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Optimization and heuristics : a comparative simulation study of management of a biological resource Matsumura, Ella Mae 1976

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OPTIMIZATION AND HEURISTICS: A COMPARATIVE SIMULATION STUDY OF MANAGEMENT OF A BIOLOGICAL RESOURCE "v by ELLA MAE MATSUMURA B.A., U n i v e r s i t y of C a l i f o r n i a , Berkeley, 1974 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department o f Mathematics and I n s t i t u t e of A p p l i e d Mathematics and S t a t i s t i c s We accept t h i s t h e s i s as conforming to the r e g u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1976 In p resent ing t h i s t he s i s in p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permiss ion fo r ex tens i ve copying of t h i s t he s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r ep re sen ta t i ve s . It i s understood that copying or p u b l i c a t i o n of t h i s t he s i s f o r f i n a n c i a l gain s h a l l not be al lowed without my w r i t t e n permis s ion. Department of Mathematics and Institute of Applied Mathematics and Statistics The Un i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 5 October 1976 Date i i A b s t r a c t Two approaches t o the f o r m u l a t i o n of r e s o u r c e management p o l i c y s ere c o n s i d e r e d , .The f i r s t was t o c o n s t r u c t a formal mathematical decision-making model o f the system and to o b t a i n o p t i m a l d e c i s i o n s a n a l y t i c a l l y . The second was t o use h e u r i s t i c s . The western t e n t c a t e r p i l l a r p o p u l a t i o n system was chosen as the resource system on which t o compare the approaches. v a r i o u s p o l i c i e s were t e s t e d on a computer s i m u l a t i o n model of the system. I t was found that a combination of the two approaches f l i n e a r programming and h e u r i s t i c s ) l e d t o s a t i s f a c t o r y h a r v e s t i n g p o l i c i e s . The r e s u l t s i n d i c a t e t h a t i g n o r i n g the b a s i c b i o l o g i c a l a t t r i b u t e s of the r e s o u r c e c o u l d l e a d to mismanagement, and p o s s i b l y even d e s t r u c t i o n of the p o p u l a t i o n . i i i TABLE OF CONTENTS C h a p t e r I . I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , 1 C h a p t e r I I . The System 6 1. The Western T e n t C a t e r p i l l a r — L i f e C y c l e . . . . 7 2. The S i m u l a t i o n Model * * . . . . . . . . . . . . . . . 12 C h a p t e r I I I . L i n e a r Programming Model ......................21 1. F o r m u l a t i o n o f t h e P r o b l e m 22 2. C o s t F u n c t i o n M o d i f i c a t i o n . . . . . . . . . . . . . . . . . 2 7 3. E s t i m a t i o n o f T r a n s i t i o n M a t r i c e s ..................34 4. D e r i v a t i o n o f A l t e r n a t i v e H a r v e s t i n g P o l i c i e s ...... 39 5. Summary . . . . . . . .-. .. .50 C h a p t e r I V . H e u r i s t i c H a r v e s t i n g P o l i c i e s ............52 1. I n t r o d u c t i o n . . . . . . . . . ... . .... . v . . . . . . . . .... ........ . .53 2. N o n - a d a p t i v e P o l i c i e s . . j , .,-.> .. , v.• i-vy. •'• # V •• •• • 56 3. A d a p t i v e P o l i c i e s w i t h P r e c i s e I n f o r m a t i o n 61 4. A d a p t i v e P o l i c i e s w i t h I m p r e c i s e I n f o r m a t i o n .......66 5. , Summary ....... .,,, . . . . . . . . . . . . . . . . . 73 C h a p t e r V. D i s c u s s i o n . ....... .... . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5 B i b l i o g r a p h y 81 A p p e n d i x . L i n e a r Programming and MPSX ............84 i v L IST OF TABLES I. T a b l e a u f o r L i n e a r C o s t F u n c t i o n s , N = 4 ••••-.../•>• , . 2 5 I I . Tab l eau f o r P i e c e s i s e - L i n e a r Co s t F u n c t i o n s , N = 4 . . . . 31 I I I . P a r amete r s f o r T r a n s i t i o n M a t r i c e s . % . . . v . . v 3 7 IV . L i n e a r Programming E f f o r t L e v e l s , R e a l i s t i c Weather . . . 4 2 V. Computed P o p u l a t i o n s B e f o r e H a r v e s t . . . . . . . . . . . . . . . . . . . 43 V-I. Mean Annua l H a r v e s t s , Yea r s 2 Through 7 . . . . . . . . . . . . . . . 4 8 V I I . Minimum Number o f C o l o n i e s B e f o r e H a r v e s t i n g , Yea r s 1 Through 8 . , ,v>.y, . , - : , :V. .»;V<v^.^y**!V»-. >,'•• •-. 48 V LIST OF FIGURES 1. Piecewise-Linear Cost Function .•,,,:.-,>•..,..>...v#•...»:.,.•,:... 28 2. Transition Function ............. 36 3. Mean Annual Harvests with Fixed E f f o r t .................. 58 Mean Annual Harvests with 200 Han-hours per Year ........60 5. Mean Annual Harvests with Fixed Escapement (Precise Information) ................63 6. Mean Annual Harvests with Fixed Escapement (Imprecise Information) .•......................68 7. Changes per Generation i n Total Populations .............71 Acknowledgements I would l i k e to thank my supervisor, Dr. I. Vertinsky, f o r his guidance, encouragement, and i n s p i r a t i o n . I would also l i k e to thank Dr. ,H,. Puterman for c r i t i c a l comments which helped me to improve the thesis, and Dr. 8. Ziemba for reading the thesis. A s p e c i a l note of thanks goes to Dr. 8. Thompson, who was an ever-present source of aid during each and every phase of t h i s study. Many other people have furnished assistance, and to a l l of them I am very g r a t e f u l . F i n a l l y , I would l i k e to thank the University of B r i t i s h Columbia for i t s f i n a n c i a l support. 1 Chapter I. I n t r o d u c t i o n 2 Managers of b i o l o g i c a l resources are confronted with the d i f f i c u l t problem of c o n t r o l l i n g an uncertain and complex environment. Even i n the best circumstances, applied b i o l o g i s t s charged with managing renewable resource systems have had to operate with only fragmentary knowledge of the r e l a t i v e importance of th e i r components and i n t e r a c t i o n s . Often there i s no information on the most basic i n d i v i d u a l attributes of the organisms they wish to co n t r o l , and managers are thus unable to predict the population conseguences of various measures. From time to time, therefore, f i s h e r i e s - , w i l d l i f e - , and pest-management programs have a l l suffered from inadvertently neglecting one of those attributes (e.g., through f a i l i n g to recognize the conseguences of changing patterns of resistance to chemical compounds introduced into the resource system) . He have considered two approaches to the formulation of resource management policy. The f i r s t approach was to construct a mathematical model of the system and to obtain optimal decisions a n a l y t i c a l l y . .Mathematical decision-making models are u t i l i z e d i n an attempt to r e l a t e knowledge about di f f e r e n t aspects of r e a l i t y and to provide insights into the r e l a t i v e merits of alt e r n a t i v e courses of action., While mathematical models enjoy the advantages of generality and ease of manipulation, a great deal of thought and care i s reguired to construct a model which accurately represents a system. It i s not always clear which elements should be 3 i n c l u d e d and which elements can be i g n o r e d . Furthermore, s o l v a b i l i t y of the problem must be borne i n mind as well as accuracy of r e p r e s e n t a t i o n . In c o n s t r u c t i n g our mathematical model, we wished to a b s t r a c t the most r e l e v a n t p r o p e r t i e s of the r e a l - w o r l d system while m a i n t a i n i n g computational f e a s i b i l i t y . In t h i s study, we chose t o focus on the q u a l i t a t i v e d i f f e r e n c e s i n the p o p u l a t i o n system and i t s h a b i t a t . Although some u n c e r t a i n t y was taken i n t o account, c o n s i d e r a t i o n of the s o l v a b i l i t y of the problem e n t a i l e d t h a t we l e a v e c o n c e n t r a t i o n on the s t o c h a s t i c i t y of the environment f o r l a t e r study. Our second approach was t o use h e u r i s t i c s . H e u r i s t i c s can be thought of as . . . r u l e s of thumb, t h a t allow us to f a c t o r , approximately, the complex per c e i v e d world i n t o h i g h l y simple components and to f i n d , approximately and reasonably r e l i a b l y , the correspondences t h a t allow us to a c t on t h a t world p r e d i c t a b l y . (Simon and Newell, 1962). , A manager i s l i k e l y t o take t h i s approach f o r a number of reasons. The system might be c o n s i d e r e d too complex to r e p r e s e n t by f o r m a l mathematical s t r u c t u r e s , or a f o r m u l a t i o n f e l t t o adeguately d e s c r i b e the system might be extremely d i f f i c u l t or impossible t o s o l v e with c u r r e n t t e c h n i q u e s . In some i n s t a n c e s , the data needed t o c o n s t r u c t a formal model 4 might be u n a v a i l a b l e . In other c a s e s , the manager might have n e i t h e r the personnel s k i l l e d i n modelling nor the d e s i r e to search f o r and h i r e such personnel. F i n a l l y , the use of h e u r i s t i c s i s appealing because s i m p l i c i t y i s a p p e a l i n g . A manager might use h e u r i s t i c s in c o n j u n c t i o n with s o l u t i o n s obtained with a formal model t o s i m p l i f y the p o l i c y or t o meet r e s t r i c t i o n s not accounted f o r i n the formal model., »e chose the western te n t c a t e r p i l l a r f o r comparing a l t e r n a t i v e approaches to management because the a v a i l a b l e i n f o r m a t i o n on i t s f u n c t i o n a l r e l a t i o n s h i p s (e.g., i t s s e n s i t i v i t y t o e x t e r n a l s t o c h a s t i c v a r i a b l e s and i t s p e r s i s t e n t i n d i v i d u a l d i f f e r e n c e s i n behaviour and development) i s so much more complete than f o r most n a t u r a l p o p u l a t i o n s (Wellington, 1957, 1960, 1964, 1965b). T h i s i n f o r m a t i o n was used t o c o n s t r u c t a s i m u l a t i o n model of the p o p u l a t i o n system (Wellington, e t - a l . , 1975). The s i m u l a t i o n model provided an experimental l a b o r a t o r y i n which we performed h a r v e s t i n g experiments t o compare managerial approaches. Q u a l i t a t i v e d i f f e r e n c e s among t e n t - c a t e r p i l l a r c o l o n i e s can be recognized by d i f f e r e n c e s i n the s i z e and shape of the communal t e n t . Tents may be one of t h r e e types: elongate, l a r g e pyramidal, or s m a l l pyramidal. C o l o n i e s with elongate t e n t s withstand bad weather much b e t t e r than the others {Wellington, 1965a). For the purpose of comparing the p o p u l a t i o n conseguences of d i f f e r e n t managerial p o l i c i e s , we 5 have designated the i n d i v i d u a l t e n t - c a t e r p i l l a r colony as a harvestable commodity. Chapter II contains a description of the resource system and of the computer simulation model used to simulate the dif f e r e n t harvesting p o l i c i e s . y Chapter III describes the development of a l i n e a r programming model to aid i n the derivation of harvesting p o l i c i e s . Chapter IV describes the he u r i s t i c harvesting p o l i c i e s simulated, and i n Chapter V se present some observations and conclusions on the res u l t s of the project. The Appendix contains d e t a i l s related to the l i n e a r programming problem. Chapter I I . The System .The Western Tent C a t e r p i l l a r — L i f e C y c l e The S i m u l a t i o n Hodel 7 1. The Western T e n t C a t e r p i l l a r — L i f e C y c l e The w e s t e r n t e n t c a t e r p i l l a r o f t h e S a a n i c h P e n i n s u l a o f V a n c o u v e r I s l a n d i n h a b i t s a h i g h l y u n s t a b l e e n v i r o n m e n t due t o f l u c t u a t i n g weather and v a r y i n g l o c a l c l i m a t i c c o n d i t i o n s { W e l l i n g t o n , 1974). The p o p u l a t i o n f l u c t u a t e s w i d e l y i n r e s p o n s e t o t h e w e a t h e r , but n e v e r becomes s o l a r g e t h a t s e v e r e d e f o l i a t i o n r e s u l t s o v e r an e n t i r e r e g i o n . The c a t e r p i l l a r y e a r l a s t s a p p r o x i m a t e l y 80 d a y s , from A p r i l t h r o u g h June. In e a r l y J u l y , f o l l o w i n g a two-»eek p u p a t i o n p e r i o d , e a c h f e m a l e moth d e p o s i t s h e r e ggs i n a s i n g l e mass on a b r a n c h o f a h o s t t r e e ( u s u a l l y r e d a l d e r , h awthorne, o r w i l l o w ) , w h i c h p r o v i d e s s u i t a b l e f o o d f o r t h e c a t e r p i l l a r s . I n A p r i l of t h e n e x t y e a r , t h e l a r v a e emerge and s p i n a communal s i l k t e n t w hich n o t o n l y c o n t r i b u t e s t o t h e g r o w t h o f t h e c a t e r p i l l a r s , b u t a l s o p r o v i d e s an e f f e c t i v e s h e l t e r a g a i n s t t h e u n c e r t a i n w e a t h e r . The s p i n n i n g o f t h e t e n t o c c u r s a s t h e c a t e r p i l l a r s c o n t i n u a l l y t r a i l s i l k f r o m t h e i r mouths. S i l k h i g h w a y s l e a d t o v a r i o u s f e e d i n g s i t e s , w h i l e a s i l k pad f o r m s a t t h e r e s t i n g s i t e . The pad becomes a t e n t a s t h e d a y s p a s s , and t h e t e n t becomes e x t e n d e d and more complex, wi t h l a y e r s e v e n t u a l l y f o r m i n g s e r i e s o f c o m p a r t m e n t s . The c h i e f a d v a n t a g e o f t h e t e n t i s i t s ' g r e e n h o u s e * e f f e c t . The t e n t b o t h a b s o r b s and r e t a i n s h e a t , e n a b l i n g t h e l a r v a e t o f e e d on e n c l o s e d l e a v e s d u r i n g c l o u d y p e r i o d s when t h e weather i s n o t warm enough f o r t h e l a r v a e t o v e n t u r e o u t 8 of the tent. I f the weather i s warm enough, the c a t e r p i l l a r s t r a v e l from the tent to eat, and return p e r i o d i c a l l y to rest as a cluster on the surface of the tent. C a t e r p i l l a r s are cold-blooded animals, hence resting on the warm tent surface accelerates t h e i r metabolic processes and thus aids i n digestion and ultimately, body growth.. The larvae, unable to survive i n very moist or very dry a i r , enjoy a further advantage provided by the tent. When rain f a l l s , the tent walls shed the water and prevent flooding of the tent. When the outside a i r i s very dry, the walls act as a vapor ba r r i e r , retarding the passage of moisture outward. There are b a s i c a l l y two classes of larvae; independent {Type I) and dependent (Type I I ) . T y p e I i n d i v i d u a l s , the most active, can t r a v e l even where there i s no s i l k t r a i l to guide them, whereas Type II in d i v i d u a l s cannot. The Type II*s exhibit d i f f e r e n t amounts of a c t i v i t y , and are therefore divided into subclasses (Types I l a , b, c ) . Type I l a c a t e r p i l l a r s are almost as active as Type I's, while Type l i e ' s are so in a c t i v e that they usually die of starvation early i n the year. Differences i n the amounts of i n d i v i d u a l a c t i v i t y are evident at b i r t h and persist throughout the development of the c a t e r p i l l a r colony. The Type I c a t e r p i l l a r s become active moths able to f l y miles before t i r i n g ; Type I I * s become l e s s active moths with reduced f l i g h t c a p a c i t i e s . The proportions of the d i f f e r e n t types of c a t e r p i l l a r s vary from colony to colony. These proportions are c r u c i a l in 9 determining the group a c t i v i t y of the colony, which i n turn af f e c t s the colony's