose costs on the resource manager for maintaining the stock at too high (or too low) a l e v e l . By s u i t a b l y manipulating the form of t h i s density cost function one could perhaps ensure that the population stays i n a range for which i t s growth i s approximately density independent. A second more serious objection to the model i s that the in t e r a c t i o n s of the sexes are modelled i n an inadequate way. The assumption that t o t a l numbers of male and female b i r t h s depend on the t o t a l population s i z e may be adequate for a d e s c r i p t i v e model of a population i n a r e l a t i v e l y stable "undisturbed" state, but i t i s quite inadequate for a c o n t r o l model, i n which the two sexes can be harvested independently. For instance i f one sex were considerably more valuable than the other, the model might well p r e d i c t , as an optimum p o l i c y , that a l l of the more valuable sex be harvested every year, leaving the less valuable sex to carry out the reproduction. Here we develop a two-sex model for the harvest of P a c i f i c Salmon based on the observations of Mathisen (1962). His experiments on Alaska Red Salmon seem to i n d i c a t e that i f there i s a preponderance 328 of female f i s h on the spawning grounds an i n d i v i d u a l male w i l l f e r t i l i z e the eggs of more than one female. He found that with a r a t i o of 15 females to one male there was a reduction of l e s s than 5% i n the number of f e r t i l i z e d eggs as compared with the s i t u a t i o n of equal males and females. In the model we s h a l l develop we s h a l l assume that male f i s h perform no r o l e s p e c i f i c to them, for the s u r v i v a l of the population other than to f e r t i l i z e eggs. Bearing i n mind our previous remarks t h i s i s possibly a f a l s e assumption. However Mathisen suggests no other e s s e n t i a l r o l e , and speaks of a " b i o l o g i c a l surplus" of males. He c a l l s t h i s surplus "a safety factor which ensured f e r t i l i z a t i o n of, a l l the a v a i l a b l e eggs", (p. 142) and suggests that t h i s surplus could be harvested to increase the y i e l d , without impairing the reproductive c a p a b i l i t y of the stock. He points out that the "success of spawning i n one year should properly be evaluated by a comparison between the number of eggs deposited and the subsequent return from the same spawning rather than be based on the number of both sexes c o n s t i t u t i n g the t o t a l escapement i n the parent year." (p. 215) In what follows we formulate a mathematical c o n t r o l model based on t h i s idea, and characterize q u a l i t a t i v e l y a p o l i c y to maximize expected discounted y i e l d . We s h a l l be concerned only with a si n g l e "race" of salmon, assumed to enter freshwater together, and subsequently to a r r i v e at common spawning grounds together. Furthermore we s h a l l assume that a l l members of the race have the same length of l i f e - c y c l e . We l e t , xlj1 be random variables denoting the number of 329 females and males j u s t p r i o r to the time at which they enter freshwater, i n period n . I t i s at t h i s time that the f i s h are harvested. We s h a l l suppose that a proportion 6 of males and of females survive the n journey to the spawning beds. In general we l e t 6^ be a random v a r i a b l e . We suppose that one male can service an average up to s females. We s h a l l assume that t h i s i s a f i x e d deterministic quantity. Thus m males are capable of s e r v i c i n g up to sm females. We assume that the eggs of no female w i l l be l e f t u n f e r t i l i z e d i f there are s t i l l potent males i n the spawning area. We assume that the male and female fecundities ( i . e . number of male or female eggs per f e r t i l i z e d female) of f e r t i l i z e d female f i s h are f ^ and f ^ re s p e c t i v e l y . Again we s h a l l assume these to be deterministic quantities although the same r e s u l t holds i f they are random v a r i a b l e s . We have then, from an escapement the numbers of f e r t i l i z e d male and female eggs B = ~n \ B 1 are given by B0 = m i n { 6 n X 0 • S 6 n X l } ' f0 B? = min{6 X n , s6 X?