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UBC Theses and Dissertations

Optimal harvest strategies for ungulate populations in relation to population parameters and environmental… Stocker, Max 1979

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OPTIMAL HARVEST STRATEGIES FOR UNGULATE POPULATIONS IN RELATION TO POPULATION PARAMETERS AND ENVIRONMENTAL VARIABILITY by MAX STOCKER B.Sc.(Agr), M c G i l l ' U n i v e r s i t y , 1973 M.Sc, U n i v e r s i t y of Guelph, 1975 A t h e s i s submitted i n p a r t i a l f u l f i l l m e n t of the requirements f o r the degree of Doctor of Phil o s o p h y i n The F a c u l t y of Graduate S t u d i e s Department of Zoology We accept t h i s t h e s i s as conforming to the r e q u i r e d standard The U n i v e r s i t y of B r i t i s h Columbia A p r i l , 1979 (c) Max Stocker, 1979 In present ing t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of 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 Co lumbia, I agree tha t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r re fe rence and s tudy . I f u r t h e r agree tha t permiss ion f o r ex tens ive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s en t a t i v e s . I t i s understood tha t copy ing or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed wi thout my w r i t t e n pe rm iss i on . Department o f 2ooio ^  The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P lace Vancouver, Canada V6T 1W5 i i ABSTRACT Optimal h a r v e s t i n g s t r a t e g i e s f o r ungulate p o p u l a t i o n s are estimated using s t o c h a s t i c dynamic programming. In the context used here, o p t i m a l s t r a t e g y r e f e r s to a s e q u e n t i a l d e c i s i o n r u l e that i s o p t i m a l with r e s p e c t to maximizing expected long term r e t u r n s from ungulate p o p u l a t i o n s . The e f f e c t s of f l u c t u a t i n g environmental c o n d i t i o n s and u n c e r t a i n t y about p o p u l a t i o n parameters were c o n s i d e r e d . Three case examples were s e l e c t e d f o r t h i s study to r e p r e s e n t c l a s s e s of r e a l ungulate systems. In e f f e c t these cases r e p r e s e n t three fragmentary views of the b a s i c f o o d - u n g u l a t e - p r e d a t i o n food c h a i n . Models i n c o r p o r a t i n g f u n c t i o n a l i n f o r m a t i o n with regard to f e c u n d i t y , s u r v i v a l , resource u t i l i z a t i o n , and p r e d a t i o n were formulated as s t o c h a s t i c dynamic programming models and optimal h a r v e s t i n g s t r a t e g i e s were d e r i v e d n u m e r i c a l l y using a d i g i t a l computer. The s t r a t e g i e s are expressed as i s o p l e t h diagrams r e l a t i n g s t a t e v a r i a b l e s and h a r v e s t r a t e s . The optimal harvest s t r a t e g i e s were g e n e r a l l y found to be i n s e n s i t i v e to environmental f l u c t u a t i o n s . On the other hand, i t was found that assumptions r e g a r d i n g b i o l o g i c a l processes have to be c a r e f u l l y i n v e s t i g a t e d f o r t h e i r e f f e c t on the f u n c t i o n a l form of the optimal h a r v e s t i n g s t r a t e g i e s . Though only simple o b j e c t i v e f u n c t i o n s were c o n s i d e r e d , i n d i c a t i o n s are that optimal h a r v e s t i n g s t r a t e g i e s are s e n s i t i v e to assumptions regarding the management g o a l s . i i i The response of the model p o p u l a t i o n s to h a r v e s t i n g , and the re t u r n s obtained from a p p l y i n g optimal h a r v e s t i n g s t r a t e g i e s as w e l l as a l t e r n a t i v e s t r a t e g i e s were explored through s i m u l a t i o n . Though the f u n c t i o n a l form of the optimal s t r a t e g i e s i s robust with regard to the u n c e r t a i n t i e s considered i n t h i s i n v e s t i g a t i o n , the r e t u r n s obtained from a p p l y i n g o p t i m a l s t r a t e g i e s are very s e n s i t i v e to these u n c e r t a i n t i e s . The e f f e c t s of decreased p r o d u c t i v i t y r e s u l t i n g from v a r y i n g the s t o c h a s t i c p o p u l a t i o n v a r i a b l e s a f f e c t e d h a r v e s t i n g r e t u r n s i n a l l cases. The most i n t e r e s t i n g r e s u l t s from t h i s study emerged from value of i n f o r m a t i o n experiments, i n v e s t i g a t i n g r e t u r n s from c o l l a p s i n g the o r i g i n a l i n f o r m a t i o n systems to y i e l d s i m p l i f i e d h a r v e s t i n g s t r a t e g i e s . E s s e n t i a l l y two types of r e s u l t s were obtained. A p p l y i n g s i m p l i f i e d h a r v e s t i n g s t r a t e g i e s e i t h e r had a negative e f f e c t or no e f f e c t on r e t u r n s obtained over long term management p e r i o d s . The best s i m p l i f i e d s t r a t e g i e s were based on ungulate p o p u l a t i o n d e n s i t y i n f o r m a t i o n . For p r a c t i c a l ungulate management t h i s i m p l i e s that e f f o r t s should be d i r e c t e d towards c o l l e c t i n g ungulate d e n s i t y i n f o r m a t i o n , while e x t r i n s i c f a c t o r s need not be r e g u l a r l y monitored. I t i s concluded t h a t f o r ungulate p o p u l a t i o n s harvested in a f l u c t u a t i n g environment, the optimal h a r v e s t i n g d e c i s i o n i n any given year must be based on the s t a t e of the system i n that year. In g e n e r a l , given the inherent u n p r e d i c t a b i l i t i e s of the r e a l world, i t i s i n d e f e n s i b l e to use non-feedback c o n t r o l p o l i c i e s , such as f i x e d h a r v e s t r a t e s or quota systems. i v Table of Contents Page A b s t r a c t i i L i s t of Fi g u r e s v i i L i s t of Tables x Acknowledgements x i Chapter 1 General I n t r o d u c t i o n 1 M o t i v a t i o n 2 Previous Work 3 Es t i m a t i n g harvest using p o p u l a t i o n models .... 3 P o p u l a t i o n a n a l y s i s 5 P r a c t i c a l ungulate p o p u l a t i o n management 7 Optimal h a r v e s t i n g s t r a t e g i e s 10 O b j e c t i v e s 13 Experimental systems 14 Scope of the i n v e s t i g a t i o n 16 Chapter 2 Ungulate P o p u l a t i o n Dynamics and O p t i m i z a t i o n Models 18 Summary 19 I n t r o d u c t i o n 20 1. The p o p u l a t i o n model 21 Pro d u c t i o n r e l a t i o n s h i p 21 Dynamic model 24 Model behavior 29 2. S t o c h a s t i c dynamic programming 36 O p t i m i z a t i o n model 36 Computational procedure 39 Optimal s t r a t e g i e s 39 •V? P r e d i c t i o n s 44 3. Values of i n f o r m a t i o n 46 S i m p l i f i e d s t r a t e g i e s 46 Computing values of i n f o r m a t i o n 54 Co n c l u s i o n s 62 Chapter 3 Dynamics of a Vegetation-Ungulate System and i t s Optimal E x p l o i t a t i o n 63 Summary 64 I n t r o d u c t i o n 65 Methods 66 Model s t r u c t u r e 66 S t o c h a s t i c dynamic programming 70 P r e d i c t e d r e t u r n s and s t a t e v a r i a b l e d i s t r i -b u t i o n s 70 E s t i m a t i n g values of i n f o r m a t i o n 72 Res u l t s and D i s c u s s i o n 76 Dynamics of the unharvested system 76 Optimal h a r v e s t i n g s t r a t e g i e s 78 P r e d i c t e d v a l u e s from optimal h a r v e s t i n g s t r a -t e g i e s 84 Comparison to a l t e r n a t i v e h a r v e s t i n g s t r a t e -g i e s 89 Co n c l u s i o n s 94 Chapter 4 O p t i m i z a t i o n Model f o r a Wolf-Ungulate System . 96 Summary 97 I n t r o d u c t i o n 98 Methods 99 Model s t r u c t u r e and parameter estimates 99 vi Environmental variation 101 Prey dynamics 103 Wolf pack dynamics 106 Predator-prey interactions I l l Optimization procedure 115 Predictions and comparisons 116 Results and Discussion 120 Model predictions 120 Optimal feedback strategies 125 Predictions using optimal strategies 134 Comparison with alternative strategies 138 Conclusions 144 Chapter 5 General Discussion 146 Comments on modelling studies 147 Discussion of the results 149 Importance of state variables 149 The e f f e c t of stochastic variation 151 Influence of population processes 153 Objective functions 154 Returns of optimal vs alternative strategies .. 155 Reducing information for decision making 156 An adaptive management system for ungulates 158 Suggestions for further work 161 Chapter 6 Conclusions 163 Literature Cited 167 v i i L i s t of F i g u r e s Figure Page 1. A d u l t and j u v e n i l e deer r e p r o d u c t i v e f u n c t i o n s ... 23 2. The impact of r a i n f a l l on deer p r o d u c t i v i t y 25 3. Three p r o b a b i l i t y d i s t r i b u t i o n s of r a i n f a l l used i n t h i s study 27 4. I s o c l i n e s , e q u i l i b r i a and t r a j e c t o r i e s f o r two component deer p o p u l a t i o n . 32 5. 50 year s i m u l a t i o n s of unharvested two component deer p o p u l a t i o n 35 6. Optimal h a r v e s t i n g d e c i s i o n s a D ° ( 2 S ) a n d yD°(jc) estimated from s t o c h a s t i c dynamic programming 41 7. Constant p o p u l a t i o n l i n e superimposed on optimal a d u l t harvest i s o p l e t h diagram 43 8. P r e d i c t e d means (+2SE) i n r e l a t i o n to r a i n f a l l p r o b a b i l i t y assumptions 45 9. 50 year s i m u l a t i o n s of o p t i m a l l y harvested deer p o p u l a t i o n s 47 10. I s o c l i n e Yt+i = Y t superimposed on optimal h a r v e s t i n g d e c i s i o n s a D ° ( 2 L ) a n d yD°(x) 49 11. S i m p l i f i e d h a r v e s t i n g s t r a t e g i e s d e r i v e d from op t i m a l h a r v e s t i n g s t r a t e g i e s using primary age r a t i o as s o l e i n f o r m a t i o n input 51 12. S i m p l i f i e d h a r v e s t s t r a t e g y Ds3 53 13. G r a p h i c a l e v a l u a t i o n of s i m p l i f i e d h a r v e s t i n g s t r a t e g y Ds2 55 14. I n i t i a l c o n d i t i o n s used to determine valu e s of i n f o r m a t i o n 57 15. Simulated response s u r f a c e s of values of i n f o r m a t i o n f o r D$i .' 59 16. Simulated response s u r f a c e of values of i n f o r m a t i o n f o r Dg2 60 17. V e g e t a t i o n growth as a f u n c t i o n of v e g e t a t i o n biomass 68 v d i i 18. Consumption per deer as a f u n c t i o n of v e g e t a t i o n biomass 69 19. I n i t i a l c o n d i t i o n s to determine valu e s of i n f o r m a t i o n 75 20. Simulated trends of unharvested deer p o p u l a t i o n and v e g e t a t i o n biomass 77 21. Harvest r a t e i s o c l i n e s f o r the standard v e g e t a t i o n p r o d u c t i o n f u n c t i o n 79 22. Harvest r a t e i s o c l i n e s f o r s i t u a t i o n s where max. p r o d u c t i o n i s at low biomass 82 23. Harvest r a t e i s o c l i n e s f o r s i t u a t i o n s where max. p r o d u c t i o n i s at high biomass 83 24. P r e d i c t e d means of harvest and s t a t e v a r i a b l e s 85 25. P r e d i c t e d p r o b a b i l i t y d i s t r i b u t i o n s of annual r e t u r n s 86 26. P r e d i c t e d p r o b a b i l i t y d i s t r i b u t i o n s of deer p o p u l a t i o n 87 27. P r e d i c t e d p r o b a b i l i t y d i s t r i b u t i o n s of v e g e t a t i o n biomass 88 28. Simulated response s u r f a c e s of values of i n f o r m a t i o n 90 29. Best f i x e d h a r v e s t r a t e s 91 30. F a c t o r s considered i n the model of a wolf-ungulate system i n Al a s k a 100 31. (a) Cumulative s n o w f a l l of Mt. McKinley N a t i o n a l Park 102 (b) P r o b a b i l i t y d i s t r i b u t i o n of winter s e v e r i t y c l a s s e s 102 32. Density dependent b i r t h r a t e f o r moose 105 33. Wolf t e r r i t o r y s i z e as a f u n c t i o n of ungulate prey d e n s i t y 109 34. (a) Food dependent wolf pack r a t e of in c r e a s e .... 110 (b) Wolf s t a r v a t i o n l o s s i n r e l a t i o n to biomass .. 35. F u n c t i o n a l response of wolf packs to moose d e n s i t y 113 i x 36. One p o s s i b l e assumption about wolf c o n t r o l c o s t as a f u n c t i o n of number of wolf packs 117 37. Parameter space of wolf search e f f i c i e n c y and other prey occurrence 121 38. Predator-prey phase planes i n r e l a t i o n to parameter cases 123 39. Optimal feedback s t r a t e g i e s f o r parameter case one 126 40. Optimal feedback s t r a t e g i e s f o r parameter case two 127 41. Optimal feedback s t r a t e g i e s f o r parameter case three 128 42. Optimal feedback s t r a t e g i e s f o r parameter case four 129 43. Optimal feedback s t r a t e g i e s f o r c w = f(Wt) 133 44. Examples of optimal paths in r e l a t i o n to wolf c o n t r o l c o s t s and parameter cases 137 45. S i m p l i f i e d h a r v e s t i n g s t r a t e g i e s 139 46. Simulated response s u r f a c e s of values of i n f o r m a t i o n f o r optimal v s . s i m p l i f i e d s t r a t e g i e s 140 47. Simulated response s u r f a c e s of values of i n f o r m a t i o n f o r optimal s t r a t e g i e s vs. f i x e d h a r v e s t r a t e p o l i c i e s 142 48. Components of a p a s s i v e adaptive ungulate management system 159 X L i s t of Tables Table Page I. Impact of r a i n f a l l on deer b i r t h s 26 I I . Parameter values used f o r deer p o p u l a t i o n model 30 I I I . S t a b l e nodes f o r unharvested deer system 34 IV. J u v e n i l e (Y) and o l d e r deer (A) values used to d e r i v e s i m p l i f i e d age r a t i o h a r v e s t s t r a t e g i e s Dgi 50 V. M o r t a l i t y (a.2) v a l u e s and p r o b a b i l i t i e s 71 VI. Summary of h a r v e s t i n g s t r a t e g i e s 74 V I I . S e n s i t i v i t y of harvest r a t e s to changes in v e g e t a t i o n biomass 81 V I I I . Impact of s n o w f a l l on moose b i r t h s 104 IX. Estimate of weighted maximum mean b i r t h r a t e 107 X. Parameter values f o r the wolf-ungulate model 114 XI. Parameter combinations used f o r o p t i m i z a t i o n 118 X I I . E q u i l i b r i a i n the absence of ha r v e s t 124 X I I I . Wolf c o n t r o l c o s t t h r e s h o l d values 131 XIV. P r e d i c t e d means i n r e l a t i o n to parameter cases and o b j e c t i v e f u n c t i o n s 135 x i ACKNOWLEDGEMENTS My s i n c e r e s t thanks are extended to P r o f e s s o r C.J. Walters under whose encouraging and h e l p f u l d i r e c t i o n t h i s study was completed. Along w i t h P r o f e s s o r Walters, I am most g r e a t f u l to the members of my s u p e r v i s o r y committee Drs. C.S. H o l l i n g , F.L. B u n n e l l , A.R.E. S i n c l a i r , C F . Wehrhahn, and R. H i l b o r n . T h e i r v a l u a b l e suggestions throughout the study, and t h e i r v a l u a b l e c r i t i c i s m s o f f e r e d from reviewing the d r a f t manuscript are deeply a p p r e c i a t e d . My a p p r e c i a t i o n and thanks are expressed to Mike S t a l e y f o r g i v i n g me advice on computational aspects of the study, and f o r p r o v i d i n g l i v e l y d i s c u s s i o n s from which many u s e f u l suggestions emerged. Thanks a l s o go to Warren K l e i n f o r h i s help regarding computer programming q u e s t i o n s . Gordon Haber, Anchorage Community C o l l e g e , provided f i e l d data from h i s mammoth study of a wolf-ungulate system i n A l a s k a . Only through h i s data c o l l e c t i o n e f f o r t was i t p o s s i b l e to develop the a n a l y s i s d e s c r i b e d i n Chapter 4 . S p e c i a l thanks are extended to Joan Anderson and Judy Maron f o r t y p i n g the t e x t of t h i s t h e s i s . As experienced e d i t o r s they took the t r o u b l e to p o i n t out many of my e r r o r s . I a l s o thank the many i n d i v i d u a l s at the I n s t i t u t e of Animal Resource Ecology, who through t h e i r s t i m u l a t i n g d i s c u s s i o n s and f r i e n d s h i p made t h i s study p o s s i b l e . I acknowledge the f i n a n c i a l support of the N a t i o n a l Research C o u n c i l of Canada f o r the d u r a t i o n of the study. x i i To my wife Margareth a l s o I want to thank f o r her p a t i e n c e and c o n t i n u i n g moral support through the many years of my s t u d i e s . 1 CHAPTER 1 General I n t r o d u c t i o n 2 MOTIVATION W i l d l i f e p o p u l a t i o n s c o n s t i t u t e a n a t u r a l resource of s u b s t a n t i a l m a t e r i a l and a e s t h e t i c v a l u e . P r o v i s i o n of a su s t a i n e d y i e l d , a f r a c t i o n of the p o p u l a t i o n that can be removed on a r e c u r r e n t b a s i s without f o r c i n g the p o p u l a t i o n to e x t i n c t i o n , i s the goal of much presen t w i l d l i f e management. The u l t i m a t e aim i s to provide the l a r g e s t p o s s i b l e s u s t a i n e d y i e l d (the maximum su s t a i n e d y i e l d ) or more s o p h i s t i c a t e d to maximize b e n e f i t s to s o c i e t y from u t i l i z a t i o n (optimum s u s t a i n e d y i e l d ) . I f p r o d u c t i o n of w i l d l i f e p o p u l a t i o n s could a c c u r a t e l y be determined, the e s t i m a t i o n of maximum su s t a i n e d y i e l d or optimal s u s t a i n e d y i e l d along with the a p p r o p r i a t e p o p u l a t i o n s i z e would be a t r i v i a l matter. However, environments f l u c t u a t e from year to year with concomitant f l u c t u a t i o n s i n p o p u l a t i o n parameters. As a r e s u l t i t i s d i f f i c u l t to determine a p p r o p r i a t e management a c t i o n s t hat w i l l m aintain p o p u l a t i o n s . Responding to the need to make management d e c i s i o n s under such complexity and u n c e r t a i n t y , t h i s i n v e s t i g a t i o n seeks to estimate optimal harvest s t r a t e g i e s f o r ungulate p o p u l a t i o n s that e x i s t i n s t o c h a s t i c environments. An innovatory aspect of the study i s to compare optimal s t r a t e g i e s with s i m p l i f i e d and t r a d i t i o n a l p o l i c i e s that demand l e s s i n f o r m a t i o n concerning p o p u l a t i o n s t a t e s and dynamics. Optimal h a r v e s t i n g s t r a t e g i e s f o r ungulate p o p u l a t i o n s i n s t o c h a s t i c environments, and t h e i r comparisons with a l t e r n a t i v e p o l i c i e s are the f o c a l p o i n t s of t h i s r e s e a r c h . However, before e l a b o r a t i n g on optimal h a r v e s t i n g s t r a t e g i e s , i t i s a p p r o p r i a t e to review p r e v i o u s work concerning the e s t i m a t i o n of h a r v e s t , u t i l i z i n g p o p u l a t i o n models; p o p u l a t i o n a n a l y s i s i n g e n e r a l ; and p r a c t i c a l ungulate p o p u l a t i o n management. Subsequently, i n Chapter one, the s p e c i f i c o b j e c t i v e s of t h i s study are s t a t e d , and three experimental ungulate p o p u l a t i o n systems, u t i l i z e d i n t h i s study, are int r o d u c e d . Chapter one concludes with o u t l i n i n g the scope of the i n v e s t i g a t i o n . Chapters two, three and fou r examine optimal h a r v e s t i n g s t r a t e g i e s of the experimental systems, and i n Chapter f i v e the obtained r e s u l t s are d i s c u s s e d . F i n a l l y i n Chapter s i x c o n c l u s i o n s of t h i s study are s t a t e d . PREVIOUS WORK Es t i m a t i n g h a r v e s t using p o p u l a t i o n models S i n g l e s p e c i e s growth models form the bulk of management models i n f i s h e r i e s and w i l d l i f e management. Much of the theory on the e x p l o i t a t i o n of f i s h p o p u l a t i o n s i s based on Baranov's y i e l d p e r - r e c r u i t model developed i n 1918 (Baranov, 1918). Beverton and H o l t (1957:327) reviewed models a p p l i e d to s u s t a i n e d y i e l d h a r v e s t i n g up to the e a r l y 1950's, and added t h e i r own v e r s i o n s . A d d i t i o n a l s u r p l u s p r o d u c t i o n models were developed by Schaefer (1954), Ric k e r (1954a), and A l l e n (1971). Dasmann (1964) and Gross (1969 and 1972) have advocated a l o g i s t i c growth management approach ( f i r s t expressed by Leopold, 1955) f o r b i g game s i m i l a r to t h a t u t i l i z e d i n f i s h e r i e s management. An attempt of communicating the e x i s t e n c e of stock r e c r u i t m e n t r e l a t i o n s h i p s i n w i l d l i f e p o p u l a t i o n s was made by Ricke r (1954b), and more r e c e n t l y m a l l a r d p o p u l a t i o n harvests were estimated using a modified Beverton-Holt r e c r u i t m e n t model (Brown, Hammack, and T i l l m a n , 1976). E s s e n t i a l l y the growth model approach i s based on e s t i m a t i n g p o p u l a t i o n parameters of r e c r u i t m e n t , growth, and m o r t a l i t y . The parameters are i n t e g r a t e d i n t o mathematical models e x p r e s s i n g the response of a p o p u l a t i o n to e x p l o i t a t i o n . Through the procedures c i t e d i n t h i s paragraph the maximum s u s t a i n a b l e y i e l d of a p o p u l a t i o n can be p r e d i c t e d . V a r i a b i l i t y of p o p u l a t i o n s may be r e p r e s e n t e d , although t h i s i s seldom done, by i n c l u d i n g s t o c h a s t i c parameters in the s i n g l e s p e c i e s growth models. However, such models cannot be considered a meaningful b a s i s f o r r e p r e s e n t i n g how ungulate p o p u l a t i o n s i n t e r a c t with food sources or p r e d a t o r s . S t i l l n e g l e c t i n g i n t e r - s p e c i e s i n t e r a c t i o n s , some authors have c o n s t r u c t e d a n a l y t i c a l models with d e t a i l e d a g e - s t r u c t u r e (e.g., Lewis, 1942; L e s l i e , 1945 and 1948; Watt, 1968; Dunkel, 1970; A l l e n and Basasibwaki, 1974; Beddington and T a y l o r , 1973). In these models dynamics of the p o p u l a t i o n are r e p o r t e d as the sum of the demographic r e a c t i o n s . o f the i n d i v i d u a l age c l a s s e s . As Caughley (1977) p o i n t s out, the d e t a i l e d a g e - s t r u c t u r e approach may seem i d e a l but with i t g e n e r a l i t y i s l o s t . A l s o the number of a g e - s p e c i f i c p o p u l a t i o n parameters that have to be estimated from o b s e r v a t i o n makes p r a c t i c a l use d i f f i c u l t ; o f t e n a n a l y s i s of s e l e c t e d i n t e r v a l s of age groups i s assumed s u f f i c i e n t to make p o p u l a t i o n management d e c i s i o n s . A v a r i a n t to the a n a l y t i c a l a g e - s t r u c t u r e approach i s the p r e d i c t i o n of p o p u l a t i o n r e a c t i o n s to e x p l o i t a t i o n by s i m u l a t i n g p o p u l a t i o n dynamics using simple or complex computer models (e.g., Niven, 1970; Walters and B u n n e l l , 1971; Walters and Gross, 5 1972; Gross, R o e l l e , and W i l l i a m s , 1973). D e t a i l e d m u l t i -v a r i a b l e s i m u l a t i o n models are not considered u s e f u l f o r o p t i m i z a t i o n procedures d e s c r i b e d h e r e i n . T h i s p o i n t i s pursued i n a subsequent s e c t i o n . P o p u l a t i o n a n a l y s i s E f f i c i e n t management, as p r o p e r l y argued by Caughley (1976), r e q u i r e s an understanding of how a managed p o p u l a t i o n works. In order to g a i n understanding of how p o p u l a t i o n s work i t i s necessary to i n v e s t i g a t e p o p u l a t i o n processes, such as f e c u n d i t y , m o r t a l i t y , and d i s p e r s a l together with the p r o p e r t i e s of the environment that may i n f l u e n c e these p r o c e s s e s . In a d d i t i o n , i n f o r m a t i o n on the abundance of p o p u l a t i o n s and numbers removed by h a r v e s t i n g p l a y s an important r o l e i n p o p u l a t i o n management. The t h e o r e t i c a l body of knowledge d e a l i n g f o r m a l l y with these numerical aspects of v e r t e b r a t e p o p u l a t i o n a n a l y s i s has r e c e n t l y been assembled i n the e x c e l l e n t book by Caughley (1977). Estimates of r e l a t i v e d e n s i t y using counts of animals or counts of animal s i g n are f r e q u e n t l y employed. The concept of c a t c h per u n i t of e f f o r t a l s o p r o v i d e s a u s e f u l index of d e n s i t y when the c a t c h i n g does not g r e a t l y reduce the p o p u l a t i o n s i z e (Caughley, 1977). Estimates of absolute d e n s i t y such as t o t a l counts or sampled counts (e.g., a e r i a l surveys) can be made when the animals are r e l a t i v e l y sedentary and when surveys are run over s h o r t p e r i o d s of time p r e c l u d i n g s i g n i f i c a n t movement. Census methods most f r e q u e n t l y used by w i l d l i f e b i o l o g i s t s are those based on removals of animals. F i r s t the method of s e l e c t i v e r e d u c t i o n or i n c r e a s e i s used where a p o p u l a t i o n i s c l a s s i f i a b l e i n t o two or more c l a s s e s (e.g., bucks and does). Through t h i s method, which was f i r s t i ntroduced by K e l k e r (1940 and 1944), the s i z e of a game p o p u l a t i o n can be estimated by knowing the number k i l l e d during the hunting season. Refinements and g e n e r a l i z a t i o n s of t h i s method were made by P e t r i d e s (1949), Chapman (1954 and 1955), Chapman and Murphy (1965), and Rupp (1966). For s i t u a t i o n s where hunting i s not d i s c r i m i n a n t with r e s p e c t to sex t h i s method cannot be a p p l i e d . For n o n - s e l e c t i v e r e d u c t i o n s of p o p u l a t i o n s i z e , the p r i n c i p l e that r a t e of capture decreases as p o p u l a t i o n s i z e decreases has been recognized by L e s l i e ( L e s l i e and D a v i s , 1939) and DeLury (1947). The l a r g e v a r i a n c e of estimates by these methods l i m i t s t h e i r a p p l i c a t i o n . Fecundity i n an animal p o p u l a t i o n i s expressed as the mean number of female l i v e b i r t h s per female over an i n t e r v a l of age (Caughley, 1977). T h i s i n f o r m a t i o n i s u s u a l l y assembled f o r a l l age c l a s s e s i n a f e c u n d i t y t a b l e . Determining hunting and t o t a l m o r t a l i t y i s h i g h l y important i n game management. Formulas or methods f o r c a l c u l a t i n g p r o p o r t i o n of p o p u l a t i o n k i l l e d by hunting or n a t u r a l causes have been presented by K e l k e r (1949-1950), Lauckhart (1950), Dasmann (1952), P e t r i d e s (1954), S e l l e c k and Hart (1957), and Hanson (1963). These r a t i o - b a s e d methods have u n i v e r s a l a p p l i c a t i o n i n determining percentage l o s s e s from hunting and n a t u r a l causes when d i f f e r e n t i a l l o s s e s by sex or age groups take p l a c e and r a t i o s i n the p o p u l a t i o n and i n the l o s s e s can be determined. As Caughley (1977) p o i n t s out most of these methods provide answers i r r e l e v a n t to most p r a c t i c a l management problems. 