chance of s u r v i v a l , through i t s e f f e c t on the length of the resting periods between feeding. The resting period begins at the tent when a few c a t e r p i l l a r s become motionless. The b r i s t l e s protruding from the i r sides interlock with those of the i r neighbours, forcing them to become motionless also. The process continues u n t i l the whole group becomes perfectly s t i l l , disbanding only when c a t e r p i l l a r s i n the outermost c i r c l e become hungry again and s t i r the other i n d i v i d u a l s . The a c t i v i t y l e v e l of individ u a l s i n the outermost c i r c l e i s thus c r u c i a l i n determining the duration of clu s t e r i n g , since active c a t e r p i l l a r s become very r e s t l e s s as they become hungry and induce the colony to disband according to their faster digestive rates instead of the slower rates of more ina c t i v e c a t e r p i l l a r s . I f a colony has many more sluggish members than active ones, the sluggish members are more l i k e l y to be on the edge of the c l u s t e r , the duration of clustering w i l l be greatly increased, and the growth of the colony w i l l be much slower than that of an active colony. The proportions of types of c a t e r p i l l a r s also a f f e c t the shape of the colony's tent, active colonies forage as f a r as 20 feet from their tent; they feed often, develop guickly, and produce several long, narrow tents. These tents are occupied and abandoned i n rapid succession, so that the active colonies continually feed on undamaged leaves as they move their tent 10 s i t e from branch t o branch. , Less a c t i v e c o l o n i e s f o r a g e over s h o r t e r d i s t a n c e s and produce one or two compact t e n t s which are n e a r l y sguare or pyramidal. I n a c t i v e c o l o n i e s are soon f e e d i n g on damaged leaves and may s t a r v e i f f r e s h f o l i a g e i s j u s t beyond t h e i r reach. I n a c t i v e c o l o n i e s show poorer s u r v i v a l than a c t i v e c o l o n i e s because o f t h e i r slower development and i n c r e a s e d s u s c e p t i b i l i t y to i n f e c t i o u s d i s e a s e s , a c t i v e c o l o n i e s g u i c k l y abandon contaminated t e n t s and t h e r e f o r e r a r e l y s u f f e r from i n f e c t i o u s d i s e a s e . The food r e s e r v e s of the female as she e n t e r s the pupal stage determine the a c t i v i t y l e v e l s of her progeny. S i n c e both the eggs and the a d u l t t i s s u e s develop dur i n g the pupal s t a t e , they compete with each other f o r the female*s accumulated food r e s e r v e s . The f i r s t eggs t o develop r e c e i v e more n u t r i e n t s than the l a s t , producing the most a c t i v e l a r v a e i n the r e s u l t i n g colony., A l l females produce a complete range of progeny, but an a c t i v e , w e l l - f e d female has much g r e a t e r food r e s e r v e s than an i n a c t i v e female and t h e r e f o r e can produce a l a r g e r number of a c t i v e o f f s p r i n g before beginning t o produce l e s s a c t i v e o f f s p r i n g . The females t h a t produce the l a r g e s t number of a c t i v e progeny are a l s o the moths t h a t f l y the f a r t h e s t from t h e i r b i r t h p l a c e before d e p o s i t i n g t h e i r eggs. L e s s a c t i v e moths, on the other hand, d e p o s i t t h e i r eggs c l o s e to t h e i r b i r t h p l a c e . T h i s d i f f e r e n t i a l d i s p e r s a l p r o t e c t s the s p e c i e s from 11 extinction by ensuring that at least some of the colonies will be in highly suitable locations. 12 2. Simulation Model Studies carried on over a number of years (Wellington, 1957, 1960, 1964, 1965; Iwao and Wellington, 1970; f o r further references see Wellington et al», 1975} indicate that the major factors c o n t r o l l i n g t h i s population system are the in d i v i d u a l differences in the population described above and the climate, both l o c a l l y and over the en t i r e Saanich Peninsula. ^ stochastic simulation model of the population system was constructed i n order to assess the e f f e c t s of these differences (Wellington et a l . , 1975). A simulation model provides an opportunity to perform experiments on the population guickly and with no damage to the r e a l system. Since parameters can e a s i l y be varied, experiments can be performed under a wide variety of i n i t i a l conditions, and i f desired, repeated. A simulation model can thus be a useful exploratory t o o l . , The model simulates the l i f e cycle of the western tent c a t e r p i l l a r on the Saanich Peninsula. The peninsula, represented by a 59 x 104 grid, i s divided into grid c e l l s .2 miles on each side. The size of the c e l l s corresponds to the maximum available resolution for l o c a l v a r i a t i o n i n climate, and also permits d i s t i n c t i o n between types of adults by t h e i r varying f l i g h t distances. Each grid c e l l i s assigned a tree density (maximum, intermediate, scarce, or zero), as well as a * local'climate* index to provide the patchwork of l o c a l conditions which comprise the c a t e r p i l l a r s * habitat. 13 The model p r i m a r i l y focuses on the l a r v a l { c a t e r p i l l a r ) stage and the a d u l t (moth) stage. I t e r a t i o n s c o n s i s t of updating each c o l o n y ' s r e c o r d s d a i l y , c a l c u l a t i n g growth as a f u n c t i o n of the p r o p o r t i o n s of each type o f c a t e r p i l l a r , the l o c a t i o n of the colony on the t r e e (high or low, i n the sun or i n the shade), the l o c a l c l i m a t e , and the o v e r a l l weather. The model a l s o k i l l s c a t e r p i l l a r s i n s i m u l a t i o n of an encounter with any of the hazards they f a c e , such as p r e d a t o r s , d i s e a s e , or s t a r v a t i o n . When the c a t e r p i l l a r s are s u f f i c i e n t l y l a r g e , they pupate. A f t e r mating, the female moths d i s p e r s e and l a y eggs according t o t h e i r l a r v a l types and weights. In the model, as i n the r e a l system, the d i s t a n c e s t r a v e l e d during d i s p e r s a l a l s o depend on the p h y s i c a l environment. A more d e t a i l e d d e s c r i p t i o n of the s i m u l a t i o n model f o l l o w s . Weather The s i m u l a t i o n model u t i l i z e s an index of t o t a l d a i l y s o l a r r a d i a t i o n , which i s an i n d i c a t o r o f the degree of c l o u d i n e s s and the l i k e l i h o o d of r a i n , to r e p r e s e n t the weather experienced by the c o l o n i e s . Since inclement weather prevents the c a t e r p i l l a r s from v e n t u r i n g from t h e i r t e n t s to forage f o r food, the index a i d s i n r e a l i s t i c a l l y s i m u l a t i n g growth and m o r t a l i t y of the c a t e r p i l l a r s . Furthermore, the 14 index i s an a p p r o p r i a t e manner i n which to represent weather experienced by the c o l o n i e s because the t e n t s serve t o take f u l l advantage of the r a d i a n t heat. The index i s the sum of 'macro-weather', indexed 0 to + 50, * l o c a l c l i m a t e * , indexed -16 t o +15, and 'mi c r o c l i m a t e * , indexed -12 t o +6. The index i s hence a number between -28 and +71; a l l n e g a t i v e i n d i c e s are rep l a c e d with an index of 0. 'Macro-weather* r e f e r s t o the weather experienced over the e n t i r e Saanich P e n i n s u l a . /The weather was modelled as t h r e e Markov c h a i n s , with d i f f e r e n t t r a n s i t i o n matrices f o r A p r i l , May, and June. These matrices were used t o generate weather which has r e a l i s t i c d a i l y f l u c t u a t i o n s and y e a r l y v a r i a t i o n s {Wellington e t a l . , 1975). Growth and s u r v i v a l of the c o l o n i e s depend on the numbers of days of o v e r c a s t and r a i n ( W e l l i n g t o n , 1964, 1965a). D a i l y weather sequences were 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 , ranging from ' e x c e l l e n t * to 'very bad* based on the i n f o r m a t i o n a v a i l a b l e on the e f f e c t s of o v e r c a s t and r a i n . Heather sequences i n v a r i o u s combinations were then used i n s i m u l a t i o n experiments. The ' l o c a l c l i m a t e 1 index expresses the d i f f e r e n c e s i n s u i t a b i l i t y o f h a b i t a t s on the p e n i n s u l a which are due to d i r e c t i o n a l a i r f l o w and p h y s i o g r a p h i c c o n d i t i o n s . The index assigned t o each g r i d c e l l i s based on the frequency of c l o u d and r a i n i n 'good' and 'poor' years, s i n c e t h a t freguency i s an i n d i c a t o r of the a i r f l o w and t e r r a i n c h a r a c t e r i s t i c s . 1 5 The l o c a t i o n o f a colony i n a t r e e a f f e c t s i t s exposure t o r a i n and s o l a r r a d i a t i o n and a l s o i t s s u s c e p t i b i l i t y t o heat l o s s . The * m i c r o c l i m a t e ' index r e f l e c t s the advantage or disadvantage o f the h e i g h t and aspect of a colony's l o c a t i o n . The r e l a t i v e advantage or disadvantage v a r i e s with the macro-weather and l o c a l c l i m a t e . In the s i m u l a t i o n model, the sum of the t h r e e i n d i c e s expresses the t o t a l weather experienced by a c o l o n y . Growth and Pupation Growth of the c o l o n i e s , accounted f o r d a i l y i n the s i m u l a t i o n model, depends on t h e i r type and i n s t a r (stage between m o l t i n g s ) , and on the weather. The average growth r a t e of a colony i s represented by that o f Type I l a i n d i v i d u a l s ; growth r a t e s of the other types o f i n d i v i d u a l s are represented as d e v i a t i o n s from the average growth r a t e of a Type I l a c o l o n y . One f a c t o r a f f e c t i n g the r a t e s o f growth dur i n g each i n s t a r i s l a r v a l weight a t the b e g i n n i n g of the i n s t a r . Bates were e x p o n e n t i a l d u r i n g t h e l i f e t i m e of the l a r v a e , with most o f the growth t a k i n g p l a c e i n the l a s t two i n s t a r s . A piecewise l i n e a r approximation can be used, so t h a t the r a t e s w i t h i n each i n s t a r can be thought of as being l i n e a r . Data obtained both under c o n t r o l l e d l a b o r a t o r y c o n d i t i o n s and i n v a r i o u s f i e l d l o c a t i o n s p rovided i n f o r m a t i o n on the 16 e f f e c t s of weather on growth. The o b s e r v a t i o n s showed t h a t a d i f f e r e n c e of up to two weeks i n the d u r a t i o n of the l a r v a l stage can be a t t r i b u t e d to the d i f f e r e n t e f f e c t s the o v e r a l l weather has on c o l o n i e s whose l o c a l c l i m a t e and m i c r o c l i m a t e are d i f f e r e n t . The e f f e c t s were i n c o r p o r a t e d i n the model by means of the weather i n d i c e s . The l a s t f a c t o r a f f e c t i n g growth r a t e s i s the range of i n s t a r s w i t h i n c o l o n i e s . D i f f e r e n c e s i n i n d i v i d u a l a c t i v i t y r e s u l t i n a range of two t o t h r e e i n s t a r s w i t h i n a colony, which i n t u r n a f f e c t s the d u r a t i o n o f the l a r v a l s t a g e . These d i f f e r e n c e s are a l s o accounted f o r i n t h e s i m u l a t i o n model. In accordance with experimental data, i n the s i m u l a t i o n , the l a r v a l stage ends and the pupal stage begins o n l y a f t e r an i n d i v i d u a l reaches a minimal age and weight. M o r t a l i t y M o r t a l i t y can occur because of a number of f a c t o r s , i n c l u d i n g the l a c k o f enough companions t o maintain a s u c c e s s f u l colony. The f a c t o r s which have v a r y i n g e f f e c t s on d i f f e r e n t t y p es of i n d i v i d u a l s i n c l u d e p r e d a t o r s , p a r a s i t e s , d i s e a s e , and s t a r v a t i o n , a l l of which have t h e i r worst e f f e c t on the l e s s a c t i v e t y p e s . The s i m u l a t i o n model t h e r e f o r e a s s i g n s m o r t a l i t y caused by these f a c t o r s i n p r o p o r t i o n to l a r v a l i n s t a r and type, a c t i v i t y l e v e l of the c o l o n y |the p r o p o r t i o n of Type I i n d i v i d u a l s i n the c o l o n y ) , and the 17 weather index. , The freguency of mortality due to predation i n a p a r t i c u l a r colony i s related to the previous history of the colony. Only ants and spiders pose a threat as predators, with spiders attacking only f i r s t ^ i n s t a r larvae, and ants k i l l i n g larvae u n t i l the end of the t h i r d i n s t a r . fints may repeatedly return to a colony to attack, while at other times they and spiders attack only those larvae which have ventured from the tent. The 'contagious* attacks which destroy whole colonies, modelled as a two-state Markov chain* and the *random* attacks which k i l l only a few larvae, constitute highly variable mortality from predation. Virus disease i s also related to the weather and to the previous history and a c t i v i t y l e v e l of the colony. , The simulation model treats i t as an i n t e r a c t i n g population whose presence i n a colony i s determined when the female moth deposits her eggs. I f the female i s infected with virus, she passes i t on to the next generation. The model assigns probablities of change in the i n i t i a l l e v e l of i n f e c t i o n on the basis of weather, colony type, and colony density i n the l o c a l i t y , as well as on the i n i t i a l l e v e l of i n f e c t i o n . The l e v e l can therefore increase and k i l l the colony, or decrease, possibly u n t i l the virus disappears from the colony. This imitates occurrences in the re a l system — active colonies may leave diseased i n d i v i d u a l s to die i n vacated tents, or carry a low l e v e l of i n f e c t i o n when they move, with subsequent 18 s u r v i v a l depending on the environment. I n a c t i v e c o l o n i e s , on the other hand, s u f f e r g r e a t e r m o r t a l i t y from i n f e c t i o n because they produce fewer t e n t s and s t a y i n c o n t a c t with d i s e a s e d members, thereby f a c i l i t a t i n g the t r a n s m i s s i o n of the v i r u s . I n a c t i v e c o l o n i e s which are i n f e c t e d s u f f e r t o t a l m o r t a l i t y i n poor weather. S t a r v a t i o n and p a r a s i t i s m were not modelled e x p l i c i t l y , s i n c e t h e i r occurrence i s not r e l a t e d to previous exposure of a colony to these f a c t o r s . The major cause of s t a r v a t i o n i s the weather, s i n c e c o o l weather p r o h i b i t s c o l o n i e s from feeding. Some p a r a s i t e s seem t o p r e f e r only f o u r t h - and f i f t h - i n s t a r l a r v a e . In good weather, a c t i v e c o l o n i e s grow r a p i d l y and may disband before the p a r a s i t e s appear. I n poor weather, the a c t i v e c o l o n i e s may s u f f e r from p a r a s i t i s m because they are the only ones with s u f f i c i e n t l y l a r g e l a r v a e . D i s p e r s a l and O v i p o s i t i o n Although the c o r r e l a t i o n between a c t i v i t y l e v e l and f l i g h t d i s t a n c e i s w e l l documented, l i t t l e i s known about the f a c t o r s determining the d i r e c t i o n of f l i g h t . The model t h e r e f o r e a s s i g n s the d i r e c t i o n randomly, with m o d i f i c a t i o n by a wind f a c t o r p o s s i b l e . A c t i v e moths can f l y f a r t h e r than i n a c t i v e moths, but may a c t u a l l y d i s p e r s e only a s h o r t d i s t a n c e f o r a v a r i e t y o f reasons, l e a d i n g to a wide range of a c t u a l f l i g h t d i s t a n c e s . In the model, as i n the r e a l system. 