} • f, 1 n 0 n 1 1 330 After the spawning season a l l of the f i s h die. The eggs hatch and a f t e r a sojourn i n fresh water the smolt migrate to the ocean. Af t e r a period i n the ocean they return to freshwater again to spawn and die. We s h a l l make the s i m p l i f y i n g assumption that a l l the young f i s h stay the same length of time i n fresh water and i n the ocean. We s h a l l suppose that the s u r v i v a l of f i s h from the stage of f e r t i l i z e d egg to returning adult i s density dependent. P r e c i s e l y we s h a l l assume a s u r v i v a l mechanism of the type .developed i n 5.3. Of B f e r t i l i z e d eggs we suppose that q(B) Z B survive to return, where q i s a 2 function from R -> [0, 1] and Z i s a diagonal matrix with random variables Z^ , Z^ d i s t r i b u t e d on [0, 1] i n the diagonal p o s i t i o n s . We s h a l l assume that the function q s a t i s f i e s the r a d i a l concavity and monotonicity condition of 5.3 that t q(tB) be increasing and concave i n t , f o r a l l B . This simply implies that as we increase the number, of f e r t i l i z e d eggs keeping t h e i r s e x - d i s t r i b u t i o n unchanged, the expected number of returning adults increases but at a decreasing rate of increase. In general the random variables Z^ , Z^ can have d i f f e r e n t d i s t r i b u t i o n s , and be non-independent. The vector ^ n + ^ represents the number of returning adults, (the time point n i s one l i f e - c y c l e length e.g. 4 years) a f t e r the time point n . We then have the dynamic equation X . = 8 min{X* sX*} q(9 min {x" , sX?}f) Z f ~n+l n 0 I n 0 1 ~ n~ (1) where , and Z i s a random matrix equal i n d i s t r i b u t i o n n 331 to Z , above. If the sequences of random variables {Z^} , are i . i . d . then {X ,.. } forms a Markov Process. We s h a l l assume t h i s n+1 to be the case, and also that the Q n' s a r e s t o c h a s t i c a l l y independent of the Z 's . n We now suppose that a harvest i s made each year as the f i s h enter freshwater. We assume that males and females can be harvested separately. We s h a l l seek to maximize expected discounted y i e l d over an i n f i n i t e time-horizon, and following the usual procedure l e t C^(x) denote the optimal expected discounted y i e l d when the current population i s x and N periods remain u n t i l a time horizon at which time the resource w i l l become valueless. Thus we l e t Cfj(x) = 0 • We l e t C(x) denote the optimal discounted y i e l d over an i n f i n i t e time horizon. Since the process i s Markovian we have the Bellman equation C N + 1 ( x ) = a max {p'-(x-y) + E[C N(F(Z, 6, y)]} (2) y:y 0lV y i - x i where p' = ( P Q , P ^ ) represents the respective y i e l d s of males and females, and F(Z,6,y) = 0 min{y 0,s y ; L} q(6 m i n l y ^ s y ^ f ) Z f The expectation i s taken over the random variables Z, 6 . We have assumed that the same proportions of male and female escapements survive the journey to the spawning ground, and that no female's eggs are l a i d i n f e r t i l e i f there i s a surviving potent male 332 to f e r t i l i z e them. From these assumptions, and from the assumption that a l l f i s h die a f t e r spawning we see that i t w i l l be optimal to allow an escapement with s times as many females as males. If the proportion of females i s greater than 1/(s+1), there w i l l be females who lay t h e i r eggs i n f e r t i l e , and i f the proportion of males i s greater than l / . ^ s i l ) , there w i l l be an i n e f f i c i e n c y i n the u t i l i z a t i o n of the males, since a smaller proportion could have f u l f i l l e d the same task. In either case harvesting the excess would have increased the y i e l d without s a c r i f i c i n g any reproductive e f f i c i e n c y . This can e a s i l y be derived mathematically from the model. For any Z, 6, we have F(Z, 0, y) = F(Z, 6, y m) where ( y 0 , y Q/s) i f y x > y Q / s y = *-m (sy, y x ) i f y ; L <_ y Q / s Thus we have that P'(x-y m) + E[C N(F(Z,0,y m))] > p'(x-y) + E[C N(F(Z,6,y))] since -p'-y > p'-y . ~ ~m — ~ We can thus r e s t r i c t the admissible set i n the Bellman equation and rewrite i t . 