7 A simple measure of a p o p u l a t i o n ' s r a t e of increase (or p r o d u c t i v i t y ) i s the r a t i o of numbers between s u c c e s s i v e years ( K e l k e r , 1947). I f a g e - s p e c i f i c i n f o r m a t i o n i s a v a i l a b l e , the most accurate i n t r i n s i c r a t e of i n c r e a s e estimates f o r an animal p o p u l a t i o n are determined from a l i f e t a b l e together with a f e c u n d i t y t a b l e (Caughley, 1977). The two are r e l a t e d by Lotka's fundamental equation of p o p u l a t i o n dynamics (Lotka, 1925). I f a g e - s p e c i f i c s u r v i v a l s and a g e - s p e c i f i c f e c u n d i t i e s are f i x e d , the p o p u l a t i o n w i l l converge (over time) to a s t a b l e age d i s t r i b u t i o n having a constant r a t e of i n c r e a s e . P r a c t i c a l ungulate p o p u l a t i o n management Managing an ungulate p o p u l a t i o n to take from i t a s u s t a i n e d y i e l d e s s e n t i a l l y i n v o l v e s two steps (1) conduct annual surveys to determine s p e c i e s p r o d u c t i v i t y , assess p o p u l a t i o n t r e n d s , and evaluate h a b i t a t c o n d i t i o n s , and (2) analyze c u r r e n t and long-term data to make h a r v e s t i n g d e c i s i o n s . Species p r o d u c t i v i t y i s o f t e n determined by employing a v a r i e t y of l i f e t a b l e s (Eberhardt, 1969). The success of deducing a p p l i c a b l e c o n c l u s i o n s from the l i f e t a b l e approach depends very much on the a v a i l a b i l i t y of exact n a t a l i t y and m o r t a l i t y data. The p r i n c i p l e l i m i t a t i o n s of l i f e t a b l e s are t h e i r i n a b i l i t y t o i n c l u d e adjustments of r a t e s r e s u l t i n g from changes of p o p u l a t i o n d e n s i t y and/or the environment, and the enormous d e t a i l of data r e q u i r e d . P o p u l a t i o n trends are assessed by c o l l e c t i n g d e t a i l e d i n f o r m a t i o n about the age and sex composition of the k i l l (eg. Aney, 1974). Whenever there i s a d i f f e r e n t i a l harvest r a t e 8 between two c l a s s e s , p o p u l a t i o n s i z e or harvest i s estimated from t h i s i n f o r m a t i o n using the p r e v i o u s l y d e s c r i b e d c h a n g e - i n - r a t i o methods ( P a u l i k and Robson, 1969). Ha b i t a t i n f o r m a t i o n i s o f t e n c o l l e c t e d on the premise t h a t i t alone i s s u f f i c i e n t f o r meaningful d e c i s i o n s , and that p o p u l a t i o n d e n s i t y can be a l t e r e d p r i n c i p a l l y by ma n i p u l a t i n g h a b i t a t . Success of t h i s approach r e s t s on an accurate knowledge of h a b i t a t , i g n o r i n g i n t e r n a l adjustments of a g e - s p e c i f i c r a t e s that generate the change, and on presence of strong compensatory adjustments i n b i r t h and/or n a t u r a l m o r t a l i t y r a t e s to i n c r e a s e d m o r t a l i t y through h a r v e s t i n g . Some advocates of t h i s approach are Denny (1944), Crawford (1950), Dasmann (1964), DeVos and Mosby (1969), Foote (1971), Yoakum (1971), and A l l e n (1977). The second s t e p i n the management process d e a l s with the q u e s t i o n of how the data base i s used f o r management d e c i s i o n s . There are at l e a s t f o u r approaches to t r a n s l a t i n g data i n t o d e c i s i o n making or r e g u l a t i o n of the ha r v e s t . F i r s t there i s the i n t e r p r e t a t i o n of observed trend data to r e l a t e them to the c u r r e n t s t a t u s of the p o p u l a t i o n ( s t a t i o n a r y , i n c r e a s i n g o r d e c r e a s i n g ) . S p e c i f i c a l l y , p r o d u c t i o n or age r a t i o i n d i c e s of game p o p u l a t i o n s are r o u t i n e l y c o l l e c t e d . For example, when pre-season i n d i c e s are lower than f o r the pre v i o u s year r e s t r i c t i v e hunting r e g u l a t i o n s are considered (Henny, Overton, and Wight, 1970). This d e c i s i o n may be c o r r e c t ; however, the e f f e c t of the r e g u l a t o r y change i n p o p u l a t i o n parameters i s o f t e n not monitored. In trend a n a l y s i s , response i s made only when d i f f i c u l t i e s become apparent. G e n e r a l l y f o r t h i s approach annual change i n har v e s t i s small u n t i l the p o p u l a t i o n i s s m a l l . U t i l i z a t i o n of feedback p o l i c i e s f o r h a r v e s t i n g d e c i s i o n s i s a second approach (eg. Walters, 1975; C l a r k , 1976). Feedback p o l i c i e s are c o n t r o l laws f o r which the d e c i s i o n i n a given year i s d i r e c t l y expressed as a f u n c t i o n of the c u r r e n t s t a t e of the p o p u l a t i o n . Such c o n t r o l s are simple to d e s c r i b e and they are a l s o capable of responding d i r e c t l y to random f l u c t u a t i o n s i n both p o p u l a t i o n s i z e and p o p u l a t i o n parameters ( C l a r k , 1976). These p o l i c i e s u s u a l l y i n v o l v e removing the lowest f e a s i b l e h a r v e s t when the p o p u l a t i o n i s below the optimum e q u i l i b r i u m and a very high h a r v e s t when above that e q u i l i b r i u m (Walters and H i l b o r n , 1978). T h i s i m p l i e s that annual change i n har v e s t i s g r e a t e s t when the p o p u l a t i o n i s near e q u i l i b r i u m . T h i s approach i s f u r t h e r d i s c u s s e d i n the next s e c t i o n . The two remaining approaches are not very w e l l documented s i n c e w i l d l i f e management d e c i s i o n s are u s u a l l y made i n an ad hoc f a s h i o n . They i n c l u d e e s t i m a t i n g a t a r g e t e q u i l i b r i u m p o p u l a t i o n s i z e , and v a r y i n g harvests i n c r e m e n t a l l y to reach i t ; and the use of harvest data trends to design h a b i t a t m a n i p u l a t i o n schemes. T h e o r e t i c a l c o n s i d e r a t i o n s of these approaches are given i n Caughley (1977). The most comprehensive ungulate management approach has been developed i n Colorado under the l e a d e r s h i p of Jack Gross (Gross, 1973; Gross e t a l . , 1973; Lipscomb, 1974; P o j a r , 1977). Gross e t a l . (1973) developed a p o p u l a t i o n s i m u l a t i o n model (ONEPOP) th a t subsequently was implemented i n the h a r v e s t i n g d e c i s i o n making process (P o j a r , 1977) . Through t h i s approach the most c r i t i c a l management i n f o r m a t i o n r e q u i r e d i s r e a d i l y i d e n t i f i e d . Some l e s s 10 s u c c e s s f u l a p p l i c a t i o n s of computer s i m u l a t i o n models to ungulate management have been made i n Ala s k a (Dean and Gallaway, 1965; Bos, 1974) and C a l i f o r n i a ( H a l t e r , Longhurst, C o n o l l y , and Anderson, 1972). OPTIMAL HARVESTING STRATEGIES Determining optimal h a r v e s t i n g s t r a t e g i e s i s b a s i c a l l y a problem of how to apply c o n t r o l s to the harvested system i n order to maximize an o b j e c t i v e . Thus three components are r e q u i r e d i n o p t i m i z a t i o n f o r m u l a t i o n s : a model d e s c r i b i n g systems dynamics, a p r e c i s e l y d e f i n e d s e t of c o n t r o l o p t i o n s (or h a r v e s t r a t e s ) , and an o b j e c t i v e f u n c t i o n . The review i n the p r e v i o u s s e c t i o n showed e s s e n t i a l l y two types of approaches to harvest o p t i m i z a t i o n . One type used feedback c o n t r o l p o l i c i e s , whereas the second type s p e c i f i e d e s s e n t i a l l y non-feedback c o n t r o l s . The absence of feedback between f u t u r e system s t a t e s and h a r v e s t p o l i c i e s , as w e l l as the pretense of p e r f e c t knowledge about the system behaviour makes the second type of approach d e c e p t i v e . In other words, the f a c t t h a t a d e c i s i o n taken at a given time a f f e c t s the behaviour of the system i n the f u t u r e (the outcome of which i s by no means c e r t a i n ) i s simply ignored. A more r e a l i s t i c approach to harvest o p t i m i z a t i o n would be: 1. to i n c l u d e u n c e r t a i n t y of the system's dynamics as a component i n the o p t i m i z a t i o n a n a l y s i s 2. that s o l u t i o n s of the o p t i m i z a t i o n s p e c i f i e s optimal 11 c o n t r o l at every p o s s i b l e s t a t e of the system 3. t h a t the s o l u t i o n s d e a l r e a l i s t i c a l l y with con-s t r a i n t s (e.g., the u n d e s i r a b i l i t y to l e t the p o p u l a t i o n drop below a s p e c i f i e d minimum l e v e l ) . P o i n t two r e q u i r e s that the s o l u t i o n i s implemented as a feedback ( c l o s e d loop) c o n t r o l because s t a t e s are not p e r f e c t l y p r e d i c t a b l e . In other words we must know what a c t i o n to take f o r whatever outcome nature throws at us. The s e q u e n t i a l o p t i m i z a t i o n technique of dynamic programming, developed by Richard Bellman (Bellman, 1957 and 1961; Bellman and Dreyfus, 1962), provides the d e s i r e d approach to the s o l u t i o n of dynamic o p t i m i z a t i o n problems ( a l s o termed o p t i m i z a t i o n problems f o r m u l t i s t a g e d e c i s i o n process; Larson, 1968; Nemhauser, 1967). B a s i c a l l y dynamic programming converts simultaneous e s t i m a t i o n of the e n t i r e optimal d e c i s i o n sequence (comput a t i o n a l l y impossible) i n t o a s e q u e n t i a l e s t i m a t i o n . T h i s i s accomplished by a p p l y i n g Bellman's p r i n c i p l e of o p t i m a l i t y . The r e s u l t i n g t o o l i s an i t e r a t i v e f u n c t i o n a l equation which can be solved very e f f i c i e n t l y using d i g i t a l computers. R e s u l t s of i n f i n i t e time h o r i z o n dynamic programming s o l u t i o n s are feedback c o n t r o l laws, and time and s t a t e dependent laws f o r f i n i t e h o r i z o n s o l u t i o n s (Walters and H i l b o r n , 1978). L u c k i l y , the feedback c o n t r o l laws even f o r f i n i t e time h o r i z o n s o l u t i o n s are o f t e n s t a t i o n a r y p o l i c i e s (time independent and s t a t e dependent). Other ways to f i n d feedback p o l i c i e s i n c l u d e the a p p l i c a t i o n of the maximum p r i n c i p l e ( C l a r k , 1976), and f i x e d form o p t i m i z a t i o n (Walters and H i l b o r n , 1978). Comparatively l i t t l e r e s e a r c h has been done using the 12 r e c u r s i v e o p t i m i z a t i o n theory of dynamic programming i n b i o l o g y . Bellman and Kalaba (1960) f i r s t suggested the a p p l i c a t i o n of t h i s technique to the optimal y i e l d problem. The technique has r e c e n t l y been in t r o d u c e d to f i s h e r i e s management ( H i l b o r n , 1976; Lord, 1973 and 1976; Walters, 1975; Walters and H i l b o r n , 1976). S t o c h a s t i c dynamic programming was a l s o used by Anderson (1974 and 1975) i n the development of 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 the North American m a l l a r d duck, p o p u l a t i o n . For other a p p l i c a t i o n s of dynamic programming to o p t i m i z a t i o n f o r e c o l o g i c a l management the reader i s r e f e r r e d to the rec e n t review by Walters and H i l b o r n (1978). U n f o r t u n a t e l y dynamic programming can onl y deal with models having few s t a t e v a r i a b l e s . Bellman (1957) termed the f a c t t h at computational requirements i n c r e a s e g e o m e t r i c a l l y with the a d d i t i o n of s t a t e v a r i a b l e s "the curse of d i m e n s i o n a l i t y " . T h i s d i f f i c u l t y has been one of the main hindrances to a p p l y i n g dynamic programming widely. For the p r a c t i c a l management problem of e s t i m a t i n g optimal h a r v e s t s t r a t e g i e s , a l i m i t of about three s t a t e v a r i a b l e s must be capable of r e a l i s t i c a l l y d e s c r i b i n g the dynamics of the system. In oth e r words the choice to pretend whether a h a r v e s t i n g system i s d e t e r m i n i s t i c and complex, or s t o c h a s t i c and simple, must be made. For t h i s i n v e s t i g a t i o n I have chosen the l a t t e r . T h i s choice does not s a c r i f i c e the i n t e r n a l model complexity represented by the number of parameters and r e l a t i o n s h i p s that are i n c l u d e d , s i n c e these do not a f f e c t the r e q u i r e d computation time. A number of other approaches to h a r v e s t i n g o p t i m i z a t i o n have 13 been taken. Davis (1967) used a l i n e a r programming model to determine o p t i m a l management pl a n s f o r w h i t e - t a i l e d deer. Lomnicki (1972) a p p l i e d n o n - l i n e a r programming techniques f o r planni n g deer p o p u l a t i o n management i n Poland. OBJECTIVES The o b j e c t i v e s of t h i s study are to estimate o p t i m a l h a r v e s t i n g s t r a t e g i e s f o r ungulate p o p u l a t i o n s e x i s t i n g under a v a r i e t y of c o n d i t i o n s , and to expl o r e the e f f e c t s of s t r a t e g i e s on model ungulate p o p u l a t i o n s . Optimal s t r a t e g i e s must cope with the randomly f l u c t u a t i n g environmental c o n d i t i o n s and the complex i n t e r a c t i o n s of p o p u l a t i o n parameters. I t w i l l be assumed throughout t h i s study that "optimal h a r v e s t r a t e s " are those which maximize the s p e c i f i c o b j e c t i v e of expected long term r e t u r n s . T h i s i n v e s t i g a t i o n w i l l s p e c i f i c a l l y examine the f o l l o w i n g s i x q u e s t i o n s : 1. How s e n s i t i v e are estimated optimal harvest s t r a t e g i e s to changes over time i n the s t a t e v a r i a b l e s that determine the behaviour of the harvested system? T h i s addresses the q u e s t i o n of how a c c u r a t e l y s t a t e v a r i a b l e s have to be estimated i n order to apply optimal harvest s t r a t -e g i e s . 2. Do the forms of the optimal harvest s t r a t e g i e s change r a d i c a l l y i f p r o b a b i l i t y d i s t r i b u t i o n s of s t o c h a s t i c 14 v a r i a b l e s are a l t e r e d ? T h i s d e a l s with the q u e s t i o n of the i n f l u e n c e of f r e q u e n c i e s of environmental p e r t u r b a t i o n s on the optimal h a r v e s t i n g s t r a t e g i e s . 3. Parameter values of p o p u l a t i o n processes such as growth and p r e d a t i o n are o f t e n d i f f i c u l t to p r e c i s e l y measure. How would the form of optimal h a r v e s t i n g s t r a t e g i e s change under a l t e r n a t i v e assumptions about such parameter values? 4 . V a r i o u s c i t i z e n s groups have a vested i n t e r e s t i n ungulate resource u t i l i z a t i o n . T h i s poses the q u e s t i o n of how s e n s i t i v e optimal h a r v e s t i n g s t r a t e g i e s are to changes i n the s p e c i f i e d o b j e c t i v e s . 5. How do r e t u r n s obtained from a p p l y i n g optimal harvest s t r a t e g i e s over long-term management p e r i o d s compare to r e t u r n s from a p p l y i n g t r a d i t i o n a l management p o l i c i e s ? T h i s addresses the q u e s t i o n of e v a l u a t i n g c u r r e n t ungulate p o p u l a t i o n h a r v e s t i n g p r a c t i c e . 6. I f we were able to reduce i n f o r m a t i o n r e q u i r e d f o r h a r v e s t i n g d e c i s i o n making by employing s i m p l i f i e d h a r v e s t i n g s t r a t e g i e s , would there be s u b s t a n t i a l r e d u c t i o n s i n r e t u r n s ? T h i s d e a l s with the qu e s t i o n of whether i n f o r m a t i o n c u r r e n t l y used i n making h a r v e s t i n g d e c i s i o n s i s a c t u a l l y v a l u a b l e . EXPERIMENTAL SYSTEMS Developing a comprehensive approach to ungulate h a r v e s t i n g management r e q u i r e s i n v e s t i g a t i o n of case examples from which 15 g e n e r a l i t i e s , c a n be d e r i v e d . Ungulates have been s t u d i e d e x t e n s i v e l y both f o r t h e i r economic value d e r i v e d from hunting as w e l l as to s a t i s f y b i o l o g i c a l c u r i o s i t y . There i s a r i c h l i t e r a t u r e from which f u n c t i o n a l i n f o r m a t i o n with regard to f e c u n d i t y , s u r v i v a l , resource u t i l i z a t i o n , and p r e d a t i o n can be e x t r a c t e d . Three case examples were s e l e c t e d f o r t h i s study to r e p r e s e n t c l a s s e s of r e a l ungulate systems. The f i r s t system (Chapter two) i s based on a long-term study of the Llano B a s i n w h i t e - t a i l e d deer (Odocoileus v i r g i n i a n u s texanus Mearns) i n Texas by Teer, Thomas, and Walker (1965). I t i s assumed to be r e p r e s e n t a t i v e of ungulate systems i n which changes i n year to year d e n s i t y are r e l a t e d to j u v e n i l e / a d u l t r a t i o , t o t a l d e n s i t y , and' some environmental f a c t o r ( i n t h i s case r a i n f a l l of the year preceding b i r t h s ) . C l a s s two systems, using Caughley's i n t e r a c t i v e growth model of a v e g e t a t i o n - d e e r system (Caughley, 1976) as an example, * r e p r e s e n t s s i t u a t i o n s where ungulates are c l o s e l y r e l a t e d t o t h e i r food source (Chapter t h r e e ) . The p a t t e r n of ungulate growth i s represented as a complex f u n c t i o n of the i n t r i n s i c dynamics of the p o p u l a t i o n i n t e r a c t i n g with the dynamics of the food supply. The t h i r d case example (Chapter four) i s based on an e i g h t year study of a wolf-ungulate system i n Alaska by Haber (1977). I t r e p r e s e n t s a c l a s s of ungulate systems i n which p r e d a t i o n p r e s s u r e may have a major i n f l u e n c e on the ungulate p o p u l a t i o n dynamics. In e f f e c t these cases r e p r e s e n t three fragmentary views of the b a s i c f o o d - u n g u l a t e - p r e d a t i o n food c h a i n . Each modelled 16 s i t u a t i o n i s a myth that can help us understand r e a l , more complex s i t u a t i o n s provided we can avoid the conceptual p i t f a l l s i n h e r e n t i n any s i m p l i f i e d views of the world. SCOPE OF THE INVESTIGATION Chapter two examines optimal harvest s t r a t e g i e s f o r a d u l t and y e a r l i n g deer i n a s t o c h a s t i c environment. A n a l y s i s of the p o p u l a t i o n system i s made to determine the response of the p o p u l a t i o n components ( j u v e n i l e s and a d u l t s ) to h a r v e s t i n g and to evalu a t e r e t u r n s from the system. S i m p l i f i e d s t r a t e g i e s based on age r a t i o s are compared to optimal s t r a t e g i e s . In Chapter three a two v a r i a b l e u n g u l a t e - v e g e t a t i o n model i s used to estimate optimal h a r v e s t s t r a t e g i e s i n r e l a t i o n to v a r y i n g assumptions about v e g e t a t i o n p r o d u c t i o n and u n c e r t a i n t y about p o p u l a t i o n parameters. Long-term y i e l d s from a s e r i e s of a l t e r n a t i v e p o l i c i e s that assume only the p o p u l a t i o n or the range can be monitored. Chapter four d e s c r i b e s a predator-prey system. The e f f e c t s of randomly f l u c t u a t i n g winter s e v e r i t y , a l t e r n a t i v e o b j e c t i v e f u n c t i o n s , and p r e d a t i o n parameters are considered i n the e s t i m a t i o n of optimal h a r v e s t i n g s t r a t e g i e s . The performance of s i m p l i f i e d s t r a t e g i e s i s again e v a l u a t e d . In Chapter f i v e the r e s u l t s are d i s c u s s e d i n the l i g h t of a c t u a l w i l d l i f e management p r a c t i c e s . Suggestions are made f o r implementing optimal s o l u t i o n s i n an adaptive management scheme. F i n a l l y , i n Chapter s i x c o n c l u s i o n s of t h i s study are s t a t e d . 18 CHAPTER 2 Ungulate P o p u l a t i o n Dynamics and O p t i m i z a t i o n Models 19 SUMMARY The o b j e c t i v e of t h i s chapter was to estimate o p t i m a l h a r v e s t i n g s t r a t e g i e s f o r a d u l t and y e a r l i n g deer in a s t o c h a s t i c environment. Data on the Llano Basin deer p o p u l a t i o n (Teer et a l . , 1965) were u t i l i z e d to c o n s t r u c t a two v a r i a b l e p o p u l a t i o n dynamics model. The model provided the b a s i s f o r e s t i m a t i n g optimal h a r v e s t i n g s t r a t e g i e s as a feedback f u n c t i o n of the c u r r e n t values of the s t a t e v a r i a b l e s (prefawning o l d e r deer and j u v e n i l e s ) , by employing s t o c h a s t i c dynamic programming. Optimal h a r v e s t s t r a t e g i e s were found to be i n s e n s i t i v e to assumptions about the p r o b a b i l i t y d i s t r i b u t i o n s of the s t o c h a s t i c v a r i a b l e ( r a i n f a l l ) . The response of the p o p u l a t i o n components to h a r v e s t i n g and the r e t u r n s obtained from a p p l y i n g optimal s t r a t e g i e s were explored through s i m u l a t i o n . S i m p l i f i e d h a r v e s t i n g s t r a t e g i e s based on a g e - r a t i o i n f o r m a t i o n as w e l l as a s i m p l i f i e d v e r s i o n based on optimal s t r a t e g i e s , but assuming p e r s i s t i n g e q u i l i b r i u m j u v e n i l e deer d e n s i t y , were compared to optimal s t r a t e g i e s through examining values of i n f o r m a t i o n . INTRODUCTION 20 Much of ungulate p o p u l a t i o n management i s aimed at p r o v i d i n g a maximum s u s t a i n e d y i e l d , a maximum crop that can be removed year a f t e r year without d r i v i n g the p o p u l a t i o n i n t o a continued d e c l i n e . A v a r i e t y of p o p u l a t i o n models are employed to p r e d i c t MSY. For example, A.S. Leopold (1955) and Dasmann (1964) advocated the use of the l o g i s t i c growth model i n b i g game management. However, most models are used o n l y to p r e d i c t optimum e q u i l i b r i u m h a r v e s t r a t e s , and few attempts have been made i n game management to develop h a r v e s t s t r a t e g i e s that s p e c i f y optimum har v e s t r a t e s under n o n - e q u i l i b r i u m c o n d i t i o n s . T h i s Chapter estimates a set of optimal h a r v e s t s t r a t e g i e s f o r an ungulate p o p u l a t i o n by using s t o c h a s t i c dynamic programming. As an example of a p p l y i n g t h i s o p t i m i z a t i o n technique to an ungulate p o p u l a t i o n model, data obtained from a long-term study of w h i t e - t a i l e d deer i n the Llano Basin of Texas (Teer e_t a l . , 1965) are used. I t i s of course not suggested that deer i n the Llano Basin should be harvested a c c o r d i n g to the f i n d i n g s o f t h i s study; the data are merely used to demonstrate that management s t r a t e g i e s can be d e r i v e d f o r t h i s and comparable r e a l world ungulate p o p u l a t i o n systems. In S e c t i o n 1, the model i s d e s c r i b e d and q u a n t i f i e d . S e c t i o n 2 d e s c r i b e s the o p t i m i z a t i o n procedure and presents the estimated o p t i m a l h a r v e s t i n g s t r a t e g i e s , under d i f f e r e n t assumptions about environmental v a r i a b i l i t y . A l s o , p r e d i c t i o n s from employing the optimal h a r v e s t i n g s t r a t e g i e s are presented. In S e c t i o n 3, values of in f o r m a t i o n are u t i l i z e d to compare the 21 optimal s t r a t e g i e s to s i m p l i f i e d and t r a d i t i o n a l h a r v e s t i n g p o l i c i e s . 1. THE POPULATION MODEL • A b a s i c s t r u c t u r a l u n i t of any ungulate p o p u l a t i o n i s the age c l a s s . At l e a s t some age c l a s s e s have measurable d i f f e r e n c e s i n r e p r o d u c t i v e and m o r t a l i t y r a t e s . Much of the computational complexity of p o p u l a t i o n models i s due to r e p r e s e n t a t i o n of age d i s t r i b u t i o n s and r e l a t e d a g e - s p e c i f i c parameters. I t i s important to ask whether t h i s complexity (and a s s o c i a t e d compounding of u n c e r t a i n t i e s through an i n c r e a s e i n the number of parameters to be estimated) has a s i g n i f i c a n t e f f e c t on the p r e d i c t i o n s . In deer p o p u l a t i o n s , the l a r g e s t a g e - s p e c i f i c d i f f e r e n c e s are between j u v e n i l e s , y e a r l i n g s , and a d u l t s , and v a r i a t i o n s of age composition w i t h i n the a d u l t pool can be expected to have r e l a t i v e l y minor e f f e c t s on o v e r a l l p o p u l a t i o n performance. While optimal open-loop p o l i c i e s can be computed f o r complex models, i t i s not known how to estimate feedback management p o l i c i e s f o r models with more than a few s t a t e v a r i a b l e s . Thus f o r the present deer p o p u l a t i o n model onl y three age groups ( j u v e n i l e s , y e a r l i n g s , and a d u l t s ) have been i n c l u d e d . Equations and parameter values used i n the model are subsequently d e s c r i b e d . P r o d u c t i o n r e l a t i o n s h i p The p r o d u c t i o n model f o r the ungulate p o p u l a t i o n s t a t e s a r e l a t i o n s h i p between the two v a r i a b l e ( o l d e r and j u v e n i l e deer) 22 breeding p o p u l a t i o n (At and Yt) and the number of fawns produced, with r a i n f a l l adding environmental ( s t o c h a s t i c ) v a r i a b i l i t y to the model system: F t = ( A t • b a ( N ) S i a ) + ( Y t • b y (N) S i y ) (1) where Ft = number of fawns born per square mile at p a r t u r i t i o n time t and s u r v i v i n g to the f a l l At = number of o l d e r deer ( y e a r l i n g s and a d u l t s ) per square mile j u s t p r i o r to p a r t u r i t i o n Yt = number of j u v e n i l e deer per square mil e having t h e i r f i r s t b i r t h d a y j u s t p r i o r to p a r t u r i t i o n s l a ' s l y = d i f f e r e n t i a l s u r v i v a l r a t e s of fawns from p a r t u r i t i o n to f a l l . I t i s assumed that deer have d e n s i t y and r a i n f a l l dependent r e p r o d u c t i o n [ b a ( N ) , b y ( N ) ] , and that p o p u l a t i o n d e n s i t y e f f e c t s on b i r t h r a t e s remain constant from year to yea r . Over the r e l e v a n t range of breeding p o p u l a t i o n d e n s i t y (N), s o c i a l e f f e c t s on r e p r o d u c t i o n are handled i m p l i c i t l y by assuming that maximum re p r o d u c t i v e p o t e n t i a l f o r an age group i s a l i n e a r l y d e creasing f u n c t i o n ( F i g u r e 1). T h i s assumption i s commonly used i n ungulate p o p u l a t i o n models (Gross, R o e l l e , and W i l l i a m s , 1973; Walters and Gross, 1972). Considerable e m p i r i c a l j u s t i f i c a t i o n f o r t h i s r e l a t i o n s h i p i s a v a i l a b l e f o r ungulate p o p u l a t i o n s (Gross, 1969; F i l o n o v and Zykov, 1974). Maximum b i r t h r a t e s to be expected at low p o p u l a t i o n d e n s i t y were estimated f o r j u v e n i l e s and o l d e r deer from Teer e_t _al. (1965) as 0.10 and 0.75 per animal r e s p e c t i v e l y . Estimates are based on pregnancy r a t e s , at average d e n s i t y , of 0.14 and 1.4 embryos per j u v e n i l e and o l d e r doe r e s p e c t i v e l y . Figure 1. Maximum older deer and juvenile deer reproductive rate i s assumed to decrease when breeding density exceeds some threshold l e v e l N Q, and cease comple-t e l y i f density exceeds N m: b (_N) =< b max a b max a i N - N 1- oa N - N ma oa i f 0<N<N oa i f N <N<N oa ma i f N>N ma b (N) =< b max y b max N - N _ „ 1 2 Z Nmy" N o y I 0 i f 0<N<N oy i f N <N<N oy my i f N>N my 23a T O T A L DENSITY (N/SQMI) 24 The i n f l u e n c e of r a i n f a l l of the preceding year on r e p r o d u c t i v e r a t e s of the deer p o p u l a t i o n has been demonstrated by Teer et a l . (1965). T h e i r data suggest an i n c r e a s e i n p r o d u c t i o n with i n c r e a s i n g r a i n f a l l i n the year preceding b i r t h s (Figure 2). Thus the d e n s i t y dependent b i r t h r a t e s are adjusted f o r the impact of r a i n f a l l of the preceding year by m u l t i p l y i n g b i r t h s by an adjustment f a c t o r dependent on r a i n f a l l (Table I ) . The p r o b a b i l i t i e s of each r a i n f a l l c l a s s o c c u r r i n g were taken from the observed f r e q u e n c i e s . Figure 3 shows three a l t e r n a t i v e r a i n f a l l p r o b a b i l i t y t e s t d i s t r i b u t i o n s used f o r t h i s study. The o p t i m i s t i c d i s t r i b u t i o n assumes a high p r o b a b i l i t y of annual r a i n f a l l exceeding 31 i nches, whereas the p e s s i m i s t i c d i s t r i b u t i o n assumes a high p r o b a b i l i t y of low r a i n f a l l . The n a t u r a l summer fawn s u r v i v a l r a t e s ( S i a = 76 percent and s l y = 74 percent) are assumed to be constant. S l i g h t l y higher m o r t a l i t y i s a t t r i b u t e d to fawns born to does having t h e i r f i r s t b i r t h d a y j u s t p r i o r to p a r t u r i t i o n (26 p e r c e n t ) . These estimates are d e r i v e d from fawn l o s s data f o r the f i v e year o b s e r v a t i o n p e r i o d reported by Teer e_t a l . (1965: 49). Dynamic Model For the two v a r i a b l e system, s t a t e t r a n s i t i o n s can be d e s c r i b e d as: At+1 = (A t • S 3 a - a H t ) S 4 a + (Y t • S 3 y - y H t ) S 4 y (2) Y t+1 = F t • S 2 (3) F i g u r e 2. The impact of r a i n f a l l of the preceding on w h i t e - t a i l e d deer p r o d u c t i v i t l y i n the Llano B a s i n . ( O r i g i n a l data from Teer et 1965) . 