19 the moths undergo one major f l i g h t , with f l i g h t distances determined by a c t i v i t y types, tree heights and d e n s i t i e s , and wind. The model accounts for the a b i l i t y of d i f f e r e n t types of females to lay t h e i r eggs at d i f f e r e n t heights i n trees; the egg mass i s equally l i k e l y to be deposited on the sunny or shady side of the tree. The model then determines the number and types of hatching individuals i n a colony from the weight and type of female that deposited the eggs., Performance of the Model The simulation model was tested to see how well i t could predict both quantitative and q u a l i t a t i v e changes i n the c a t e r p i l l a r population on the Saanich Peninsula. To test the prediction of quantitative changes, the simulation was run with the actual weather sequence experienced on the peninsula from 1959 to 1964. The census of fourth-instar colonies taken annually i n the simulation agreed quite closely with the actual census taken on the peninsula during those years (Wellington et a l . , 1975). Since f i f t h - i n s t a r colonies disband, surveys taken while most of the population i s i n the f i f t h - r i n s t a r would almost surely contain errors because of the d i f f i c u l t y of deciding whether an empty tent at that point represents a l i v e , but disbanded colony, or a dead one. Surveys taken during the 20 t h i r d i n s t a r , on the other hand, would over-estimate the number of l i v e c o l o n i e s because some o f the r e l a t i v e l y i n a c t i v e c o l o n i e s which would probably not s u r v i v e t o pupate would be counted. F i e l d surveys, which take approximately two weeks, were t h e r e f o r e timed so t h a t most of the c o l o n i e s counted would be counted while i n t h e i r f o u r t h i n s t a r . Timing o f the survey was harder when p o p u l a t i o n s were l a r g e than when they were s m a l l , and hence g r e a t e r e r r o r s i n counting probably o c c u r r e d when the p o p u l a t i o n s were l a r g e . Counts i n the s i m u l a t i o n were of f o u r t h - i n s t a r c o l o n i e s . The above remarks might l e a d one t o expect t h a t the s i m u l a t i o n counts would d i f f e r most from the a c t u a l counts when the p o p u l a t i o n s were l a r g e . T h i s i s indeed what happened. , The q u a l i t a t i v e changes o f i n t e r e s t were the changes i n d i s t r i b u t i o n of the c o l o n i e s on the Saanich P e n i n s u l a , and a l s o the changes i n p o p u l a t i o n q u a l i t y . The simulated d i s t r i b u t i o n s d u r i n g p e r i o d s o f l a r g e s t and s m a l l e s t p o p u l a t i o n s match the d i s t r i b u t i o n s a c t u a l l y recorded on the peni n s u l a d u r i n g 1959 and 19.63. Furthermore, the simulated changes i n q u a l i t y o f the p o p u l a t i o n correspond t o those observed i n the a c t u a l p o p u l a t i o n . The s i m u l a t i o n model, c l o s e l y i m i t a t i n g the r e a l system, w i l l h e r e a f t e r be used i n place of the r e a l system. I t provi d e s the l a b o r a t o r y i n which we conduct our experiments and o b t a i n data f o r e s t i m a t i o n . 21 Chapter I I I . Linear Programming Model 1. Formulation of the Problem 2. Cost Function Modification 3. Estimation of Transition Matrices 4 . Derivation of Alternative Harvesting P o l i c i e s 5. Summary 22 1. Formulation o f the Problem A v a r i e t y of managerial approaches t o the c o n s t r u c t i o n of a mathematical decision-making model e x i s t s . , T h i s chapter assumes t h a t the manager o f t h e resource system t a k e s f u l l advantage of the knowledge a v a i l a b l e about the dynamics of the system. To u t i l i z e the i n f o r m a t i o n about the r e l a t i v e advantages of c e r t a i n area types and a l s o of d i f f e r e n c e s between c o l o n i e s i n terms of r e p r o d u c t i v e p o t e n t i a l , both the area types and the colony types ( i . e . a c t i v i t y l e v e l s ) were sub d i v i d e d . S i n c e the a c t i v i t y l e v e l s of c o l o n i e s are d i s t i n g u i s h a b l e by t h e i r t e n t types, c o l o n i e s were c a t e g o r i z e d by t e n t type (elongate, l a r g e pyramidal, and s m a l l p y r a m i d a l ) . The area types were a l s o broken down i n t o three c a t e g o r i e s , corresponding t o poor, medium, and e x c e l l e n t l o c a l c l i m a t e s . A s t a t e v e c t o r f o r any year c o n s i s t s of nine v a r i a b l e s , each r e p r e s e n t i n g one combination of an area type and an a c t i v i t y l e v e l . The manager thus has nine d e c i s i o n v a r i a b l e s per year. The v e c t o r components 1,2,...,9 correspond to the f o l l o w i n g combinations: AREA TYPES Poor Medium E x c e l l e n t I n a c t i v e 1 2 3 COLONY Medium a c t i v e 5 6 TYPES A c t i v e 7 8 9 2 3 The objective He w i l l consider i s the maximization of p r o f i t over a period of H years. If H i s large, the number of variables w i l l be large; i n t h i s case a l i n e a r programming model i s the only type of mathematical model which i s c e r t a i n to y i e l d computational r e s u l t s . Be therefore chose to develop a l i n e a r programming model. One constraint i s the preservation of a given minimum population a f t e r harvesting each year. This constraint, which guards against the extinction of the population, implies that the number of colonies harvested w i l l be l e s s than or egual to the number of colonies available immediately before the harvest. A further constraint i s that the number of colonies harvested be nonnegative. In the simplest case, where costs of harvesting and revenue are l i n e a r functions of the numbers harvested, our problem i s the following: Maximize t = i subject to " H T > P (Problem I) X^, Ht > 0, t=1 N, where & = discount factor, 24 M* = transpose of the vector o f weights which i n d i c a t e the net value of a colony of a given a c t i v i t y l e v e l and area type, H t = v e c t o r of harvest d e c i s i o n v a r i a b l e s f o r year t , X t = p o p u l a t i o n v e c t o r immediately a f t e r h a r v e s t i n g i n year t , (X 0 i s the i n i t i a l p o p u l a t i o n v e c t o r ) , P = v e c t o r of f i x e d minimum po p u l a t i o n s to be maintained a f t e r h a r v e s t i n g each year. W, H^, X^_ , and P are column v e c t o r s with nine components, corresponding to the p r e v i o u s l y d e s c r i b e d combinations of area and c o l o n y types. X t i s a l i n e a r f u n c t i o n of X t_ ( and H^. The i " component of i s a l i n e a r combination of the components of X^ ._( and t h e x*^ component of H^ .. The f u n c t i o n takes i n t o account the s u r v i v a l from the day o f h a r v e s t u n t i l the end o f the year, d i s p e r s a l o f the moths a f t e r pupation, and s u r v i v a l of the p o p u l a t i o n u n t i l the day of harvest f o r the next year. The f u n c t i o n i s a 9 x 9 matrix S t m u l t i p l i e d by the p o p u l a t i o n v e c t o r , minus the h a r v e s t v e c t o r H t. Rewriting the c o n s t r a i n t s , and adopting the convention t h a t H- = 0 f o r i < 0 y i e l d s the g e n e r a l c o n s t r a i n t H i + S, H, , • • • • • S L S • • • S J t S S, • • • S B < S ±---S X -P t t -t-l t -t-i 3 2. -t fc-i 4. i t i o f o r t = 1,...,N. For example, the c o n s t r a i n t f o r t = 3 i s H, + S H + S„S H < S S S X - P. 3 3 2. 3 2. I 3 - 1 1 ° The t a b l e a u f o r the f i r s t problem, where the o b j e c t i v e f u n c t i o n assumes t h a t the c o s t o f h a r v e s t i n g i s l i n e a r , i s given i n Table I . Since s u r v i v a l depends h e a v i l y on weather, the matrices 25 H, f Hurl J; ConS+rainW' IM roj 1 0+ S». row 1 ef Sx. I 1 O O. r««< I 0+ S"3ST root <\ #f SiS,. rem 1 of Sj rou) 1 of S 3 l 1 1 1 0 1 1 0 Year 4 ConS-fraMts )l .tn rou) [ of S^ SjS! n>tJ 1 of- SHSJSJ. roio 1 of S^ Sj r w 9 of SfSj row 1 »f 5M I &01WDS COST ~w' • • • -w1 -Ci-tfw' • • • -fi-tfj* T a b l e I . Ta b l e a u f o r L i n e a r C o s t F u n c t i o n s , N = 4 26 S^ . w i l l depend on the weather from the day o f h a r v e s t i n year t-1 u n t i l the day o f harvest i n year t . The d e t e r m i n a t i o n of how these matrices vary as a f u n c t i o n o f weather w i l l be d i s c u s s e d i n the next s e c t i o n . For the purposes of the l i n e a r programming model, the manager i s assumed to r e l y on past experience as he/she remembers i t i n making N-year weather p r o j e c t i o n s . Since memory i s o f t e n i m p e r f e c t , t h e behaviour of managers with o p t i m i s t i c , p e s s i m i s t i c , and r e a l i s t i c assessments of the weather w i l l be s i m u l a t e d . Day 31 of the c a t e r p i l l a r year was chosen as the day of harvest. At t h a t stage, the c o l o n i e s are e a s i l y i d e n t i f i a b l e and the shapes of t h e i r t e n t s serve as accurate i n d i c a t o r s of q u a l i t y . Since data has t r a d i t i o n a l l y been c o l l e c t e d a f t e r 31 days, day 31 r e p r e s e n t s the only point i n time f o r which r e a l data i s a v a i l a b l e , and hence r e p r e s e n t s the p o i n t f o r which p o p u l a t i o n counts are the most a c c u r a t e . Moreover, the s i m u l a t i o n model was found t o c l o s e l y reproduce the a c t u a l 'day 31* counts. Day 31 i s a l s o approximately the l a s t day i n the c a t e r p i l l a r year that we can be c e r t a i n no c o l o n i e s w i l l have pupated. The p o s s i b i l i t y of h a r v e s t i n g pupated c o l o n i e s would thus be circumvented. 27 2. Cost Function Modification Experience with the resource system indicates that moths disperse randomly and are not prone to pick a p a r t i c u l a r type of s i t e more often than another. The egg masses may therefore be assumed to be d i s t r i b u t e d randomly, and since the colonies do not move very far from the s i t e of t h e i r egg masses, the colonies themselves may be assumed to be d i s t r i b u t e d randomly. Furthermore, tents of a given type are equally conspicuous regardless of the number of colonies present. The implication t h i s c a r r i e s i s that costs of harvesting w i l l be density-independent, i . e . regardless of the actual size of a population of a given colony type on a given area type, the cost of harvesting a given proportion of the population w i l l be approximately the same., Empirical evidence shows that an i n i t i a l sweep of the entire harvesting area requires approximately 250 man-hours, during which we may expect to find the most conspicuous half of the population. To locate approximately half the remaining population requires twice the e f f o r t l e v e l needed for the previous half. The process can be repeated, continually d i v i d i n g the remaining population i n half. Harvesting w i l l continue as long as i t i s p r o f i t a b l e . Since the assumption of l i n e a r i t y of costs i n harvesting i n t e n s i t y i s r e a l i s t i c only for l o c a l approximation, we might reasonably select a large number of i n t e r v a l s and choose an appropriate l i n e a r function f o r each i n t e r v a l . The number of i n t e r v a l s should be chosen so 2 8 t h a t i t i s u n p r o f i t a b l e t o harvest the l a s t few c o l o n i e s i n a p o p u l a t i o n , s i n c e we expect t h a t i t would be p r o h i b i t i v e l y expensive t o har v e s t them. Our h a r v e s t i n g cost f u n c t i o n s r e f l e c t the behaviour d e s c r i b e d above and were chosen t o be piecewise l i n e a r f u n c t i o n s of the p o p u l a t i o n s i z e immediately b e f o r e h a r v e s t i n g . H a r v e s t i n g the f i r s t h a l f of a p o p u l a t i o n o f s i z e K i n c u r s a given average c o s t per colony o f <x ; h a l f of the remaining h a l f c o s t s 2ct per colony, e t c . Each category of c o l o n i e s (X*,... ,X *) has a c o s t f u n c t i o n o f t h i s shape, with p o s s i b l y d i f f e r e n t s l o p e s QCL . F i g u r e 1 d e p i c t s the c o s t f u n c t i o n where J , the number o f i n t e r v a l s chosen, i s f o u r . Cost P o p u l a t i o n F i g u r e 1. P i e c e w i s e - L i n e a r Cost F u n c t i o n In order to s o l v e t h i s new problem, j new v a r i a b l e s are i n t r o d u c e d f o r each of the v a r i a b l e s i n the o r i g i n a l problem {Problem I ) . Each o f the J v a r i a b l e s corresponds t o a p r o p o r t i o n o f the o r i g i n a l v a r i a b l e , and the valu e s of the J new v a r i a b l e s must sum t o the value of the v a r i a b l e f o r which 29 they have been s u b s t i t u t e d . T h i s r e p r e s e n t s one new group of c o n s t r a i n t s . Another new group of c o n s t r a i n t s i s necessary because each o f t h e new v a r i a b l e s must take up no more than i t s a l l o t t e d p r o p o r t i o n of the corresponding o r i g i n a l v a r i a b l e . Because each c o s t f u n c t i o n i s convex and s t r i c t l y i n c r e a s i n g , and because of the p r o p e r t i e s of the l i n e a r programming a l g o r i t h m , the two groups o f c o n s t r a i n t s are s u f f i c i e n t t o ensure t h a t the v a r i a b l e c o r r e s p o n d i n g t o the f i r s t h a l f of the p o p u l a t i o n i s a l l o t t e d i t s maximum value before the v a r i a b l e c o r r e s p o n d i n g t o the next quarter o f the p o p u l a t i o n i s a l l o t t e d a non-zero value, e t c . The v e c t o r w» • now r e p r e s e n t s the r e t u r n s f o r each category o f colony (X|,...,X|) and i s s t i l l m u l t i p l i e d by H t. The s l o p e s m u l t i p l i e d by the new v a r i a b l e s account f o r the c o s t of h a r v e s t i n g . Hhereas Problem I has 9 r o w s and 9N columns, t h i s new problem has 9(8-1) • 98 rows and 9N(1*J) columns. The t a b l e a u i s given i n Table I I . 30 Key to Table II 1. Block A consists of the c o e f f i c i e n t s given i n Table I. 2. Block Bi represents the d i v i s i o n of each constraint represented by a roa of AL i n t o J new constraints. The c o e f f i c i e n t s i n each row of B-L are a proportion of the c o e f f i c i e n t s of a row of A L. 3. Each box i n block indicates a J x J i d e n t i t y matrix. Each 1 i n each of the blocks E L in block A becomes a J x J i d e n t i t y i n block CL. 4. Each dash i n block D indicates J consecutive -1's. 5. Block E - indicates a 9 x 9 i d e n t i t y matrix. A l l other blank spaces i n d i c a t e c o e f f i c i e n t s of zero. 31 (or.g.Val varieties) (new V a r i a b l e s ) W columns 1J columns 9J"columns Ucolumns Utolumns <?