333 C N + I ( x ) = a m a x { p o ( x o " y ) + P i ( x r y / s ) + E[ c N(? ( z' e' ( y' y / s ) ) ) ] } y<min{xQ,sx^} (3) We see that i n determining the optimal escapement we need consider only a one-dimensional problem, concerning the number of f e r t i l i z e d females. We l e t it C N (x) = C N(x, x/s) . Then from (3) we have C* + 1(x) = a max{(p 0+ P'/s)(x-y) + E[C N(F(Z,6,(y,y/s)))]} (4) y<x We now express ^ ( z ) i n terms of for any vector z = ( Z Q , Z ^ ) . We know, from (3), that when N periods remain an optimal harvest w i l l be to a l e v e l (y, y/s) f o r some y . Thus we have a ( z r V s ) + V V ' s z i - z o CN (5> = \ a ( z Q - s z 1 ) + ^ ( s z ^ , s z 1 <_ z Q We w i l l use t h i s i n equation (4) to derive a fun c t i o n a l equation of the Bellman type for C^(x) • F i r s t l y we note, from the d e f i n i t i o n of F that F(Z,e,(y, y/s)) = 6y q(6yf)Zf . Thus assuming the independence on 6 and Z we have 334 E[C N(F(Z, 6,(y, y/s)))] = E[C*(eyq(0yf)min{sZ ; L , Z ^ } ) ] + a E[eyq(eyf)]{p 1E[Z 1f 1 If we denote the quantity i n s i d e the curly brackets by B , and the random v a r i a b l e m i n f s Z ^ ^ , Zgfg} by W , we can write the above as where the expectations are taken over 6 and W j o i n t l y . We note that B i s independent of y . Putting t h i s i n (4) we get C N + l ( x ) = a ™*x{(p +p /s)(x-y) + aBE[0yq(0yf)] + E[C (0yq(0yf)W)]} (5) that C^(x) 1 S concave and increasing i n x . Since q i s r a d i a l l y concave and increasing we have from Lemma 3.3.2 that C^(yq(6yf)0W) i s concave and increasing i n y , f o r each r e a l i z a t i o n of 0 and W . It follows that E[C^(yq(0yf))0W] has the same property. Also E[0yq(0yf)] has t h i s property and hence the whole expression i n curly brackets i n (5) i s concave i n y . a E[0yq(0yf)]B + E[C (0yq(6yf)W)] , y<x From here we proceed by induction i n the usual„way, and assume There i s then a l e v e l k. 'N+l at which t h i s expression attains i t s maximum, and we have C N + I ( x H a BE[0xq(0xf)] + aE[C (9xq(9xf)-W)-] , x <_ x ^ a(p 0 + P l/s)(x-X n + 1) + a 2 B E [ 0 x N + 1 q ( 0 x N + 1 f ) ] + a E t C N ( 9 x N + i q ( 9 W ) W ) ] ' X > X N + 1 We now show that (-'JT+ I( x) ^ s concave and increasing on [0, °°) . F i r s t l y we note that i t has t h i s property on the i n t e r v a l s [0, x ^ ^ - l and ( x J T + I » °°) • Furthermore i t i s continuous at ^§+2. ' Thus i t i s increasing on [0, 00) . To show concavity on [0, 0 0) we s h a l l show that df V l ( x ) ^ a ( p 0 + P l / s ) f ° r X - X N + 1 -The concavity w i l l follow from t h i s since d* CN+1 ( X ) = a ( P 0 + P l / s ) f o r X > ^ N+l • (NB The d i f f e r e n t i a b i l i t y of c a n he proved by induction i f we assume d i f f e r e n t i a b i l i t y of a l l functions involved). Since the expression i n curly brackets i n (5) i s concave i n y and a t t a i n s i t s maximum at » w e have -(p 0+ P ; L/s) + {a 9Bxq(9xf) + E[C^(0xq(9xf )W) ] } > 0 for x < ^+1 Rearranging we get dx" CN+1 ( X )  y- ^ P 0 + P l / S ) 336 Thus we have that C i s concave and increasing on [0,.°°).. • * Since C Q ( X ) = 0 has t h i s property we have that i t holds f o r a l l N . By the uniform convergence of (x) to C (x) we have that C i s concave and non-decreasing. I t follows that an optimal i n f i n i t e period p o l i c y when the current l e v e l i s (x, x/s) i s to harvest down to a l e v e l (x, x/s) i f x > x , and not to harvest i f x <^  x . When x = ( X Q , X ^ ) , an optimal i n f i n i t e period p o l i c y to maximize discounted y i e l d i s to the following (a) If X Q > x and x^ > x/s harvest down to (x, x/s) '' X 0 (b) If eit h e r X Q <_ x or x^ <_ x/s harvest down to m i n { — , x^}(s,l) This p o l i c y i s represented i n the following diagram, X l no. of males In t h i s region harvest only males down to l i n -In t h i s region harvest male/EJ and f>emales down to / P / / In t h i s region harvest only females down to l i n e X Q , no. of females F i g . 