25a. 40 H 2 i -z LU O cr LU CL 20H 10 A ° / / / / O / / o — i 1 1 1 r 10 20 30 RAINFALL IN YEAR PRECEDING BIRTH (in.) ' i 40 26 Table I . Impact of r a i n f a l l of the preceding year on deer b i r t h s ( O r i g i n a l data from Teer e t al.,1965). R a i n f a l l Annual r a i n - F r e q . % fawns i n k^ 1 c l a s s f a l l (in.) f a l l p o pln. High >31 0.375 37 1.28 Average 21 - 31 0.375 29 1.00 Low <21 0.250 20 0.69 s t o c h a s t i c b i r t h adjustment f a c t o r F i g u r e 3 . Three p r o b a b i l i t y d i s t r i b u t i o n s of r a i n f a l l i n year preceding b i r t h s used f o r t h i s study. 0.8 A OPTIMISTIC 0.6 A CD 0.4 A < CO. o °- 0.2 0.0 0.8 A 0.6 H 0.4 A 0.2 A 0.0 OBSERVED 0.8 0.6 A 0.4 J 0.2 A CO-PESSIMISTIC >3I 21-31 *20 >3I 21-31 <20 >3I 21-31 <20 RAINFALL IN YEAR PRECEDING BIRTH (in.) 28 where S3a = Summer s u r v i v a l r a t e of a d u l t s S3y = Summer s u r v i v a l r a t e of y e a r l i n g s s 4 a = Winter s u r v i v a l r a t e of a d u l t s S4y = Winter s u r v i v a l r a t e of y e a r l i n g s S2 = Winter s u r v i v a l r a t e of fawns a H t = Number of a d u l t s per square mile removed by h a r v e s t i n g yHt = Number of y e a r l i n g s per square mile removed by h a r v e s t i n g The extent of n a t u r a l m o r t a l i t y of y e a r l i n g s and a d u l t s that occurs a n n u a l l y was d i v i d e d i n t o summer m o r t a l i t y and posthunting winter m o r t a l i t y . Only a f r a c t i o n of the a d u l t s (96 percent) and y e a r l i n g s (93 p ercent) w i l l s u r v i v e to the f a l l forming the f a l l p o p u l a t i o n . A number are removed by hunters ( aHt and yHt) from each age group. Of the remainder about 85 percent w i l l s u r v i v e the winter to form the f o l l o w i n g s p r i n g ' s prefawning o l d e r deer p o p u l a t i o n ( A t + i ) . M o r t a l i t y estimates were d e r i v e d from d i f f e r e n c e s i n the Llano B a s i n p o p u l a t i o n standing crops from year to year (Teer e_t a_l., 1965: 50). Equation (2) assumes in e f f e c t that hunting m o r t a l i t y occurs over such a s h o r t p e r i o d of time (15-46 day hunting season) t h a t o t h e r sources of m o r t a l i t y during that p e r i o d may be n e g l e c t e d . T h i s assumption i s commonly made i n the development of game management models (Brown e_t a_l., 1976). Of the fawns ( F t ) , only a f r a c t i o n s u r v i v e the winter to form next year's prefawning j u v e n i l e p o p u l a t i o n ( Y t + i ) . The n a t u r a l winter fawn m o r t a l i t y (I - S 2 ) i s assumed to be about 18 29 percent. Table II summarizes parameter values used f o r the deer p o p u l a t i o n model. Model behaviour One way to v i s u a l i z e the combined e f f e c t s of p r o d u c t i o n and p o p u l a t i o n components on the behaviour of the model system i s by using phase plane a n a l y s i s ( H o l l i n g , 1973). For the model above t h i s i s simply a p l o t of the numbers of o l d e r deer (A) a g a i n s t the numbers of j u v e n i l e s (Y), with s u c c e s s i v e c o o r d i n a t e p a i r s r e p r e s e n t i n g d i f f e r e n t times. As time proceeds the changing combinations of v a l u e s t r a c e a t r a j e c t o r y on the phase p l a n e . The phase plane can be p a r t i t i o n e d by two c r i t i c a l l i n e s , or i s o c l i n e s , t h a t r e p r e s e n t no change from one year to the next of the two p o p u l a t i o n components. I n t e r s e c t i o n of the i s o c l i n e s d e f i n e s the e q u i l i b r i u m v a l u e s f o r the model system. No change from one year to the next of the p o p u l a t i o n components i m p l i e s : At = A t + i = A or Y t = Y t+1 = Y. To f i n d the o l d e r animal i s o c l i n e , we s u b s t i t u t e the f i r s t of these c o n d i t i o n s i n equation (2) to g i v e : Rearranging terms a l g e b r a i c a l l y , the equation f o r the o l d e r animal i s o c l i n e becomes A = A • S 3 a • S 4 a + Y t • S 3 y • S 4 y (4) (5) 3 0 Table I I . Parameter v a l u e s used f o r deer p o p u l a t i o n model ( o r i g i n a l data from Teer e t al.,1965) Parameter Fawns Age group J u v e n i l e s Older deer Max. fawn prod-u c t i o n r a t e per animal b max=0.10 b max=0.75 Density/sqmi above which r e p r o d u c t i o n begins t o decrease N =20.0 N =50.0 oy oa Density/sqmi above which r e p r o d u c t i o n ceases N =80.0 my N =170.0 ma D i f f e r e n t i a l fawn summer s u r v i v a l r a t e S.v.. =0.738 S, =0.74 9 l a Summer s u r v i v a l r a t e S, =0.930 3y S 0 =0.960 3a Winter s u r v i v a l r a t e S„=0.823 S, =0.846 S. =0.853 4y 4a 31 Since the survival fractions are constants we can define j_ S 3 y . S 4 y J r i - s 3 a . S 4 a l and substitute into equation (5) we have c • A, or A = Yt/c ( 6 ) This equation for the i s o c l i n e states that the equilibrium number of older deer (A) is proportional to the number of juveniles (Yt) with the proportionality constant (1/c) being a function of survival and independent of b i r t h s . The l a t t e r property implies also that the i s o c l i n e for A is independent of the stochastic variable k r. Sim i l a r l y substituting equation (1) into equation (3), and setting Yt = Yt+1 = Y, the juvenile i s o c l i n e equation becomes Y = [A t • b a(N) • S i a + Y • b y(N) • S i y] • S 2 (7) Rearranging the terms the equation for the juvenile c r i t i c a l l i n e i s stating that the equilibrium number of juveniles (Y) depends both on survival and density dependent births of older and juvenile deer, making i t c u r v i l i n e a r and also dependent on the r a i n f a l l variable k r. Solutions for equations 6 and 8 in r e l a t i o n to the r a i n f a l l variable k r are shown in Figure 4 . The older animal i s o c l i n e is the same for a l l three r a i n f a l l situations. The juvenile i s o c l i n e s , since they are dependent on births and therefore r a i n f a l l , s h i f t to a lower peak as r a i n f a l l conditions get worse or F i g u r e 4. I s o c l i n e s ( A t + i = A t and Y t + ^ = Y t ) , e q u i l i b r i a (A* and Y*), and t r a j e c t o r i e s (heavy l i n e s ) f o r d e t e r m i n i s t i c two component deer popu-l a t i o n i n r e l a t i o n to r a i n f a l l c o n d i t i o n s a) k r=1.28: b) k r=1.00 and c) k r=0.69. 32a 33 (Figure 4a, 4b, and 4 c ) . The t r a j e c t o r i e s f o r the three d e t e r m i n i s t i c s i t u a t i o n s were obtained through computer s i m u l a t i o n by c a l c u l a t i n g each value of A and Y from preceding v a l u e s , using annual increments. The r e s u l t s show t r a j e c t o r i e s l e a d i n g to a s t a b l e node ( H o l l i n g , 1973; May, 1974) A*, Y* i n which there are no o s c i l l a t i o n s and the node i s approached m o n o t o n i c a l l y . In b i o l o g i c a l terms t h i s i n d i c a t e s that a f t e r an i n i t i a l p e r i o d the p o p u l a t i o n a t t a i n s a s t a b l e age d i s t r i b u t i o n . T h i s o b s e r v a t i o n i s c o n s i s t e n t with p o p u l a t i o n age s t r u c t u r e a n a l y s i s employing age s t r u c t u r e m a t r i c e s (e.g., L e s l i e , 1945; Lewis, 1942; Beddington, 1974; Beddington and T a y l o r , 1973). E q u i l i b r i u m v a l u e s (A* and Y*, Table I I I ) f o r parameter v a l u e s used i n t h i s study depend on the r a i n f a l l v a r i a b l e k r; as expected, h i g h e s t values are a t t a i n e d when r a i n f a l l c o n d i t i o n s are f a v o u r a b l e ( k r = 1.28). Simulations i n c l u d i n g s t o c h a s t i c v a r i a t i o n (by employing a random number procedure to generate r a i n f a l l c o n d i t i o n s with a p p r o p r i a t e p r o b a b i l i t i e s from F i g u r e 3) i n d i c a t e that the speed wi t h which the p o p u l a t i o n approaches i t s s t a b l e age d i s t r i b u t i o n depends on the p r o b a b i l i t y d i s t r i b u t i o n ( F i g u r e 5). S t a r t i n g from low i n i t i a l c o n d i t i o n s n a t u r a l l e v e l s are reached f a s t e s t with o p t i m a l r a i n f a l l p r o b a b i l i t y c o n d i t i o n s ( F i g u r e 5a). The converse i s true f o r the p e s s i m i s t i c r a i n f a l l p r o b a b i l i t y d i s t r i b u t i o n ( F i g u r e 5 c ) . Absolute s t a b i l i t y i s of course never reached due to the randomly f l u c t u a t i n g r a i n f a l l c o n d i t i o n s i n d i c a t e d by the jumps i n the a d u l t and j u v e n i l e time p l o t s ( F i g u r e 5 ) . 34 Table I I I . S t a b l e nodes f o r unharvested deer system i n r e l a t i o n t o r a i n f a l l v a r i a b l e k (determi-n i s t i c c a s e s ) . r k r A Y 1.28 1.00 0.69 100.24 89. 02 67.72 23.08 20.63 15.60 F i g u r e 5. 50 year s i m u l a t i o n s of unharvested two component deer p o p u l a t i o n i n r e l a t i o n t o p r o b a b i l i t y d i s -t r i b u t i o n of r a i n f a l l i n year p r e c e d i n g b i r t h s a) o p t i m i s t i c , b) observed, and c) p e s s i m i s t i c . q MRX= 100.0 Y MRX= 28.7 a) TIME fl MflX= 95.5'.. Y MAX= 28.7" b) TIME fl MRX= 80.0 Y MRX= 29.0 c) 11 21 31 41 51 TIME 36 2. STOCHASTIC DYNAMIC PROGRAMMING Dynamic programming i s an approach to o p t i m i z a t i o n f o r systems i n which a s e r i e s of d e c i s i o n s must be made s e q u e n t i a l l y , and where each d e c i s i o n a f f e c t s f u t u r e system s t a t e s and thus f u t u r e d e c i s i o n s . The b a s i c concept of r e c u r s i v e o p t i m i z a t i o n was f i r s t i n t roduced by Bellman who s t a t e d the p r i n c i p l e of o p t i n i a l i t y (Bellman, 1957 :83): "An optimal p o l i c y has the pro p e r t y t h a t whatever the i n i t i a l s t a t e and i n i t i a l d e c i s i o n are, the remaining d e c i s i o n s must c o n s t i t u t e an optimal p o l i c y with regard to the s t a t e r e s u l t i n g from the f i r s t d e c i s i o n . " In the context used here, dynamic programming estimates optimal harvest r a t e s f o r o l d e r deer and j u v e n i l e s f o r any s t a t e combination (At/Yt) f o r the s p e c i f i e d o b j e c t i v e of maximizing r e t u r n s (number of animals harvested) t o t a l l e d across age groups and over time. In the present problem, f u t u r e s t a t e s are u n c e r t a i n and thus optimal s o l u t i o n s are found i n the face of u n c e r t a i n t y ; the goal becomes to maximize the expected t o t a l l e d r e t u r n s . F o l l o w i n g the example of Anderson (1975), the d i s c u s s i o n of dynamic programming i s based on Nemhauser's n o t a t i o n f o r m u l t i - s t a t e v a r i a b l e , s t o c h a s t i c dynamic programming (Nemhauser, 1967). O p t i m i z a t i o n model The two s t a t e v a r i a b l e model developed i n S e c t i o n 1 was employed to estimate optimal h a r v e s t s t r a t e g i e s f o r a d u l t and y e a r l i n g deer. To summarize: the o l d e r deer p o p u l a t i o n d e n s i t y At+i i n year t+1 was expressed as a f u n c t i o n of o l d e r deer 37 d e n s i t y i n year t , j u v e n i l e s i n year t , and the l e v e l s of e x p l o i t a t i o n i n year t , a D t and y D t . The j u v e n i l e deer d e n s i t y Y^+i i n year t+1 was expressed as a f u n c t i o n of fawns born i n year t , F t / which i n turn i s a f u n c t i o n of At/ Yt/ and the s t o c h a s t i c r a i n f a l l v a r i a b l e Rt-l« L e v e l s of e x p l o i t a t i o n f o r each s t a t e v a r i a b l e i n a given year are d e f i n e d as the d e c i s i o n v a r i a b l e s . Time r e s o l u t i o n of deer p o p u l a t i o n dynamics was de f i n e d i n terms of annual time p e r i o d s from prefawning i n one year to prefawning of the f o l l o w i n g year. For the dynamic programming model, these annual p e r i o d s are the stages numbered from 1 t o L where L i s a f i n i t e time planning h o r i z o n . The f o l l o w i n g stage t r a n s f o r m a t i o n equations (Nemhauser, 1967:26) based on the p r e v i o u s l y d e f i n e d dynamic model (equations 2 and 3) were used: A n+1 = ( A n * 0.96 - a H n ) • 0.853 + ( Y n 0.93 - y H n ) . 0.846 (9) Y n+1 = Fn ' 0.823 (10) where aHn = ( aD n - 1) • 0.05 yHn = (yD n " D * 0.05 F n = (A n • b a(N) • 0.749 • k n ) + ( Y n . b y(N) • 0.738 • k n ) k n = b i r t h adjustment f a c t o r as a f u n c t i o n of the s t o c h a s t i c r a i n f a l l v a r i a b l e L e t us denote xn as the two dimensional s t a t e vector c o n t a i n i n g A n and Y n, and t n (Nemhauser, 1967: 26) as the s t a t e t r a n s f o r m a t i o n f u n c t i o n [ x n + i = t n(j£ n)]. Assuming that the management o b j e c t i v e i s to maximize the expected r e t u r n s of a d u l t s and y e a r l i n g s we can d e f i n e the short-term o b j e c t i v e 38 f u n c t i o n s r e s u l t i n g from d e c i s i o n s made i n stage n as: a r n = ^ P n ( k n ) • a r n ( i i n f a D n ' kn) k and y r n = .S?n( kn) * yrn(*nry'Dnr k n ) k s t a t i n g t h at f o r the s t o c h a s t i c s i t u a t i o n the expected annual r e t u r n s ( a f n , y r n ) are determined by simply weighting each r e t u r n ( a r n / y r n ) by the p r o b a b i l i t y of the random outcome k n . The r e t u r n s ( a r n , y r n ) r e s u l t i n g from d e c i s i o n s ( aD n, yD n) at stage n are a f u n c t i o n of the system s t a t e v e c t o r xn, the d e c i s i o n v a r i a b l e ( aD n, y D n ) , and the random outcome k n . The long-term o b j e c t i v e to maximize the expected f u t u r e v a l u e s ( i n a s t a t i s t i c a l sense) o f summed annual r e t u r n s can be w r i t t e n as the fundamental r e c u r s i o n equations (as a f u n c t i o n of the s t a t e v e c t o r ) a f t e r Nemhauser (1967: 155): a f n ( x n ) = max 1L- p n ( k n ) • a Q n ( x n / a Dn» kn) 1 £ n £ L a D n k n (over a l l a D n ) and y f n ( x n ) = max 2. p n ( k n ) • yQn(£n' y D n ' kn) 1 ^ n z L y D n k n (over a l l y D n ) where the expected f u t u r e values are: aQntiinf a D n ' k n l = a rnt2£n' a Dn* k n J + a r n - l t t n ( i i n ' a D n ' k n ) ] yQn^iinf y D n ' k n l = y rnt x.nr yDn' k n] + y f n - ± [ t n ()c n, y D n f k n ) ] 2 Z n z_ L and aQlUl* a D l ' k l ) = a r l ( i i l r a D l ' k l ) + a vl(i£l) y Q 1 ( x 1 , y D l f k i ) = y r 1 ( x 1 , y D l r k i ) + y V 1 ( x 1 ) where aVi(2£l) and yVi(jci) are a r b i t r a r y endpoint values assigned across the i n i t i a l s t a t e v e c t o r x i . 39 Computational procedure A computer a l g o r i t h m was w r i t t e n i n FORTRAN to f i n d numerical s o l u t i o n s of the optimal r e t u r n f u n c t i o n s -,f T i(x T i) and y f l / i i L ) a n ^ t n e a s s o c i a t e d optimal d e c i s i o n sequences a D ° L ' a D ° L - l " - - ' a D ° n " - " a D l a n d y D ° L ' y ^ L - l ' - ' - i y D ° n , . . . , yD^. The numerical procedure r e q u i r e d d i s c r e t i z a t i o n of the continuous s t a t e and d e c i s i o n v a r i a b l e s . The idea behind d i s c r e t i z a t i o n i s the same as used f o r s o l v i n g d i f f e r e n t i a l equations t a k i n g s h o r t time s t e p s . For t h i s study best r e s u l t s were obtained by using 13 d i s c r e t e o l d e r deer l e v e l s and 11 j u v e n i l e deer l e v e l s . Thus the o l d e r deer d e n s i t y (A) was v a r i e d between zero and 120 deer per square mile i n increments of 10. J u v e n i l e s (Y) v a r i e d between zero and 30 per square mile i n increments of three. L e v e l s of e x p l o i t a t i o n between zero and 50 percent (D n = 1, 2, 3,..., 11) i n increments of f i v e percent were used f o r both age groups. A f t e r s e v e r a l backward r e c u r s i o n s t e p s , the endpoint values ceased to have an e f f e c t , and the optimal d e c i s i o n s f o r each o l d e r d e e r - j u v e n i l e deer combination became independent of the time step. In other words the optimal c o n t r o l p o l i c i e s became " s t a t i o n a r y " , or s t a b i l i z e d (e.g., D ° L = D ° L _ i ) . In the p r e s e n t study t h i s occurred a f t e r 10-15 stages. T h i s phenomenon im p l i e s t h a t optimal d e c i s i o n s are s o l e l y a feedback f u n c t i o n of the s t a t e v e c t o r , D°(x), r a t h e r than time. A key f e a t u r e of the o u t l i n e d procedure i s t h a t a l l p o s s i b l e f u t u r e s are e x p l i c i t l y c o n s idered (Walters, 1975; Anderson, 1975). Optimal s t r a t e g i e s m, ^. n , . . „ n°(x) and „D°(x) can be presented The optimal d e c i s i o n s a u K—' Y — ^ 4 0 g r a p h i c a l l y as i s o p l e t h s on a plane with prefawning o l d e r deer (A) on the X a x i s and prefawning j u v e n i l e s (Y) on the Y a x i s . The o p t i m a l h a r v e s t r a t e s (h.r.) f o l l o w from the o p t i m a l d e c i s i o n s : h.r. = (D° - 1) • 0.05. Figure 6 p r e s e n t s optimal d e c i s i o n s aD°(50 and yD°()c) in r e l a t i o n to p r o b a b i l i t y d i s t r i b u t i o n s of r a i n f a l l i n the year preceding b i r t h s ( F i g u r e 3). The l i n e s r e p r e s e n t the unsmoothed h a r v e s t i n g d e c i s i o n i s o p l e t h s as contoured by the computer. The exact s o l u t i o n s should be smooth i s o p l e t h s , the jumps are due to the d i s c r e t i z a t i o n approximation used i n the dynamic programming c a l c u l a t i o n s . I t can be seen that optimal d e c i s i o n s are almost independent of the p r o b a b i l i t y d i s t r i b u t i o n s , as the forms of the i s o p l e t h s do not change s u b s t a n t i a l l y as the r a i n f a l l assumptions are changed from o p t i m i s t i c to p e s s i m i s t i c . For p r a c t i c a l ungulate p o p u l a t i o n management t h i s i m p l i e s that i t i s not necessary to be concerned with the exact frequency of occurrence of environmental c o n d i t i o n s ( i n t h i s case r a i n f a l l ) i n order to make a h a r v e s t i n g d e c i s i o n . For any given d e n s i t y of j u v e n i l e s , the s t r a t e g i e s f o r a d u l t deer [ aD°(x)] e s s e n t i a l l y c a l l f o r a f i x e d escapement of o l d e r deer or an o l d e r deer d e n s i t y below which no harvest i s taken. The f i x e d escapement i s s e n s i t i v e to the j u v e n i l e deer d e n s i t y , being around 20 per square mile i f j u v e n i l e s are abundant and 40 per square m i l e i f j u v e n i l e s are s c a r c e . I f o l d e r deer are more abundant than the f i x e d escapement l e v e l , i n c r e a s i n g h a r v e s t r a t e s up to the h i g h e s t f e a s i b l e e x p l o i t a t i o n l e v e l (50 percent i n the example) are c a l l e d f o r . F i g u r e 6 . Optimal h a r v e s t i n g d e c i s i o n s a D ° Q<) a n c ^ yD° (x) , estimated from s t o c h a s t i c dynamic programming, i n r e l a t i o n t o r a i n f a l l p r o b a b i l i t y assumptions. Optimal h a r v e s t i n g l e v e l s are based on the s i z e bf the prefawning o l d e r deer d e n s i t y (A/sqmi) and j u v e n i l e d e n s i t y (Y/sqmi). C i r c l e s r e f e r t o o p t i -mal e q u i l i b r i a . a D ° ( x ) OPTIMISTIC PESSIMISTIC 42 The optimal yearling harvest [vD°(x)] i s almost independent of juvenile deer density (indicated by the near v e r t i c a l harvest isopleths) for a l l r a i n f a l l p r o b a b i l i t y d i s t r i b u t i o n s . The strategies for yearlings c a l l for a fixed escapement of juveniles that is dependent on the density of older deer. As expected the escapement being lowest i f older deer density drops off and juveniles are abundant. The optimal e q u i l i b r i a (Figure 6) refer to the values the older deer and juvenile deer w i l l reach by s t a r t i n g at any i n i t i a l condition of the phase plane and applying the optimal feedback strategies [ aD°(x) and yD°(x)], while maintaining the respective r a i n f a l l p r o b a b i l i t y assumptions. Of course for the present stochastic s i t u a t i o n the equilibrium point w i l l not be reached because of the randomly fluctuating r a i n f a l l v ariable. If we superimpose the li n e Y = 60 - A on the optimal strategy isopleth diagram (Figure 7), an interesting phenomenon emerges. Any point along this l i n e represents t o t a l population size of 60 ( i . e . , A+Y = 60). The optimal adult harvest strategy remains constant along this l i n e (20 percent), indicating that one should harvest adults at the same rate no matter what the juvenile to older deer r a t i o i s , provided simply that the t o t a l population is 60. This observation is contrary to the popular b e l i e f of w i l d l i f e managers that i f r a t i o s get small, harvest rates should be altered. With regard to the two variable optimal harvesting strategies, the observation implies that by collapsing the state information (At/Yt) into a one variable ( t o t a l population) F i g u r e 7. Constant p o p u l a t i o n l i n e (A+Y=60) superimposed on o p t i m a l a d u l t h a r v e s t i s o p l e t h diagram. 44 system, we can do j u s t as w e l l . T h i s p o i n t i s f u r t h e r pursued in S e c t i o n 3. P r e d i c t i o n s To e v a l u a t e performance of the optimal h a r v e s t i n g s t r a t e g i e s , very long s i m u l a t i o n runs (100 times 50 years) were a p p l i e d to the s t a t e t r a n s f o r m a t i o n equations 9 and 10. I n i t i a l c o n d i t i o n s f o r each 50 year run were se t near (A,Y) e q u i l i b r i u m . A random number procedure that generated r a i n f a l l d i s t r i b u t i o n s with the a p p r o p r i a t e p r o b a b i l i t i e s (Figure 3) was u t i l i z e d . F i g u r e 8 p r e s e n t s p r e d i c t e d means (+ 2SE) from 5000 year s i m u l a t i o n t r i a l s , d e s c r i b e d above, in r e l a t i o n to p r o b a b i l i t y d i s t r i b u t i o n s f o r a d u l t and y e a r l i n g h a r v e s t as w e l l as o l d e r deer and j u v e n i l e d e n s i t i e s a s s o c i a t e d with the h a r v e s t . These r e s u l t s show t h a t harvest r e t u r n s and j u v e n i l e d e n s i t y are q u i t e s e n s i t i v e to the r a i n f a l l p r o b a b i l i t y assumptions. For example the mean annual a d u l t h a r v e s t f o r the p e s s i m i s t i c assumption i s o n l y two-thirds of the mean annual a d u l t harvest obtained under the o p t i m i s t i c r a i n f a l l assumption. T h i s i s a t t r i b u t a b l e to the poor r e c r u i t m e n t r e s u l t i n g from frequent fawn crop r e d u c t i o n s . The o l d e r deer d e n s i t y on the other hand i s completely i n s e n s i t i v e to the r a i n f a l l p r o b a b i l i t y assumptions. In other words, the optimal p o l i c y maintains a f a i r l y constant o l d e r deer d e n s i t y r e g a r d l e s s of the environmental c o n d i t i o n s , by m anipulating h a r v e s t r a t e s as a f u n c t i o n of the s t a t e v a r i a b l e s . The average t o t a l ( a d u l t p l u s y e a r l i n g ) h a r v e s t f o r the observed r a i n f a l l assumption i s 8.9 per square mile or 15 percent F i g u r e 8 . P r e d i c t e d means Qf2 S.E.) i n r e l a t i o n t o r a i n f a l l p r o b a b i l i t y a s s u m p t i o n s f o r a) mean a n n u a l a d u l t h a r v e s t , b) mean o l d e r d e e r d e n s i t y ( A / s q m i ) , c) mean a n n u a l y e a r l i n g h a r v e s t , a n d d) mean j u v e n i l e d e n s i t y ( Y / s q m i ) . V a l u e s w e r e o b t a i n e d f r o m 100 f i f t y - y e a r s i m u l a t i o n s e m p l o y i n g o p t i m a l h a r v e s t s t r a t e g i e s . MEAN ANNUAL YEARLING HARVEST/SQ M 01 b OJ bi b 4> bi cn b _ L _ b > > o H ,o C D . CO l-CM "0 ;o O CD > CD H -< O CO H CD cz "0 J m co MEAN SPRING JUVENILE DEER/SQMI Ol b o "D H b O _L_ 00 b to o o b ro b I—O—I o 03. CO H C H m' co t-CH MEAN ANNUAL ADULT HARVEST/SQMI OJ b OJ bi b cn b I—O—I MEAN SPRING OLDER DEER/SQMI OJ K J> 8 § 8 5 S 46 of the t o t a l average prefawning p o p u l a t i o n s i z e (Figure 8 ) . T h i s i s s l i g h t l y below the d e t e r m i n i s t i c maximum s u s t a i n a b l e y i e l d r a t e of 16.5 p e r c e n t . In other words optimal h a r v e s t i n g s t r a t e g i e s f o r the s t o c h a s t i c s i t u a t i o n are s l i g h t l y more c o n s e r v a t i v e than MSY l e v e l s . P r e d i c t e d p o p u l a t i o n trends under optimal h a r v e s t i n g s t r a t e g i e s f o r d i f f e r e n t r a i n f a l l assumptions are shown in Figure 9. E s s e n t i a l l y the model p r e d i c t s that optimal h a r v e s t i n g s t r a t e g i e s should hold the o l d e r deer d e n s i t y (A) r e l a t i v e l y s t a b l e , with i n c r e a s i n g f l u c t u a t i o n of j u v e n i l e s (Y) as the p r o b a b i l i t y of fawn r e d u c t i o n s i s i n c r e a s e d . 3. VALUES OF INFORMATION An e s s e n t i a l q u e s t i o n f o r p o p u l a t i o n management i s how much in f o r m a t i o n i s needed each year to decide on a good p o l i c y f o r that year. Can one compress the i n f o r m a t i o n of a management system to produce s i m p l i f i e d management s t r a t e g i e s , and i f so, how do these s i m p l i f i e d s t r a t e g i e s compare with o p t i m a l s t r a t e g i e s t h at are i n f o r m a t i o n r i c h ? These q u e s t i o n s are addressed i n t h i s s e c t i o n . S i m p l i f i e d h a r v e s t i n g s t r a t e g i e s Age r a t i o s are o f t e n used i n w i l d l i f e management to i n d i c a t e the p r o d u c t i v i t y of a p o p u l a t i o n . I t i s t h e r e f o r e c o n c e i v a b l e to d e r i v e s i m p l i f i e d h a r v e s t i n g s t r a t e g i e s based on optimal s t r a t e g i e s (from S e c t i o n 2) t h a t use age r a t i o s as i n f o r m a t i o n input to make h a r v e s t i n g d e c i s i o n s . Three c l a s s e s of animals are d e f i n e d by Hanson (1963): "(1) j u v e n i l e s are l e s s than f u l l y grown animals; (2) subadults are F i g u r e 9. 