(w-0 roWS 9J" ro\w'5 BOUNDS COST(Mi'n) ^.5,SxS,X0 - P f (proportions) Table I I . for Piecewise-Linear Cost Functions, N = 4 32 I d e a l l y , the i n i t i a l s l o p e s of the c o s t f u n c t i o n s would vary so that the t o t a l c o s t of h a r v e s t i n g h a l f the p o p u l a t i o n of a given type would be c o n s t a n t . However, t h i s poses a major c o m p l i c a t i o n because the p o p u l a t i o n each year depends on the h a r v e s t s of the p r e v i o u s years. /To o b t a i n the proper i n i t i a l s l o p e of the c o s t f u n c t i o n , we must d i v i d e by l i n e a r combinations of d e c i s i o n v a r i a b l e s . An a l t e r n a t e approach f o r determining the i n i t i a l s l o p e s was taken. Slopes were chosen based on i n f o r m a t i o n about the environment of the r e s o u r c e system. S p e c i f i c a l l y , r e l a t i v e c o s t s were c a l c u l a t e d based on a f i x e d c o s t to search a u n i t area o f l a n d , plus a c o s t per t r e e searched. T a k i n g i n t o account t h e f a c t t h a t t a l l t r e e s are more common i n areas o f high h o s t - t r e e d e n s i t y , we s e l e c t e d the f o l l o w i n g v e c t o r of r e l a t i v e s l o p e s : i n a c t i v e a c t i v e (ST^TVr 3, 2, 1,;ST^Tl) • Since a c t i v e c o l o n i e s are g e n e r a l l y s i t u a t e d r e l a t i v e l y high i n t r e e s , the l a s t t h r e e s l o p e s were i n c r e a s e d , r e s u l t i n g i n the v e c t o r oc = <3, 2, 1, 3, 2, 1, 5, 3, 1.33). An i t e r a t i v e procedure was a p p l i e d i n an attempt to improve our estimates o f the c o s t s f o r each category (X*,...,X«). The l i n e a r programming model was run f o r e i g h t t -t years using a c o s t f u n c t i o n c o n s i s t i n g of seven p i e c e s , with pc as the v e c t o r of i n i t i a l s l o p e s . The c o l o n i e s were assumed to 33 be e g u a l l y v a l u a b l e , with r e l a t i v e worth 11.75. The worth was chosen so t h a t the l a s t p r o p o r t i o n o f the p o p u l a t i o n , which was l e a s t expensive t o h a r v e s t , would not be worth h a r v e s t i n g , based on an expected p o p u l a t i o n s i z e o f 16 c o l o n i e s (16 i s approximately the geometric mean of our t o t a l i n i t i a l p o p u l a t i o n ) . The worth of the l e a s t expensive type of colony would have t o be i n the range (8,16) i n order to s a t i s f y the above c r i t e r i o n , so again the geometric mean (11.75) was chosen. The second i t e r a t i o n used i n i t i a l s l o p e s of 16c<L/l!I^, where E-t i s the average p o p u l a t i o n b e f o r e h a r v e s t i n g f o r category i and i s the i n i t i a l s l o p e chosen f o r category i . The l i n e a r programming model was run ag a i n , with the new s l o p e s and the same worth (11.75) f o r each co l o n y type. The procedure was a p p l i e d t o a p e s s i m i s t i c weather p r o j e c t i o n , an o p t i m i s t i c p r o j e c t i o n , and two s t a t i s t i c a l l y average p r o j e c t i o n s . For each p r o j e c t i o n , the average p o p u l a t i o n s i n each category a f t e r the second i t e r a t i o n c l o s e l y agreed with the averages a f t e r the f i r s t i t e r a t i o n . The i t e r a t i v e procedure was t h e r e f o r e terminated, with d i f f e r e n t c o s t v e c t o r s oC f o r each type of weather p r o j e c t i o n . The two r e a l i s t i c weather p r o j e c t i o n s y i e l d e d v i r t u a l l y the same c o s t v e c t o r . , 34 3. E s t i m a t i o n of T r a n s i t i o n Matrices The data to e s t i m a t e the t r a n s i t i o n m atrix was o b t a i n e d from runs of the s i m u l a t i o n model on the northern Saanich P e n i n s u l a . Because colony growth i s h e a v i l y dependent on weather, data was obtained f o r f i v e d i f f e r e n t c a t e g o r i e s of weather: very good, good, average, bad, and very bad. Two d i f f e r e n t weather sequences and two d i f f e r e n t random number seeds {input f o r the s i m u l a t i o n model) were used w i t h i n each weather category.^The data obtained from each s i m u l a t i o n run was the p o p u l a t i o n d i s t r i b u t i o n r e s u l t i n g i n year three a f t e r a l l but one of the nine c o l o n y c l a s s i f i c a t i o n s had been k i l l e d o f f on day 31 of year two. I t was hypothesized t h a t the p o p u l a t i o n s r e s u l t i n g from s i m u l a t i o n s w i t h i n a given weather category would not be s i g n i f i c a n t l y d i f f e r e n t . To t e s t t h i s h y p o t h e s i s , p a i r w i s e comparisons were performed f o r each weather category. F o r each comparison, e i t h e r the weather sequence or the random number seed was i d e n t i c a l . For each weather c a t e g o r y , row-by-row comparisons were performed on the f o u r 9 x 9 matrices of o b s e r v a t i o n s , using a c h i - s q u a r e t e s t , the h y p o t h e s i s t h a t the p o p u l a t i o n s had the same d i s t r i b u t i o n c o u l d not be r e j e c t e d at the .05 s i g n i f i c a n c e l e v e l f o r a l l but two o f the comparisons. However, at the .01 l e v e l , the hypothesis c o u l d never be r e j e c t e d . Each element (Ty- ) i n the t r a n s i t i o n matrix r e p r e s e n t s the expected number of c o l o n i e s of type j i n g e n e r a t i o n n+1, 35 given one colony o f type i i n generation n* He can expect t h a t given s u f f i c i e n t l y poor weather, the T t J w i l l be n e a r l y z e r o . S i m i l a r l y , with s u f f i c i e n t l y good weather, each TL- w i l l approach some maximum value ( d i f f e r e n t f o r d i f f e r e n t combinations of i and j ) . The graph of a s p e c i f i c T^j a g a i n s t macro-weather (0-50) w i l l show an s-shaped curve. The procedures f o r e s t i m a t i n g these curves w i l l now be d e s c r i b e d . , Each f u n c t i o n T Lj (w) was approximated as a piecewise l i n e a r f u n c t i o n o f w (weather) with :at most three p i e c e s . The f o l l o w i n g f o u r f u n c t i o n s were c o n s i d e r e d : 1) y = a + bw ( 1-piece) 2) y = f o f o r w< y (2-piece) b (w-v) f o r Y <w 3) y = fb(w-p) *c f o r w<p (2-piece) \c f o r ^  <w 4) y = f0 f o r w<v (3-piece) ib (W-Y) f o r V<w<^ b(^~ i f) f o r p<w where y i s some p a r t i c u l a r T-j and w i s some measure of weather. Eguation 1 r e q u i r e s f i t t i n g two parameters, e q u a t i o n s 2 and 3 r e q u i r e t h r e e parameters, and eguation 4 r e q u i r e s f o u r paramters. Parameters were determined by minimizing the sum of the 36 squared errors. Bhile t h i s procedure i s straightforward for equation 1, i t i s not so when the break-points are parameters to be determined (as i n equations 2, 3, and 4). This d i f f i c u l t y was surmounted by an enumerative procedure. F i r s t the wK were arranged i n ascending order (w, < w^  < ... < wK} , where samples were taken at k values of ». Then, using equation 2 as an example, best values for b and Y were found under each of the following assumptions: X < w( , Y = W | , w, < v < w2, Y = ... , w K < Y . The solution with the minimum sum of squares was thus found by a process of enumeration. Final s e l e c t i o n aiongst the four models was settled by comparing the F-test values. Equation 4 was most commonly adopted, with occasional best f i t s for equations 1 or 2. Since w i s bounded (0-50), a l l 4 equations can be represented as three-piece equations, with the extra pieces outside the domain of w i f necessary {Figure 2). Derived values for y , p, and b are given i n the matrices in Table I I I . y i Figure 2. Transition Function 37 VALUES FOR ~35.0 35.0 35.0 35.0 35.0 34.0 30.0 29.0 18.8 35.0 32.0 32.0 32.0 31.0 30.0 32.0 30.0 14.3 34.0 32.0 26.0 33.0 28.0 17.2 27.0 — 22.0 9.2 VALUES FOR b 0.163 0.012 0.004 0.003 0.004 0.008 0.001 0.001 0.004 0. 162 0.026 0.018 0.026 0.098 0.032 0.017 0.019 0.058 0.310 0.069 0.035 0.098 0.153 0.026 0.020 0.030 0. 114 VALUES FOR 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 35.0 35.0 35.0 50.0 35.0 33.0 32.0 22.0 19.0 35.0 34.0 2 8.0 32.0 27.0 26.0 24.0 17.9 7.9 31.0 27.0 26.0 26.0 20.3 14.6 21.0 12.0 12. 0 0. 008 0.002 0 .001 0. 0 0.003 0 .002 0. 001 0.000 0 .001 0. 047 0.011 0 .006 0. 011 0.016 0 .012 0. 006 0. 005 0 .016 0. 081 0. 024 0 .020 0. 031 0.036 0 .015 0. 013 0.014 0 .070 50.0 50.0 50. 0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 40.9 41.2 35.0 35.0 35.0 50.0 35.0 33.0 35.0 23.0 22.0 19.0 34.0 33.0 31.0 28.0 24.0 26.0 17.3 12. 0 11.9 29.0 28.0 29.0 21.0 15.2 12.2 19.0 12.0 5.8 0.004 0.001 0.0 0.002 0.001 0.001 0.001 0.001 0.001 0.019 0.015 0.006 0.009 0.008 0.013 0.007 0.004 0.010 0.051 0.035 0.029 0.026 0.023 0.022 0.035 0. 016 0.041 50.0 50.0 50.0 37.2 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 35.5 50.0 50.0 50.0 50.0 50.0 50.0 50.0 39.8 38.9 39. 8 Table I I I . Parameters for Transition Matrices 38 When an N-year weather sequence i s chosen f o r the l i n e a r programming model, a matrix T (v) i s computed f o r each year. The matrices S t are t h e t r a n s p o s e s o f t h e T-matrices f o r years 1 through N. 39 4. D e r i v a t i o n of A l t e r n a t i v e H a r v e s t i n g P o l i c i e s H a r v e s t i n g p o l i c i e s were d e r i v e d under a v a r i e t y of assumptions about weather, r e l a t i v e worth o f each of the c a t e g o r i e s of c o l o n i e s , and d i s c o u n t r a t e s . An o p t i m i s t i c , a p e s s i m i s t i c , and t h r e e r e a l i s t i c e i g h t - y e a r weather p r o j e c t i o n s were used. The three r e a l i s t i c weather p r o j e c t i o n s were generated from one 'normal* weather p r o j e c t i o n * A normal { s t a t i s t i c a l l y average) weather p r o j e c t i o n was generated using the Harkov ch a i n s d e s c r i b e d i n Chapter I I , then the weather of that p r o j e c t i o n was c y c l i c a l l y permuted by one year, and f i n a l l y , the weather f o r the e i g h t years was t o t a l l y scrambled. Weather w i t h i n a year, however, remained unchanged. The p e s s i m i s t i c p r o j e c t i o n r e p l a c e d a l l good years i n the normal p r o j e c t i o n with average years, and the o p t i m i s t i c p r o j e c t i o n r e p l a c e d a l l bad years with average y e a r s . For each weather p r o j e c t i o n , p o l i c i e s were d e r i v e d under the assumption t h a t a l l types o f c o l o n i e s were e g u a l l y v a l u a b l e , with r e l a t i v e worth 11.75. The only c o n s t r a i n t on the p o p u l a t i o n was t h a t at l e a s t f i v e a c t i v e c o l o n i e s be l e f t on refuge l o c a t i o n s a f t e r h a r v e s t i n g , to ensure the s u r v i v a l of the s p e c i e s . Discount r a t e s c o n s i d e r e d were 0% and 10%. I t i s of i n t e r e s t to d i s c u s s the r e s u l t s of experiments i n terms of e f f o r t r e g u i r e d to a t t a i n a c e r t a i n harvest, s i n c e a manager must decide the number of people to h i r e . We t h e r e f o r e r e q u i r e a f u n c t i o n a l r e l a t i o n s h i p between e f f o r t , number of c o l o n i e s , and harvest. A modified d i s k f u n c t i o n 40 {Rolling, 1959) was used to represent the expected harvest H(E,K) i n terms of e f f o r t E and population si z e K. The r e l a t i o n i s H(E,K) ••= {AEK)/{AE • K •/<>, \ , p > 0. The value of A i s the maximum harvest rate for an i n f i n i t e population and hence may he estimated from the r e l a t i o n H(E,«>)/E = X . For the northern Saanich Peninsula, A = 20 colonies per hour. Solving f o r p i n the r e l a t i o n f o r H (1,1) yields ft - 10,000 for the northern peninsula., The harvest decision values were therefore transformed into e f f o r t l e v e l s using the eguation E- = r.Hl(Kl • 10,000) M20(K L - H L ) , where H- i s the number of colonies of category i to be harvested, K -L i s the population of category i immediately before harvesting, and E; i s the e f f o r t to be applied i n harvesting category i . The following discussion i s i n terms of the e f f o r t l e v e l s computed from the l i n e a r programming solutions. The e f f o r t l e v e l s were not chosen to be the decision variables i n the o r i g i n a l l i n e a r programming problem because no h l i n e a r i t i e s would have resulted. The l i n e a r programming model was f i r s t run with a r e a l i s t i c ( s t a t i s t i c a l l y average) weather projection. The costs of harvesting used (<<.= (20.5, 2.92, 0.11, 8.23, 1.64, 0. 22, 3.95, 1.98, 0.35}) were those determined by the r e a l i s t i c weather projection and the i t e r a t i v e procedure 41 p r e v i o u s l y d e s c r i b e d . When r e l a t i v e w e i g h t s were a l l 11.75 and t h e d i s c o u n t r a t e was 0%, e f f o r t l e v e l s were a s shown i n T a b l e I V . The t a b l e i n d i c a t e s t h a t t h e manager would do w e l l t o a p p l y r e l a t i v e l y c o n s t a n t e f f o r t f o r e a c h c a t e g o r y , b u t d i f f e r e n t e f f o r t l e v e l s f o r d i f f e r e n t c a t e g o r i e s . , . C o m p a r i s o n w i t h computed p o p u l a t i o n s j u s t p r i o r t o h a r v e s t i n g ( T a b l e V) i n d i c a t e s t h a t o c c a s i o n a l s h a r p d i p s o r e v e n z e r o s i n o t h e r w i s e c o n s t a n t n o n - z e r o e f f o r t l e v e l s a r e due t o e i t h e r a z e r o p o p u l a t i o n t h a t y e a r o r a much l a r g e r p o p u l a t i o n t h e f o l l o w i n g y e a r , i n w h i c h c a s e t h e manager s h o u l d p o s t p o n e h a r v e s t i n g f o r t h a t c a t e g o r y . I n c a t e g o r i e s 2, 4, 5, and 7, t h e y e a r s o f z e r o o r r e d u c e d e f f o r t l e v e l s o c c u r r e d i n t h e two y e a r s w i t h t h e w o r s t w e a t h e r o f t h e e i g h t - y e a r s e g u e n c e . T h i s s u g g e s t s t h a t t h e manager c o u l d employ a s i m p l e d e c i s i o n r u l e b a s e d on t h e a v e r a g e weather from day 31 o f t h e p r e v i o u s y e a r t o day 31 o f t h e c u r r e n t h a r v e s t y e a r . I f t h e weather has been v e r y p o o r , t h e h a r v e s t i n g c o u l d be c o n c e n t r a t e d on c a t e g o r y 3 a l o n e , w hich c o n s i s t e n t l y has t h e h i g h e s t p o p u l a t i o n o f a l l t h e c a t e g o r i e s . A l t e r n a t i v e l y , t h e manager c o u l d expend a s m a l l amount o f e f f o r t a t t e m p t i n g t o l o c a t e c o l o n i e s i n c a t e g o r i e s 2, 4, 5, and 7, and t h e n make h i s d e c i s i o n a b o u t h a r v e s t i n g t h e s e c a t e g o r i e s b a s e d on t h e numbers f o u n d . I t i s n e v e r p r o f i t a b l e t o h a r v e s t c a t e g o r y 1 b e c a u s e o f i t s c o n s i s t e n t l y s m a l l p o p u l a t i o n s and r e l a t e d h i g h c o s t o f h a r v e s t i n g . C a t e g o r i e s 6, 8, and 9 a r e n o t h a r v e s t e d b e c a u s e CATEGORY 1 2 3 4 5 6 7 8 9 YEAR 1 0 3788 64919 500 3507 0 1501 0 0 2 0 3306 65185 472 3581 0 1473 0 0 3 0 1633 66551 0 0 0 0 0 0 4 0 3541 64644 496 3476 0 1513 0 0 5 0 3641 66982 502 3521 0 1490 0 0 6 0 0 66766 0 3361 0 1502 0 0 7 0 3457 63245 496 3446 0 504 0 0 8 0 3487 64340 499 3543 31369 1515 3545 10692 Table IV. Linear Programming Eff o r t Levels, R e a l i s t i c Weather CATEGORY 1 2 3 4 5 6 7 8 9 YEAR 1 1 6 65 1 20 28 9 34 35 2 0 4 90 2 14 47 14 17 42 3 0 3 103 2 11 51 12 14 42 4 15 42 195 19 29 97 39 29 72 5 0 12 231 9 28 114 34 31 88 6 0 0 153 0 12 77 6 19 67 7 0 10 199 7 23 98 28 25 76 8 6 37 301 19 3 9 149 52 41 110 T a b l e V. Computed P o p u l a t i o n s B e f o r e H a r v e s t , R e a l i s t i c Hea the r they are extremely valuable as reproductive agents, with category 9 the most valuable. This i s indicated by the relative costs obtained from the linear programming solution. If even one colony of type 9 aere harvested in the f i r s t year, for instance, the long-run profit would decrease by over 18 times the already high worth per colony of 11.75. As one might expect, the relative costs decrease as the end of the time horizon i s approached, f i n a l l y becoming zero in year eight. Since future populations