5.3 337 This i s exactly the r e s u l t we would expect i f we consider the quantity of f e r t i l i z e d females as the quantity which we have to c o n t r o l . The number of f e r t i l i z e d females i s s t a b l i z e d , with any excess males or females not able to take part i n a f e r t i l i z a t i o n process for lack of a mate also being harvested. If as the r e s u l t s of Mathisen i n d i c a t e , there i s r u s u a l l y a b i o l o g i c a l surplus of male f i s h , most sample points would be i n the upper l e f t hand part of region A, or i n region C . As we indicated e a r l i e r , i t i s not necessary, to assume that the fecundities f ^ , f ^ are deterministic. A s i m i l a r analysis can be performed with and f ^ as random v a r i a b l e s . However i t i s necessary to assume that s , the number of females that a s i n g l e male can service i s non-stochastic. Without t h i s assumption we cannot know that an escapement with males and females i n the r a t i o l : s w i l l be optimal. Also we need to assume that the same proportion of males and females survive the freshwater journey to the spawning ground, or at l e a s t that they are always i n a constant r a t i o . Without these assumptions i t i s l i k e l y that an optimal p o l i c y w i l l allow for a safety margin of surplus males. In the model we have assumed that any f l u c t u a t i o n s i n the male-female r a t i o are the r e s u l t of f l u c t u a t i n g s u r v i v a l rates of the two sexes, from hatching to return. As we have mentioned i t i s possible also to consider f l u c t u a t i o n s i n male and female f e c u n d i t i e s . The optimal p o l i c y removes the surplus of e i t h e r sex caused by these f l u c t u a t i o n s . 338 The b i o l o g i c a l assumptions of t h i s model are open to c r i t i c i s m . Even assuming that the males of the species perform no e s s e n t i a l task, s p e c i f i c to them, other than f e r t i l i z i n g eggs, i t i s questionable whether the perfect mating we assume i s possible, even for a very l o c a l i z e d race of salmon. Also we have assumed that the l i f e - c y c l e of a l l f i s h has the same length. There are p r a c t i c a l problems associated with adopting a harvest strategy based on sex. F i r s t l y there would be the problem of i s o l a t i n g a race of f i s h when they enter freshwater. For the assumption of perfect mating to be approximately true we would require a spawning area of very l i m i t e d geographic extent, and also a spawning period of l i m i t e d temporal extent. This would mean that a race of f i s h considered would have to be rather small. There would l i k e l y be problems of i d e n t i f y i n g such a race as i t enters fresh water. Secondly there would be the problem of estimating the abundance of males and females i n the given race, and t h i r d l y the p r a c t i c a l problems of achieving a s e x - s p e c i f i c harvest. I t may w e l l be that these p r a c t i c a l problems are too formidable to be overcome economically. As Mathisen says, " I t remains f o r future research to devise ways and means by which such a surplus can be harvested." F i n a l l y we look at the case i n which a sing l e harvest e f f o r t acts equally on both age-classes. This i s a s p e c i a l case of Theorem 5.3.1, since from (1) we can write X ^ = 0 min 1,< n+1 n 1 n x i : i s — x n X0 } q ( 0 n min{xj}, sX*} 339 If the proportion G , of survivors of the freshwater n journey, i s deterministic (non-random), then t h i s equation of the form (1) of 5.3, where X _ l and the matrix •1 g(x) = 6 min{l, s —} q(0 min{X A, sX,}) z = zh 1 ' °1 If q i s r a d i a l l y concave and increasing then so i s g . It follows that the r e s u l t of Theorem 5.3.1 holds, i . e . the p o l i c y which maximizes t o t a l expected discounted y i e l d among a l l p o l i c i e s i n which the harvest e f f o r t acts equally on both sexes i s a s t a b i l i z a t i o n of escapement p o l i c y as shown i n F i g . 5.2. Mann (1970) shows that an optimal p o l i c y f or h i s population and harvesting model, which allows for s u r v i v a l of parents a f t e r breeding i s of the following form. 1 no. of males II In t h i s region harvesl only males jlown to cu^rve r t / / 1 In t h i s regigjt'harvest both males ana femgLers' to P / III IV In t h i s region don't harvest In j ^ i i s region harvest only females down to curve x F i g . 5.4 0 no. of females 340 References Beddington, J . R. and Taylor, D. B. (1973). Optimum Age S p e c i f i c Harvesting of a Population. Biometrics 29, 801-809. Beverton, R. J . H. and Holt, S. J . (1957). The Dynamics of Exploited F i s h Populations. 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