50 year s i m u l a t i o n s of o p t i m a l l y h a r v e s t e d deer p o p u l a t i o n s f o r d i f f e r e n t p r o b a b i l i t y d i s t r i b u -t i o n s of r a i n f a l l i n year preceding b i r t h s , a) o p t i m i s t i c , b) observed, and c) p e s s i m i s t i c r a i n f a l l d i s t r i b u t i o n s . fl MflX= 48.7 Y MflX= 25.8 <+7cx a) 1 11 21 31 41 51 TIME fl MflX= 48. r. Y MflX= 24.3 . .5 » ; / » 1 u / \ A i , ' i \ i i " 11 i 1 >'i / i u i' i ' 1 { x c) 1 11 21 31 41 51 TIME 48 e s s e n t i a l l y f u l l y grown, but the m a j o r i t y of t h e i r cohort have not completed t h e i r f i r s t breeding season; (3) a d u l t s are f u l l y grown and the m a j o r i t y of t h e i r cohort have completed one or more breeding seasons." F o l l o w i n g Hanson (1963) and Caughley (1974) combining a d u l t s and subadults to form mature animals, the primary age r a t i o i s d e f i n e d as the r a t i o of j u v e n i l e s to mature animals, or j u v e n i l e s (Y) to o l d e r deer (A). Two s i m p l i f i e d h a r v e s t i n g s t r a t e g i e s were developed t h a t use the primary age r a t i o as s o l e i n f o r m a t i o n i n p u t . The f i r s t set of s i m p l i f i e d s t r a t e g i e s (Dgi) was based on a d u l t and y e a r l i n g o p t i m a l s t r a t e g i e s (Figure 6) along the c r i t i c a l l i n e Yt+1 = Yt ( F i g u r e 4). Superimposing Yt+i = Yt on the optimal d e c i s i o n i s o p l e t h diagrams, the s i m p l i f i e d s t r a t e g i e s are d e f i n e d as the h a r v e s t r a t e s of the Y/A r a t i o s where op t i m a l h a r v e s t s t r a t e g y i s o p l e t h s i n t e r s e c t the c r i t i c a l l i n e s (Figure 10). In s i m u l a t i o n t r i a l s i t was noted t h a t ( Y t f A t ) t r a j e c t o r i e s tend to move onto and along the (Yt+i = Yt) i s o c l i n e i n the phase space (F i g u r e 4); p r o v i d e d t h a t the Y/A r a t i o v a r i e s c o n s i d e r a b l y and m o n o t o n i c a l l y along t h i s i s o c l i n e , the r a t i o w i l l (most of the time) a c t as a good s t a t e index. The a p p r o p r i a t e v a l u e s i n r e l a t i o n to the r a i n f a l l assumptions are given i n Table IV. P l o t t i n g h a r v e s t r a t e s a g a i n s t the age r a t i o y i e l d s the r e l a t i o n s h i p presented i n F i g u r e 11a. Depending on the r a i n f a l l p r o b a b i l i t y d i s t r i b u t i o n s two t h r e s h o l d s emerge d i c t a t i n g to harvest hard (h.r. = 0.50) i f the r a t i o i s below the lower r a t i o t h r e s h o l d , or not to h a r v e s t ( h . r . = 0.00) i f the r a t i o i s above the higher r a t i o t h r e s h o l d . The second set of s i m p l i f i e d s t r a t e g i e s (Dg2) were based F i g u r e 10. I s o c l i n e Y t +]_=Y t superimposed on op t i m a l har-v e s t i n g d e c i s i o n s aD°(x) and yD°(x) f o r the observed r a i n f a l l p r o b a b i l i t y d i s t r i b u t i o n . I n t e r s e c t i o n s (dots) form b a s i s f o r s i m p l i -f i e d h a r v e s t s t r a t e g i e s Dsi and Ds2' Y/SQMI Y/SQMI b 50 Table IV. J u v e n i l e (Y) and o l d e r deer (A) va l u e s used t o d e r i v e s i m p l i f i e d age r a t i o h a r v e s t s t r a t e g i e s Y from A from Y/A Harvest r a t e y D ° C x ) p Y t + 1 = Y t a D ° ( x ) n Y t + 1 = Y t - o p t i m i s t i c : ' 18.1 30.0 0.60 0.00 19.8 33.8 0.59 0.10 20.1 37.5 0.54 0.20 20.8 40.0 0.52 0.30 21.6 45.0 0.48 0.40 22.3 50.0 0.45 0.50 observed 14.8 30.0 0.49 0.00 15.8 35.0 0.45 0.10 16.5 40.0 0.41 0.20 17.2 46.3 0.37 0.30 17.5 52.5 0.33 0.40 18.0 60.0 0.30 0.50 P e s s i m i s t i c 10.0 30.0 0.33 0.00 11.3 41.3 0.27 0.10 12.0 45.6 0.26 0.20 12.8 50.6 0.25 0.30 13.3 57.5 0.23 0.40 14.1 70.0 0.20 0.50 F i g u r e 11. S i m p l i f i e d h a r v e s t i n g s t r a t e g i e s d e r i v e d from o p t i m a l h a r v e s t i n g s t r a t e g i e s u s i n g primary a g e - r a t i o as s o l e i n f o r m a t i o n i n p u t . S t r a t e -g i e s are shown i n r e l a t i o n to r a i n f a l l assump-t i o n s . a) Bsi based on aD°(x) and yD°(x) b) D S2 based on aD°(x) o n l y HARVEST RATE HARVEST RATE H on the optimal a d u l t harvest s t r a t e g i e s aD°()0 on l y (Figure l i b ) . Superimposing Y^+i = Y t on the a d u l t o p t i m a l d e c i s i o n i s o p l e t h diagram. Dg2 i s d e f i n e d as the h a r v e s t r a t e s of the Y/A r a t i o s where a d u l t optimal harvest s t r a t e g y i s o p l e t h s i n t e r s e c t the i s o c l i n e . Again depending on the r a i n f a l l assumption, two h a r v e s t i n g t h r e s h o l d values are e v i d e n t . Dgi and Dg2 d i f f e r i n t h a t the former i s based on both optimal h a r v e s t i n g d e c i s i o n s [ aD°(x) a n (3 yD°(_x)] while the l a t t e r , c o n t a i n i n g l e s s i n f o r m a t i o n , i s based s o l e l y on aD°(}c). Based on optimal a d u l t h a r v e s t i n g s t r a t e g i e s aD°(>c) a t h i r d s e t of s i m p l i f i e d s t r a t e g i e s (Dg3) might use input i n f o r m a t i o n o n l y with regards to o l d e r deer d e n s i t y ( A ) , i m p l i c i t l y assuming j u v e n i l e d e n s i t y remains constant at the e q u i l i b r i u m l e v e l or i s p r e d i c t a b l e from A. Dg3 was obtained by expanding the optimal h a r v e s t d e c i s i o n s at the e q u i l i b r i u m j u v e n i l e deer d e n s i t y over the e n t i r e Y a x i s ( F i g u r e 12). The s i m p l i f i e d age r a t i o s t r a t e g i e s imply that the same har v e s t r a t e i s a p p l i e d f o r a g i v e n Y/A r a t i o no matter what the a c t u a l age group d e n s i t i e s are that make up t h i s p a r t i c u l a r r a t i o . Thus the two t h r e s h o l d r a t i o values of no harvest and maximum harv e s t can be represented as two s t r a i g h t l i n e s in the o l d e r d e e r - j u v e n i l e deer phase pla n e : A = Y / r a t i o of zero h a r v e s t r a t e A = Y / r a t i o of 0.50 harvest r a t e For example Fig u r e 13 shows D s 2 f o r the observed r a i n f a l l p r o b a b i l i t y d i s t r i b u t i o n superimposed on the optimal a d u l t h a r v e s t s t r a t e g i e s aD°(x.). I t can be seen that the s i m p l i f i e d d e c i s i o n i s o p l e t h s run i n the opposite d i r e c t i o n compared to the F i g u r e 12. S i m p l i f i e d h a r v e s t i n g s t r a t e g y Ds3 based on aD°(x) a t e q u i l i b r i u m j u v e n i l e deer d e n s i t y (Y ). S t r a t e g y shown i s f o r the observed r a i n f a l l p r o b a b i l i t y d i s t r i b u t i o n . D.D o CD Y/SQMI 10.0 20, _J 30.0 _J O 01 (0 CD CD ' O -< VJ1 p 54 optimal d e c i s i o n i s o p l e t h s . T h i s means that along a constant p o p u l a t i o n l i n e (as i l l u s t r a t e d i n F i g u r e 7) the d e c i s i o n would be to harvest not at a l l or to harvest very hard (50 p e r c e n t ) . T h i s i s i n c o n t r a s t to the optimal s t r a t e g i e s which have constant ( i . e . , apply same harvest rate) h a r v e s t r a t e s along constant p o p u l a t i o n l i n e s . In e f f e c t the phase plane ( F i g u r e 13) can be d i v i d e d i n t o three r e g i o n s : (1) no d i f f e r e n c e between Dg2 and a D ° ( 3 c ) , (2) a r e g i o n where o v e r e x p l o i t a t i o n takes p l a c e i f age r a t i o s t r a t e g i e s are a p p l i e d , and (3) a re g i o n where p o p u l a t i o n components are u n d e r e x p l o i t e d i f such s i m p l i f i e d s t r a t e g i e s are a p p l i e d . From these simple g r a p h i c a l examples we can conclude t h a t h a r v e s t i n g d e c i s i o n s based only on age r a t i o s can le a d to s e r i o u s problems. The subsequent paragraphs develop numerical comparisons of r e t u r n s obtained from a p p l y i n g optimal vs. s i m p l i f i e d s t r a t e g i e s . Computing values of i n f o r m a t i o n As H o l l i n g (1973) p o i n t s out, use of dynamic resources i n v o l v e s s h i f t s i n e q u i l i b r i u m s t a t e s and a l s o movement of p o p u l a t i o n (or p o p u l a t i o n components) away from e q u i l i b r i a . I t i s i n the context of s h i f t s i n e q u i l i b r i u m s t a t e s , and the a p p l i c a t i o n of the a l t e r n a t i v e s t r a t e g i e s (which do not use a l l a v a i l a b l e information) t h a t r e t u r n s of long term (50 year) management p e r i o d s are compared. For these comparisons the concept of value of i n f o r m a t i o n w i l l be u t i l i z e d . In the context used here i t i s a v a r i a n t of the value of i n f o r m a t i o n term a p p l i e d i n economics l i t e r a t u r e . Economists r e f e r to the value of i n f o r m a t i o n as p r o f i t s gained through a l l o c a t i o n of resources F i g u r e 13. G r a p h i c a l e v a l u a t i o n of s i m p l i f i e d h a r v e s t i n g s t r a t e9Y D S 2 ' k > a s e d o n a g e - r a t i o s , f o r the observed r a i n f a l l p r o b a b i l i t y d i s t r i b u t i o n . (Dotted l i n e s r e p r e -sent s i m p l i f i e d s t r a t e g y ) . REGION OF OVER HARVESTING REGION OF UNDER HARVESTING 56 for obtaining information about various p r o b a b i l i t y d i s t r i b u t i o n s needed for the solution of stochastic decision problems (e.g., Marschalk, 1971; Ziemba and Butterworth, 1975; Huang, Vertinsky and Ziemba, 1977; A v r i e l and Williams, 1970). Economic value of research has also been evaluated in f i s h e r i e s management ( S i l v e r t , 1977; Huang, Vertinsky, and Wilimovsky, 1976). Collapsing the information system in the present two component ungulate population system was done by using (1) s i m p l i f i e d age r a t i o strategies (Dgi and Dg2)f an<3 (2) information only with regard to older deer density (D S3). Differences of average f i f t y - y e a r returns between applying optimal harvest strategies [ aD°(x) and yD°(x)] and applying s i m p l i f i e d strategies for each of 120 i n i t i a l conditions (Figure 14) were computed from 500 year (ten times f i f t y - y e a r ) simulation runs. These runs employed a random number procedure to generate stochastic b i r t h rates as described in previous sections. Value of information was defined as the sum of computed differences: 50 50 VI = ^ (Hi I D°(x)) - ^ (Hi I D s) i=l 1 i=l 1 where V I = value of information (deer harvest/square mile) per 50 year period H = annual harvest (deer/square mile) D°(}c) = optimal harvesting strategies D S = s i m p l i f i e d harvesting strategies H | D = harvest given policy D is used Relative or percent value of information was defined as: F i g u r e 14. I n i t i a l c o n d i t i o n s used to determine v a l u e s of i n f o r m a t i o n f o r u s i n g o p t i m a l h a r v e s t i n g s t r a t e -g i e s compared to s i m p l i f i e d s t r a t e g i e s . 57a £ - | o o o o o o o o 0 o o o o o o o o o o o o ° o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o CM CO >—<=! C3 . 0.0 _ 1 1 — 40.0 80.0 fi/SQMI 120.0 58 50 50 ^ (Hi | D-O(x)) - ^ (Hi I D s) PVI = i ^ l i = l x 100 50 ^ (Hi I DO(X)) i = l Values of i n f o r m a t i o n and percent values of i n f o r m a t i o n were p l o t t e d as i s o p l e t h s on the o l d e r d e e r - j u v e n i l e deer phase p l a n e . Each p o i n t on a value of i n f o r m a t i o n i s o p l e t h s u r f a c e i s the expected g a i n i n y i e l d over a 50 year management p e r i o d by using the optimal h a r v e s t i n g s t r a t e g y as opposed to the s i m p l i f i e d s t r a t e g y . F i g u r e 15 and 16 present value of i n f o r m a t i o n (VI) and percent value of i n f o r m a t i o n (PVI) i s o p l e t h diagrams in r e l a t i o n to r a i n f a l l p r o b a b i l i t y assumptions and s i m p l i f i e d age r a t i o h a r v e s t i n g s t r a t e g i e s . For example the i n t e r s e c t i o n i n i t i a l A per square mile = 40 and i n i t i a l Y per square mile = 10 i n Figure 15 f o r the observed r a i n f a l l p r o b a b i l i t y d i s t r i b u t i o n shows an expected g a i n i n r e t u r n of 1 0 0 deer per square mile f o r a 50 year h a r v e s t i n g p e r i o d i f a D ° ( 2 £ ) a n d y D ° ( 2 £ ) are used as opposed to a p p l y i n g s i m p l i f i e d age r a t i o s t r a t e g y Dgi. The r e s u l t s i n d i c a t e (as expected) t h a t the r e t u r n s from using the optimal h a r v e s t i n g s t r a t e g i e s are i n v a r i a b l y higher than the r e t u r n s from a p p l y i n g s i m p l i f i e d age r a t i o s t r a t e g i e s . N e g l i g i b l e l o s s e s (maximum 15 percent) are i n c u r r e d i n the s i t u a t i o n where p r o d u c t i v i t y i s low ( p e s s i m i s t i c r a i n f a l l p r o b a b i l i t y d i s t r i b u t i o n ) and the s i m p l i f i e d s t r a t e g y i s based on both optimal s t r a t e g i e s (Dgi, F i g u r e 1 5 ) . I f the s i m p l i f i e d s t r a t e g y i s based only on o l d e r deer optimal s t r a t e g i e s [ aD°(x.)], the l o s s e s are g e n e r a l l y higher i n d i c a t e d by the higher values of F i g u r e 15. Simulated response s u r f a c e s of v a l u e s of i n f o r -mation (VI) and percent v a l u e s of i n f o r m a t i o n (PVI) f o r Dsi i n r e l a t i o n t o r a i n f a l l p r o b a b i l i t y assumptions. C i r c l e s i n d i c a t e o p t i m a l e q u i l i b r i a . 0 S9< V I PVI 8~| 3TS350 275 I33S R~l ssso « 1 1 1 0 0 40.0 80 .0 120.0 I N I T I A L fl/SQMI 1 : — i 1 0.0 10 .0 80 .0 120.0 I N I T I A L fl/SQMI OPTIMISTIC 8n C5C3-O S 1 1 1 0.0 10.0 80 .0 120.0 I N I T I A L A/SQMI op. A v~>r 3035 20 \ \ 1 1 1 o.o io.o eo.o 120.0 I N I T I A L fl/SQMI OBSERVED 0.0 10.0 80 .0 I N I T I A L fl/SQMI 120.0 CJo . in" >s~\ 1 1 1 o.o io.o eo.o i20.o I N I T I A L fl/SQMI PESSIMISTIC F i g u r e 16. Simulated response s u r f a c e s of v a l u e s of i n f o r -mation (VI): and percent v a l u e s of i n f o r m a t i o n (PVI) f o r Ds2 i n r e l a t i o n to r a i n f a l l p r o b a b i l i t y assumptions. C i r c l e s r e f e r t o o p t i m a l e q u i l i b r i a . P V I 60 «. R~| 375 750 cnr cr 0.0 7 40.0 80.0 INITIAL A/SQMI -1 120.0 R~l 60S! 50 45 C J o . C O ™ 0.0 40.0 80.0 IN IT IAL A/SQMI 120.0 OPTIMISTIC R~| TOO 17S cor CC 0.0 40.0 80.0 INITIAL A/SQMI 120.0 OEM 0.0 40.0 80.0 INITIAL A/SQMI 1 120.0 OBSERVED C J O _ | R l 0.0 40.0 80.0 INITIAL A/SQMI -1 120.0 C 3 o . C O ™ d M O I— d. r - 1 1 0.0 40.0 80.0 120.0 INITIAL A/SQMI PESSIMISTIC 61 i n f o r m a t i o n (Figure 16). For both s i m p l i f i e d s t r a t e g i e s , values of i n f o r m a t i o n i n c r e a s e d as o l d e r deer s t a r t i n g d e n s i t i e s were decreased. Another emergent f e a t u r e i s that value of i n f o r m a t i o n decreased from the most p r o d u c t i v e r a i n f a l l p r o b a b i l i t y assumption to the l e a s t p r o d u c t i v e assumption. These low values of i n f o r m a t i o n ( e s p e c i a l l y near e q u i l i b r i a ) f o r s i t u a t i o n s where high frequency of fawn crop l o s s e s occur imply that s i m p l i f i e d (age r a t i o ) s t r a t e g i e s might be a reasonable a l t e r n a t i v e . However, f o r both the o p t i m i s t i c and observed r a i n f a l l p r o b a b i l i t y d i s t r i b u t i o n s , age r a t i o s t r a t e g i e s are a poor s u b s t i t u t e f o r optimal h a r v e s t i n g s t r a t e g i e s . T h i s c o n c l u s i o n was a l r e a d y suspected through the g r a p h i c a l e v a l u a t i o n of s i m p l i f i e d s t r a t e g i e s . Comparison of the optimal h a r v e s t i n g s t r a t e g i e s with the s i m p l i f i e d s t r a t e g i e s Dg3 (based on optimal s t r a t e g i e s at e q u i l i b r i u m j u v e n i l e deer d e n s i t i e s ) i n d i c a t e s that no gain i n r e t u r n s can be made. In other words, no l o s s e s are i n c u r r e d through c o l l a p s i n g the d e c i s i o n system to use a s i n g l e input v a r i a b l e . These value of i n f o r m a t i o n experiments i n d i c a t e that c u r r e n t l y used age r a t i o h a r v e s t i n g d e c i s i o n s are inadequate. As Caughley (1974), through a n a l y s i n g model p o p u l a t i o n s , p o i n t s out the i n t e r p r e t a t i o n of age r a t i o s i s a hazardous undertaking. 62 CONCLUSIONS Although the a n a l y s i s was centered around the Llano B a s i n deer p o p u l a t i o n , the methods should apply to a number of ungulate systems. The r e c u r s i v e o p t i m i z a t i o n procedure of dynamic programming i s u s e f u l to estimate optimal h a r v e s t s t r a t e g i e s f o r simple two a g e - c l a s s p o p u l a t i o n models i n c o r p o r a t i n g random f l u c t u a t i o n s of the environment. Optimal feedback s t r a t e g i e s i n t h i s study are i n s e n s i t i v e to p r o b a b i l i t y d i s t r i b u t i o n s of the s t o c h a s t i c v a r i a b l e . The f i x e d escapement s t r a t e g i e s f o r a d u l t s are somewhat dependent on the j u v e n i l e d e n s i t i e s , whereas the optimal y e a r l i n g s t r a t e g i e s are almost independent of j u v e n i l e deer d e n s i t y . Applying o p t i m a l h a r v e s t i n g s t r a t e g i e s i n d i c a t e s a decrease i n the p o p u l a t i o n component s i z e s . A d u l t s drop to about one-half the n a t u r a l unharvested l e v e l , whereas j u v e n i l e s drop by o n l y about ten percent. Returns from a p p l y i n g optimal h a r v e s t i n g s t r a t e g i e s are g e n e r a l l y higher than those obtained from a p p l y i n g s i m p l i f i e d s t r a t e g i e s based on age r a t i o i n f o r m a t i o n . Yet, r e t u r n s from a p p l y i n g s i m p l i f i e d s t r a t e g i e s based on one s t a t e v a r i a b l e only (o l d e r deer) are as high as those obtained through a p p l y i n g optimal s t r a t e g i e s . 6 3 CHAPTER 3 Dynamics of a Vegetation-Ungulate System and i t s Optimal E x p l o i t a t i o n 64 SUMMARY Optimal h a r v e s t s t r a t e g i e s f o r an u n g u l a t e - v e g e t a t i o n system are estimated using s t o c h a s t i c dynamic programming. The e f f e c t s of a randomly f l u c t u a t i n g p o p u l a t i o n parameter and a l t e r n a t i v e assumptions about v e g e t a t i o n p r o d u c t i o n are considered i n the e s t i m a t i o n . In the context used here, optimal s t r a t e g y r e f e r s to a s e q u e n t i a l d e c i s i o n r u l e , optimal with r e s p e c t to maximizing h a r v e s t . Values of i n f o r m a t i o n were computed, and these i n d i c a t e d that c o n s i d e r a b l y higher r e t u r n s can be produced by using o p t i m a l p o l i c i e s as compared to t r a d i t i o n a l approaches. I m p l i c a t i o n s f o r ungulate p o p u l a t i o n management are d i s c u s s e d . 65 INTRODUCTION W i l d l i f e management has not had a u n i f y i n g theory or paradigm of p o p u l a t i o n e x p l o i t a t i o n . T r a d i t i o n a l l y , s u s t a i n e d - y i e l d h a r v e s t i n g was approached i n a pragmatic manner by e i t h e r manipulating p o p u l a t i o n s or t h e i r h a b i t a t . Leopold's sigmoid management p r i n c i p l e , forming a cornerstone of w i l d l i f e management phil o s o p h y , provided a more s t r i n g e n t approach (Leopold, 1955). T h i s approach has been employed by a v a r i e t y of b i g game workers (Dasmann, 1964; Gross, 1969 and 1972; Caughley, 1976). However, the aim of p r o v i d i n g a maximum s u s t a i n e d y i e l d , t h a t can be taken year a f t e r year without f o r c i n g the p o p u l a t i o n i n t o d e c l i n e , has a number of i n h e r e n t l y dangerous assumptions. F i r s t , i t pretends that p o p u l a t i o n s can be maintained at e q u i l i b r i u m . But we know that there e x i s t s i n c r e d i b l e v a r i a t i o n i n p o p u l a t i o n abundance; p o p u l a t i o n s are r a r e l y at e q u i l i b r i u m . Second, most d e r i v a t i o n s assume some d e t e r m i n i s t i c r e l a t i o n s h i p between environmental v a r i a b l e s and p o p u l a t i o n parameters. Yet, examination of any time s e r i e s of v a r i a b l e w i l d l i f e abundance r e v e a l s the h i g h l y u n p r e d i c t a b l e nature of environmental e f f e c t s on p o p u l a t i o n s i z e . T h i r d , i t i s pretended that the optimum ha r v e s t r a t e can be employed year a f t e r year (termed open loop c o n t r o l ) without causing changes i n the p o p u l a t i o n abundance. I f t h i s l a t t e r assumption were true i t would be d i f f i c u l t to e x p l a i n p o p u l a t i o n f l u c t u a t i o n s of e x p l o i t e d p o p u l a t i o n s , so commonly observed. In l i g h t of these d e f i c i e n c i e s , a new methodology s t r e s s i n g 66 the requirement to express harvest r a t e s as a f u n c t i o n of the v a r i a b l e s t a t e of the system (termed feedback c o n t r o l ) has r e c e n t l y been introduced to n a t u r a l resource management. Walters (1975) and H i l b o r n (1976) used the technique of s t o c h a s t i c dynamic programming to determine optimal e x p l o i t a t i o n of f i s h s t o c k s . Anderson (1975) used dynamic programming i n the management of waterfowl p o p u l a t i o n s . Other examples are reviewed i n Walters and H i l b o r n (1978). The r e c u r s i v e o p t i m i z a t i o n theory of dynamic programming, f i r s t developed by Bellman (1957), o f f e r s the p o s s i b i l i t y of e s t i m a t i n g o p t i m a l feedback p o l i c i e s f o r w i l d l i f e p o p u l a t i o n s . The purpose of the prese n t study was twofold: (1) to develop a s e t of optimal harvest s t r a t e g i e s f o r an ungulate p o p u l a t i o n i n r e l a t i o n to v a r y i n g assumptions about v e g e t a t i o n p r o d u c t i o n and u n c e r t a i n t y about p o p u l a t i o n m o r t a l i t y ; and (2) to compare long-term y i e l d s using these o p t i mal h a r v e s t s t r a t e g i e s to y i e l d s from a s e r i e s of a l t e r n a t i v e s t r a t e g i e s . The i n t e r a c t i v e growth model of a ve g e t a t i o n - d e e r system d e s c r i b e d by Caughley (Caughley, 1976) was used as an example f o r t h i s i n v e s t i g a t i o n . METHODS Model s t r u c t u r e The example used to re p r e s e n t the dynamics of a veg e t a t i o n - u n g u l a t e system i s a d i f f e r e n c e equation v e r s i o n of the p l a n t - h e r b i v o r e model d e s c r i b e d by Caughley. For a d e t a i l e d d e s c r i p t i o n of model d e f i n i t i o n s and assumptions see Caughley (1976: 206-209). The model i s : 67. v t - d i V t Vt+1 = V t + a i V t d - ) - C ! N t d - e ) (1) K - d 2 V t Nt+1 = N t + N t [ - a 2 + c 2 d - e )] (2) where = v e g e t a t i o n biomass a v a i l a b l e to ungulate as food i n year t a i = i n t r i n s i c growth r a t e of v e g e t a t i o n (Figure 17) K = maximum s u s t a i n a b l e biomass of v e g e t a t i o n a v a i l a b l e to ungulate as food c i = maximal r a t e of v e g e t a t i o n consumption per deer (F i g u r e 18) d i = g r a z i n g e f f i c i e n c y of deer when v e g e t a t i o n i s at low d e n s i t y ( F i g u r e 18) Nt = ungulate p o p u l a t i o n s i z e i n year t a 2 = m o r t a l i t y r a t e i n the absence of v e g e t a t i o n ( s t o c h a s t i c v a r i a b l e ) c 2 = term d e c r e a s i n g m o r t a l i t y r a t e ( a 2) at high v e g e t a t i o n biomass d 2 = a b i l i t y of ungulate p o p u l a t i o n to reproduce at low v e g e t a t i o n biomass I v l e v ' s (1961) equation i s used to model the r e l a t i o n s h i p s between v e g e t a t i o n biomass and r a t e of v e g e t a t i o n consumption by the ungulate (the f u n c t i o n a l response of H o l l i n g , 1961), and between v e g e t a t i o n biomass and a b i l i t y of the deer p o p u l a t i o n to reproduce (the numerical response of H o l l i n g , 1961) . To i n v e s t i g a t e the q u a l i t a t i v e behaviour of the veg e t a t i o n - u n g u l a t e system, a nominal parameter s e t was chosen: a i = 0.8 a 2 = 1.1 - 1.3 (see Table V) K = 3000 c 2 = 1.5 c i = 1.2 d 2 = 0.001 <3l = 0.001 F i g u r e 17. V e g e t a t i o n growth (_G) as a f u n c t i o n of v e g e t a t i o n biomass .(V).. G m - maximal growth a) l o g i s t i c growth; K/2 = biomass a t which growth i s maximal; G=ajV(l-V/K). b)_ and c) a l t e r n a t i v e growth assumptions. K /4 K/2 3 K/4 F i g u r e 18. Consumption per deer (c) as a f u n c t i o n of v e g e t a t i o n biomass (V) . c^ - maximal consumption. Gradual s a t i a -t i o n curve -d,V\ c = c^ (1 - e 1 ) ( A f t e r I v l e v , 1961). 