are then no longer considered, categories 6, 8, and 9 are harvested. Not unexpectedly, the values of the harvest decision variables when the population constraint was removed remained virtually the same as their values i n the previous solution. Also, changing the discount rate to 1051 did not significantly change the values of the variables. parametric programming was performed to assess the sensitivity of the policy to the particular worth chosen. Reducing the weights from 11.75 to 6.75 in the objective function yielded a similar pattern of effort levels within categories, although lower than those of the f i r s t solution because of the decreased worths. Category 4 became unprofitable to harvest. When weights were decreased to 1.75, the policy was to harvest only category 3 with f a i r l y constant effort, occasionally harvesting category 5, in years followed by a year of poor weather. The relative costs from the linear programming model 45 indicate what the p o l i c i e s would be under the assumption that active colonies were more valuable than the others. The worths of categories 8 and 9 would have to be increased phenomenally before they are harvested consistently, since t h e i r value l i e s i n t h e i r production of colonies in category 3, which would be harvested consistently as long as i t i s profitable to take any type 3 colonies. The r e l a t i v e costs f o r category 9 were a minimum of 2.5 times those for category 8. Under the assumption that b i o l o g i c a l preferences vary inversely with p r i c e differences, so that i n a c t i v e colonies are more valuable than the others, category 3 would s t i l l be harvested heavily, with categories 8 and 9, and probably also category 7 remaining unharvested, depending on the worths of colonies of those types. The worths of the i n a c t i v e colonies would determine whether or not categories 1 and 2 were harvested. The solutions provided by the l i n e a r programming model using the alternate normal weather projections yielded s i m i l a r p o l i c i e s . The l i n e a r programming model was next run under the same pessimistic weather projection used in the i t e r a t i v e procedure to determine costs of harvesting. These costs, and r e l a t i v e weights of 11.75 yielded the same general policy as that of the normal weather seguences, except that category 4 became unprofitable to harvest because i t s expected population size was small under the pessimistic weather projection. A discount 46 rate of 10% leads to a s i m i l a r p o l i c y , with occasional harvesting of categories 6 and 8 toward the end of the time horizon, and smaller harvests i n every category at the end of the time horizon. The optimistic weather projection also yielded the same type of p o l i c y . Because of the assumed good weather, i t was at times even prof i t a b l e to harvest colonies i n category 1.( Each of the harvest p o l i c i e s s p e c i f i e d by the l i n e a r programming solutions for the three r e a l i s t i c weather projections was tested i n the simulation model with the weather seguence for which i t was designed. The expected number harvested was computed by the formula B = Kp, where H i s harvest, K i s population s i z e , and p i s the probability that a colony w i l l be harvested. The r e l a t i o n used was p = AE/(AE + K where E i s e f f o r t , A i s the maximum harvest rate, and A/(A * f +1} i s the probability of finding a colony with one unit of e f f o r t and one colony present. In this system, A = 20 colonies per hour and j4 ranges from 2500 to 10,000 colonies, depending upon the type of colony and size of area searched. Although the l i n e a r programming model overestimates the population s u b s t a n t i a l l y , i t does well at protecting the su r v i v a l of the species (at l e a s t 69 colonies were a l i v e each year before harvesting). The three exact p o l i c i e s were also each tested on the r e a l i s t i c weather seguences for which they were not designed. In a l l but one case, the p o l i c i e s protected 47 the s p e c i e s w e l l . , No p o l i c y c o u l d be judged best among the three (Tables VI and V I I ) . The l i n e a r programming model g u i t e e v i d e n t l y has l i m i t a t i o n s . I t i s a r i g i d d e t e r m i n i s t i c model of a s t o c h a s t i c system. I t must deal wih the d i f f i c u l t problem of p o p u l a t i o n e s t i m a t i o n from year t o year. P e r f e c t i n f o r m a t i o n about the system and the f u t u r e weather must be assumed f o r each p a r t i c u l a r problem. ,< Never the l e s s , the l i n e a r programming model may be of value i n determining t h e g e n e r a l shape of p o l i c i e s , and thus a i d i n the d e r i v a t i o n of h e u r i s t i c h a r v e s t procedures. The need f o r h e u r i s t i c s might a r i s e i n any case i f . t h e r e are c o n s t r a i n t s which do not appear e x p l i c i t l y or i m p l i c i t l y i n the l i n e a r programming model. One such c o n s t r a i n t of r e a l importance i s an upper bound on the e f f o r t expended each year. As remarked e a r l i e r , t he l i n e a r programming s o l u t i o n s i n d i c a t e d t h a t g e n e r a l l y , constant e f f o r t w i t h i n c a t e g o r i e s should be used. Category 3 c o n s t i t u t e s a subpopulation which i s both p l e n t i f u l and expendable, and t h e r e f o r e can s a f e l y be harvested u n t i l i t i s no longer p r o f i t a b l e t o do so. The high worth of c a t e g o r i e s 6, 8, and 9 as r e p r o d u c t i v e agents suggests that i n good weather, they should be l e f t t o reproduce. However, i n bad weather, only c o l o n i e s i n category 9 are h i g h l y l i k e l y t o s u r v i v e , so t h a t i t might be worthwhile to harvest c o l o n i e s from c a t e g o r i e s 6 and/or 8. Harvest experiments using a v a r i e t y of h e u r i s t i c s , i n c l u d i n g some of 48 WEATHER SEQUENCE USED FOR LINEAR PROGRAMMING 1 2 3 WEATHER SEQUENCE 1 50 56 64 USED 18 SIMULATION 2 93 111 101 3 69 42 77 Table VI. Mean Annual Harvests, Years 2 Through 7 R e a l i s t i c Weather Sequences WEATHER SEQUENCE USED FOB 1 LI NEAR PROGRAMMING 2 3 WEATHER SEQUENCE 1 69 68 69 USED IN SIMULATION 2 79 107 108 3 95 19 95 Table VII. Minimum Number of Colonies Before Harvesting, Years 1 Through 8 R e a l i s t i c Weather Sequences those mentioned above, are discussed i n the next chapter. 50 5..Summary a l i n e a r programming model was formulated which takes i n t o account d i f f e r e n c e s i n g e o g r a p h i c a l area types as w e l l as d i f f e r e n c e s i n a c t i v i t y l e v e l s o f c o l o n i e s . The nine r e s u l t i n g c a t e g o r i e s of c o l o n i e s correspond t o nine d e c i s i o n v a r i a b l e s f o r each of the » years over which p r o f i t from h a r v e s t i n g c o l o n i e s was maximized. The c o n s t r a i n t s were the n o n n e g a t i v i t y c o n s t r a i n t s and the p r e s e r v a t i o n of a given minimum po p u l a t i o n of c o l o n i e s a f t e r h a r v e s t i n g every year. The o b j e c t i v e f u n c t i o n i n c o r p o r a t e d l i n e a r r e t u r n s and p i e c e w i s e - l i n e a r {as a f u n c t i o n of p o p u l a t i o n size) c o s t f u n c t i o n s . D i f f e r e n t c o s t f u n c t i o n s were d e r i v e d f o r o p t i m i s t i c , r e a l i s t i c , and p e s s i m i s t i c weather p r o j e c t i o n s . Data f o r the e s t i m a t i o n o f the c o e f f i c i e n t matrix was obtained from the s i m u l a t i o n model. Si n c e colony growth i s h e a v i l y dependent on weather, data was obtained f o r f i v e d i f f e r e n t types of weather ranging from very good t o very bad. The data was used to e s t i m a t e matrices r e p r e s e n t i n g the s u r v i v a l , r e p r o d u c t i o n , and d i s p e r s a l from one year to the next, given a p a r t i c u l a r weather p a t t e r n f o r the year. These t r a n s i t i o n m a t r i c e s were used to generate a c o e f f i c i e n t matrix f o r each l i n e a r programming problem. The l i n e a r programming model was used to d e r i v e e i g h t - y e a r h a r v e s t i n g p o l i c i e s under a v a r i e t y of assumptions about weather, r e l a t i v e worth of each o f the nine c a t e g o r i e s of c o l o n i e s , and d i s c o u n t r a t e s . P o l i c i e s were d i s c u s s e d i n 51 terms of e f f o r t l e v e l s computed from the actual values f o r harvesting indicated by the li n e a r programming solutions. The l i n e a r programming solutions obtained under a r e a l i s t i c weather projection and with no discounting of p r o f i t indicated that r e l a t i v e l y constant e f f o r t l e v e l s should be applied f o r each category, but d i f f e r e n t levels f o r d i f f e r e n t categories. The greatest amount of e f f o r t should be spent on category 3 (small pyramidal colonies i n excellent l o c a t i o n s ) . A comparatively small amount of e f f o r t should be expended on categories 2, 4, 5, and 7. Categories 6, 8, and e s p e c i a l l y 9 are valuable as reproductive agents and should be l e f t alone. Category 1 should never be harvested because i t i s never pr o f i t a b l e to do so. Solutions obtained with di f f e r e n t weather projections and a discount rate of 10% indicated s i m i l a r p o l i c i e s . The harvesting p o l i c i e s s p e c i f i e d by the li n e a r programming solutions for the r e a l i s t i c weather projections were tested i n the simulation model with the weather seguence for which i t was designed. The l i n e a r programming model overestimates the population s u b s t a n t i a l l y , but does well at protecting s u r v i v a l of the population. The p o l i c i e s were also tested on alternate r e a l i s t i c weather projections. No policy could be judged best among the three tested. 52 Chapter IV. Heuristic Harvesting P o l i c i e s 1. Introduction 2. Non-adaptive P o l i c i e s 3. Adaptive P o l i c i e s with Precise Information 4. Adaptive P o l i c i e s with Imprecise Information 5. Summary 53 1. I n t r o d u c t i o n The h e u r i s t i c approach i n v o l v e s r e d u c i n g the number of a l t e r n a t i v e s so t h a t search f o r an o p t i m a l p o l i c y i s s i m p l i f i e d . For the h e u r i s t i c experiments performed on the s i m u l a t i o n model, we chose to use some of the same s i m p l i f i c a t i o n s t h a t we employed i n the l i n e a r programming model. For example, the annual day o f h a r v e s t i n g was a g a i n chosen to be day 31. a l l types of c o l o n i e s were co n s i d e r e d e g u a l l y v a l u a b l e , so that the o b j e c t i v e e s s e n t i a l l y became t o maximize the number o f c o l o n i e s harvested. The experiments d e s c r i b e d i n t h i s c h apter are h e u r i s t i c a d a p t a t i o n s of the f i x e d - e f f o r t p o l i c i e s i n d i c a t e d by the l i n e a r programming model, and fixed-escapement p o l i c i e s advocated i n the l i t e r a t u r e . a l l simulated h a r v e s t s were s t a r t e d under the same c o n d i t i o n s , using 1960 survey data to e s t a b l i s h numbers and l o c a t i o n s of egg masses i n the n o r t h e r n s i x miles o f the Saanich P e n i n s u l a (Wellington, 1964 and u n p u b l i s h e d ) . P r e v i o u s work (Thompson e t a l . , 1976) had shown that the northern p a r t of the p e n i n s u l a was a s u i t a b l y s e l f - c o n t a i n e d u n i t f o r any type of s i m u l a t i o n study. With the e x c e p t i o n of one s e r i e s o f t e s t s , the s i m u l a t i o n s were r e s t r i c t e d t o e i g h t years, with a randomly c o n s t r u c t e d average d a i l y weather sequence o p e r a t i o n d u r i n g t h a t p e r i o d . E v a l u a t i o n of management p o l i c i e s i n v o l v e d c o n s i d e r a t i o n of the numbers harvested as w e l l as the maintenance of the 54 s t o c k . To a v o i d u n r e a l i s t i c e f f e c t s at the beginning and end of the runs, the h a r v e s t s i n the f i r s t and l a s t of the e i g h t simulated years were ignored i n computing the mean y e a r l y h a r v e s t . The problem of s t o c k maintenance could not be handled so simply* ,. I d e a l l y , long runs of about 100 years would g i v e the most complete i n f o r m a t i o n on p o l i c i e s t h a t might l e a d to e x t i n c t i o n , but l a r g e numbers o f such runs are p r o h i b i t i v e l y expensive to operate on the computer. Our s o l u t i o n was based on a p r e v i o u s o b s e r v a t i o n that t o t a l p o p u l a t i o n s maintaining more:than 50 c o l o n i e s f o r e i g h t c o n s e c u t i v e years c o u l d s u r v i v e almost i n d e f i n i t e l y i n the northern p e n i n s u l a , whereas p o p u l a t i o n s f a l l i n g below 20 c o l o n i e s i n one generation u l t i m a t e l y became e x t i n c t . Consequently, we d e f i n e d a c c e p t a b l e management p o l i c i e s as those i n which the r e s i d u a l annual p o p u l a t i o n s exceeded 50 c o l o n i e s throughout the e i g h t simulated years. ;. Two c l a s s e s o f i n f o r m a t i o n play an important r o l e i n the management of b i o l o g i c a l r esources. I n f o r m a t i o n concerning the s t a t e of the system c o n s t i t u t e s the f i r s t c l a s s ; knowledge of f u n c t i o n a l r e l a t i o n s h i p s i n the system forms th e second. Harvest p o l i c i e s formulated as f u n c t i o n s o f the s t a t e of the resource system (or estimates thereof) w i l l be c a l l e d a d a p tive. Those not contingent upon s t a t e i n f o r m a t i o n are l a b e l e d non-adaptive. During the f o l l o w i n g t e s t s , both kinds of p o l i c i e s w i l l be f u r t h e r s u b d i v i d e d a c c o r d i n g to the assumed l e v e l s of knowledge concerning f u n c t i o n a l 55 relationships and population a t t r i b u t e s . 