69a. These parameter values are intended to r e p r e s e n t a deer p o p u l a t i o n occupying a mosaic of g r a s s l a n d and f o r e s t (Caughley, 1976: 208). The standard form of v e g e t a t i o n growth (Figure 17a) i s w e l l e s t a b l i s h e d , both t h e o r e t i c a l l y and e m p i r i c a l l y (Noy-Meir, 1975; Donald, 1961; Brougham, 1955 and 1956). By v a r y i n g a\, q u a l i t a t i v e l y d i f f e r e n t assumptions about v e g e t a t i o n growth can be modelled ( F i g u r e s 17b and 17c). The s t o c h a s t i c v a r i a b l e ( a 2 ) can assume a value of 1.1 or 1.3, with f r e q u e n c i e s r e p r e s e n t i n g r a t e s at which d i s t u r b a n c e i s introduced i n t o the system. Disturbance i n the form of a higher m o r t a l i t y r a t e i s e q u i v a l e n t to a burn or severe winter. S i x p o s s i b l e frequency d i s t r i b u t i o n s of d i s t u r b a n c e were t e s t e d (Table V ) . S t o c h a s t i c dynamic programming To answer the c e n t r a l q u e s t i o n (what sequence of h a r v e s t r a t e s w i l l maximize the expected f u t u r e returns?) the growth model (eqns. 1 and 2) was formulated as a two s t a t e v a r i a b l e ( N t / v t ) i s t o c h a s t i c dynamic programming problem. For a d e t a i l e d d e s c r i p t i o n of the computational procedure see Chapter 2 and Walters (1975). The continuous s t a t e v a r i a b l e s (Nt, Vt) , c o n t r o l s (utW and random v a r i a b l e were d i s c r e t i z e d as f o l l o w s : 21 l e v e l s f o r each s t a t e v a r i a b l e (N t = 0, 51, 1 0 2 , 1 0 2 0 ; V t = 0, 150, 300,... ,3000), 11 harvest r a t e s ( u t = 0 .00, 0 .05, 0 .10 ,... ,0.50) and two s t o c h a s t i c outcomes ( a 2 = 1.1, 1.3). P r e d i c t e d r e t u r n s and s t a t e v a r i a b l e d i s t r i b u t i o n s To e v a l u a t e the performance of the optimal h a r v e s t i n g s t r a t e g i e s , and t h e i r e f f e c t on the deer p o p u l a t i o n and Table V. M o r t a l i t y (a 2) v a l u e s and p r o b a b i l i t i e s used f o r t h i s study. Case P r o b a b i l i t i e s a2 1 1.0 0.0 1.1 1.3 2 0.9 0.1 1.1 1.3 3 0.6 0.4 1.1 1.3 4 0.3 0.7 1.1 1.3 5 0.1 1.1 0.9 1.3 6 0.0 1.0 1.1 1.3 72 v e g e t a t i o n biomass, very long s i m u l a t i o n runs (5000 years) were a p p l i e d to the dynamic model (eqns. 1 and 2). S i m u l a t i o n runs employed a random number procedure to generate m o r t a l i t y values with a p p r o p r i a t e p r o b a b i l i t i e s . Runs were executed f o r the three v e g e t a t i o n growth f u n c t i o n s i l l u s t r a t e d i n F i g u r e 17. E s t i m a t i n g value of i n f o r m a t i o n To compare the performance of the optimal h a r v e s t i n g s t r a t e g i e s to a l t e r n a t i v e h a r v e s t i n g s t r a t e g i e s , a s e r i e s of s h o r t e r s i m u l a t i o n runs (500 years) were c a r r i e d out. Runs were done f o r one v e g e t a t i o n growth f u n c t i o n o n l y ( G m a t K/2). S p e c i f i c a l l y three types of a l t e r n a t i v e s were explo r e d . F i r s t , b e s t f i x e d h a r v e s t i n g s t r a t e g i e s (Pp) were devised f o r each case l i s t e d i n Table V, where best f i x e d h arvest r a t e was d e f i n e d as the one maximizing the average of ten f i f t y - y e a r r e t u r n s beginning at e q u i l i b r i u m p o p u l a t i o n s i z e and v e g e t a t i o n biomass. Secondly, two d i s t u r b a n c e response s t r a t e g i e s based on the f i x e d h a r v e s t r a t e of 20 percent were t r i e d : f o l l o w i n g a d i s t u r b a n c e ( a 2 = 1.3), reduce h a r v e s t from ( i ) 20 p e r c e n t to 15 p e r c e n t ( P R I ) , or ( i i ) from 20 percent to 10 percent ( P R 2 ) . T h i s type of responsive management approach has been p r a c t i c e d i n e a s t e r n North America. E s p e c i a l l y f o r w h i t e - t a i l e d deer p o p u l a t i o n s , reduced h a r v e s t r a t e s are a p p l i e d a f t e r severe w i n t e r s . F i n a l l y comparisons were made between o p t i m a l h a r v e s t i n g s t r a t e g i e s and s i m p l i f i e d h a r v e s t i n g s t r a t e g i e s (Pgrj and Pgy) • Based on optimal h a r v e s t i n g s t r a t e g i e s , one s e t of s i m p l i f i e d s t r a t e g i e s ( P S D ) uses only i n f o r m a t i o n r e g a r d i n g deer p o p u l a t i o n (assuming v e g e t a t i o n biomass remains constant at the e q u i l i b r i u m v e g e t a t i o n biomass). T h i s set of s i m p l i f i e d s t r a t e g i e s was obtained by expanding the o p t i m a l h a r v e s t r a t e s (from P 0 ) at the e q u i l i b r i u m biomass l e v e l over the e n t i r e v e g e t a t i o n biomass a x i s . The other set of s i m p l i f i e d s t r a t e g i e s (Psv) u t i l i z e s v e g e t a t i o n i n f o r m a t i o n only, assuming deer p o p u l a t i o n remains constant at the e q u i l i b r i u m d e n s i t y . T h i s s e t was d e r i v e d from expanding the optimal h a r v e s t r a t e s at the e q u i l i b r i u m deer p o p u l a t i o n s i z e over the e n t i r e deer p o p u l a t i o n a x i s . A summary of the v a r i o u s h a r v e s t i n g s t r a t e g i e s i s g iven i n Tab l e V I . The d i f f e r e n c e s of average f i f t y - y e a r r e t u r n s between a p p l y i n g optimal h a r v e s t s t r a t e g i e s and a p p l y i n g a l t e r n a t i v e s t r a t e g i e s f o r each of 100 i n i t i a l c o n d i t i o n s ( F i g u r e 19) are de f i n e d as values of i n f o r m a t i o n : 50 50 VI = 2 (Hi I P 0 ) - Z ( H i | p A ) i = l 1 i = l 1 where VI = value of i n f o r m a t i o n measured i n r e t u r n s of ungulates per 50 year p e r i o d H = annual h a r v e s t (number of ungulates per year) PQ = optimal feedback s t r a t e g y P A = a l t e r n a t i v e h a r v e s t i n g s t r a t e g i e s (Table VI) H | P = h a r v e s t given s t r a t e g y P i s used The v a l u e s of i n f o r m a t i o n were p l o t t e d as contour maps or value of i n f o r m a t i o n i s o p l e t h diagrams f o r each of the s i x cases i n Table V. Each p o i n t on the value of i n f o r m a t i o n i s o p l e t h s u r f a c e r e p r e s e n t s expected y i e l d s per 50 year h a r v e s t i n g p e r i o d T a b l e V I . Summary o f h a r v e s t i n g s t r a t e g i e s c o n s i d e r e d i n t h i s s t u d y . S y m b o l PO A p p l y i n g O p t i m a l f e e d b a c k s t r a t e g y e s t i m a t e d by two v a r i a b l e s t o c h a s t i c d y n a m i c p r o g r a m m i n g . Pp B e s t f i x e d h a r v e s t r a t e , d e t e r m i n e d by one v a r i a b l e ( u n g u l a t e ) maximum s u s t a i n e d y i e l d l e v e l . p R l R e d u c e d h a r v e s t r a t e ( f r o m 20% t o 15%) f o r one y e a r f o l l o w i n g e a c h d i s -t u r b a n c e ( a 2 = 1 . 3 ) . PR2 Same a s P R i b u t r e d u c t i o n f r o m 20% t o 10% f o l l o w i n g d i s t r u b a n c e . Pgo S i m p l i f i e d h a r v e s t s t r a t e g y b a s e d on o p t i m a l s t r a t e g y , b u t u s i n g o n l y u n g u l a t e p o p u l a t i o n i n f o r m a t i o n ( v e g e t a t i o n b i o m a s s assumed t o r e m a i n a t e q u i l i b r i u m ) . P g V S i m p l i f i e d h a r v e s t s t r a t e g y b a s e d on o p t i m a l s t r a t e g y , b u t u s i n g o n l y v e g e t a t i o n b i o m a s s i n f o r m a t i o n ( u n g u l a t e p o p u l a t i o n assumed t o r e m a i n a t e q u i l i b r i u m ) . F i g u r e 19. I n i t i a l c o n d i t i o n s ( c i r c l e s ) used to determine values of i n f o r m a t i o n i n comparison of o p t i m a l h a r v e s t s t r a t e g i e s w i t h a l t e r n a t i v e s t r a t e g i e s , UJ Q I i . O m z e> z i— cr. < I020-| 918 816-714-612-510-40 8 306 204 102 0 O O O O O O O O O O O O O O O O O O O O O O O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o ~i r T r 30 60 90 120 150 ISO 210 240 270 300 STARTING BIOMASS OF VEGETATION (xlO) 76 t h a t i s to be gained by using the two v a r i a b l e feedback s t r a t e g y as opposed to the a l t e r n a t i v e s t r a t e g y . RESULTS AND DISCUSSION Dynamics of the unharvested system P r e d i c t e d deer p o p u l a t i o n and v e g e t a t i o n biomass trends f o r the three a l t e r n a t i v e v e g e t a t i o n p r o d u c t i o n f u n c t i o n s and a randomly f l u c t u a t i n g m o r t a l i t y r a t e (case two) are de p i c t e d i n Fi g u r e 20. B a s i c a l l y three types of q u a l i t a t i v e behaviour are p r e d i c t e d , depending on the v e g e t a t i o n p r o d u c t i o n f u n c t i o n used. For the standard p r o d u c t i o n case ( G m i n F i g u r e 17 at K/2) the deer p o p u l a t i o n e r u p t s , then approaches an e q u i l i b r i u m with the v e g e t a t i o n through dampened o s c i l l a t i o n s (Caughley, 1976; Noy-Meir, 1975). Concomitantly, v e g e t a t i o n crashes and then approaches an e q u i l i b r i u m with g r a z i n g pressure (Figure 20a). In the phase p o r t r a i t (Figure 20b) f o r t h i s case, the t r a j e c t o r y s p i r a l s i n t o an e q u i l i b r i u m . The o c c a s i o n a l jumps are caused by the randomly f l u c t u a t i n g m o r t a l i t y r a t e . Another s t a b l e s i t u a t i o n occurs when maximum v e g e t a t i o n p r o d u c t i o n i s at low v e g e t a t i o n biomass ( G m at K/4). However, no o s c i l l a t i o n s occur (the e q u i l i b r i u m i s reached m o n o t o n i c a i l y as i n F i g u r e s 20c and 20d). A l s o , the maximum deer p o p u l a t i o n a t t a i n e d i s only about h a l f the s i z e of the deer p o p u l a t i o n i n the standard case. Evidence f o r t h i s growth form i s provided by a study of shrubby understory on Vancouver I s l a n d ( B u n n e l l , unpublished d a t a ) . For the s i t u a t i o n where maximum v e g e t a t i o n p r o d u c t i o n i s at Figure 20. Simulated trends of unharvested deer population R and vegetation biomass V (case two) during an ungu-late eruption, and corresponding phase p o r t r a i t s i n r e l a t i o n to vegetation production functions: a), b) G m at K/2 c ) , d) G m at K/4 e), f) G m at 3K/4 77a. 1 100 0 1500 3000 TIME V R MFIX= 537.2183 V MRX= 3000. 537, \ ,' \ ; \; J. 268 100 TIME 1500 V 3000 78 . a high biomass l e v e l ( G m a t 3K/4) the deer p o p u l a t i o n erupts to a maximum value and then crashes to e x t i n c t i o n . The v e g e t a t i o n biomass crashes d r a s t i c a l l y and then very slowly recovers (Figure 2 0 e ) . The t r a j e c t o r y ( F i g u r e 2 0 f ) appears to be an example of an unstable system, whereby one component (deer p o p u l a t i o n ) becomes e f f e c t i v e l y e x t i n c t . Mathematically, the system behaviour i n t h i s case i s a l i m i t c y c l e with very long p e r i o d ; e v e n t u a l l y the v e g e t a t i o n would recover enough to permit another ungulate e r u p t i o n , thus r e p e a t i n g the p a t t e r n . Optimal h a r v e s t i n g s t r a t e g i e s S t o c h a s t i c dynamic programming computes optimal h a r v e s t i n g s t r a t e g i e s (harvest r a t e s ) as a f u n c t i o n of the s t a t e v a r i a b l e s (deer p o p u l a t i o n s i z e , v e g e t a t i o n biomass), and of time. F o r t u n a t e l y , the o p t i m a l s t r a t e g i e s turn out to be s t a t i o n a r y (time independent), so only the s t a t e dependency need be considered f u r t h e r . The o p t i m a l harvest s t r a t e g i e s can be presented as h a r v e s t r a t e i s o p l e t h s on a plane with v e g e t a t i o n biomass on the X a x i s and deer p o p u l a t i o n on the Y a x i s . F i g u r e 21 p r e s e n t s i s o p l e t h s of optimal harvest r a t e s f o r the standard v e g e t a t i o n p r o d u c t i o n f u n c t i o n ( G m a t K/2). I t can be seen that optimal harvest s t r a t e g i e s are almost independent of the m o r t a l i t y value ( a 2 ) p r o b a b i l i t i e s , as the form of the i s o p l e t h s does not change s u b s t a n t i a l l y as the frequency of d i s t u r b a n c e i s i n c r e a s e d . For any f i x e d v e g e t a t i o n l e v e l , the s t r a t e g i e s c a l l f o r a f i x e d escapement of deer ( i . e . , a p o p u l a t i o n s i z e below which no h a r v e s t i s taken). T h i s escapement remains the same f o r about h a l f the p o s s i b l e range of F i g u r e 21. Harvest r a t e i s o c l i n e s d e r i v e d from s t o c h a s t i c dynamic programming f o r the standard v e g e t a t i o n p r o d u c t i o n f u n c t i o n CG m a t K/2). E n c i r c l e d numbers r e f e r t o p r o b a b i l i t y cases l i s t e d i n T a b l e V. C i r c l e s i n d i c a t e o p t i m a l e q u i l i b r i a . Harvest r a t e = (number - 1)0.05. 8 0 , v e g e t a t i o n biomasses, and then i n c r e a s e s r a p i d l y over the l e s s p r o d u c t i v e upper h a l f of the v e g e t a t i o n biomass a x i s . A l s o , the optimal s t r a t e g i e s are s e n s i t i v e to changes in v e g e t a t i o n biomass and deer p o p u l a t i o n s i z e near the optimal e q u i l i b r i a i n d i c a t e d i n F i g u r e 21. For example, the optimal harvest r a t e can change as much as 37.5 percent i f v e g e t a t i o n biomass i s changed from 300 u n i t s below the e q u i l i b r i u m to 300 u n i t s above the e q u i l i b r i u m u n i t s of v e g e t a t i o n biomass (Table V I I ) . S i m i l a r r e s u l t s are obtained f o r the a l t e r n a t i v e v e g e t a t i o n p r o d u c t i o n f u n c t i o n s ( F i g u r e s 22 and 23). However, at low deer p o p u l a t i o n s i z e s and low v e g e t a t i o n biomass, the p r e d i c t i o n i s to h a r v e s t hard i n order to allow the v e g e t a t i o n to r e c o v e r . A l s o , c o n s i d e r a b l e s h i f t s i n the optimal e q u i l i b r i a of the deer p o p u l a t i o n s are e v i d e n t . The e q u i l i b r i u m i s lower f o r the case of maximal v e g e t a t i o n p r o d u c t i o n being at low biomass (Figure 22), compared to the standard ( F i g u r e 21). The converse i s true i f maximal v e g e t a t i o n p r o d u c t i o n i s at high biomass ( G m a t 3K/4) ( F i g u r e 23). A number of p r o p e r t i e s make s t o c h a s t i c dynamic programming a b e t t e r method f o r producing optimal h a r v e s t i n g s t r a t e g i e s , than the maximum s u s t a i n e d y i e l d methods c u r r e n t l y employed by some w i l d l i f e managers (e.g. Dasmann, 1964). F i r s t l y , the o p t i m a l c o n t r o l law or harvest s t r a t e g y expresses the best harvest r a t e s d i r e c t l y as a f u n c t i o n of the c u r r e n t s t a t e v a r i a b l e s (N^, v t ) • Perhaps more important, the c o n t r o l law accounts f o r u n c e r t a i n t y with r e s p e c t to the s t o c h a s t i c m o r t a l i t y v a r i a b l e . The c o n t r o l laws i n F i g u r e s 21, 22 and 23 are optimal only with r e s p e c t to the d e f i n e d o b j e c t i v e of maximizing s u s t a i n e d Table VII., S e n s i t i v i t y of harvest rates to changes i n vege tatio n biomass at equilibrium deer population size, i n r e l a t i o n to vegetation production function, (see Figure 17). , Vegetation Production Function Case G at K/.4 G m at K/2 G at 3K/4 m 2 m m Ahr Ahr Ahr 1 .350 .375 .400 2 .250 .375 .400 3 .250 .300 .350 4 .200 .350 .250 5 .300 .300 .250 6 .100 .200 .200 Refer to Table V. Change of harvest rate of going from 300 units below to 300 units above equilibrium vegetation biomass. Figure 22. Harvest rate i s o c l i n e s derived from stochastic dynamic programming for situations where maximal vegetation production i s at low vegetation biomass (G m at K/4). Encircled numbers refer to p r o b a b i l i t y cases l i s t e d in Table V. C i r c l e s indicate optimal e q u i l i b r i a . F i g u r e 23. Harvest r a t e i s o c l i n e s d e r i v e d from s t o c h a s t i c dynamic programming f o r s i t u a t i o n s where maximal v e g e t a t i o n p r o d u c t i o n i s a t hig h e r v e g e t a t i o n biomass ( G m a t 3K/4). E n c i r c l e d numbers r e f e r to p r o b a b i l i t y cases l i s t e d i n Table V. C i r c l e s i n d i c a t e o p t i m a l e q u i l i b r i a . 84 deer y i e l d . I t should be noted that a l t e r n a t i v e or m u l t i p l e goal o b j e c t i v e s can j u s t as e a s i l y be i n c o r p o r a t e d (Chapter 4 ) . The form of the feedback c o n t r o l laws in F i g u r e s 21, 22 and 23 i s c a l l e d a "bang-bang" c o n t r o l (Bushaw, 1958; C l a r k , 1976). For system dynamics that are continuous i n time, t h i s i m p l i e s t h a t the c o n t r o l v a r i a b l e , u n l i k e i n a c t u a l management p r a c t i c e , takes on low extreme, high extreme, or e q u i l i b r a t i n g v a l u e s . I f the dynamics are d i s c r e t e , as i n the p r e s e n t study, the bang-bang c o n t r o l i s e q u i v a l e n t to having a f i x e d escapement f o r the deer p o p u l a t i o n below which no harvest i s taken. The o p t i m a l e q u i l i b r i a ( F i g u r e s 21, 22 and 23) r e f e r to the values each s t a t e v a r i a b l e w i l l reach by s t a r t i n g anywhere i n the phase plane and a p p l y i n g the c o n t r o l law. Of course f o r s t o c h a s t i c s i t u a t i o n s (cases 2-5) these e q u i l i b r i a are never r e a l l y reached as random f l u c t u a t i o n s d i s t u r b the v a r i a b l e s . P r e d i c t e d v a l u e s from optimal h a r v e s t i n g s t r a t e g i e s F i g u r e 24 p r e s e n t s p r e d i c t e d mean values from 5000 year s i m u l a t i o n t r i a l s . Applying the optimal h a r v e s t i n g s t r a t e g i e s d e c r i b e d above, the values are shown in r e l a t i o n to frequency of d i s t u r b a n c e and the v e g e t a t i o n p r o d u c t i o n f u n c t i o n . The mean annual h a r v e s t (Figure 24b) d e c l i n e s p r o g r e s s i v e l y as the frequency of d i s t u r b a n c e i s i n c r e a s e d . On the other hand, as the v e g e t a t i o n d e n s i t y g i v i n g maximum pr o d u c t i o n i s i n c r e a s e d , the mean annual h a r v e s t i n c r e a s e s . The u n c e r t a i n t y represented i n s t o c h a s t i c optimal s o l u t i o n s i s best expressed by p r e s e n t i n g the above values as p r e d i c t e d p r o b a b i l i t y d i s t r i b u t i o n s . F i g u r e s 25, 26 and 27 are p r e d i c t i o n s SSo. 8.0-1 70^ CO 6.0 cr < UJ >-o tn tr. o 5 .0T 4.0J. CO 3.0-^ UJ > < x 2.0H 1.0-0.0+-a) 0.0 0.2 0.4 0.6 0.8 1.0 CO UJ > QC < < UJ 16 N 1.4 o 1.2 1.0 0.8 < 0.6 Z z < 0.4 0.2H o.o-0.0 0.2 b) — i 1 1 i 0.4 0.6 0 .8 1.0 • G m ot K / 4 O G „ at K/2 m A G m at 3 K / 4 d) 8CH 70H ? 6.0-r 9 50H <* o a. o a. cr ui UJ a < ui 2 40H 3CH 2CH 1.0 -0.0 c) 0.0 0.2 0.4 i i i 0.6 0.8 1.0 2.6-2.5 4 A ro 2.4 O CO CO 2.3 Q .„ OQ 2.2 d> 0 O g " r— UJ S 2 0 - i > Ul 2 1.9 I.8H 1.7 0 . 0 0.2 0.4 0.6 0.8 1.0 FREQUENCY OF DISTURBANCE Figure. 25. P r e d i c t e d p r o b a b i l i t y d i s t r i b u t i o n s of annual r e t u r n s from f i f t y year h a r v e s t i n g p e r i o d s , u s i n g o p t i m a l h a r v e s t i n g s t r a t e g i e s , f o r the standard v e g e t a t i o n p r o d u c t i o n f u n c t i o n CGm a t K/2). E n c i r c l e d numbers r e f e r to p r o b a b i l i t y cases i n Table V. Figure 26. Predicted p r o b a b i l i t y d i s t r i b u t i o n s of deer popu-l a t i o n associated with annual returns for the stan-dard vegetation production function (G m at K/2). Encircled numbers refer to p r o b a b i l i t y cases i n Table V. F i g u r e 27. P r e d i c t e d p r o b a b i l i t y d i s t r i b u t i o n s of v e g e t a t i o n biomass a s s o c i a t e d with annual r e t u r n s f o r the standard v e g e t a t i o n p r o d u c t i o n f u n c t i o n CG m a t K/2) E n c i r c l e d numbers r e f e r t o p r o b a b i l i t y cases i n Table V. V E G E T A T I O N BIOMASS 89 a s s o c i a t e d with the standard v e g e t a t i o n p r o d u c t i o n model (G m a t K/2). These r e s u l t s show that there i s no a p p r e c i a b l e e f f e c t on the v a r i a n c e of harvest as the frequency of d i s t u r b a n c e i s i n c r e a s e d . A c c o r d i n g to the above r e s u l t s , ( F i g u r e 24), the w i l d l i f e manager's task of p r o v i d i n g a s u s t a i n e d y i e l d depends not o n l y on the c u r r e n t deer p o p u l a t i o n s i z e , but a l s o on the form of the v e g e t a t i o n p r o d u c t i o n f u n c t i o n ( F i g u r e 17). T h i s suggests t h a t i n c r e a s i n g emphasis should be placed on measuring f u n c t i o n a l r e l a t i o n s h i p s between food p r o d u c t i o n and v e g e t a t i o n d e n s i t y , r a t h e r than on h a b i t a t e v a l u a t i o n on a s t r i c t l y s t a t i c b a s i s . Comparison to a l t e r n a t i v e h a r v e s t i n g s t r a t e g i e s F i g u r e 28 p r e s e n t s s i x value of i n f o r m a t i o n i s o p l e t h diagrams, s e l e c t e d from among a t o t a l of 30 ( f i v e a l t e r n a t i v e h a r v e s t s t r a t e g i e s times s i x c a s e s ) . Each p o i n t on the value of i n f o r m a t i o n contour s u r f a c e r e p r e s e n t s y i e l d s gained over a 50 year h a r v e s t i n g p e r i o d by using the optimal h a r v e s t i n g s t r a t e g y i n s t e a d of the a l t e r n a t i v e s t r a t e g y . For example the i n t e r s e c t i o n v e g e t a t i o n biomass = 120, deer p o p u l a t i o n = 50 i n F i g u r e 28a shows an expected gain of 300 deer f o r a f i f t y - y e a r h a r v e s t i n g p e r i o d i f the two v a r i a b l e feedback s t r a t e g y i s a p p l i e d as opposed to a p p l y i n g an annual f i x e d h a r v e s t r a t e of 19 percent. Best f i x e d harvest r a t e s i n r e l a t i o n to p r o b a b i l i t y cases are d e p i c t e d i n F i g u r e 29. The most important c o n c l u s i o n to be drawn from these r e s u l t s i s t h a t i n a l l s i t u a t i o n s (except near e q u i l i b r i a ) the r e t u r n s from using the optimal h a r v e s t i n g s t r a t e g i e s were g r e a t e r than F i g u r e 28. Simulated response s u r f a c e s of v a l u e s of informa-t i o n f o r v a r i o u s combinations o f s t a r t i n g deer p o p u l a t i o n and s t a r t i n g v e g e t a t i o n d e n s i t i e s with regards to a l t e r n a t i v e h a r v e s t i n g s t r a t e g i e s . (Numbers on i s o p l e t h s r e f e r to mean v a l u e s of i n f o r -mation f o r ten f i f t y - y e a r h a r v e s t i n g p e r i o d s ) . Values of i n f o r m a t i o n d e r i v e d from a) and b) o p t i m a l vs f i x e d h a r v e s t r a t e p o l i c i e s (Pp); c) and d) op t i m a l vs r e s p o n s i v e h a r v e s t i n g p o l i c i e s ( P R ) ; e) and f) opt i m a l vs s i m p l i f i e d h a r v e s t i n g s t r a t e g i e s ( P S ) . E n c i r c l e d numbers r e f e r t o p r o b a b i l i t y cases l i s t e d i n Table V. 30a. b) F i g u r e 29. Rest f i x e d h a r v e s t r a t e s f o r the standard v e g e t a t i o n p r o d u c t i o n ffunction (Gj^ a t K/2) , i n r e l a t i o n t o pro-b a b i l i t y cases l i s t e d i n Table V. Best h a r v e s t r a t e i s l o c a t e d a t the e q u i l i b r i u m v e g e t a t i o n biomass l e v e l producing maximum s u s t a i n a b l e deer r e t u r n s f o r a 50 year h a r v e s t i n g p e r i o d . 0 . 4 | • • 1 I ) • I i • | • • • 1 j 1 1 1 1 i 5 10 15 2 0 2 5 HARVEST RATE (%) the returns from applying alternative harvesting strategies. For a l l s i t u a t i o n s , the values of information increased as extremes were reached ( i . e . , moving away from e q u i l i b r i a ) . However, the degree of benefit (higher returns) varied considerably among the a l t e r n a t i v e s . Comparison of the optimal harvesting strategies with the best fixed harvest rates indicates that substantial improvements can be made by having information about deer population size and vegetation biomass. This is indicated by the high values of information obtained as extremes are reached (Figure 28a). These improvements decrease with an increase in the frequency of disturbance (Figure 28b). Of the two responsive harvesting p o l i c i e s (PR]_, PR2 ) t n e more conservative policy PR2 (reducing harvest rate from 20 percent to 10 percent for one year following a disturbance) shows higher returns than P R j at st a r t i n g vegetation biomass ranging between 600 and 1200. This is shown by the lower values of information in Figure 28d compared to Figure 28c over the indicated biomass range. At lower vegetation densities, returns from using either responsive policy are equally low, as the high values of information indicate (Figure 28c and 28d). The most interesting isopleth diagrams are Figures 28e and 28f. The contours e s s e n t i a l l y indicate that no substantial improvements can be made by having information about the vegetation biomass. This is shown by the maximum value of only 110 animal units gained over a 50 year harvesting period (a loss of only two percent). On the other hand substantial gains are made, as s t a r t i n g vegetation biomass increases, by having i n f o r m a t i o n about the c u r r e n t s i z e of the deer p o p u l a t i o n . Comparison of the op t i m a l c o n t r o l law with r e s p o n s i v e h a r v e s t i n g p o l i c i e s ( F i g u r e s 28c and 28d) r e v e a l s that i t i s not a u t o m a t i c a l l y best to reduce harvest r a t e s f o l l o w i n g any d i s t u r b a n c e . The b e t t e r approach i s to determine the harvest r a t e a c c o r d i n g to c u r r e n t c o n d i t i o n s of the s t a t e v a r i a b l e s . Once an a p p r o p r i a t e v e g e t a t i o n p r o d u c t i o n f u n c t i o n has been e s t a b l i s h e d (previous s e c t i o n ) i t does not pay to i n v e s t more time and money i n monitoring the c u r r e n t v e g e t a t i o n biomass ( F i g u r e 28e) which i s a f a v o u r i t e preoccupation of many w i l d l i f e managers. I t appears that monitoring resources would be b e t t e r a l l o c a t e d to determining as a c c u r a t e l y as p o s s i b l e the c u r r e n t s i z e of the ungulate p o p u l a t i o n (Figure 2 8 f ) . 94 CONCLUSIONS Although the a n a l y s i s was centered around an imaginary d e e r - v e g e t a t i o n system, the computational methods should be a p p l i c a b l e to r e a l ungulate systems. I t should be po i n t e d out tha t i t i s p o s s i b l e to design and i n c o r p o r a t e m u l t i p l e o b j e c t i v e s . S p e c i f i c c o n c l u s i o n s that f o l l o w from t h i s study are: 1. S t o c h a s t i c dynamic programming appears to be a b e t t e r method f o r e s t i m a t i n g optimal c o n t r o l laws than c u r r e n t l y employed s u s t a i n a b l e y i e l d c a l c u l a t i o n s . P a r t i c u l a r l y r e c o g n i t i o n of the s t o c h a s t i c environment i n e s t i m a t i n g s t r a t e g i e s i s a d e s i r a b l e aspect of the presented computational method. 2. Optimal c o n t r o l laws appear to be i n s e n s i t i v e to the p r o b a b i l i t y d i s t r i b u t i o n f o r the m o r t a l i t y value ( a 2 ) . The form of the optimal h a r v e s t i s o p l e t h s does not change s u b s t a n t i a l l y as the frequency of occurrence of higher a 2 v a l u e s i s i n c r e a s e d . 3. Optimal c o n t r o l laws appear to be s e n s i t i v e to assumptions regarding v e g e t a t i o n p r o d u c t i o n . E s s e n t i a l l y the r e g i o n of maximum h a r v e s t i n g (u = 0.50) becomes sm a l l e r as the v e g e t a t i o n biomass producing maximally i s in c r e a s e d . The converse i s true f o r the no h a r v e s t i n g r e g i o n (u = 0 .00 ) . 4 . Returns from a p p l y i n g optimal c o n t r o l laws are g e n e r a l l y higher compared to r e t u r n s from c o n v e n t i o n a l h a r v e s t i n g 95 methods, except near e q u i l i b r i u m . Best f i x e d harvest r a t e p o l i c i e s , as w e l l as responsive h a r v e s t i n g p o l i c i e s showed c o n s i d e r a b l e l o s s e s i n harvest r e t u r n s when compared to optimal c o n t r o l laws. The l o s s e s , as i n d i c a t e d by high values of in f o r m a t i o n were p a r t i c u l a r l y l a r g e when extreme i n i t i a l c o n d i t i o n s are assumed. 