56 2. Non-adaptive P o l i c i e s The f i r s t t h r e e p o l i c i e s c o n s i d e r e d were non-adaptive, with s e v e r a l f i x e d l e v e l s of h a r v e s t i n g e f f o r t . In g e n e r a l , such p o l i c i e s have been shown t o r e s u l t i n low r a t e s of e x p l o i t a t i o n ( c f . Anderson, 1975) but i n some s i t u a t i o n s they may s t i l l be a t t r a c t i v e because they are easy t o manage. They were i n c l u d e d here t o p r o v i d e b a s e l i n e s f o r comparing other types of p o l i c i e s . P o l i c y 1; Fixed E f f o r t , No A r e a l R e s t r i c t i o n A s e r i e s of f i x e d l e v e l s of e f f o r t ( h a r v e s t i n g i n t e n s i t i e s ) v a r y i n g from 25-100 man-hours per year (mhpy) was s e l e c t e d and a p p l i e d i n d i s c r i m i n a t e l y t o the whole northern p e n i n s u l a . T h i s p o l i c y was examined under f i v e weather p r o j e c t i o n s : one o p t i m i s t i c , three s t a t i s t i c a l l y average, and one p e s s i m i s t i c . The o p t i m i s t i c p r o j e c t i o n was generated from an average one (Wellington e t a l . , 1975) by r e p l a c i n g poor with average years. C o n v e r s e l y , the p e s s i m i s t i c p r o j e c t i o n was d e r i v e d from the same average p r o j e c t i o n by r e p l a c i n g the good years with average ones. With P o l i c y 1, no f i x e d l e v e l o f e f f o r t more than 50 mhpy was a c c e p t a b l e d u r i n g poor weather. Bad weather kept numbers so low t h a t even modest ha r v e s t s of approximately 15 c o l o n i e s per year t h a t c o u l d be o b t a i n e d with f i x e d e f f o r t s of 200-300 mhpy exterminated the p o p u l a t i o n i n as l i t t l e as f o u r years. With average ( i . e . , more r e a l i s t i c ) weather under P o l i c y 57 1, some 200-300 mhpy were s t i l l r e q u i r e d t o o b t a i n maximal h a r v e s t s . C o l l e c t i o n s of more than 30 c o l o n i e s with : e f f o r t s g r e a t e r than 200 mhpy u s u a l l y l e d to e x t i n c t i o n . S u s t a i n e d y i e l d s were most o f t e n obtained with e f f o r t s of 100-150 mhpy {Fig. ,3, Curve A) • , O p t i m i s t i c weather p r o j e c t i o n s permitted s u s t a i n e d y i e l d s with e f f o r t s as high as 300 mhpy, with a peak harvest of 60 c o l o n i e s s u s t a i n e d near 225 mhpy. Harvests approximating 45 c o l o n i e s c o u l d s t i l l be b r i e f l y maintained with e f f o r t s as high as 350 mhpy, but p o p u l a t i o n s r e g u l a r l y s u b j e c t e d to such o v e r a l l pressure u l t i m a t e l y became e x t i n c t . P o l i c y 2: Fi x e d E f f o r t , R e s t r i c t e d Harvest Area For a common property resource (such as a f i s h e r y ) one would expect more concentrated e f f o r t i n areas t h a t seem t o o f f e r c o n s i s t e n t l y higher y i e l d s . P o l i c y 2 p r o v i d e s an extreme example of such . behaviour by r e s t r i c t i n g h a r v e s t i n g t o t h a t 25% of the northern p e n i n s u l a comprising the ref u g e areas (those l o c a l i t i e s i n which the p o p u l a t i o n was observed to p e r s i s t , even i n the most c l i m a t i c a l l y adverse p e r i o d s ) . Reducing the area searched by 75% a l s o reduced /* t o 2500 c o l o n i e s . By r e s t r i c t i n g h a r v e s t s t o l o c a l i t i e s with the h i g h e s t d e n s i t i e s , annual r e t u r n s i n c r e a s e d as much as 50% over the more s c a t t e r e d e f f o r t s of P o l i c y 1. S u b s t a n t i a l l y l e s s e f f o r t thus was r e g u i r e d t o o b t a i n a s p e c i f i e d y i e l d under P o l i c y 2 58 EFFORT (MAN HOURS/YEAR) Figure 3. Hean Annual Harvests with Fixed E f f o r t P o l i c y A: Fixed e f f o r t over the entire northern peninsula P o l i c y B: Fixed e f f o r t confined to refuges S o l i d l i n e s i n d i c a t e e f f o r t l e v e l s which maintained the population; broken l i n e s i n d i c a t e e f f o r t l e v e l s leading to extinc t i o n . Average weather was used. 59 ( F i g . 3, Curve B). As might be expected, however* the r i s k o f o v e r e x p l o i t a t i o n with P o l i c y 2 was c o r r e s p o n d i n g l y h i g h e r than with P o l i c y 1. For example, e f f o r t s of o n l y 150 mhpy i n poor or average weather under P o l i c y 2 exterminated the whole p o p u l a t i o n even more r a p i d l y than e f f o r t s of 250 mhpy under P o l i c y 1. C o n c e n t r a t i n g on areas of high d e n s i t y without knowing why they e x i s t thus may reduce h i g h l y v u l n e r a b l e s t o c k s so q u i c k l y t h a t t h e r e i s not enough time to d i s c o v e r the e r r o r and a l t e r the p o l i c y b e f o r e the resource i s destroyed. P o l i c y 3: Fixed E f f o r t P a r t i t i o n e d Among Types of C o l o n i e s With the two preceding p o l i c i e s , peak h a r v e s t s , whether temporary or s u s t a i n e d , tended to occur near e f f o r t - l e v e l s of 200 mhpy with a l l f i v e weather p r o j e c t i o n s . The t h i r d p o l i c y t h e r e f o r e p a r t i t i o n e d t h a t amount o f e f f o r t between s m a l l pyramidal {SP) and l a r g e pyramidal (LP) c o l o n i e s i n the refuge areas. Elongate (E) c o l o n i e s everywhere were untouched. A l l d i v i s i o n s of e f f o r t under t h i s p o l i c y were a c c e p t a b l e . Maximal ha r v e s t s (approximately 110 c o l o n i e s ) were obtained by devoting 75^90% of the 200 mhpy t o the poorest q u a l i t y (SP) c o l o n i e s i n the refuge areas ( F i g . 4). Two f a c t o r s account f o r t h i s dramatic improvement over the preceding r e s u l t s : (a) no e f f o r t was expended where colony d e n s i t y was lowest, and (b) those c o l o n i e s with the h i g h e s t r e p r o d u c t i v e p o t e n t i a l were spared everywhere. 60 120 T-0 50 100 PERCENT EFFORT F i g u r e 4. Hean Annual Harvests with 200 Han-hours per Year H a r v e s t i n g took p l a c e o n l y i n r e f u g e a r e a s , with the percentage o f e f f o r t on the a b s c i s s a devoted to SP c o l o n i e s and the remainder of the e f f o r t devoted to LP c o l o n i e s . 61 3. Adaptive P o l i c i e s with P r e c i s e I n f o r m a t i o n The remaining s i x p o l i c i e s are a l l a d a p t i v e , with some p r o v i s i o n f o r f i x e d escapement. P a r t i c u l a r l y i n f i s h e r i e s ( L a r k i n and S i c k e r , 1964; Tautz et a l . , 1969) such p o l i c i e s a re expected to provide maximal harvests over a long p e r i o d . Indeed, Jaguette (1972, 1974) and o t h e r s have shown t h a t f i x e d escapement can be o p t i m a l with a s t o c h a s t i c a l l y v a r y i n g system under the a p p r o p r i a t e c o n d i t i o n s . N e v e r t h e l e s s , p o l i c i e s based on f i x e d escapement may a l s o produce high v a r i a n c e i n y e a r l y h a r v e s t s ( A l l e n , 1973; B a i t e r s , 1975)., In adapting such p o l i c i e s f o r our purposes we f i r s t assumed that the a c t u a l numbers of h a r v e s t a b l e c o l o n i e s were known ( P o l i c i e s 4-6). In c o n t r a s t , we assumed t h a t the managers of P o l i c i e s 7-9 never had p r e c i s e i n f o r m a t i o n on the numbers a v a i l a b l e . Dnder a l l s i x p o l i c i e s , i n i t i a l h a r v e s t t a r g e t s were determined from the a c t u a l or estimated p o p u l a t i o n s i z e and the d e s i r e d maintenance l e v e l (escapement) f o r the s t o c k . / F o r d e s i r e d harvest, H*, and p o p u l a t i o n , K, the p r o b a b i l i t y t h a t a c o l o n y w i l l be harvested i s p = H*/K. C o l o n i e s were harvested s t o c h a s t i c a l l y , each with p r o b a b i l i t y p. 62 P o l i c y 4a: F i x e d Escapement, U n l i m i t e d Area and E f f o r t With t h i s v a r i a n t , d e s i r e d escapement l e v e l s from 100-300 c o l o n i e s per year proved t o be a c c e p t a b l e ( F i g . 5, Curve A). Harvests were maximal with escapements of 200-225 per year, and the ha r v e s t per u n i t e f f o r t i n c r e a s e d with i n c r e a s i n g escapement, s u g g e s t i n g t h a t higher escapement l e v e l s might give b e t t e r economic r e t u r n s i f l a b o r c o s t s {effort} were high. On the other hand, v a r i a n c e i n y e a r l y h a r v e s t a l s o i n c r e a s e d with i n c r e a s i n g escapement, so i f reducing the v a r i a b i l i t y i n y e a r l y harvest has the highest p r i o r i t y , somewhat lower escapements might be p r e f e r a b l e . P o l i c y 4b: F i x e d Escapement with C o n s t r a i n e d E f f o r t Over U n l i m i t e d Area T h i s p o l i c y combined two management s t y l e s . Harvesting e f f o r t had an upper l i m i t t o maintain a d e s i r e d breeding s t o c k . F i g . 5, Curves B and C, show two f i x e d l e v e l s of e f f o r t , 500 and 200 mhpy. As i n P o l i c y 4a, the best h a r v e s t s were obtained near escapements of 225 c o l o n i e s per year. The maximal a l l o w a b l e l e v e l s of e f f o r t s e l e c t e d f o r P o l i c y 4b d i d not reduce peak h a r v e s t s a p p r e c i a b l y below val u e s obtained with the u n l i m i t e d e f f o r t of P o l i c y 4a ( F i g . 5, Curve A). In f a c t , the more severe c o n s t r a i n t o f 200 mhpy o f t e n provided l a r g e r h a r v e s t s as w e l l as a g r e a t e r chance of stock maintenance a t the lowest escapement l e v e l (the s o l i d vs. Broken l i n e s i n F i g . 5). LU > cr < x A — 0 te-0 100 200 300 ESCAPEMENT (COLONIES/YEAR) Figure 5. Mean Annual Harvests with Fixed Escapement (Precise Information) Policy A (black c i r c l e s ) : Unbounded e f f o r t Policy B (t r i a n g l e s ) : E f f o r t r e s t r i c t e d to 500 mhpy Policy C (squares): E f f o r t r e s t r i c t e d to 200 mhpy Broken l i n e s indicate p o l i c i e s leading to ex t i n c t i o n . 64 Policy 5: Fixed Escapement, Restricted Harvest area with t h i s p o l i c y , as with Policy 2, harvesting was r e s t r i c t e d to the refuges. Desired escapements were calculated only for those areas, so they are not s t r i c t l y comparable with those of Policy 4. Escapements of 100-300 colonies were acceptable under Policy 5, with those between 125-20 0 colonies giving the best harvests, approximately 90 colonies per year. That harvest, however, was s t i l l some 15% lower than the maximal y i e l d under Policy 4, with i t s lack of areal r e s t r i c t i o n . As with Policy 4, constraints on e f f o r t were most valuable when escapement l e v e l s were set very low. Policy 6: Fixed Escapement with Protected Classes i n Restricted Areas When there are recognizable differences i n the reproductive a b i l i t y of survivors from dif f e r e n t types of colonies (Wellington, 1964, 1965b) there are incentives to develop p o l i c i e s aimed at preserving those colonies with the highest reproductive value, by harvesting only those with low reproductive a b i l i t y . In addition to imposing t h i s constraint, Policy 6, l i k e the l a s t policy, r e s t r i c t e d harvesting to high-yield refuges. In those areas, a l l type E colonies were conserved, LP colonies had fixed escapements, and a l l SP colonies were harvested. Escapement l e v e l s of 1-100 LP colonies per year were 65 examined. The policy proved superior to Policy 5, with a peak return of more than 110 colonies f o r an escapement of 35 LP colonies; i . e . , harvest per unit e f f o r t was higher than f o r Policy 5. In addition, numbers harvested varied only s l i g h t l y over a wide range of escapements (e.g., from 110-95 colonies for escapements from 10 to approximately 150 colonies) in d i c a t i n g that the policy would be r e l a t i v e l y i n s e n s i t i v e to errors i n estimates of population s i z e . , 66 4. Adaptive P o l i c i e s with Imprecise Information In real s i t u a t i o n s , the pre- and post-harvest census data by which the escapement l e v e l s of adaptive p o l i c i e s are p e r i o d i c a l l y adjusted contain a number of e r r o r s . The following three adaptive p o l i c i e s therefore were subjected to more stringent tests than the former p o l i c i e s by adjusting t h e i r escapement rates i n r e l a t i o n to unsophisticated projections of population s i z e derived from pooled data. Projections included only the weather immediately following the preceding harvest, and were based on the replacement curve shown in F i g . 2 of Thompson et a.1*, 1976. Policy 7: Fixed Escapement with Imprecise Projections of Population Size The shapes of the y i e l d curves from several runs under Policy 7 were s i m i l a r to those shown i n Fig. 5 for Policy 4. The inaccurate population estimates of Policy 7 often led to over-exploitation, however, when the l e v e l of e f f o r t was set too high on the basis of an overly optimistic prediction. Then the high harvest decimated the population, bringing i t hear extinction and ushering i n a period of n i l harvests. Such disasters could only be avoided by i n s t i t u t i n g a sub-policy of fixed e f f o r t s that r e s t r i c t e d harvesting i n t e n s i t y to less than 500 mhpy. That change in policy had several e f f e c t s ; (a) harvests were generally as large as those obtained with unconstrained 67 e f f o r t s ; (b) usually no more than half the population was taken, even i n bad years, and (c) the r e s t r i c t e d policy functioned as well with imperfect as with perfect information. (A v i r t u a l l y i d e n t i c a l policy (Walters, 1975} has been recommended f o r harvesting sockeye salmon i n the Skeena River). Policy 8: Fixed Escapement with Imprecise Population Projections i n Restricted Areas P o l i c i e s of t h i s type gave the harvests i l l u s t r a t e d i n Fig. 6. They were s i m i l a r i n most respects to y i e l d s obtained with Policy 5. Fa i l u r e to l i m i t e f f o r t , however, generally threatened the population with extinction, just as with Policy 7. Because of the danger of inadvertently harvesting too many E colonies when t o t a l populations i n the refuge areas were very low, i t was necessary t o set e f f o r t s at l e s s than 200 mhpy. Policy 9: Fixed Escapement with Imprecise Population Projections f o r Protected Classes i n Restricted Areas Osing inaccurate population estimates to set desired escapement l e v e l s separately for dif f e r e n t types of colonies i n d i f f e r e n t areas almost invariably produced unacceptable r e s u l t s . When forecasts were pessimistic, e f f o r t l e v e l s were overly r e s t r i c t e d and the harvests were poor — usually no better than those from the fixed e f f o r t s of Policy 1. Overly 68 0&: 1 . 1 1 0 100 200 300 D E S I R E D E S C A P E M E N T ( C O L O N I E S / Y E A R ) Figure 6. Mean Annual Harvests with Fixed Escapement (Imprecise Information) Policy A (black c i r c l e s ) : Unbounded e f f o r t Policy B (black c i r c l e s ) : E f f o r t r e s t r i c t e d to 500 mhpy Policy C (squares): E f f o r t r e s t r i c t e d to 200 mhpy Curves A and B were i d e n t i c a l . As indicated by the broken l i n e s , the p o l i c i e s always l e d to ext i n c t i o n . 69 o p t i m i s t i c f o r e c a s t s p e r m i t t e d u n d e s i r a b l y high l e v e l s of e f f o r t which g u i c k l y e l i m i n a t e d the p o p u l a t i o n . Conseguently, even though P o l i c y 9 u t i l i z e d i n f o r m a t i o n on the whereabouts of d i f f e r e n t kinds o f c o l o n i e s i n an attempt t o p r o t e c t s e l e c t e d c l a s s e s at p a r t i c u l a r times, the f l u c t u a t i n g l e v e l s o f e f f o r t generated by i n a c c u r a t e f o r e c a s t s made t h i s p o l i c y f a r l e s s e f f e c t i v e than, say. P o l i c y 3, which c a r e f u l l y p a r t i t i o n e d a f i x e d e f f o r t among d i f f e r e n t types o f c o l o n i e s i n d i f f e r e n t p l a c e s . A modified v e r s i o n of the l a t t e r p o l i c y t h e r e f o r e was t e s t e d f o r a lon g e r p e r i o d t o determine the e f f e c t s of more v a r i e d d i v i s i o n s o f e f f o r t than previous t e s t s had allowed. A Twenty-year S i m u l a t i o n : F i x e d E f f o r t P a r t i t i o n e d Among A l l Types of C o l o n i e s and Areas Except i n P o l i c y 9, type E c o l o n i e s had not been s e l e c t i v e l y harvested when h a r v e s t i n g e f f o r t was d i v i d e d among d i f f e r e n t types o f c o l o n i e s d u r i n g the e i g h t - y e a r t e s t s . None of those t e s t s was s u f f i c i e n t l y long t o provide an adeguate assessment of the e f f e c t s of very complicated d i v i s i o n s of e f f o r t among a l l the kinds o f c o l o n i e s i n a l l p l a c e s . Some p r e l i m i n a r y t r i a l s suggested t h a t 20 years would be adeguate f o r the s i m u l a t i o n experiments r e g u i r e d t o examine such complex d i v i s i o n s , so the f o l l o w i n g t e s t s were run f o r that p e r i o d . Four l e v e l s o f e f f o r t ranging from 50-500 mhpy were 70 d i v i d e d nine ways among E, LP and SP c o l o n i e s w i t h i n and o u t s i d e the r e f u g e areas. Each t r i a l began with a t o t a l of 199 c o l o n i e s , 35 o f which were type E c o l o n i e s s c a t t e r e d through th e r e f u g e s . The s t a r t i n g group of c o l o n i e s was i n i t i a l l y d i s t r i b u t e d by s i m u l a t i n g a p a t t e r n of s u r v i v a l -and d i s p e r s a l t y p i c a l of a year with good s p r i n g weather (Wellington e t a l . , 1975).. The simulated weather was generated as a s t a t i s t i c a l l y average sample f o r the p e r i o d . Each t r i a l was r e p l i c a t e d t h r e e times. Two extreme and two i n t e r m e d i a t e examples are shown i n F i g . 