5. Returns from s i m p l i f i e d c o n t r o l laws using only deer p o p u l a t i o n i n f o r m a t i o n are n e a r l y o p t i m a l , whereas c o n t r o l based only on v e g e t a t i o n ( h a b i t a t ) data i s an i n v i t a t i o n to d i s a s t e r . CHAPTER 4 O p t i m i z a t i o n Model f o r a Wolf-Ungulate System SUMMARY 97 P u b l i c o u t c r i e s a g a i n s t predator c o n t r o l c r e a t e a need to de v i s e management p o l i c i e s that o p t i m a l l y balance the cost (managerial and environmental) of predator c o n t r o l a g a i n s t the b e n e f i t of ungulate h a r v e s t i n g . To address t h i s problem, an o p t i m i z a t i o n procedure u t i l i z i n g s t o c h a s t i c dynamic programming i s d e s c r i b e d . Through t h i s approach optimal feedback s t r a t e g i e s f o r a wolf-ungulate system i n Alaska are estimated. The dynamic predator-prey model used i n the a n a l y s i s i s based on parameter estimates from data c o l l e c t e d over an e i g h t year p e r i o d i n D e n a l i (Mt. K i n l e y ) N a t i o n a l Park. S t a b i l i t y a n a l y s i s of the system r e v e a l e d s t a b i l i t y p r o p e r t i e s to depend on pr e d a t o r search e f f i c i e n c y . The e f f e c t s of randomly f l u c t u a t i n g winter s e v e r i t y and a l t e r n a t i v e o b j e c t i v e f u n c t i o n s are considered i n the e s t i m a t i o n of optimal feedback s t r a t e g i e s . The s t r a t e g i e s are compared to c u r r e n t and s i m p l i f i e d management p o l i c i e s . INTRODUCTION 98 A c o n s i d e r a b l e body of theory now e x i s t s on p r e d a t o r - p r e y systems (e.g., Rosenzweig and MacArthur, 1963; Rosenzweig, 1973; H o l l i n g , 1965; Noy-Meir, 1975). The e x i s t e n c e of m u l t i p l e e q u i l i b r i a i s one of the more u s e f u l d i s c o v e r i e s f o r n a t u r a l resource management. A number of re c e n t papers i n d i c a t e t h a t t h i s phenomenon may e x i s t i n a v a r i e t y of e c o l o g i c a l systems through d i f f e r e n t mechanisms (e.g., H o l l i n g , 1973 and 1978; Peterman, 1977; May, 1974; Bazykin, 1974; Tanner, 1975). One of the mechanisms i s the e f f e c t of depensatory m o r t a l i t y ( R i c k e r , 1954a), whereby an e f f i c i e n t predator i n c r e a s e s the p r o p o r t i o n of a prey p o p u l a t i o n k i l l e d as the prey p o p u l a t i o n decreases. T h i s can r e s u l t i n the predator h o l d i n g the prey at low d e n s i t y f o r extended p e r i o d s , or even d r i v i n g i t e x t i n c t . An i n c r e a s i n g number of o b s e r v a t i o n s i n the ungulate l i t e r a t u r e suggests e f f i c i e n t p r e d a t o r s such as wolves may keep t h e i r prey at low d e n s i t i e s (Haber, 1977; Bergerud, 1974; Mech and Karns, 1977; P i m l o t t , 1967). T h i s suggests that the f u n c t i o n a l and numerical responses of predators ( H o l l i n g , 1965) should be considered i n developing ungulate management p o l i c i e s . The aim of t h i s chapter i s t h r e e f o l d . F i r s t , i t d e s c r i b e s the s t r u c t u r e and behaviour of a s t o c h a s t i c wolf-ungulate model based on a l a r g e s c a l e s i m u l a t i o n model developed from data of an e i g h t - y e a r f i e l d study conducted by Dr. Gordon Haber i n the D e n a l i r e g i o n , A l a s k a (Haber, 1977; Haber, Walters, and Cowan, 1976). Second, i t prese n t s optimal moose h a r v e s t i n g and wolf 99 c o n t r o l s t r a t e g i e s estimated through s t o c h a s t i c dynamic programming f o r a number of a l t e r n a t i v e o b j e c t i v e s , i n response to the need f o r c o u p l i n g predator c o n t r o l with ungulate h a r v e s t i n g . These s t r a t e g i e s are of feedback c h a r a c t e r as d e s c r i b e d i n previous c h a p t e r s , s p e c i f y i n g the next a c t i o n (moose h a r v e s t , wolf c o n t r o l ) f o r any combination of wolf pack numbers and moose p o p u l a t i o n d e n s i t y . F i n a l l y , comparisons of long term y i e l d s obtained from a p p l y i n g o p t i m a l , s i m p l i f i e d , and f i x e d h a r v e s t r a t e s t r a t e g i e s are made. Har v e s t i n g i s of course not advocated f o r the D e n a l i N a t i o n a l Park; the data are merely used to d e r i v e management p o l i c i e s f o r comparable predator-ungulate systems. METHODS Model s t r u c t u r e and parameter estimates By compressing a d e t a i l e d computer s i m u l a t i o n model of wolf-moose-sheep i n t e r a c t i o n s i n D e n a l i N a t i o n a l Park (Haber, 1977: 431-520; Haber, Walters and Cowan, 1976) a d i s c r e t e time, s t o c h a s t i c s i m u l a t i o n model was developed. The model c o n s i d e r s events and changes over a time s c a l e of decades and uses a b a s i c time step of one year, beginning on May 1. To r e p r e s e n t dynamic changes (numerical responses) i n the t e r r i t o r i a l mosaic of wolf packs over l a r g e areas the model c o n s i d e r s a s p a t i a l area of 5,000 square m i l e s (roughly corresponding to the D e n a l i " r e g i o n " as d e f i n e d by Haber, 1977:8). General f a c t o r s considered i n the model are shown in F i g u r e 30. The model has two s t a t e v a r i a b l e s F i g u r e 30. F a c t o r s c o n s i d e r e d i n the model of a wolf-ungulate system i n A l a s k a . 100a, ENVIRONMENTAL WOLF POPULATION 101 M t = moose densi t y / s q u a r e mile Wt = number of wolf packs over the 5000 square m i l e area and a s t o c h a s t i c v a r i a b l e St = s n o w f a l l i n the p r e v i o u s winter Mt i s assumed to have d e n s i t y and s n o w f a l l dependent reproduc-t i o n , d e n s i t y independent n a t u r a l m o r t a l i t y (not due to wolf p r e d a t i o n ) , p r e d a t i o n l o s s e s , and harvest. Changes in Wt are modelled as a process of t e r r i t o r y s i z e adjustment. S t o c h a s t i c e f f e c t s are i n c l u d e d by a t h r e e - l e v e l s n o w f a l l v a r i a b l e St = (low, average, h i g h ) . I t i s assumed t h a t St+i i s independent of S f The f o l l o w i n g s e c t i o n s give d e t a i l e d d e s c r i p t i o n s of the f u n c t i o n a l r e l a t i o n s h i p s of the four model components: environmental v a r i a t i o n ( s n o w f a l l ) , prey dynamics, wolf pack dynamics, and predator-prey i n t e r a c t i o n s . Haber"s f i e l d data (Haber, 1977) provided almost a l l parameter estimates used i n the model. Environmental v a r i a t i o n Cumulative s n o w f a l l i s considered to be the best o v e r a l l index of environmental v a r i a t i o n i n the Mt. McKinley r e g i o n (Figure 31a). Each winter of the h i s t o r i c a l r ecord was assigned to a winter s e v e r i t y c l a s s : low, average, or high, depending on the cumulative s n o w f a l l (Figure 31b). The p r o b a b i l i t y of each type of winter o c c u r r i n g i n a given year corresponds to the observed f r e q u e n c i e s . Moose b i r t h r a t e s are assumed to respond i n v e r s e l y to cumulative s n o w f a l l d u r i n g the winter of pregnancy. Thus b i r t h s of moose are adjusted according to the r e l a t i o n s h i p F i g u r e 31. a) Cumulative s n o w f a l l recorded a t Mt. McKinley N a t i o n a l Park headquarters, w i n t e r s 1925—26 to 1973-74 Cfrom Haber, 1977:12). h)_ P r o b a b i l i t y d i s t r i b u t i o n o f winter s e v e r i t y c l a s s e s S = {low,average,high} w i t h correspon-d i n g cumulative s n o w f a l l ( i n . ) , d e r i v e d from the above r e c o r d . 102a a) 1930-31 1940-41 1950-51 1960-61 1970-71 b) 0.5 -j C L I 1 I 1 — 0.4->-H 0.3-_J § 0.2-m i — , ° 0 I -cc Q_ o.o - » — M — M — M — LOW AVER. HIGH CLASS 0-56 57-112 >II2 CUM.SNOWFALL (in.) 10 3 i n Table V I I I . Haber's data a l s o i n d i c a t e that s n o w f a l l i n f l u e n c e s the winter p u r s u i t success of wolves, but t h i s e f f e c t i s ignored. Prey dynamics Only the dynamics of the moose p o p u l a t i o n i s considered i n t h i s model. I t i s assumed that sheep, c a r i b o u , and other prey that frequent the r e g i o n p r o v i d e a f i x e d source of p o t e n t i a l food f o r wolves (O). In systems terms, other prey occurrence (0) i s t r e a t e d as a d r i v i n g v a r i a b l e that a f f e c t s p a t t e r n s of wolf t e r r i t o r y adjustment but the impact of wolves on 0 i s ign o r e d . D e n s i t y dependent r e p r o d u c t i v e r a t e s are assumed to operate i n the moose p o p u l a t i o n ( F i g u r e 32). This assumption has been commonly made i n ungulate p o p u l a t i o n models (Gross et a_l., 1973; Walters and Gross, 1973; Walters and B u n n e l l , 1971). E m p i r i c a l evidence i s a v a i l a b l e from a number of ungulate s t u d i e s (Teer et a l . , 1965; H e s s e l t o n , Severinghaus, and Tanck, 1965; O'Roke and Hamerstrom, 1948; Swank, 1958; F i l o n o v and Zykov, 1974). To o b t a i n a mean p o p u l a t i o n b i r t h r a t e (b) i t i s assumed that the age s t r u c t u r e of y e a r l i n g s and o l d e r moose i s r e l a t i v e l y s t a b l e and can thus be ignored. More p r e c i s e l y i t i s assumed t h a t the p r o p o r t i o n of moose v^ of age i > 1 i s independent of time. I f the number of c a l v e s C born a n n u a l l y can be expressed as C = 2 b i N i i where b i = age s p e c i f i c b i r t h r a t e N i = number of age i moose 104 Table VIII. Impact of snowfall on moose b i r t h s . Winter rating Cumul. snow- Calves/100 cows"'" B i r t h adjust-class f a l l (in.) ment factor low 0 - 56 125 1.00 average 57 - 112 95 0.7 6 high >112 60 0.48 From Haber 1977: F i g . 46 105 F i g u r e 32. _ _ De n s i t y dependent b i r t h r a t e f o r moose b=kL = v-0.045M. MOOSE/SQMI 106 The argument of f i x e d p o r t i o n s v-^  i m p l i e s C = Z b i v i M i i _ = Mb where b=^LbiVi i where M i s the t o t a l moose p o p u l a t i o n and b i s a weighted mean b i r t h r a t e per animal. Density dependent changes in a l l the b^ p a r a l l e l one another and do not r e s u l t i n a l a r g e change i n V J _ . Thus b as a f u n c t i o n of Mt i s not h e a v i l y dependent on the age composition of M f The mean b i r t h r a t e i s adjusted f o r w i n t e r s e v e r i t y by m u l t i p l y i n g t o t a l b i r t h s by the a p p r o p r i a t e b i r t h adjustment f a c t o r k s (Table V I I I ) . From age s p e c i f i c f e r t i l i t y r a t e s a mean maximum p o p u l a t i o n b i r t h r a t e at low d e n s i t y ( b m a x ) was estimated from Markgren, (1969). The estimate of b m a x (Table IX) assumes that age s p e c i f i c o v u l a t i o n r a t e s are an i n d i c a t o r of b i r t h r a t e s , r e c o g n i z i n g the f a c t t h at not a l l o v u l a t i o n s w i l l r e s u l t i n l i v e b i r t h s . The net r e s u l t i s that about 60 percent of pregnant female moose produce twin c a l v e s (LeResche and Hinman, 1973). Separate non-predator m o r t a l i t y r a t e s were a p p l i e d w i t h i n the year f o r c a l v e s , and a d u l t moose. From age s p e c i f i c m o r t a l i t y estimates a weighted mean m o r t a l i t y r a t e was estimated f o r ages > 1 (m a = 0.06). C a l f m o r t a l i t y was assumed to be 0.4 (Haber, 1977: 442). Wolf pack dynamics Based on e m p i r i c a l evidence that f u n c t i o n a l response parameters (handling time and p u r s u i t success) are independent of pack s i z e (Haber, 1977: F i g u r e s 56 and 58), the wolf pack i s T a b l e I X . E s t i m a t e o f w e i g h t e d maximum mean b i r t h r a t e b m a x ( f r o m M a r k g r e n , 1 9 6 9 ) . Age i P r o p o r t i o n O v u l a t i o n r a t e V i b ± v i b i 2 0.22 0.29 0.06 3 0.17 0.29 0.05 4 0.14 0. 34 0.05 5 0.12 0. 34 0.04 6 0.08 0.58 0.05 7 0.08 0.58 0.05 8 0.02 0.59 0.01 9 + 0.17 0.50 0.09 1.00 8 bm a x = S I v i b i = 0.40 "From M a r k g r e n 1969: 'From M a r k g r e n 1969: 265 202 108 assumed to be the f u n c t i o n a l u n i t of p r e d a t i o n i n the model. Changes i n number of wolf packs i n r e l a t i o n to changing prey d e n s i t y are represented as a process of t e r r i t o r y adjustment. From v a r i o u s s t u d i e s of wolf t e r r i t o r y s i z e i n North America, (Haber, 1977) suggested that t e r r i t o r y s i z e s are adjusted to prevent d e p l e t i o n of prey w i t h i n t e r r i t o r i e s ( F i g u r e 33). In the model the e q u i l i b r i u m t e r r i t o r y s i z e f o r any prey d e n s i t y i s estimated a c c o r d i n g to the r e l a t i o n s h i p : T ~ Tmax 1 + .001 B where T = e q u i l i b r i u m t e r r i t o r y s i z e / w o l f pack (square mile) Tmax = maximum t e r r i t o r y s i z e (square mile) B = ungulate prey biomass (lbs./square mile) When Wt times the t a r g e t area T i s l e s s than the t o t a l a v a i l a b l e area and i f B i s not too low, Wt+i i s in c r e a s e d to f i l l the t o t a l area provided t h i s does not exceed.the wolf's maximum i n t r i n s i c r a t e of i n c r e a s e (estimated to be 0.5 Wt)• The wolf pack r a t e of in c r e a s e i s f u r t h e r assumed to depend on the annual food intake per pack f o r small values of B (F i g u r e 34a). P o p u l a t i o n i n c r e a s e i s reduced g r a d u a l l y as food intake decreases below twice the maintenance requirement o f 40,000 l b s . per pack per year. A sharp drop i n the i n c r e a s e r a t e occurs i f food intake i s l e s s than 1.5 times the maintenance requirement e s t a b l i s h e d as 4-6 l b s . per wolf per day (Haber, 1977: 455). When the e q u i l i b r i u m t e r r i t o r y s i z e times Wt i s g r e a t e r than the a v a i l a b l e t o t a l area ( i n d i c a t i n g overcrowding) Wt w i l l 109 F i g u r e 33. Wolf t e r r i t o r y s i z e as a f u n c t i o n of ungulate prey d e n s i t y Cfrom Haber, 1977:483). 1500, I00£H I *• \ \ \ «^02 \ \ \ \ \ 03 N 500H . ? o « SOURCE OF DATA FOR ESTIMATES I I - BAFFIN ISLAND (CLARK, 1971) 2- TOKLAT PACK (PRESENT STUDY) 3" SAVAGE PACK (PRESENT STUDY) 4- ALBERTA - JASPER (CARBYN, 1974) 5- ONTARIO (KOLENOSKY, 1972) 6 " ISLE ROYALE, 1959-1961 (PETERSON, 1974) 7 - ISLE ROYALE, 1970-1974 (PETERSON, 1974) 8 " ONTARIO (PIML0TT,l967 t PIMLOTT ET AL, B69) 9 - MINNESOTA (VAN BALLENBERGHE ET AL.197S) BIOMASS CALCULATIONS ASSUME THE FOLLOWING1 MEAN LIVE WEIGHTS: MOOSE BOO LB. DALL SHEEP 125 BIGHORN SHEEP 160 MOUNTAIN GOAT 160 BARREN GROUND CARIBOU 225 WOODLAND CARIBOU 350 ELK 650 WHITE-TAILED DEER I 25 MULE DEER 150 or or • 5 S. •8 ""?Cr # 7 i 5 2500 O NUMBER OF PREY • BIOMASS (LBSJ 10 5000 15 7500 SO. MILE F i g u r e 34. a) Food dependent wolf pack r a t e of i n c r e a s e , measured r e l a t i v e to the i n n a t e c a p a c i t y f o r i n c r e a s e and assum-ing the area a v a i l a b l e f o r new t e r r i t o r i e s i s not l i m -i t i n g . b) Wolf s t a r v a t i o n l o s s i n r e l a t i o n to t o t a l biomass eaten. 110a,. BIOMASS EATEN/PACK/YR (xlO3) 112 NAi = A • Ni/(1 + A JEthjNj) • F j where NAi = expected number of k i l l s of prey category i per month Ni = d e n s i t y of prey category i (numbers/square mile) th-i = han d l i n g time f o r each prey of category i that was k i l l e d A = wolf pack search e f f i c i e n c y c onstant F = t o t a l wolf pack t e r r i t o r y over which search i s d i s t r i b u t e d The three prey c a t e g o r i e s included are: moose c a l v e s , o l d e r moose, and other prey. The m u l t i s p e c i e s d i s c equation parameters were estimated from Haber's data on wolf movements and behaviour. D i s a g g r e g a t i o n of p r e d a t i o n components r e l a t i n g number of prey k i l l e d by a predator to i t s time budget and prey d e n s i t y was f i r s t developed by H o l l i n g (1959). The search e f f i c i e n c y parameter (A) i s a combination of the p r o p o r t i o n of prey encounters that are s u c c e s s f u l ( P i ) , and the r a t e of e f f e c t i v e search f o r each prey ( a i ) . I f only moose were preyed upon (other prey a b s e n t ) , and moose d e n s i t i e s were low, the moose k i l l r a t e i s approximately A *N m/F, where A i s the slope of the f u n c t i o n a l response ( F i g u r e 35) at very low prey d e n s i t i e s , and F i s the wolf pack's t e r r i t o r y s i z e over which search i s d i s t r i b u t e d . To determine q u a l i t a t i v e behaviours of the system, s i m u l a t i o n s were c a r r i e d out i n the wolf pack search e f f i c i e n c y (A) — ot h e r prey d e n s i t y (O) parameter space. Table X summarizes parameter v a l u e s used f o r the d i s c r e t e time, s t o c h a s t i c s i m u l a t i o n model. I l l decrease by 10 p e r c e n t from t to t+1. There i s u n f o r t u n a t e l y no e m p i r i c a l data on how f a s t wolf t e r r i t o r y s i z e s can be a d j u s t e d i n nature. Thus the f o l l o w i n g j u s t i f i c a t i o n s are based on c i r c u m s t a n t i a l evidence. Mech (1977) studying a seven year deer d e c l i n e i n Minnesota noted that at l e a s t one wolf pack c o n t r a c t e d i n s i z e to the dominant mating p a i r while s t i l l m a i n t a i n i n g i t s t e r r i t o r y . Thus i t i s c o n s e r v a t i v e to assume t e r r i t o r i e s to be maintained u n t i l the dominant wolves d i e (about 10 y e a r s ) . In e f f e c t the model assumes that prey a v a i l a b i l i t y v a r i e s between wolf t e r r i t o r i e s such that a few packs are b a r e l y h o l d i n g onto t e r r i t o r i e s ( f i r s t 10 percent to drop out) w h i l e other wolf packs w i l l not f e e l the e f f e c t s of an o v e r a l l prey d e c l i n e f o r a s u s t a i n e d p e r i o d (Walters, Stocker, and Haber, 1979). The assumed slow (10 percent/year) r a t e s of p r e d a t o r p o p u l a t i o n adjustment a f t e r a prey d e c l i n e r e s u l t s i n exaggeration of any depensatory p r e d a t i o n e f f e c t s , and i s thus c r i t i c a l f o r the system's s t a b i l i t y p r o p e r t i e s . The model a l s o s u b j e c t s wolf packs to l o s s e s due to s t a r v a t i o n using the f u n c t i o n i n F i g u r e 34b. T h i s f u n c t i o n i s based on estimates of maintenance requirement per wolf pack per year; i t models l o s s e s as i n c r e a s i n g g r a d u a l l y as food intake drops below twice the maintenance l e v e l , then i n c r e a s i n g s h a r p l y i f food intake f a l l s below a lower t h r e s h o l d . Predator-prey i n t e r a c t i o n s The model estimates short-term f u n c t i o n a l response of wolf packs to prey d e n s i t y using a modified m u l t i s p e c i e s d i s c , e q u a t i o n (Charnov, 1973; Murdoch, 1973): 113 F i g u r e 35. F u n c t i o n a l response of wolf packs to moose d e n s i t y . 113a, MOOSE/SQMI (N m ) T a b l e X. P a r a m e t e r v a l u e s f o r t h e w o l f - u n g u l a t e m o d e l . 114 P a r a m e t e r D e s c r i p t i o n V a l u e m. m_ T. max -max max t h . t h . t h r A NA, •m K c a l f m o r t a l i t y r a t e 0.40 o l d e r moose m o r t a l i t y r a t e 0.06 a v e r a g e maximum b i r t h r a t e f o r moose p o p u l a t i o n 0.40 s l o p e o f d e n s i t y d e p e n d e n t b i r t h -0.045 r a t e f u n c t i o n maximum w o l f p a c k t e r r i t o r y s i z e 1500 (nri.2) maximum i n t r i n s i c r a t e o f i n c r e a s e . 0.50 o f w o l f p a c k numbers maximum f o o d d e p e n d e n t w o l f p a c k 0.20 m o r t a l i t y r a t e p r e d a t o r h a n d l i n g t i m e f o r o l d e r 0.01 moose p r e d a t o r h a n d l i n g t i m e f o r moose " I 0.00 5 c a l v e s p r e d a t o r h a n d l i n g t i m e f o r o t h e r 0.005 p r e y w o l f p a c k s e a r c h e f f i c i e n c y c o n s t a n t 200-3000 maximum number o f moose e a t e n p e r 100-300 w o l f p a c k p e r y e a r h a l f s a t u r a t i o n moose d e n s i t y 0.1-0.5 ( F i g u r e 35) t o t a l w o l f p a c k t e r r i t o r y o v e r w h i c h 5000 s e a r c h i s d i s t r i b u t e d (mi^) 115 O p t i m i z a t i o n procedure State increment, s t o c h a s t i c dynamic programming (Larson, 1968) was a p p l i e d to the two s t a t e v a r i a b l e dynamic model to estimate o p t i m a l feedback s t r a t e g i e s f o r both moose h a r v e s t i n g and wolf c o n t r o l . Dynamic programming estimates optimal s t r a t e g i e s (moose h a r v e s t , wolf c o n t r o l ) as a f u n c t i o n of the s t a t e v a r i a b l e s (wolf pack numbers, moose per square m i l e ) . The management o b j e c t i v e was to maximize the expected value of annual r e t u r n s over an a r b i t r a r y f u t u r e time planning h o r i z o n T or: T V t = 2 R t t where V t = t o t a l r e t u r n s from t to T Rt = annual r e t u r n s Assuming that annual r e t u r n s are a f u n c t i o n of both moose harv e s t and wolf c o n t r o l , the annual r e t u r n s can be w r i t t e n as: R t = H t - c w P t where Ht = moose h a r v e s t (moose/square mile) Pt = number of wolf packs removed by wolf c o n t r o l per year c w = r e l a t i v e c o s t of wolf c o n t r o l per wolf pack removed measured r e l a t i v e to moose harvest per square m i l e . For t h i s study three a l t e r n a t i v e c o s t f a c t o r s , assumed to be p e n a l t i e s f o r removing wolf packs, were c o n s i d e r e d : c w = 0 (No p e n a l t i e s were assessed f o r removing wolf packs) c w = constant ( P e n a l t i e s are assessed such that no wolf c o n t r o l i s deemed necessary) 116 c w = f(W t) (The p e n a l t i e s assessed are a f u n c t i o n of the number of wolf packs p r e s e n t ; F i g u r e 36). The l a t t e r two assumptions imply d i f f e r e n t types of c o n s e r v a t i o n a c t i o n taken a g a i n s t wolf c o n t r o l , whereas the f i r s t assumption i m p l i e s no a v e r s i o n to wolf c o n t r o l . For a d e t a i l e d d e s c r i p t i o n of the f o r m u l a t i o n of a two v a r i a b l e s t o c h a s t i c dynamic programming problem, the reader i s r e f e r r e d to Chapter 2. The continuous s t a t e and c o n t r o l v a r i a b l e s were approximated by the f o l l o w i n g d i s c r e t e v a l u e s : 17 l e v e l s of moose d e n s i t i e s (M t = 0.0, 0.5, 1.0,..., 8.0), 11 l e v e l s of wolf pack numbers (Wt = 0.0, 2.0, 4.0,..., 20.0), 9 h a r v e s t l e v e l s (H t = 0.0, 0.5, 1.0,..., 4.0), 3 wolf c o n t r o l s (P^ = 0.0, 2.0, 4 . 0 ) , and 3 s t o c h a s t i c outcomes ( S t = low, average, high) were used. The optimal moose s t r a t e g i e s are mapped as harvest r a t e i s o p l e t h s on the wolf-moose phase plane, whereas the optimal wolf c o n t r o l s t r a t e g i e s are represented as number of wolf packs to be removed ( a l s o on the phase p l a n e ) . Table XI shows parameter combinations of wolf pack search e f f i c i e n c y constants (A) and other prey d e n s i t i e s (0) f o r which optimal feedback s t r a t e g i e s were estimated. I t was assumed t h a t H t «== 0.5 and that P t < 4 packs per year. Optimal feedback s t r a t e g i e s were a l s o computed f o r a wide range of wolf c o n t r o l c o s t f a c t o r s ( c w ) . P r e d i c t i o n s and comparisons Optimal feedback s t r a t e g i e s were a p p l i e d to the dynamic model f o r very long s i m u l a t i o n runs (5000 years) to p r e d i c t r e t u r n s , as w e l l as e f f e c t s on s t a t e v a r i a b l e s . I n i t i a l 117 F i g u r e 36. One p o s s i b l e assumption about wolf c o n t r o l c o s t ( c w ) as a f u n c t i o n of number of wolf packs present. 117a.. Table XI. Parameter combinations f o r which o p t i m a l feedback s t r a t e g i e s were estimated. Parameter Search constant Other prey case-. (A) d e n s i t y ( 0 ) 1 550 3.50 2 500 1.30 3 900 0.60 4 300 0.05 119 c o n d i t i o n s were set near e q u i l i b r i u m c o n d i t i o n s , and the s i m u l a t i o n runs were c a r r i e d out using a random number procedure to generate winter c o n d i t i o n s with a p p r o p r i a t e p r o b a b i l i t i e s ( F i g u r e 30). To determine the s t a b i l i t y of the harvested system under the a p p l i c a t i o n of optimal s t r a t e g i e s , short-term s i m u l a t i o n s were performed by s y s t e m a t i c a l l y v a r y i n g i n i t i a l s t a t e values to extreme combinations. The r e s u l t i n g optimal paths were mapped on the wolf-moose phase plane. Comparisons were made between the optimal feedback s t r a t e g i e s , s i m p l i f i e d feedback s t r a t e g i e s , and f i x e d harvest r a t e p o l i c i e s . For t h i s purpose a s e r i e s o f 500 year (10 times 50 years) s i m u l a t i o n runs were c a r r i e d out f o r parameter cases one and three (Table XI) f o r o b j e c t i v e f u n c t i o n s where c w i s constant ( i . e . , Pt = 0). The d i f f e r e n c e s of average f i f t y - y e a r r e t u r n s (moose/square mile) between a p p l y i n g optimal s t r a t e g i e s and a p p l y i n g a l t e r n a t i v e s t r a t e g i e s or p o l i c i e s f o r each of 160 i n i t i a l c o n d i t i o n s were p l o t t e d as value of i n f o r m a t i o n i s o p l e t h diagrams as d e f i n e d i n previous c h a p t e r s . I n i t i a l c o n d i t i o n s were d e f i n e d at the i n t e r s e c t i o n s o f 16 l e v e l s of moose d e n s i t i e s (M = 0.5, 1.0, 1.5,... 8.0) and 10 l e v e l s of wolf pack numbers (W = 2.0, 4.0, 6.0,... 20.0) i n the wolf-moose phase pla n e . The s i m p l i f i e d moose h a r v e s t i n g s t r a t e g i e s f o r cases one and three were based on the e q u i l i b r i u m l e v e l of wolf packs (W* = 12 f o r case one and W* = 10 f o r case three; Figures 39 and 41) of the optimal s t r a t e g i e s . T h i s i m p l i e s that the s i m p l i f i e d s t r a t e g i e s u t i l i z e moose p o p u l a t i o n i n f o r m a t i o n only, assuming the wolf pack numbers remain constant at the e q u i l i b r i u m l e v e l . 120 The f i x e d h a r v e s t r a t e p o l i c i e s a p p l i e d best f i x e d h a r v e s t r a t e s , estimated by s i m u l a t i o n t r i a l s with the s t o c h a s t i c v e r s i o n of the model, to moose every year of the 50 year s i m u l a t i o n s . The best f i x e d h a r v e s t r a t e (0.05 f o r case one; 0.03 f o r case three) i s the one that maximizes the average of ten f i f t y year r e t u r n s at e q u i l i b r i u m wolf pack numbers and moose d e n s i t y . RESULTS AND DISCUSSION Model p r e d i c t i o n s S t a r t i n g a number of 100 year s i m u l a t i o n runs from very low moose d e n s i t i e s and high wolf pack numbers, and i g n o r i n g s t o c h a s t i c e f f e c t s , i t was found that the parameter space of search e f f i c i e n c y (A) and other prey occurrence (O) could be d i v i d e d i n t o four regions (Figure 37). For regions one and two, s i m i l a r p o p u l a t i o n trends were p r e d i c t e d . The only d i f f e r e n c e was a s l i g h t r e d u c t i o n i n moose d e n s i t y followed by recovery in r e g i o n two, whereas moose d e n s i t y i n c r e a s e d from the o u t s e t i n r e g i o n one. Region three parameter combinations p r e d i c t e d e x t i n c t i o n f o r the moose p o p u l a t i o n , with wolves being able to p e r s i s t on a l t e r n a t i v e prey. Region four parameter combinations p r e d i c t e d wolves to d r i v e moose down and to e f f e c t i v e l y go e x t i n c t with subsequent recovery of the moose (the system behaviour i n t h i s case appears to be a l i m i t c y c l e with very long p e r i o d ) . One parameter case was chosen from each r e g i o n (dots i n Figu r e 37) f o r subsequent s t a b i l i t y and o p t i m i z a t i o n a n a l y s e s . Of the four cases the model leads to two q u a l i t a t i v e l y F i g u r e 37. Parameter space of wolf search, e f f i c i e n c y (A) and ot h e r prey,occurrence CO) i n d i c a t i n g f o u r r e g i o n s t h a t e x h i b i t d i s t i n c t q u a l i t a t i v e behaviors of moose and wolf dynamics. Dots i n d i c a t e parameter combina-t i o n s f o r which o p t i m i z a t i o n s were c a r r i e d out. 121a, 0-1 . - 1 . r 0.0 1.0 2.0 3.0 4.0 OTHER PREY/SQ Ml ( 0 ) 1 2 2 d i f f e r e n t types of predator-prey behaviour between wolves and moose (Figure 38; Table.XII). If wolves are assumed to have r e l a t i v e l y low search e f f i c i e n c y (A •«? 550), then for a wide range of other prey densities (O) the model has a single stable equilibrium point where the i s o c l i n e s Wt+^ = Wt and M^+i = Mj-cross (cases one, two and four; Figure 38). In other words the equilibrium is globally stable (Holling, 1973; Hassel, Lawton, and May, 1976). If the wolves are e f f i c i e n t searchers (A = 900) and s u f f i c i e n t a lternative prey are available, then for low moose densities (Mt <r 0.5) a second equilibrium occurs at low wolf pack numbers and with moose extinct. Parameter case three defines two domains of a t t r a c t i o n , one around a high equilibrium point (as for the other cases) and the other involving extinction of moose. In t h i s s i t u a t i o n , perturbations such as harvesting or habitat deterioration can lead to crossing the boundary between the two domains of attraction,* creating a dangerous management p o s s i b i l i t y . Similar predictions were made from a large scale simulation model of the wolf-ungulate system which included representation of prey age classes (Haber, 1977; Haber et a l , 1976); the present simple model captures the essential ecological behaviour stemming from s i m i l a r non-linear relationships in the more complex system. The assumption that Wt w i l l begin to respond immediately to reduction in Mt i s an o v e r s i m p l i f i c a t i o n . Mech (1977) has observed a wolf pack in Minnesota that did not give up i t s t e r r i t o r y during a seven year deer decline although the pack was reduced in size. However, to include response lags in the model, F i g u r e 38. I s o c l i n e s of zero moose growth CMt+]_=Mt)_ and i s o -c l i n e s of zero wolf pack growth CWt+i=Wt) i n the wolf pack-moose phase plane. E n c i r c l e d numbers r e f to parameter case combinations l i s t e d i n Table XI. 124 Table XII. E q u i l i b r i a i n the absence of h a r v e s t r e l a t e d - t o -search and other prey parameters ( k g = .76) Parameter Search constant Other prey S t a b l e Unstable case (A) d e n s i t y W* (0/sq.mi..) M* W* M* 1 550 3.50 3.8 16.1 — 2 500 1.30 3.6 13.4 — 3 900 0.60 3.4 12.3 0.5 4 300 0.05 3.5 12.0 125 at l e a s t one a d d i t i o n a l s t a t e v a r i a b l e would be needed to t r a c k accumulated s t r e s s on the wolves (Walters e t al., 1979). S i m i l a r arguments could be made f o r a number of other p o s s i b l e s t a t e v a r i a b l e s t h a t might have been i n c l u d e d . The choice made here was to remain simple to the p o i n t where e x i s t i n g o p t i m i z a t i o n techniques such as dynamic programming could be u t i l i z e d . Optimal feedback s t r a t e g i e s Optimal moose har v e s t r a t e s and optimal wolf c o n t r o l i n r e l a t i o n to the four parameter cases and f o r va r i o u s wolf c o n t r o l c o s t s ( c w ) are presented i n F i g u r e s 39-42. I f no wolf c o n t r o l c o s t i s assessed ( c w = 0.0), the optimal moose harvest i s almost independent of wolf d e n s i t y ( i n d i c a t e d by the near v e r t i c a l h a r v e s t i s o p l e t h s ) f o r a l l parameter cases. For a l l values of Wt, these s t r a t e g i e s c a l l f o r a f i x e d escapement of about 2.5-3.5 moose per square m i l e . For parameter case one, t h i s c o n c l u s i o n i s not changed when wolf c o n t r o l c o s t s are inc r e a s e d to a l e v e l where wolf c o n t r o l i s excluded ( c w = 0.8). I f wolf c o n t r o l c o s t s are between the two prev i o u s s i t u a t i o n s ( c w = 0.4) the harvest r a t e i s o p l e t h s s h i f t to higher moose d e n s i t i e s . T h i s i n d i c a t e s t h a t moose t h a t would have been k i l l e d by wolves can now be harvested, s i n c e wolf packs are removed i f Wt < 4. A d i f f e r e n t p i c t u r e emerges when wolves are assumed to be more e f f i c i e n t s e a r c h e r s (case three) and/or other prey d e n s i t y i s reduced (cases two, three, and f o u r ) . When wolf pack numbers are g r e a t e r than about 16, and when no wolf c o n t r o l i s o p t i m a l , 126 Figure 39. Optimal feedback strategies D° estimated by sto-chastic dynamic programming for d i f f e r e n t wolf con-t r o l costs ( c w ) , dynamic parameter case one i n Fig-ure 38. Dots refer to optimal e q u i l i b r i a . 19.6a co • C C a. o cc L U 2:° ZD in ' MOOSE HARVEST RATES C =0.0 w 0.0 a .100 .200 .300 . 400 .300 2.0 4.0 6.0 8.0 M00SE/SQMI CO • cr C L 0= S o . U_ a ce L U WOLF CONTROL 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 "•0 2^0 4*0^  JuT M00SE/SQMI 8 0 CO • (->" C C a. 0 ° o cc L U 2:° =)in' 0 .0 1 1 r— 2.0 4.0 6.0 M00SE/SQMI CO -^ i n . cc Q_ 0 ° C C L U ca =3 i n ' 8.0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 *0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00-. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o o o o o o o o o o o o o o a o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I 1 1 1 0.0 2.0 4.0 6.0 8.0 M00SE/S0MI C w =0 . 4 CO • v^ tn _ (->" cc a. OR cc L U 2:R S i n " V „ „ 1 • 1 1 0.0 2.0 4.0 6.0 8.0 MO0SE/SQMI cc C L . L l -o in 3 m ' 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 • 0 0 0 0 0 0 0 0 0 0 0 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 — - w* — 0 /4 / 4 4 4 4 4 4 4 4 4 4 4 4 4 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 T 1 1 0.0 2.0 4.0 6.0 8.0 M03SE/SQMI 127 Figure 40. Optimal feedback strategies D° estimated by sto-chastic dynamic programming for d i f f e r e n t wolf con-t r o l costs ( c w ) , dynamic parameter case two i n F i g -ure 38. Dots refer to optimal e q u i l i b r i a . F i g u r e 41. Optimal feedback s t r a t e g i e s D° estimated by s t o -c h a s t i c dynamic programming f o r d i f f e r e n t wolf con-t r o l c o s t s ( c w ) , dynamic parameter case three i n F i g u r e 38. Dots r e f e r to o p t i m a l e q u i l i b r i a . I28cu MOOSE HARVEST RATES WOLF CONTROL cw=0.0 "T * 1 1 0.0 2.0 4.0 6.0 8.0 M00SE/S0MI r\j 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 co°- 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 RC 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 Q. LF 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 UJ 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 Z 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 2 2 Z 2 2 2 2 2 2 2 2 2 2 2 2 2 I T 1 0.0 2.0 4.D 6.0 MOaSE/SQMI B.O n V I . I. co • cr Q_ 3 D . U_ a ce UJ •.JOUDO.SJO . « ! .991 1 1 1 — 0.0 2.0 4.0 6.0 MOOSE/sawi CO -(_>"* cn a. OR o ce L U 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o o o o - o T T o T o~o-o-o o o o ^ — . 0 0^0 0 0 0 0 0 0 0 0 0 0 0 0 0 o / o o o o o o o o o o o o o o o 0' 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8.0 I 1 1 — 0.0 2.0 4.0 6.0 M00SE/SQMI 8.0 cw=0.55 2.0 4.0 6.0 M00SE/SQMI CO • CJ" CE Q_ 3 o . o ce LU 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ,0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 • 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 /A 4 4 4 4 w* 4 4 4 4 4 4 4 4 4 4 / 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 8.0 1 T r— 0.0 2.0 4.0 6.0 M00SE/SQMI 8.0 Figure 42. Optimal feedback strategies D° estimated by sto-chastic dynamic programming for d i f f e r e n t wolf con-t r o l costs ( c w ) , dynamic parameter case four i n Figure 38. Dots re f e r to optimal e q u i l i b r i a . MOOSE HARVEST RATES WOLF CONTROL I25c i cn ^in _ (_)" cc CL O R CC LU CQ c^O.O .100OT.300 ao 0LJ00.2do .3M .4D0 .900 1 1 ? r— 0-0 2.0 4.0 6.0 MOOSE/SQMI 7.0 CO • CC a . o°. L l _ O tt CU r a m -4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 2 Z 2 2 Z 2 2 2 2 2 2 2 2 2 2 2 1 • — I 1 0.0 2.0 4.0 6.0 a.o MOOSE/SQMI cn (_>-cc O R S o . O CC L U CO 7Za. ZDm" cw=I.O 0.0 ~~I 1 I — 2.0 4.0 6.0 MOOSE/SQMI 8.0 cn • <->~" CC CL O R u_ o ce L U i= R S i n ' 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I I 1 1 0-0 2.0 4.0 6.0 a 0 MOOSE/SQMI cn • c_>~ CC CL O R : * o -ce LU 2 : R Z J m " c w = 0 5 Q .100.200 .300 .400 .300 I T 1— 0-0 2.0 4.0 6 0 MOOSE/SQMI a.o CO • <_> cc CL O R a cc LU CD 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 u O O O J O O O O O O O O O O O 0 0 O^CTTT 0 O ^ ^ O ^ D - O ^ O 0 0 0^0 0 0 0 0 0 0 0 0 0 0 0 0 0 / 0 / 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o ' o o o o *o o o o o o o o o o o / 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b o o o o o o o o o o o o o o o ta/4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 V 2222 i 2 2 2 2 2 2 2 2 2 2 2 1 T 1 0.0 2.0 4.0 6.0 MOOSE/SQMI 8.0 130 the best moose s t r a t e g y i s to harvest the moose very hard (harvest r a t e = 0.50; F i g u r e s 40-42). For parameter case three where wolves are very e f f i c i e n t searchers (A = 900), t h i s high h a r v e s t reduces moose to e x t i n c t i o n ; they would not recover from wolf p r e d a t i o n anyway (F i g u r e 41). On the other hand, the i m p l i c a t i o n of hard h a r v e s t i n g i n case two (fewer a l t e r n a t i v e prey than case one) i s merely to expedite the approach to the optimal f i x e d escapement ( F i g u r e 40). I f other prey d e n s i t y i s very low ((O) = 0.05 and c w = 0.50), hard h a r v e s t i n g at high wolf pack numbers i n d i c a t e s t h a t i t i s optimal to d r i v e the moose to very low l e v e l s l e a d i n g to wolf e x t i n c t i o n and subsequent moose recovery (case 4). T h i s c o n c l u s i o n i s probably the most i n t e r e s t i n g , as i t p r o v i d e s a means f o r a p p l y i n g i n d i r e c t wolf c o n t r o l through s t a r v a t i o n ( F i g u r e 42). For a l l cases, the optimal wolf c o n t r o l s t r a t e g i e s are completely i n s e n s i t i v e to moose d e n s i t y . Furthermore, these s t r a t e g i e s are not s e n s i t i v e to wolf search e f f i c i e n c y and other prey d e n s i t y . Wolf c o n t r o l s t r a t e g i e s are however very dependent on the c o s t of wolf c o n t r o l . Table XIII l i s t s t h r e s h o l d values f o r wolf c o n t r o l c o s t s above which c o n t r o l should never be a p p l i e d . T h i s t a b l e i n d i c a t e s t h r e s h o l d values to be somewhat dependent on predator e f f i c i e n c y and other prey d e n s i t y . For c w lower than t h r e s h o l d v a l u e s , i t i s optimal to e x e r t wolf c o n t r o l o n l y when Wt i s l e s s than W* ( F i g u r e s 39-42), where W* r e p r e s e n t s a c r i t i c a l wolf pack number above which no c o n t r o l should be a p p l i e d . By e s t i m a t i n g optimal feedback s t r a t e g i e s f o r d i f f e r e n t v alues of c w , i t was found that W* i n c r e a s e d as c w decreased. A l s o , as W* tended towards zero, c w approached u n i t y . These 131 Table X I I I . Wolf c o n t r o l c o s t t h r e s h o l d v a l u e s i n r e l a t i o n to parameter cases. Parameter Search, co n s t a n t Other prey : : : C r i t i c a l case . (A) d e n s i t y C (0/sq.mi.) 1 550 3.50 0.8 2 500 1.30 1.1 3 900 0.60 1.1 4 300 0.05 1.0 132 r e s u l t s are c o u n t e r i n t u i t i v e s i n c e i t was expected that optimal wolf c o n t r o l s t r a t e g i e s would c a l l f o r removing wolf packs o n l y when moose are at low d e n s i t i e s . A d d i t i o n a l c o m p l i c a t i o n s r e s u l t when wolf c o n t r o l c o s t s are assumed to be a f u n c t i o n of wolf pack numbers (Figure 43). Using t h i s o b j e c t i v e f u n c t i o n , an i n t e r e s t i n g optimal moose h a r v e s t i n g diagram emerges f o r parameter case one. The v e r t i c a l p a t t e r n i s e s s e n t i a l l y r e t a i n e d as f o r the a l t e r n a t i v e o b j e c t i v e f u n c t i o n (e.g., c w = 0.0), but a l o c a l d e p ression of harvest r a t e s i s optimal at r e l a t i v e l y high wolf pack numbers and moose a t 3.5 per square m i l e . For cases three and four i t i s apparently optimal to h a r v e s t moose to e x t i n c t i o n f o r very low moose d e n s i t i e s and wolf pack numbers g r e a t e r than ten. The hard h a r v e s t i n g s o l u t i o n s found f o r the a l t e r n a t i v e o b j e c t i v e f u n c t i o n ( c w = constant) f o r cases two, three and fo u r no longer hold (except f o r moose > &6 per square mile) s i n c e c w i s near zero at high wolf pack numbers (implying wolf c o n t r o l ) . For t h i s o b j e c t i v e f u n c t i o n the optimal wolf c o n t r o l s t r a t e g i e s are dependent on moose d e n s i t y i n a l l cases (Figure 43). I t i s optimal to apply wolf c o n t r o l only when i s l e s s than W* or g r e a t e r than W** (F i g u r e 43), where W** re p r e s e n t s an upper wolf pack number above which wolf c o n t r o l i s again a p p l i e d . The s t a t e space r e g i o n over which no wolf c o n t r o l i s the op t i m a l s o l u t i o n ( i . e . , W*<Wt<W**) i s s e n s i t i v e to wolf pack search e f f i c i e n c y and other prey d e n s i t y , not s u r p r i s i n g l y being s m a l l e s t at high search e f f i c i e n c y (A = 900; case t h r e e ) . The optimal s t r a t e g i e s can be summarized as f o l l o w s : F i g u r e 43. Optimal feedback s t r a t e g i e s D° estimated by s t o -c h a s t i c dynamic programming f o r c w = f ( W t ) . E n c i r -c l e d numbers r e f e r to parameter cases. Dots i n d i cate o p t i m a l e q u i l i b r i a . MOOSE HARVEST RATES WOLF CONTROL © O..1O0.2OD -SCO .«o .xs 3.0 4.0 6.0 MOOSE/SQMI d= 53^ 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 4 ^ 4 4 4 4 2 4 4 4 2 2 2 / 0 0 0 0 0 0 0 0 0 0 2 2 / 0 0 * 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o o o o o o o o o o o o o o o o _ w* 0 0 / ^ 4 4 4 4 4 4 4 4 4 4 4 4 4 0 ' 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 - i • r — 2.0 4.0 6.0 MOOSE/SQMI S i a.ioa jaa ,WD ,«o 2.0 4.0 6.0 MOOSE/SQMI 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 ; 4 4 4 4 4 4 _W*K 4 4 4 4 4 4 4 4 4 4 A; 0 0 0 0 0 4 4 4 2 2 2 2 2 2/0 0 0 0 0 0 0 2 2/0 0 * 0 0 0 0 0 0 0 0 0 0 0 0 o o o o o o o o o o o o o o o o w * . _ 0 0 0 0 / / 4 4 4 4 4 4 4 4 4 4 4 4 0 ^ 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 I 0 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 —1 <? 1 2.0 4.0 6.0 MOOSE/SQMI : * o . a.iao.zdo .300 .ao .300 ^ T 1— 2.0 4.0 6.0 MOOSE/SQMI 0 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 0 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 0 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 0 4 4 4 4 4 4 4 4 4 4 4 4 ' 0 0 0 0 4 4 2 2 2 2 2 2 2 2 , '0 0 0 0 0 ^-"« ' 0 2 2./ 0 0 0 0 0 0 0 0 0 0 0 0 0 — W* 0 0 0 , ' 4 4 4 4 4 4 4 4 4 4 4 4 4 0 / 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 -r 2.0 4.0 6.0 MOOSE/SQMI 8.0 3 m ' 0-0 2.0 4.0 6.0 MOOSE/SQMI 0 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 0 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 0 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 0 4 4 4 4 4 4 4 4 4 4 4 4 / 0 0 0 0 4 4 4 2 2 2 2 2 2 2 M) 0 0 0 0 . W M ' 0 4 2 ^ 0 0 * 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 0 0 . w* 0 0 / 4 4 4 4 4 4 4 4 4 4 4 4 4 4 / 0 / 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 / 2 2 7 . 2 2 2 2 2 2 2 2 2 2 2 2 2 T 2.0 4.0 6.0 MOOSE/SQMI 6.0 134 h a r v e s t the moose p o p u l a t i o n to leave an optimal escapement of two to three moose per square m i l e , and e s s e n t i a l l y do nothing to the wolf p o p u l a t i o n i f c o n t r o l c o s t s are r e l a t i v e l y h i gh. On the other hand i f c o n t r o l c o s t s are zero or near zero remove wolf packs to i n c r e a s e the moose h a r v e s t . The c o n t r o l s t r a t e g i e s i n F i g u r e s 39-43 are op t i m a l o n l y with r e s p e c t to the d e f i n e d o b j e c t i v e of maximizing r e t u r n s in the l i g h t of wolf c o n t r o l c o s t s . A l t e r n a t i v e optimal s t r a t e g i e s would be found i f o b j e c t i v e f u n c t i o n s were changed. For example o b j e c t i v e f u n c t i o n s could be d e f i n e d that allow at l e a s t some moose harvest even at very low moose d e n s i t i e s . Fixed escapement of breeding stock has been advocated i n f i s h e r i e s work ( L a r k i n and R i c k e r , 1964; Tautz, L a r k i n , and R i c k e r , 1969; Walters, 1975). S p e c i f i c a l l y i t has been shown that higher y i e l d s can be obtained from s t o c h a s t i c a l l y v a r y i n g p o p u l a t i o n s such as the moose i n the present study, by ma i n t a i n i n g a f i x e d breeding stock r a t h e r than a p p l y i n g a f i x e d e x p l o i t a t i o n r a t e . So f a r such s t r a t e g i e s have not been developed f o r ungulate p o p u l a t i o n s . To employ optimal feedback s t r a t e g i e s i n a c t u a l management of wolf-ungulate systems, i t would be necessary to an n u a l l y determine the values of the two s t a t e v a r i a b l e s (Mt, Wt). P r e d i c t i o n s using optimal s t r a t e g i e s Table XIV prese n t s p r e d i c t e d mean values from 5000 year s i m u l a t i o n t r i a l s (100 f i f t y - y e a r runs s t a r t e d near e q u i l i b r i u m ) when optimal feedback s t r a t e g i e s are a p p l i e d , of annual moose r e t u r n s , wolf c o n t r o l , moose d e n s i t y , and wolf pack numbers. Table XIV. P r e d i c t e d means from long s i m u l a t i o n runs{(5000 years) i n r e l a t i o n to parameter cases and a l t e r n a t i v e o b j e c t i v e f u n c t i o n s . Parameter case Mean annual r e t u r n / m i 2 Mean annual wolf c o n t r o l Mean moose p o p u l a t i o n / m i 2 Mean wolf pack numbers Obj. f u n c t i o n 1 PM- = U 4 . - c w w t 1 2 3 4 1 2 3 4 1 2 3 4 0.285 0.285 0.285 0.285 0.133 0.118 0.093 0.104 0.133 0.118 0.093 0.104 c w w t 0.120 0.120 0.120 0.120 Obj. f u n c t i o n 2 = ut -0.000 0.000 0.000 . 0.000 Obj. f u n c t i o n 3 R t = ut - C w w t 0.000 0.000 0.000. 0.000 <cw = 0.0) 4.039 0.280 4. 039 0.280 4. 039 0 .280 4.039 0.280 ( c w = constant; Table XI 2. 363 12.571 2.348 10.750 2. 804 11.047 2.825 10.428 <cw = f ( W t ) ; F i g u r e 36) 2. 363 12.571 2. 348 10.750 2. 804 11.047 2.825 10.428 VJ1 136 Values are shown f o r a l t e r n a t i v e o b j e c t i v e s and parameter cases. The most important c o n c l u s i o n i s that the r e t u r n s as w e l l as the a s s o c i a t e d moose and wolf p o p u l a t i o n s are q u i t e s e n s i t i v e to the o b j e c t i v e s , but are l e s s s e n s i t i v e to the parameter combinations of wolf search e f f i c i e n c y and other prey d e n s i t y . For example i n case one, the average h a r v e s t s obtained i f no wolf c o n t r o l i s a p p l i e d are p r e d i c t e d to be l e s s than h a l f than would be expected i f wolf packs were removed on a r e g u l a r b a s i s . T h i s r e s u l t i s not s u r p r i s i n g s i n c e moose that would otherwise have been preyed upon by wolves could be harvested. By s y s t e m a t i c a l l y s t a r t i n g 50 year s i m u l a t i o n s at d i f f e r e n t p o p u l a t i o n s i z e s , i t was p o s s i b l e to map the optimal paths on the wolf-moose phase plane. S t o c h a s t i c s i m u l a t i o n s of the optimal s t r a t e g i e s being a p p l i e d to a l l parameter cases, f o r wolf c o n t r o l c o s t being zero, i n d i c a t e d that the optimal s t r a t e g i e s tend to s t a b i l i z e the moose p o p u l a t i o n at about 4 per square m i l e , while the wolves are wiped out (Figure 44a). If the c r i t i c a l c w values are used f o r the s t o c h a s t i c s i m u l a t i o n s (Table XIII) the optimal s t r a t e g i e s s t a b i l i z e the moose at about 2 to 3 per square m i l e and the wolves at about 10 to 12 packs ( F i g u r e 44b). In comparison the d e t e r m i n i s t i c unharvested n a t u r a l d e n s i t i e s averaged about 4 moose per square m i l e . T h i s o b s e r v a t i o n i s c o n s i s t e n t with one of the p o i n t s of the present theory of e x p l o i t a t i o n of animal p o p u l a t i o n s summarized by Anderson (1975) . For the high wolf search e f f i c i e n c y case (A=900), two e q u i l i b r i a emerge as i n the unharvested system. T h i s i n d i c a t e s that two domains of a t t r a c t i o n are pr e s e n t ( H o l l i n g , 1973), which can be separated by 137 F i g u r e 44. Examples of o p t i m a l paths d e r i v e d from 50 year simu-l a t i o n s a p p l y i n g o p t i m a l feedback s t r a t e g i e s , i n r e -l a t i o n to wolf c o n t r o l c o s t s ( c w ) and parameter cases. Dotted l i n e s i n d i c a t e domains of s t a b i l i t y f o r o p t i m a l t r a j e c t o r i e s l e a d i n g to c o e x i s t e n t versus s i m p l i f i e d system (wolves or moose absent) e q u i l i b r i a . 138 a boundary (dotted l i n e s i n F i g u r e s 41 and 44c). An a d d i t i o n a l domain of a t t r a c t i o n i s found i n a l l cases when c w i s l e s s than the c r i t i c a l c w l i s t e d i n Table X I I I . For these s i t u a t i o n s the lower domain boundaries are d e f i n e d by W* ( F i g u r e s 39-42, and 44d) . M u l t i p l e s t a b i l i t y r e g i o ns are a l s o i n d i c a t e d i f the wolf c o n t r o l c o s t i s a f u n c t i o n of wolf pack numbers. For example i n parameter case f o u r a l a r g e r e g i o n of the phase plane leads to s t a b l e moose p o p u l a t i o n s i z e s at 4 per square m i l e , while wolves are e l i m i n a t e d ( F i g u r e s 43 and 44e). Comparison with a l t e r n a t i v e s t r a t e g i e s The s i m p l i f i e d moose h a r v e s t i n g s t r a t e g i e s f o r cases one and three are shown i n Fig u r e 45. The s t r a t e g i e s e s s e n t i a l l y c a l l f o r a h a l t to e x p l o i t a t i o n once the moose d e n s i t y drops to 2.0 per square mile f o r case one (Figure 45a) and 2.5 per square mile f o r case three ( F i g u r e 45b) r e g a r d l e s s of the number of wolf packs present. The d i f f e r e n c e s i n r e t u r n s (moose/square mile) obtained between a p p l y i n g the optimal moose h a r v e s t i n g s t r a t e g y and a p p l y i n g the s i m p l i f i e d h a r v e s t i n g s t r a t e g y f o r parameter case one are n e g l i g i b l e . T h i s i s i n d i c a t e d by the extremely low values of i n f o r m a t i o n d e p i c t e d i n Fig u r e 46a. T h i s r e s u l t suggests t h a t no s u b s t a n t i a l gains can be made from i n c l u d i n g knowledge about the s i z e of the wolf p o p u l a t i o n i n the moose ha r v e s t d e c i s i o n i f search e f f i c i e n c y of wolves i s r e l a t i v e l y low (A=550). Th i s c o n c l u s i o n i s s l i g h t l y changed f o r the s i t u a t i o n where wolf pack search e f f i c i e n c y i s high (A=900). S l i g h t but Figure 45. Simplified harvesting strategies estimated from op-timal feedback strategies, assuming wolves remain near equilibrium, a) parameter case one; b) para-meter case three. F i g u r e 46. Simulated response s u r f a c e s of valu e s o f informa-t i o n f o r 160 combinations of moose d e n s i t i e s and wolf pack numbers d e r i v e d from comparing o p t i m a l to s i m p l i f i e d s t r a t e g i e s a) parameter case one; b) parameter case t h r e e . (Contours r e p r e s e n t mean values o f i n f o r m a t i o n f o r ten f i f t y - y e a r manage-ment p e r i o d s ) . I N I T I A L NO. OF WOLF PACKS Q.O 5.0 10.0 15.0 <=> j I I I o I cr 141 c o n s i s t e n t gains are expected (up to 1.8 moose/square mile f o r a 50 year management period) i f the number of wolf packs are in c l u d e d i n the moose ha r v e s t d e c i s i o n ( i . e . , using optimal feedback s t r a t e g y ) . However t h i s i s only the case i f wolf packs are above 16 (Figure 46b). A l s o , these gains are d e c e p t i v e , s i n c e at high wolf pack numbers i t i s optimal to e l i m i n a t e the moose s i n c e they would not recover from wolf p r e d a t i o n i n any case (as i n d i c a t e d i n F i g u r e s 41 and 44c). Comparisons of the optimal s t r a t e g i e s with the best f i x e d h a r v e s t r a t e p o l i c i e s i n d i c a t e t hat s u b s t a n t i a l improvements can be made by u t i l i z i n g i n f o r m a t i o n about both the moose d e n s i t y and the wolf pack numbers i f moose are above 4.0 per square m i l e f o r both parameter cases (Figure 47). T h i s i s shown by the values of in f o r m a t i o n ranging from 0.3 to 3.3 f o r parameter case one and from 0.4 to 4.4 (moose/square mile f o r 50 year management period) f o r parameter case t h r e e . For the 5000 square mile area under c o n s i d e r a t i o n f o r t h i s study, the maximum value of 4.4 moose per square m i l e corresponds to an annual g a i n of 440 moose. The most important c o n c l u s i o n to be drawn from these r e s u l t s i s that the r e t u r n s from using s i m p l i f i e d s t r a t e g i e s are not d r a m a t i c a l l y d i f f e r e n t from those of using optimal feedback s t r a t e g i e s . For p r a c t i c a l management, t h i s i m p l i e s that e f f o r t s should be d i r e c t e d towards the r o u t i n e c o l l e c t i o n of ungulate p o p u l a t i o n d e n s i t y i n f o r m a t i o n , while the predator p o p u l a t i o n s i z e need not be monitored a n n u a l l y , except as needed to e s t a b l i s h b a s i c f u n c t i o n a l r e l a t i o n s h i p s (model parameters). Furthermore, the ungulate h a r v e s t i n g d e c i s i o n should be based on c u r r e n t ungulate d e n s i t y (feedback c o n t r o l ) r a t h e r than on a F i g u r e 47. Simulated response s u r f a c e s of valu e s o f informa-t i o n f o r 160 combinations of moose d e n s i t i e s and wolf pack numbers d e r i v e d from comparing o p t i m a l moose h a r v e s t s t r a t e g i e s to f i x e d h a r v e s t r a t e p o l i c i e s a) parameter case one; b) parameter case t h r e e . (Contours r e p r e s e n t mean v a l u e s of informa-t i o n f o r ten f i f t y - y e a r management p e r i o d s ) . 143 p r e v i o u s l y c a l c u l a t e d , f i x e d h arvest r a t e (open loop c o n t r o l ) as documented by the s u b s t a n t i a l l o s s e s ( i n d i c a t e d by high values of information) from a p p l y i n g f i x e d h arvest r a t e s . 144 CONCLUSIONS S t o c h a s t i c dynamic programming seems an a p p r o p r i a t e method f o r e s t i m a t i n g optimal feedback c o n t r o l s t r a t e g i e s f o r preda t o r - p r e y systems, p r o v i d e d the system dynamics can be captured i n r e l a t i v e l y simple models (having few s t a t e v a r i a b l e s ) . While the pr e s e n t study has concentrated only on the Alaskan wolf-ungulate system, the o p t i m i z a t i o n procedure should be a p p l i c a b l e i n many other s i t u a t i o n s , f o r which the necessary parameter values can be estimated. I t must be emphasized that the d e s c r i b e d optimal feedback c o n t r o l s t r a t e g i e s apply only when the o b j e c t i v e s are (as defined) to maximize average r e t u r n s . I f other f a c t o r s , such as f o r example hunting q u a l i t y , number of hunters, or season length were i n c l u d e d i n the o b j e c t i v e f u n c t i o n , the optimal feedback s t r a t e g i e s might be r a d i c a l l y d i f f e r e n t . S p e c i f i c c o n c l u s i o n s that f o l l o w from the study of the wolf-ungulate system are: 1. Optimal moose h a r v e s t i n g s t r a t e g i e s appear to be dependent on wolf c o n t r o l c o s t s . I f no wolf c o n t r o l c o s t i s assessed, o p timal moose harvest i s a l s o independent of wolf d e n s i t y . Where wolf c o n t r o l i s excluded optimal moose h a r v e s t i n g s t r a t e g i e s a l s o depend on the pr e d a t o r search e f f i c i e n c y and other prey occurrence. 