7. The r e s u l t s of a very r e s t r i c t e d e f f o r t (50 mhpy) devoted e n t i r e l y t o the h a r v e s t i n g of E c o l o n i e s w i t h i n the refuges are shown i n F i g . 7, Curve A. Other types of c o l o n i e s i n the refuges were not h a r v e s t e d , nor was any type of colony c o l l e c t e d o u t s i d e . T h i s very l i m i t e d e f f o r t s c a r c e l y a f f e c t e d the r e p r o d u c t i v e c o n t r i b u t i o n of the E c o l o n i e s t o the p o p u l a t i o n , with the r e s u l t t h a t the t o t a l p o p u l a t i o n i n c r e a s e d from 199 t o 1699 c o l o n i e s over 20 years. F i g . 7, Curve B shows the r e s u l t s when 500 mhpy were expended on a l l types of c o l o n i e s o u t s i d e the r e f u g e s , and on t h e LP and SP c o l o n i e s w i t h i n them. E c o l o n i e s i n s i d e the r e f u g e s were untouched under t h i s p o l i c y . Conserving t h a t b e s t s t o c k i n the best l o c a l c l i m a t e s allowed the p o p u l a t i o n t o maintain i t s e l f throughout the p e r i o d , d e s p i t e the manor e f f o r t devoted t o i n d i s c r i m i n a t e h a r v e s t i n g o u t s i d e , and t o s e l e c t i v e h a r v e s t i n g w i t h i n the refuges. The annual r e s i d u a l 71 1 5 10 . 1 5 20 G E N E R A T I O N S F i g u r e 7. Changes per Generation i n T o t a l P o p u l a t i o n s (20-year Runs) P o l i c y A (black c i r c l e s ) : E f f o r t s o f 50 mhpy were used, devoted e x c l u s i v e l y t o E c o l o n i e s i n re f u g e s P o l i c y B ( s q u a r e s ) : 500 mhpy were devoted t o a l l types of c o l o n i e s o u t s i d e the r e f u g e s , but o n l y SP and LP c o l o n i e s i n s i d e the r e f u g e s P o l i c y C ( t r i a n g l e s ) : 50 mhpy were devoted t o r e f u g e - a r e a E c o l o n i e s , and 150 mhpy were devoted to a l l types of c o l o n i e s o u t s i d e the r e f u g e s and SP and LP c o l o n i e s i n s i d e P o l i c y D (open c i r c l e s ) : E f f o r t s of 500 mhpy were used, devoted e x c l u s i v e l y to E c o l o n i e s i n r e f u g e s In t h i s f i g u r e , broken l i n e s do not i n d i c a t e e x t i n c t i o n . 72 population in any r e p l i c a t e rarely exceeded 200 colonies under t h i s p o l i c y ; e.g., there were 121 colonies in the twentieth generation. Fig. 7, Curve C shows the r e s u l t of an e f f o r t of only 200 mhpy, with 50 man-hours devoted exclusively to c o l l e c t i n g E colonies i n the refuge areas. The remaining 150 mhpy were divided between harvesting LP and SP colonies i n the refuges, and every kind of colony outside. Despite the comparatively modest e f f o r t under t h i s p o l i c y , r e p l i c a t e populations approached extinction in 13-20 years. In the example shown, there were only 38 colonies remaining i n the twentieth generation. Fig. 7, Curve D i l l u s t r a t e s the rapid collapse of a whole population i n which only the refuge-area E colonies were harvested with an i n t e n s i t y of 500 mhpy. Replicates under t h i s p o l i c y collapsed after approximately 6 generations, even though none of the colonies outside the refuges was harvested. 73 5. Summary The simulation model was used to perform h e u r i s t i c harvesting procedures u t i l i z i n g d i f f e r e n t amounts of information on the locations and reproductive a b i l i t i e s of the colonies. The f i r s t set of procedures were f i x e d - e f f o r t p o l i c i e s . Best among the three tested was the one which partitioned the e f f o r t between small and large pyramidal colonies i n refuge (excellent l o c a l climate) areas, with most of the e f f o r t devoted to small pyramidal colonies. The next set of procedures were fixed-escapement p o l i c i e s . As might be expected, these p o l i c i e s were se n s i t i v e to overestimation of the population. The p o l i c i e s worked well with accurate population counts, r e s t r i c t e d e f f o r t to protect against over-exploitation, and protection of elongate colonies i n refuge areas. The above procedures were a l l tested on eight-year runs. Twenty-year tests were performed i n order to examine the eff e c t s of fixed e f f o r t partitioned among a l l types of colonies and areas. In agreement with some of the re s u l t s obtained from the eight-year runs, the twenty-year runs showed that even populations subjected to guite intense harvesting were not affected adversely as long as only expendable stock was taken and the reproductively superior colonies i n refuge areas were protected. On the other hand, intense harvesting of the superior stock i n refuge areas drove the entire population to extinction even when the rest of the population was t o t a l l y i g n o r e d . Chapter V. C o n c l u s i o n s 76 .& r e a l i s t i c simulation model of the western tent c a t e r p i l l a r population system provided us with an opportunity to compare d i f f e r e n t approaches to management of a b i o l o g i c a l resource system. Each of the two approaches we chose {linear programming and heuristics) had both advantages and drawbacks. In most cases of management of b i o l o g i c a l resources, i t i s a l l but impossible to experiment with the r e a l system. Not only would the experiments be very time-consuming, but also undesirable because of the danger of producing i r r e v e r s i b l e changes i n the population system. A manager who desires to experiment with d i f f e r e n t h e u r i s t i c p o l i c i e s i n order to discover which of several p o l i c i e s i s best therefore requires an experimental laboratory. For resource managers, the laboratory would very l i k e l y be a simulation model, but a note of caution i s i n order: Because simulation models are deceptively easy to conceptualize and because fewer abstractions need be made i n constructing simulation models, simulation may be chosen over other methods of analysis. However, the costs of designing, programming, accumulating data for, and running a simulation model are often much higher than anticipated, simulation would not, i n many cases, be selected as the technigue for analysis i f these costs were estimated c o r r e c t l y . {Meier, Mewell, and 77 Pazer, 1969). Meier, Newell, and Pazer go dn to p o i n t out another of the d i f f i c u l t i e s with the h e u r i s t i c approach: In s i m u l a t i o n models th e r e are no f u n c t i o n a l r e l a t i o n s h i p s t h a t can be manipulated by t e c h n i q u e s such as l i n e a r programming or c a l c u l u s to o b t a i n optimum values of the d e c i s i o n v a r i a b l e s . T h e r e f o r e , o p t i m i z a t i o n through experimentation with a s i m u l a t i o n model n e c e s s i t a t e s employment of some s o r t of search technique. In many cases t h e r e i s l i t t l e d i f f i c u l t y i n p l a n n i n g the search f o r an optimum, s i n c e t h e r e are only a few a l t e r n a t i v e s t o be i n v e s t i g a t e d . But i n many o t h e r i n s t a n c e s , there may be s e v e r a l d e c i s i o n v a r i a b l e s and many p o s s i b l e values of each d e c i s i o n v a r i a b l e . T h i s was the case i n our problem and undoubtedly a l s o i n other problems of resource management. There are a number of a t t r i b u t e s which might be c o n t r o l l e d , such as time of h a r v e s t i n g , l o c a t i o n of h a r v e s t i n g , amount o f e f f o r t t o be devoted t o h a r v e s t i n g , e t c . There are s e v e r a l c h o i c e s f o r each of these a t t r i b u t e s . There i s one great advantage with the h e u r i s t i c approach. 78 I f t h e s i m u l a t i o n model i s a v a l i d one ( i . e . , i t a c c u r a t e l y r e p r e s e n t s the system under s t u d y ) , i t g i v e s a much more r e a l i s t i c p i c t u r e of the e f f e c t s o f d i f f e r e n t d e c i s i o n s . As an example, the s i m u l a t i o n model of 'the western t e n t c a t e r p i l l a r p o p u l a t i o n takes i n t o account the d a i l y f l u c t u a t i o n s i n weather, as w e l l as a d d i t i o n a l e f f e c t s o f l o c a t i o n on the p e n i n s u l a , and even p o s i t i o n on a t r e e (e.g., high or low on the t r e e ) . The l i n e a r programming model was r e l a t i v e l y easy to c o n s t r u c t , i n terms of the amount of data r e q u i r e d . Each l i n e a r programming problem r e q u i r e d at most a few minutes of computer time, and the computer output provided a d d i t i o n a l i n f o r m a t i o n about the nature of the s o l u t i o n s . The l i n e a r programminq model was mathematically t r a c t a b l e , which i s not an advantage t o be taken l i g h t l y , but the model had a number of shortcomings. One of the s i m p l i f i c a t i o n s was to compress the 80 days of weather i n d i c e s per year i n t o one i n d i c a t o r of the weather f o r each year. We then attempted t o estimate the change i n p o p u l a t i o n as a f u n c t i o n of t h a t i n d i c a t o r . I f the e s t i m a t i o n procedure c o u l d be improved, i t i s p o s s i b l e t h a t t h e l i n e a r programming s o l u t i o n s might y i e l d more r e a l i s t i c answers. One way we might improve the e s t i m a t i o n i s t o break t h e year up i n t o s t a g e s , and e s t i m a t e from stage t o stage. T h i s would h o p e f u l l y improve the e s t i m a t i o n because i t would allow us t o d i s t i n g u i s h , f o r example, between good weather e a r l y i n the year and good 7 9 weather l a t e r i n the year. However, t h i s refinement would g r e a t l y i n c r e a s e the s i z e of the problem. In e s t i m a t i n g the t r a n s i t i o n s from year t o year, we assumed t h a t the nine separate c a t e g o r i e s were independent. Whether or not t h i s was a v a l i d assumption i s not c l e a r , bat the g u e s t i o n i s worthy of f u r t h e r c o n s i d e r a t i o n . One p o s s i b i l i t y o f i n t e r e s t i s to choose a d i f f e r e n t mathematical model t o r e p r e s e n t the system. An obvious c h o i c e i s a s t o c h a s t i c model, which would i n c o r p o r a t e what the l i n e a r programming model, being d e t e r m i n i s t i c , l a c k s . , However, to maintain s o l v a b i l i t y i t might be necessary to i g n o r e the q u a l i t a t i v e d i f f e r e n c e s i n the p o p u l a t i o n , which both the l i n e a r programming and h e u r i s t i c programming i n d i c a t e d were important. A s t o c h a s t i c model might a l s o encounter some of the same d i f f i c u l t i e s t h a t the l i n e a r programming model faced, such as how to d e a l with the f l u c t u a t i n g d a i l y weather as one of the parameters a f f e c t i n g growth and s u r v i v a l of the c a t e r p i l l a r s . S t o c h a s t i c dynamic programming i s p a r t i c u l a r l y a t t r a c t i v e t h e o r e t i c a l l y , and has been a p p l i e d t o problems with r e l a t i v e l y few s t a t e v a r i a b l e s {Anderson, 1 9 7 5 ; Walters, 1 9 7 5 ; Walters and H i l b o r n , 1 9 7 6 ) . T h i s approach seems t o be c o m p u t a t i o n a l l y i n f e a s i b l e f o r problems with many v a r i a b l e s per stage. Anderson r e q u i r e d 9 0 minutes of computer time to s o l v e a t w o - v a r i a b l e problem; Walters and H i l b o r n r e g u i r e d 5 hours of computer time to s o l v e a t h r e e - v a r i a b l e problem. : 8 0 I t i s g e n e r a l l y d i f f i c u l t t o decide which approach i s the most a p p r o p r i a t e f o r any g i v e n problem. I n our c a s e , the l i n e a r programming model i n d i c a t e d t h a t s m a l l pyramidal c o l o n i e s i n refuge areas should be harvested h e a v i l y , with r e l a t i v e l y c o n s t a n t e f f o r t . The h e u r i s t i c d e r i v e d from that p o l i c y was t o use a f i x e d amount of e f f o r t each year, with the amount bounded to a r e a s o n a b l e l e v e l . Of the two approaches we c o n s i d e r e d , e i t h e r by i t s e l f might have proved i n s u f f i c i e n t , but i n c o n j u n c t i o n with each other, l e d t o the development of s a t i s f a c t o r y h a r v e s t i n g p o l i c i e s . The p o l i c y determined by the mathematical model i n d i c a t e d simple r u l e s to keep i n mind while attempting to optimize through s i m u l a t i o n . The r e s u l t s i n d i c a t e t h a t i g n o r i n g the b a s i c b i o l o g i c a l a t t r i b u t e s of the resource c o u l d l e a d to mismanagement, and p o s s i b l y even d e s t r u c t i o n of the p o p u l a t i o n . 81 B i b l i o g r a p h y A l l e n , K.B. (1973) The i n f l u e n c e o f random f l u c t u a t i o n s i n the s t o c k - r e c r u i t m e n t r e l a t i o n on the economic r e t u r n from salmon f i s h e r i e s . Cons. .Int.- E x p l o r . 8er fiapp. 164:: Anderson, D.R. (1975) Optimal e x p l o i t a t i o n s t r a t e g i e s f o r an animal p o p u l a t i o n i n a Markovian environment: a theory and an example. Ecology 56: 1281-1297. H o l l i n g , C.S. (1959) Some c h a r a c t e r i s t i c s o f simple types of p r e d a t i o n . Can. Ent. 96: 385-398., Iwao, S., And H.G. Wellington {1970) The western t e n t c a t e r p i l l a r : q u a l i t a t i v e d i f f e r e n c e s and the a c t i o n of n a t u r a l enemies. Res. Popul. E c o l . 12: 81-99. J a c q u e t t e , D.L. (1972) A d i s c r e t e time p o p u l a t i o n c o n t r o l model. Hath. Bio s c i . 15: 231-252. Jacq u e t t e , D.L. (1974) A d i s c r e t e - t i m e p o p u l a t i o n - c o n t r o l model with setup c o s t . Op. Res. 22: 298-303. L a r k i n , P.A. And W.E. Ricker (1964) F u r t h e r i n f o r m a t i o n on s u s t a i n e d y i e l d s from f l u c t u a t i n g environments. J . F i s h . Res. Bd. Can. 21: 1-7. ~* Meier, R.C., W.T. Newell, and H.L. Pazer (1969) S i m u l a t i o n i n Business and Economics. New J e r s e y : P r e n t i c e - H a l l , Inc., 1969, pp. 23 and 314. Simon, H.A., and A.. Newell (1962) S i m u l a t i o n of human t h i n k i n g , i n Management - and the Computer of the Future. Martin Greenberger, ed., N. Y.: John Wiley S Sons, I n c . , p. 113. Tautz , A., P.A. L a r k i n , and W.E. Ri c k e r (1969) Some e f f e c t s of simulated long-term environmental f l u c t u a t i o n s on maximum s u s t a i n e d y i e l d . J . F i s h . Res. Bd. Can. 26: 351-359. 2715-2726. 82 Thompson, W.A., P.J. Cameron, W.G. W e l l i n g t o n , and I.B. V e r t i n s k y (1976) Degrees of h e t e r o g e n e i t y and the s u r v i v a l of an i n s e c t p o p u l a t i o n . Hes. Pogul. E c o l . , 18 {in p r e s s ) . Walters, C.J. (1975) Optimal h a r v e s t s t r a t e g i e s f o r salmon i n r e l a t i o n t o environmental v a r i a b i l i t y and u n c e r t a i n p r o d u c t i o n parameters. J . F i s h . Res. , Bd« . Can.. 32: 1777-1784. Walters, C.J. , And B. H i l b o r n (1976) Adaptive c o n t r o l of f i s h i n g systems. J . F i s h . Res. Bd. Can. 33: 145-159. We l l i n g t o n , W.G. (1957) I n d i v i d u a l d i f f e r e n c e s as a f a c t o r i n p o p u l a t i o n dynamics: the development of a problem.; Can. J . Z o o l . 35: 293-323. We l l i n g t o n , S.G. (1960) Q u a l i t a t i v e changes i n n a t u r a l p o p u l a t i o n s d u r i n g changes i n abundance. Can* J . Z o o l . 38: 289-314. We l l i n g t o n , W.G. (1964) Q u a l i t a t i v e changes i n p o p u l a t i o n s i n u n s t a b l e environments. Can. Ent. 96: 436-451., We l l i n g t o n , W.G. (1965a) The use of cloud p a t t e r n s to o u t l i n e areas with d i f f e r e n t c l i m a t e s during p o p u l a t i o n s t u d i e s . Can. Ent. 97: 617-631. W e l l i n g t o n , W.G. (1965b) Some maternal i n f l u e n c e s on progeny q u a l i t y i n the western t e n t c a t e r p i l l a r , Malacosoma p l u v i a l e (Dyar). Can. Ent. 97: 1-14. W e l l i n g t o n , W.G. (1974) Tents and t a c t i c s of c a t e r p i l l a r s . Nat. H i s t . 83: 64-72. We l l i n g t o n , W.G., P.J. Cameron, W.A. Thompson, I.B. , V e r t i n s k y , and A.S. Landsberg (1975) A s t o c h a s t i c model f o r a s s e s s i n g the e f f e c t s of e x t e r n a l and i n t e r n a l h e t e r o g e n e i t y on an i n s e c t p o p u l a t i o n . Res. Popul. E c o l . 17: 1-28. Appendix. L i n e a r Programming and J3PSX 84 L i n e a r programming i s one of many mathematical models which can be used to represent systems i n which a l t e r n a t i v e c o u r s es o f a c t i o n must be analyzed. The l i n e a r programming model i s a d e t e r m i n i s t i c model, i . e . there i s no u n c e r t a i n t y i n the system. P r e r e q u i s i t e s f o r a c t i o n s and the r e s u l t s of a l l a c t i o n s a re known. Few systems i n the r e a l world are d e t e r m i n i s t i c , but t h i s type of model can be used i f i t i s a reasonable approximation o f the r e a l - w o r l d system. The go a l i s s i m p l i f i c a t i o n while maintaining a model which adequately d e s c r i b e s the r e a l - l i f e s i t u a t i o n . Perhaps the one overwhelming advantage i n using l i n e a r programming i s the e x i s t e n c e of a h i g h l y e f f i c i e n t method (the simplex algorithm) t o s o l v e the problems. Many of the computers commercially a v a i l a b l e have codes embodying the simplex technique. In p a r t i c u l a r , IBM has produced a powerful program c a l l e d MPSX (Mathematical Programming System—Extended) which i s composed of a s e t of procedures, some of which d e a l with the s o l u t i o n o f l i n e a r programming problems. A s e r i e s of these procedures comprises a user*s s t r a t e g y f o r s o l v i n g a l i n e a r programming problem. MPSX can handle extremely l a r g e problems and has many u s e f u l f e a t u r e s . These i n c l u d e the c a p a b i l i t y t o do f u l l p o s t - o p t i m a l i t y a n a l y s i s and parametric programminq. MPSX r e q u i r e s i n p u t t o be i n very r i g i d format. B r i e f l y , the i n p u t i s d i v i d e d i n t o s e c t i o n s as f o l l o w s : (1) Name f o r the data s e t 85 (2) Bows Se c t i o n Each row, i n c l u d i n g the c o s t row, must be given a name. The row names must be l i s t e d , and UPSX must be t o l d how each i s c o n s t r a i n e d . (3) Columns S e c t i o n The c o e f f i c i e n t s of the c o n s t r a i n t matrix must be giv e n column by column. A column name must be s p e c i f i e d , then each non-zero element i n that: column must be given by s p e c i f y i n g which row i t i s a s s o c i a t e d with. iU) B i g h t Hand Side (BHS) S e c t i o n The elements are given as i n the Columns S e c t i o n . A name i s s p e c i f i e d f o r a RHS v e c t o r , then each non-zero element i s given by a s s o c i a t i n g i t with the row i n which i t appears. (5) Bounds S e c t i o n The type of bound f o r each bounded v a r i a b l e ( i f any) must be specified.... C a l c u l a t i n g c o e f f i c i e n t m a t r i c e s and t y p i n g i n p u t f o r each new problem would not only be t e d i o u s , but a l s o very r i s k y , s i n c e the p r o b a b i l i t y o f making an e r r o r i s very l a r g e , while the p r o b a b i l i t y of d e t e c t i n g t h a t e r r o r q u i c k l y i f p e c u l i a r r e s u l t s a r i s e i s very s m a l l . I t i s t h e r e f o r e w e l l worthwhile t o take the time and e f f o r t t o c a r e f u l l y w r i t e a computer program which w i l l take data, perform a l l the necessary c a l c u l a t i o n s , and write the in p u t f o r HPSX i n p r e c i s e l y the format MPSX r e q u i r e s . T h i s was done ( i n FORTRAN) f o r t h e comparatively simple 86 f i r s t problem i n which the o b j e c t i v e f u n c t i o n c o n s i s t s of a d iscounted r e t u r n , with l i n e a r c o s t s of h a r v e s t i n g . The c o s t row i s nonconstrained, and the remaining c o n s t r a i n t s are < c o n s t r a i n t s . The c o n s t r a i n t s on the nine f i r s t - y e a r v a r i a b l e s H * , . . . a r e merely bounds on the v a r i a b l e s . Because HPSX handles bounded v a r i a b l e s more e f f i c i e n t l y than i t does an o r d i n a r y c o n s t r a i n t which r e p r e s e n t s a bound on a v a r i a b l e , the f i r s t nine i n e q u a l i t i e s are i n c l u d e d i n the Bounds S e c t i o n of the i n p u t t o HPSX. Another program was w r i t t e n so t h a t the p i e c e w i s e - l i n e a r (as a f u n c t i o n of p o p u l a t i o n s i z e ) c o s t f u n c t i o n s c o u l d be i n c o r p o r a t e d . , In t h i s case, the c o s t row i s n o n constrained, the next group of c o n s t r a i n t s c o n s i s t s of < c o n s t r a i n t s , and the l a s t group c o n s i s t s of e q u a l i t y c o n s t r a i n t s . Both the number of v a r i a b l e s and the number of c o n s t r a i n t s i n c r e a s e as J , the number o f i n t e r v a l s chosen f o r the c o s t f u n c t i o n s , i n c r e a s e s . T h i s problem r e q u i r e s a l l the p r e v i o u s v a r i a b l e s and c o n s t r a i n t s , p l u s new v a r i a b l e s and c o n s t r a i n t s which a r i s e from s p l i t t i n g each of the p r e v i o u s v a r i a b l e s i n t o J new v a r i a b l e s and each of the p r e v i o u s c o n s t r a i n t s i n t o J new c o n s t r a i n t s * Further new c o n s t r a i n t s a r i s e because each p r e v i o u s v a r i a b l e must be assigned the sum o f the J new v a r i a b l e s c o r r e s p o n d i n g to i t . Both problems were presented t o MPSX as mini m i z a t i o n problems. S i n c e the c o e f f i c i e n t matrices are n e a r l y square, the problems were s o l v e d by the p r i m a l method. An e i g h t - y e a r 87 problem with eight i n t e r v a l s resulted i n a l i n e a r programming problem with 640 rows, including the cost row, and 648 columns. Density af t e r the addition of slack variables was less than 2%. The t o t a l solution time, including generation of the input f o r MPSX, ranged from 110 to 125 seconds on an IBM 370/168. LISTING OF FILE M 1 2 88 3 FILE MT62:MPSXIN 4 5 C PBOGRAM TO CBE&TE MPSX INPUT DATA FILE FOB HARVESTING 6 C CATERPILLAR COLONIES. 7 C BEAD FBOH UNIT 5, WRITE MPSX DATA INPUT ON UNIT 1, 8 C OTHER OUTPUT ON UNIT 6. 9 C 10 C SAMPLE INPUT FOB MPSXIN IS IN MT62:EXAMPLE 11 C SAMPLE INPUT FOB MPSX GENEBATED IS IN MT62:SAMPIN. 12 EXTERNAL CONVBT 13 BEAD (5,2)DUMMY 14 2 FORMAT (A4) 15 CALL FREAD(5,»I: 1,N¥EABS) 16 CALL GSPACE(S,9*9*NYBABS*4) 17 N¥1=NYEARS-1 18 CALL GSPACE(T,9*9*NY1*4) 19 CALL GSPACE{COST,9*NYEARS*4) 20 CALL CALLEB(CGNVBT,S,T,COST,IPTS (N YEARS) ,IPTR{NY1}) 21 CALL SYSTEM 22 END 23 C 24 C 25 C 26 SUBROUTINE CONVRT(S,T,COST,NYEARS,NY1) 27 DIMENSION S{9,9,NYEABS) ,T(9,9,NY 1) ,COST(9,NYEARS) 28 DIMENSION POPMIN (9) ,SX{9) ,B{9) ,RCHA8 (4) ,CCHAR{4) 29 DIMENSION WGT{9) ,NAME(8), XIN{9) 30 LOGICAL*1 BCHAR,CCHAB,NAME 31 1 F01MAT{4A1) 32 BEAD (5,1) DUMMY 33 CALL FREAD(5, *B V:»,XIN,9) 34 BEAD (5, 1) DUMMY 35 CALL FBEAD{5,'R V: », POPMIN, 9) 36 READ (5,1) DUMMY 37 CALL FfiEAD(5, 'B V:*,HGT,9) 38 READ (5,1) DUMMY 39 CALL FBEAD{5,*R: *,DELTA) 40 BEAD{5,1}DUMMY 41 CAJLL FREAD (5, » STRING:', NAME, 8) 42 C 43 C READ IN TRANSPOSE OF S MATRICES BY COLUMNS, I.E. BEAD IN 44 C SOWS OF S MATRICES. 45 DO 5 J=1,NYEARS 46 BEAD{5,1)DUMMY 47 DO 5 1=1,9 48 5 CALL FREAD{5,* B V: ' , S { 1,1, J) , 9) 49 C 50 C WRITE OUT INPUT 51 WRITE (6, 3) NAME 52 3 FORMAT {*0NAME:',8A1) 53 WRITE{6,10)XIN 54 10 FORMAT{* OXIN ',/,3X,9F10.1) 55 WBITE (6 , 20) POPMIN 56 20 FOBMAT{*0POPMIN«,/,3X,9F10.1) 57 58 LISTING OF FILE M 59 60 89 61 WRITE (6 , 30) WGT 62 30 FORNAT(»0WGT',/,3X,9F10.3} 63 WRITE {6 , 40) DELTA 64 40 FORMAT{*0DELTA=',F7.4) 65 DO 11 K=1,NYEARS 66 WRITE(6,50)K 67 50 FORMAT(» OS* ,13) 68 DO 12 J= 1,9 69 12 WRITE (6, 60) (S ( I , J , K) ,1= 1, 9) 70 60 FORMAT (' 0' ,9F10. 7) 71 11 CONTINUE 72 C 73 C 74 C BEGIH TO WRITE DATA FILE, ASSIGNED TO ONIT 1.. 75 C BLANK LINE MOST BE INGLODED. 76 WRITE (1,100) 77 100 FORMAT {* ') 78 WRITE(1, 110) NAME 79 110 FORMAT (» NAME*, T0X,8A1) 80 C 81 C DO ROWS SECTION. 82 WRITE (1,120) 83 120 FORMAT (* ROWS1) 84 WRITE (1,130) 85 130 FORMAT{* N*,2X,»COST *) 86 LIM=9*(NYEARS-1) 87 DO 15 1=1,LIM 88 CALL BTD ( I , RCHAR {1) , 4, USD, * 0* ) 89 15 WRITE (1,140) RCHAR 90 140 FORMAT{" L*,2X, ,R',4A1) 91 C 92 C COMPUTE COST COEFFICIENTS. , 93 D1=1./(1.-DELTA) 94 DO 25 I=1,NYEARS 95 D1=D1*(1.-DELTA) 96 DO 25 J=1,9 97 25 COST{J,I)=-(D1*WGT(J)) 98 C 99 C 100 C DO COLUMNS SECTION. 101 C WRITE COEFFICIENTS OF TABLEAU, BY COLUMNS. 102 WRITE (1,150) 103 150 FORMAT{*COLUMNS 1) 104 NUM=9*NY EARS 105 M=2 106 INTL=-8 107 UNIT=1.0 108 C 109 C DC LOOP FOR COLUMNS; 9 COLUMNS PER YEAR. 110 DO 45 IC0L=1,NUM 111 CALL BTD (ICOL,CCHAR {.1) , 4 , NN, «0 ») 112 K C= M 0 D (I COL ,9) 113 C KC=1 MEANS THE FIRST OF ANOTHER YEAR'S VARIABLES. 114 IF(KC.NE.I) GO TO 46 115 116 LISTING OF FILE M 117 118 90 119 INTL-INTL+9 120 C 121 C COMPUTE MATRICES FOR ANOTHER YEAR., 122 C T MATRICES ARE TEMPORARY * HOLDING* MATRICES FOR PRODOCT 123 C OF S MATRICES. 124 CALL COPY{S (1, 1,M) ,T (1, 1, 1) * 9, 9) 125 JY1=NYEARS-M 126 IF(JYR.LE.O) GO TO 47 127 DO 55 INNER=1,JYR 128 55 CALL MOLT (S {1, 1, M+INN ER) , T (1 , 1,1 NN ER) , T { 1, 1,INNER* 1) 129 1,9,9) 130 47 M=M+1 131 46 IF(KC.NE.O) GO TO 48 132 KC=9 133 C 134 C GENERATE IDENTITY MATRIX FOR COEFFICIENTS IF ICOL.GE.10. 135 48 IM=ICOL-9 136 IF{IMi.LE.O) GO TO 49 137 CALL BTD {IM,RCH AR{1) , 4,NSD,* 0 *) 138 WRITE(1.160)CCHA8,RCHA8,UNIT 139 C 140 C KT IS INDEX FOR T MATRICES. 141 49 KT=0 142 IF (INTL.GT.LIM) GO TO 43 143 DO 65 IROW=INTL,LIH 144 KR=MOD(IROW,9) 145 IF (KR.NE.O) GO TO 68 146 KR=9 147 68 IF (KR.EQ.1) KT=KT+1 148 C 149 C WRITE NON-ZERO ROW COEFFICIENTS. 150 IF{T (KR,KC,KT) . EQ.O. 0) GO TO 65 151 CALL BTD(IROW,fiCHAB (1) ,4,NSD,* 0*} 152 WRITE(1, 160)CCHA8,RCHAR,T(KR,KC* KT) 153 160 FORMAT(4X,»C»,4A1,5X, ,R* , 4A1, 5X, E11. 5) 154 65 CONTINUE 155 C 156 C WHITE COST COEFFICIENTS. 157 43 IYEAR=M—2 158 WRITE{1,170) CCHARjCOST(KC,IYEAR) 159 170 F0RMAT{«X,*C*,4A1,5X,«C0ST*,6X,E11.5) 160 45 CONTINUE 161 C 162 C 163 C DO RHS SECTION. 164 WRITE{1,180) 165 180 FORMAT(*BHS«) 166 IR=1 167 CALL MATVEC (S {1, 1, 1) ,XIN,SX, 9,9) 168 DO 75 MNUH=2,NYEARS 169 C 170 C COPY SX INTO R. 171 DO 85 L=1,9 172 85 R(L)=SX(L) 173 174 LISTING OF FILE M 175 176 91 177 C 178 c ACCUMULATE PRODUCT OF S»S * X INTO SX. 179 CALL MATVEC(S(1,1,MNUM),R,SX,9,9) 180 DO 95 LL=1,9 181 R (LL)=SX (LL) -POPMIN(LL) 182 IF(R(LL) .GE.O.O) GO TO 444 183 WRITE (6, 230) IR 184 230 FORMAT (»0 PROBLEM IS IN FEASIBLE. RHS VALUE IN ROW*, 185 114,* IS LESS THAN 0.0') 186 444 CALL BTD( IR,RCHAR(1),4,NSD,»0«) 187 WRITE (1, 190) RCHAR,B (LL) 188 190 FORMAT(4X,»RHS1*,6X,»R*,4A1,5X,E11.5) 189 95 1R=IR+1 190 75 CONTINUE 191 c 192 c 193 c DO BOUNDS SECTION. 194 c BOUNDS USED INSTEAD OF FIRST-YEAR CONSTRAINTS. ; 195 WRITE(1,200) 196 200 FORMAT (• BOUNDS *) 197 CALL MATVEC(S{1, 1, 1) ,XIN,SX,9,9) 198 DO 105 J=1,9 199 R (J) =SX (J) - POPMIN (J) 200 IF (R (J) .GE. 0.0) GO TO 1 05 201 WRITE(6,240) J 202 240 FORMAT(* 0 PROBLEM IS INFEASIBLE. UPPER BOUND IN 203 TROW*,12,* i s LESS THAN 0.0») 204 105 WRITE (1, 210) J,R(J) 205 210 FORMAT{• UP YR1*,7X, ,COO0 ,,I1,5X,E11.5) 206 c 207 c 208 c ENDATA CARD SIGNALS END OF DATA. 209 WRITE (1,220) 210 220 FORMAT{*ENDATA*) 211 STOP 212 END LISTING OF FILE E 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 92 FILE MT62:EXAMPLE NOMBEE OF YEARS 2 INITIAL POPULATION VECTOR 1 6 65 1 20 28 9 34 35 CONSTRAINT POPULATION VECTOR 0 0 0 0 0 0 0 0 5 HEIGHTS FOR RETURN FUNCTION 3 2 1 3 2 1 5 3 1.33 DISCOUNT FACTOR 0 NAME FOR DATA SET SAMPLE S1 1 0 0 0 0 0 0 0 0 S2 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0.0268 0 0 0 0 0 0.6513 0.0095 0 0 0.2140 0 0.1053 1.8505 0.0571 0 0 0 0 1 0 0 0 0 0 0 0.0010 0 0 0.0395 0 0.1858 0.1885 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0.0076 0.0021 0. 0026 0. 0 0 0 0 0 0 0. 1155 0 0.2842 0.0606 0. 0525 0. 0 0 0 0 0.1667 0.1178 0.2381 0.2909 0.9424 0.2275 0.2087 0.8049 LISTING OF FILE S 2 93 3 4 FILE MT62: SAMPIN 5 6 NAME SAMPLE 7 ROWS 8 N COST 9 L R0001 10 L R0002 11 L R0003 12 L R0004 13 L R0005 14 L R0006 15 L R0007 16 L ROO08 17 L R0009 18 COLUMNS 19 C0001 COST -.30000E*01 20 C0002 COST 20000E+01 21 C0003 R0003 0.26800E-01 22 C0003 R0005 0. 10000E-02 23 C0003 R0006 0.76000E-02 24 C0003 R0007 0.21000E-02 25 C0003 R0008 0.26000E-02 26 C0003 R0009 0.71000E-02 27 C0003 COST -.10000E*01 28 C0004 COST -.30000E+01 29 C0005 R0008 0.11550E+00 30 C0005 COST -.20000E+01 31 C0006 R0003 0.65130E+00 32 COO 06 R0004 0.95000E-02 33 C0006 R0005 0.39500E-01 34 C0006 R0006 0.28420E+00 35 C0006 R0007 0.60600E-01 36 C0006 R0008 0.525O0E-01 37 C0006 R0009 0. 1346 OE* 00 38 C0006 COST -.10000E+01 39 C0007 COST -.50000E*01 40 C0008 R0003 0.21400E+00 41 C0008 R0005 0. 18580E+00 42 CO 00 8 R0006 0.1667GE+00 43 C0008 R0007 0.11780E*00 44 C0008 R0008 0.23810E+00 45 C0O08 R0009 0.29090E+00 46 C0008 COST -.30000E+01 47 CO 00 9 R0002 0.10530E+00 48 C0009 R0003 0.18505E+01 49 CO 009 R0004 0.57100E-01 50 C0009 R0005 0.18850E+00 51 C0009 R0006 0. 94240E+00 52 C0009 R0007 0.22750E+00 53 C0009 R0008 0.20870E+00 54 C0009 R0009 0.80490E+00 55 C0009 COST 13300E+01 56 C0010 R0001 0. 10000E+01 57 L I ST ING OF F I L E S 59 60 61 CO010 COST - . 3 0 0 0 0 E * 0 1 62 C0011 R000 2 0 . 10000E+01 63 CO011 COST 20000E+01 64 CO012 R0003 0.10000E+01 65 CO012 COST 10000E+01 66 C0013 R0004 0.10000E+01 67 C0013 COST - . 3 0 0 0 0 E + 0 1 68 CO014 R0005 0. 10000E+01 69 CO014 COST - . 20000E+01 70 C0015 R0006 0.10000E+01 71 CO 0-1S COST - . 10000E+01 72 C0016 R0007 0.10000E+01 73 CO 016 COST - . 50000E+01 74 C0017 BOO 08 0. 10000E+01 75 C0017 COST - . 3 0 0 0 0 E * 0 1 76 C0018 B0009 0 . 10000E+01 77 CO018 COST 13300E+01 78 RHS 79 RHS1 BOO 01 0 . 0 80 RHS 1 B0002 0 . 3 6 8 5 5 E * 0 1 81 RHS 1 B0003 0 .92022E + 02 82 8HS1 R0004 0.22645E+01 83 RHS1 S0005 0.14086E+02 84: BHS1 R0006 0.47103E+02 85 RHS1 R0007 0.13801E+02 86 RHS1 R00 08 0 .19349E+02 87 RHS1 R0G09 0 . 3 7 2 9 2 E * 0 2 88 BOUNDS 89 UP ¥ 1 1 coooi 0.1O000E+01 90 UP ¥ R 1 C0002 0.60000E+01 91 UP YB1 C0003 0.65000E+02 92 UP XR 1 COO 04 0.10000E+01 93 UP ¥ 1 1 C0005 0.20000E+02 94 UP YR1 C0006 0.28000E+02 95 UP YR 1 COO 07 0.90000E+01 96 DP YB1 COO 08 0 . 3 4 0 0 0 E * 0 2 97 UP YR 1 CO 009 0.30000E+02 98 ENDATa 

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