2. Optimal wolf c o n t r o l s t r a t e g i e s are completely i n s e n s i t i v e to moose d e n s i t y . They are, as expected, dependent on wolf c o n t r o l c o s t assumptions. 145 3. Expected r e t u r n s from best f i x e d harvest r a t e p o l i c i e s are s u b s t a n t i a l l y lower than expected r e t u r n s from optimal s t r a t e g i e s . 4 . Expected r e t u r n s from using s i m p l i f i e d s t r a t e g i e s (based on optimal s t r a t e g i e s ) are not d r a m a t i c a l l y d i f f e r e n t from those of using optimal feedback s t r a t e g i e s . 146 CHAPTER 5 G e n e r a l D i s c u s s i o n 147 COMMENTS ON MODELLING STUDIES The c o n s t r u c t i o n of models has the p o t e n t i a l to play a powerful r o l e i n the development of management s t r a t e g i e s and p o l i c y f o r m u l a t i o n . Since the foundations f o r a coherent theory of a p p l i e d ecology has been l a i d , there has been a growing i n t e r e s t i n a p p l y i n g models to h a r v e s t i n g and pes t c o n t r o l problems (see r e c e n t review by Conway, 1977). The e a r l y p e r c e p t i o n that resource management f i e l d s are r e l a t e d to one another by a common dependence on ecology, and by a common problem of o p t i m i z a t i o n , has l e d to a p r o l i f e r a t i o n of models a p p l i e d to a number of areas of resource management. Converting i n f o r m a t i o n i n t o a c t i o n r e q u i r e s some form of m o d e l l i n g . The f o l l o w i n g d i s c u s s i o n i s centered around the issue of whether s u b s t a n t i a l b e n e f i t s r e s u l t from p r e c i s e q u a n t i f i c a t i o n of v e r b a l and i n t u i t i v e models that would otherwise provide the b a s i s f o r ungulate management. I t i s q u i t e e v i d e n t that no model of an e c o l o g i c a l system can capture a l l of i t s behaviours. Forces f o r which no r e l a t i o n s h i p s are determined that a ct on the system from the o u t s i d e , such as changes i n "other prey" i n the wolf-ungulate model, w i l l e v e n t u a l l y make mistakes i n e v i t a b l e . The same i s true f o r unmodelled g e n e t i c changes i n ungulate p o p u l a t i o n s t h a t a c t on the system from w i t h i n . A l l that can be done i s to compare a l t e r n a t i v e frameworks that can be e n v i s i o n e d ( i n t u i t i v e versus simple versus d e t a i l e d models) i n terms of p r e d i c t i v e power and t r a c t a b i l i t y f o r o p t i m i z a t i o n a n a l y s i s , while always 148 t r y i n g to design p o l i c i e s that are robust to the i n e v i t a b l e f a i l u r e s . Or as H o l l i n g , Jones, and C l a r k (1976:3) put i t : "The aim of sound e c o l o g i c a l p o l i c y i s not to p r e d i c t and e l i m i n a t e f u t u r e s u r p r i s e s , but r a t h e r to design systems that can absorb and s u r v i v e unexpected events when they occur." A s i m u l a t i o n model r e p r e s e n t s an a b s t r a c t i o n to r e a l i t y , and i t i s q u i t e easy to i n c l u d e an immense ar r a y of i n t e r a c t i n g v a r i a b l e s and thus make i t i n t r a c t a b l e . A c r u c i a l and unresolved q u e s t i o n i s how d e t a i l e d ( r e a l i s t i c and p r e c i s e ) a model f o r o p t i m i z a t i o n needs to be? I t was shown (Chapter 2) how some d e t a i l s , such as age s t r u c t u r e can be aggregated without l o s i n g too much p r e d i c t i v e power. However, s p a t i a l and g e n e t i c composition e f f e c t s , which were not i n c l u d e d , could a l t e r p r e d i c t i o n s c o n s i d e r a b l y . What was hoped i n the p r e s e n t management systems was that the e s s e n t i a l aspect of behaviour i n the complex s t a t e space w i l l l a r g e l y occur i n some r e s t r i c t e d subspace. Examples of t h i s would be the subspace along the Yt+i = Yt i s o c l i n e i n the deer p o p u l a t i o n model (Chapter 2), or the assumptions that one s t a t e v a r i a b l e remains near e q u i l i b r i u m (Chapters 3 and 4 ) . The m o n o t o n i c a i l y i n c r e a s i n g a g e - r a t i o along the i s o c l i n e produces apparently simple o v e r a l l system behaviour. I t i s commonly assumed that the p r i n c i p a l l i m i t a t i o n of employing models in p r a c t i c a l h a r v e s t i n g management i s t h e i r p r e c i s e q u a n t i f i c a t i o n . Yet, i n a l l the examples used i n t h i s study i t was found that b a s i c a l l y the same s t r a t e g i e s were optimal i n s p i t e of c o n s i d e r a b l e v a r i a t i o n i n parameter v a l u e s . Returns from the managed system, not s t r a t e g i e s are dependent on p r e c i s e parameter v a l u e s . Thus choosing a s t r a t e g y does not 149 depend on p r e c i s e q u a n t i f i c a t i o n of the model. A p a r t i c u l a r l y d e s i r a b l e p r o p e r t y of a model i s i t s g e n e r a l i t y ( H o l l i n g , 1966). The three models employed in t h i s study are r e l a t i v e l y g e n e r a l . The p o p u l a t i o n processes d e s c r i b e d f o r the d i f f e r e n t c l a s s e s of ungulate p o p u l a t i o n s a r e • u n i v e r s a l , and with d i f f e r e n t parameter estimates and some m o d i f i c a t i o n s the models could be employed f o r managing a l a r g e number of ungulate systems throughout the world. DISCUSSION OF THE RESULTS Importance of s t a t e v a r i a b l e s E x p l o i t i n g ungulate p o p u l a t i o n s demands a t t e n t i o n to f a c t o r s that r e g u l a t e p r o d u c t i v i t y . Watt (1968) l i s t s s e v e r a l c a u s a l pathways by which a p o p u l a t i o n ' s p r o d u c t i v i t y can be r e g u l a t e d . S e v e r a l f a c t o r s can operate through these c a u s a l pathways to r e g u l a t e p r o d u c t i v i t y , and the f a c t o r s may be c l a s s i f i e d as i n t r i n s i c p o p u l a t i o n f o r c e s ( n a t a l i t y , m o r t a l i t y , d i s p e r s a l ) and e x t r i n s i c f o r c e s (food, weather, p r e d a t i o n ) . T h i s study i n v e s t i g a t e d some of these f a c o t r s i n r e l a t i o n to the p r o d u c t i o n of ungulates. In the three ungulate case systems i n v e s t i g a t e d , the s t a t e v e c t o r s contained both e x t r i n s i c and i n t r i n s i c v a r i a b l e s . The two component deer p o p u l a t i o n s t a t e v e c t o r (Chapter 2) c o n s i s t e d of i n t r i n s i c v a r i a b l e s o n l y , whereas the s t a t e v e c t o r s of the d e e r - v e g e t a t i o n system (Chapter 3) and wolf-ungulate system (Chapter 4) contained e x t r i n s i c ( v e g e t a t i o n 150 biomass, number of wolf packs) and i n t r i n s i c ( s i z e of ungulate p o p u l a t i o n ) v a r i a b l e s . The a n a l y s i s showed that the importance of the two kinds of f a c t o r s i n the h a r v e s t i n g process v a r i e d c o n s i d e r a b l y . In the two component deer system, the s i z e of the o l d e r deer p o p u l a t i o n (A) i s more important. For example, near the e q u i l i b r i u m , the optimal a d u l t harvest r a t e can change by 20 percent i f o l d e r deer p o p u l a t i o n s i z e i s changed by 25 p e r c e n t . In c o n t r a s t , i f j u v e n i l e deer d e n s i t y i s changed by 25 percent the optimal a d u l t h a r v e s t r a t e changes only by 10 p e r c e n t . The s i z e of the o l d e r deer p o p u l a t i o n i s a l s o more important f o r making y e a r l i n g h a r v e s t i n g d e c i s i o n s . For t h i s system p r o d u c t i v i t y i s i n f l u e n c e d by both the r a i n f a l l c o n d i t i o n s of the year preceding b i r t h s and the s i z e of the breeding p o p u l a t i o n , y e t h a r v e s t r a t e s are mostly dependent on the s i z e of the o l d e r deer p o p u l a t i o n . In the d e e r - v e g e t a t i o n system, the two v a r i a b l e s are e q u a l l y important near and above the e q u i l i b r i u m v e g e t a t i o n biomass f o r the standard v e g e t a t i o n p r o d u c t i o n f u n c t i o n ( G m a t K/2). At lower than e q u i l i b r i u m v e g e t a t i o n biomasses, the number of ungulates become more important i n the harvest d e c i s i o n ; the best response to changing ungulate d e n s i t y becomes independent of v e g e t a t i o n biomass. While p r o d u c t i v i t y i s p r i n c i p a l l y i n f l u e n c e d by v e g e t a t i o n c o n d i t i o n and s i z e of the breeding ungulate p o p u l a t i o n , h a r v e s t d e c i s i o n s are more dependent on the s i z e of the ungulate p o p u l a t i o n over most of the d e e r - v e g e t a t i o n phase plane (Figure 21). T h i s c o n c l u s i o n i s changed f o r the s i t u a t i o n where maximal v e g e t a t i o n p r o d u c t i o n i s at higher v e g e t a t i o n biomass (G m a t 3K/4). There the harvest d e c i s i o n s are e q u a l l y 151 dependent on both s t a t e v a r i a b l e s (Figure 23). In the Alaskan wolf-ungulate system, the moose d e n s i t y i s g e n e r a l l y more important than number of wolf packs near the e q u i l i b r i u m . The optimal harvest r a t e does not change a p p r e c i a b l y i f number of wolf packs i s changed. In c o n t r a s t , i f the moose d e n s i t y i s a l t e r e d , the optimal moose harvest r a t e changes s u b s t a n t i a l l y . The importance of the s t a t e v a r i a b l e s i n making ha r v e s t d e c i s i o n s i s however dependent on e x t r i n s i c f a c t o r s (wolf search e f f i c i e n c y , other prey occurrence and wolf c o n t r o l ) f o r system s t a t e s away from e q u i l i b r i u m . S p e c i f i c a l l y , wolf pack number i s more important i n s i t u a t i o n s where no wolf c o n t r o l i s exerted and the number of packs i s high ( F i g u r e s 40-42). As was the case f o r the d e e r - v e g e t a t i o n system, p r o d u c t i v i t y i n the wolf-ungulate system i s i n f l u e n c e d by e x t r i n s i c f a c t o r s as w e l l as the ungulate breeding p o p u l a t i o n , whereas h a r v e s t i n g d e c i s i o n s are more dependent on the s i z e of the ungulate d e n s i t y over most of the wolf-moose phase plane. Thus, f o r t h i s study, I conclude that i n general the i n t r i n s i c p o p u l a t i o n f a c t o r s are more important i n making h a r v e s t i n g d e c i s i o n s than are e x t r i n s i c f a c t o r s . T h i s i s a c o u n t e r i n t u i t i v e r e s u l t ; many w i l d l i f e agencies spend c o n s i d e r a b l e money and e f f o r t on a c t i v i t i e s r e l a t e d to e x t r i n s i c f a c t o r s , such as v e g e t a t i o n monitoring, and predator c o n t r o l . The e f f e c t of s t o c h a s t i c v a r i a t i o n Random environmental e f f e c t s were represented d i r e c t l y in the f o r m u l a t i o n of o p t i m i z a t i o n s f o r the three ungulate p o p u l a t i o n systems. I t was found that the f u n c t i o n a l forms of 152 the optimal h a r v e s t i n g s t r a t e g i e s were robust to changes in the p r o b a b i l i t y d i s t r i b u t i o n s . S p e c i f i c a l l y , optimal d e c i s i o n s i n the two component deer system were shown to be r e l a t i v e l y independent of the r a i n f a l l p r o b a b i l i t y d i s t r i b u t i o n s ( F i g u r e 6 ) . The h a r v e s t i n g i s o p l e t h s kept the same form as the r a i n f a l l assumption was changed from o p t i m i s t i c to p e s s i m i s t i c . S i m i l a r l y , h a r v e s t r a t e i s o p l e t h s d i d not change s u b s t a n t i a l l y when p r o b a b i l i t i e s of m o r t a l i t y were changed in the deer v e g e t a t i o n system (Figure 21). For the d e e r - v e g e t a t i o n system, i t was a l s o e v i d e n t that o p t i m a l feedback s t r a t e g i e s assuming s t o c h a s t i c dynamics are s i m i l a r to p o l i c i e s estimated from d e t e r m i n i s t i c dynamics. T h i s o b s e r v a t i o n i s c o n s i s t e n t with a few s t u d i e s of the e f f e c t s of d i f f e r e n t p r o b a b i l i t y d i s t r i b u t i o n s of s t o c h a s t i c v a r i a b l e s on the form of o p t i m a l s t r a t e g i e s (Reed, 1975; W a l t e r s , 1975; Walters and H i l b o r n , 1976). In g e n e r a l , i t can be concluded that o p t i m a l h a r v e s t i n g s t r a t e g i e s estimated from s t o c h a s t i c dynamics are very s i m i l a r to those estimated from d e t e r m i n i s t i c dynamics, p a r t i c u l a r l y when the d e s i r e d o b j e c t i v e i s to maximize f u t u r e r e t u r n s (Walters and H i l b o r n , 1978). A l s o , as Walters and H i l b o r n (1978) p o i n t out, the a b i l i t y to p r e d i c t systems behaviour over time i s not r e q u i r e d f o r good management, r a t h e r reasonably accurate and temporally s t a b l e p r o b a b i l i t y d i s t r i b u t i o n s are r e q u i r e d so we can p r e d i c t p o s s i b l e outcomes of a p p l y i n g p o l i c i e s . S t o c h a s t i c dynamic programming computes optimal harvest s t r a t e g i e s as a f u n c t i o n of the s t a t e v a r i a b l e s . The f u n c t i o n a l form of the optimal feedback s t r a t e g i e s in cases where the system dynamics are d i s c r e t e and the o b j e c t i v e i s a simple sum of 153 harvests i s commonly r e f e r r e d to as a f i x e d escapement. The o p t i m a l s t r a t e g y i s to leave a f i x e d ungulate p o p u l a t i o n below which no h a r v e s t i s taken. In other words, i f the p r e - h a r v e s t p o p u l a t i o n i s l e s s than the f i x e d escapement, the o p t i m a l d e c i s i o n i s not to harvest at a l l . The o p t i m a l i t y of f i x e d escapement s o l u t i o n s i s a very lucky outcome of the o p t i m i z a t i o n procedure. Such h a r v e s t i n g s t r a t e g i e s are e a s i l y understandable and might be more economical and more p r a c t i c a l to implement i n c o n j u n c t i o n with p o p u l a t i o n monitoring than would more complex, time v a r y i n g h a r v e s t schemes. In f l u e n c e of p o p u l a t i o n processes V a r i o u s e c o l o g i c a l processes have been the s u b j e c t of much re s e a r c h (e.g., Watt, 1968; H o l l i n g , 1959). In t h i s study, v e g e t a t i o n growth and wolf p r e d a t i o n i n p a r t i c u l a r were examined i n r e l a t i o n to t h e i r i n f l u e n c e on optimal h a r v e s t i n g s t r a t e g i e s . The a n a l y s i s i n Chapter 3 i n d i c a t e s that the f u n c t i o n a l form of optimal deer h a r v e s t i n g s t r a t e g i e s i s robust with r e s p e c t to a l t e r n a t i v e assumptions r e g a r d i n g the v e g e t a t i o n p r o d u c t i o n f u n c t i o n . The e s s e n t i a l d i f f e r e n c e between maximum v e g e t a t i o n p r o d u c t i o n being at low or intermediate versus high v e g e t a t i o n biomass i s a down- and l e f t - w a r d s h i f t i n the ascending limb of the convex f i x e d escapement l e v e l ( F i g u r e s 21-23). According to t h i s r e s u l t , even though the f u n c t i o n a l form i s robust, the w i l d l i f e managers' task of p r o v i d i n g a s u s t a i n e d y i e l d depends on d e t e r m i n a t i o n of process parameters which in turn determine f u n c t i o n a l r e l a t i o n s h i p s between food p r o d u c t i o n and v e g e t a t i o n d e n s i t y . He must at l e a s t know the q u a l i t a t i v e form of the 154 p r o d u c t i o n f u n c t i o n . The f u n c t i o n a l form of the optimal h a r v e s t i n g s t r a t e g i e s estimated f o r the moose p o p u l a t i o n , i n the absence of wolf c o n t r o l , was i n f l u e n c e d by wolf search e f f i c i e n c y through lowering occurrence of a l t e r n a t i v e prey. The v e r t i c a l f i x e d escapement l e v e l ( F i g u r e 39) at high wolf pack numbers changed to a concave form when other prey d e n s i t y was s u b s t a n t i a l l y reduced. Thus parameters of the p r e d a t i o n process i n f l u e n c e the form of the optimal h a r v e s t i n g s t r a t e g i e s . O b j e c t i v e f u n c t i o n s For t h i s study only simple o b j e c t i v e f u n c t i o n s were c o n s i d e r e d . The s p e c i f i c o b j e c t i v e employed was to maximize long-term ungulate r e t u r n s , except that c o s t f a c t o r s were cons i d e r e d i n r e l a t i o n to wolf c o n t r o l (Chapter 4 ) . The o b j e c t i v e f u n c t i o n s that accommodated wolf c o n s e r v a t i o n groups (by a s s e s s i n g p e n a l t i e s f o r removing wolf packs) l e a d to concave f i x e d escapement moose h a r v e s t i n g s t r a t e g i e s f o r high wolf pack numbers, but only f o r the cases where a l t e r n a t i v e prey occurrence was r e l a t i v e l y low ( F i g u r e s 40-42). In c o n t r a s t , o b j e c t i v e f u n c t i o n s having no a v e r s i o n to wolf pack removal produced v e r t i c a l , f i x e d escapement moose h a r v e s t i n g s t r a t e g i e s at high wolf pack numbers f o r these cases. S e n s i t i v i t y of optimal h a r v e s t i n g s t r a t e g i e s to d i f f e r e n t management goals was a l s o i n d i c a t e d by a study of optimal h a r v e s t s t r a t e g i e s f o r salmon (Walters, 1975). Dome-shaped c o n t r o l laws were obtained when the o b j e c t i v e was to minimize the v a r i a n c e around a d e s i r e d c a t c h , whereas convex f i x e d escapement c o n t r o l 155 laws emerged for the objective of maximizing mean catch. The objective function used in the optimization procedure can of course be more complex. It does not have to be long-term ungulate harvest, i t could also be economic return of harvest, t o t a l recreation days generated from hunting, or any combination of factors. There are some d i f f i c u l t i e s in quantifying more complex objective functions. Attempts to formally establish complex objectives for ecosystem management have recently been made by employing multiattribute u t i l i t y functions (Keeney, 1977; Hilborn and Walters, 1977; Powers and Lackey, 1976; Keeney and Ra i f f a , 1976). Management p o l i c i e s are often sensitive not only to variables included in objective functions but also to discounting rates and r i s k aversion (Walters and Hilborn, 1978). Discounting rates can be included very easily in the outlined optimization procedure, by reducing the value vector for each year of backward i t e r a t i o n . Discounting was not included in this study, since i t would be very d i f f i c u l t to determine an appropriate s o c i a l discounting rate for ungulate populations. Furthermore, the technique could be applied to situations where i t is undesirable to close the resource to hunting for a number of successive years (aversion to low harvests), by including appropriate constraints or boundary conditions. Returns of optimal vs alternative strategies Though the functional form of the optimal strategies is robust with regard to the uncertainties used in the investigation, the returns obtained from applying optimal 156 s t r a t e g i e s are very s e n s i t i v e to these u n c e r t a i n t i e s . S p e c i f i c a l l y , f o r the two component deer system, both the mean a d u l t and mean annual y e a r l i n g harvest dropped c o n s i d e r a b l y as net p r o d u c t i v i t y decreased due to l e s s f a v o r a b l e r a i n f a l l p r o b a b i l i t y d i s t r i b u t i o n s . The e f f e c t of decreased p r o d u c t i v i t y r e s u l t i n g from v a r y i n g the s t o c h a s t i c m o r t a l i t y v a r i a b l e i n the de e r - v e g e t a t i o n system a l s o a f f e c t e d h a r v e s t i n g r e t u r n s . The re t u r n s f o r extended management p e r i o d s decreased s u b s t a n t i a l l y as the frequency of d i s t u r b a n c e i n c r e a s e d . Thus while the f u n c t i o n a l form of the optimal h a r v e s t i n g s t r a t e g i e s do not d i f f e r due to u n c e r t a i n t i e s , the r e t u r n s are s u b s t a n t i a l l y a f f e c t e d by these u n c e r t a i n t i e s . H a r v e s t i n g of ungulate p o p u l a t i o n s can e s s e n t i a l l y be based on two concepts: (1) an average best h a r v e s t r a t e can be a p p l i e d each year; termed open loop s t r a t e g y , or (2) an optimal s t r a t e g y based on the observed system s t a t e s can be a p p l i e d each year; termed " c l o s e d loop" or "feedback" s t r a t e g y (Walters and H i l b o r n , 1978). The r e s u l t s from the value of i n f o r m a t i o n experiments i n d i c a t e that a p p l y i n g feedback c o n t r o l s i n v a r i a b l y produce b e t t e r r e t u r n s than do open loop s t r a t e g i e s . By a p p l y i n g open loop s t r a t e g i e s , such as f i x e d quota ( t o t a l h a r v e s t ) , the p o s i t i v e p r o b a b i l i t y t h at a sequence of poor p r o d u c t i o n years (due to s t o c h a s t i c weather c o n d i t i o n s ) can s e r i o u s l y j e o p a r d i z e the p o p u l a t i o n i f c o n t i n u o u s l y harvested at an average best l e v e l , i s t o t a l l y ignored. Reducing i n f o r m a t i o n f o r d e c i s i o n making The most i n t e r e s t i n g r e s u l t s from t h i s study emerged from 157 the value of i n f o r m a t i o n experiments i n v e s t i g a t i n g r e t u r n s from c o l l a p s i n g the o r i g i n a l i n f o r m a t i o n systems. E s s e n t i a l l y two types of r e s u l t s were obtained. Reducing the i n f o r m a t i o n f o r the h a r v e s t i n g d e c i s i o n e i t h e r had a negative e f f e c t or no e f f e c t on r e t u r n s obtained over long-term management p e r i o d s . S p e c i f i c a l l y , c o l l a p s i n g the i n f o r m a t i o n system i n the two component ungulate p o p u l a t i o n (Chapter 2) was done by using s i m p l i f i e d age r a t i o s t r a t e g i e s and s i m p l i f i e d s t r a t e g i e s based on o l d e r deer d e n s i t y i n f o r m a t i o n o n l y . The a n a l y s i s of the value of i n f o r m a t i o n experiments i n d i c a t e d t h a t the mangement p r a c t i c e of basing harvest d e c i s i o n s only on age r a t i o s i s i n d e f e n s i b l e . These r a t i o s are not adequate s u b s t i t u t e s f o r absolute p o p u l a t i o n estimates or d e n s i t y i n d i c e s from which feedback h a r v e s t i n g d e c i s i o n s can be made. On the o t h e r hand the r e s u l t s i n d i c a t e d that reducing the p o p u l a t i o n system to a s i n g l e s t a t e v a r i a b l e ( o l d e r deer d e n s i t y ) " i s adequate f o r d e c i s i o n making. The a n a l y s i s of the value of i n f o r m a t i o n experiments f o r the v e g e t a t i o n - d e e r system (Chapter 3) i n d i c a t e d that no s u b s t a n t i a l improvements c o u l d be made by i n c l u d i n g i n f o r m a t i o n about the v e g e t a t i o n biomass. Yet, s u b s t a n t i a l gains are made by having i n f o r m a t i o n about the c u r r e n t s i z e of the deer p o p u l a t i o n , p a r t i c u l a r l y i f the system i s not at i t s e q u i l i b r i u m s t a t e . Or put d i f f e r e n t l y , r e t u r n s from s i m p l i f i e d s t r a t e g i e s u s i n g deer p o p u l a t i o n i n f o r m a t i o n are n e a r l y o p t i m a l , whereas c o n t r o l based onl y on v e g e t a t i o n ( h a b i t a t ) data can l e a d to s e r i o u s mismanagement. S i m i l a r r e s u l t s were obtained f o r the wolf-ungulate system. Returns f o r long-term management p e r i o d s r e s u l t i n g from the 158 application of s i m p l i f i e d strategies based on moose information, while ignoring wolf pack numbers, were not dramatically d i f f e r e n t from returns obtained using optimal feedback strategies. For p r a c t i c a l ungulate population management this again implies that e f f o r t s should be directed towards c o l l e c t i n g moose density information, while the predator population size need not be regularly monitored. AN ADAPTIVE UNGULATE MANAGEMENT SYSTEM Adaptive management implies a process of extending knowledge about system responses through experience gained from managing the system. Adaptive control is generally c l a s s i f i e d as active or passive, depending on whether action is deliberately taken to gain insight into the system (active) or whether this insight is gained through chance events (passive). Components of a passive adaptive ungulate management system are shown in Figure 48. The managed ungulate system is observed to produce time series information of state indicators such as r e l a t i v e abundance before and after hunting, k i l l by sex and age, and tag recoveries. "These observations are never a complete or accurate r e f l e c t i o n of the true system state" (Walters and Hilborn, 1978). The p a r t i a l information is subsequently used to estimate parameter values and system states. A number of sophisticated tools are now available for systems i d e n t i f i c a t i o n and parameter estimation (e.g., Bard, 1974; Young, 1974; Astrom and Eykoff, 1971). One of the fundamental problems of ungulate population management is that, unlike in f i s h e r i e s , there is never a large portion of the 159 F i g u r e 48. Components of a p a s s i v e a d a p t i v e ungulate management system. ENVIRONMENTAL DISTURBANCES PROCESS MISRE-PRESENTATION MEASUREMENT ERRORS * UNGULATE STATE RESPONSE MODEL 7 K CONTROL OR HARVEST RATE! 7R ±— OBSERVATION MODEL OPTIMAL HARVEST STRATEGY PARTIAL STATE OBSERVATION STATE RECONSTRUCTION! 8 PARAMETER ESTIMATION 160 population removed. This makes i t extremely d i f f i c u l t to use data on k i l l and e f f o r t in the estimation procedure, since estimation requires that harvesting causes s i g n i f i c a n t changes in system state. The most productive substitutes to k i l l and e f f o r t data are estimates of r e l a t i v e abundance before and after hunting. The focus of this study has been on the estimation of optimal harvesting strategies and the state-control linkage of the passive adaptive ungulate management system. The passive adaptive approach involves v a r i a t i o n over time not only due to the extreme actions that were taken (no harvest, maximum harvest) but also due to the environmental disturbances (stochastic weather variables) and to changes in parameter estimates (and thus assessed p o l i c i e s ) . Thus observations w i l l be made not only under equilibrium conditions but also above and below these conditions. From these observations, more knowledge can be gained about functional relationships used in the ungulate state dynamic model thus improving the decision making process. An obvious extension of this concept is to those situations where we have l i t t l e insight into the dynamics of the system to be managed, but gain this insight by deliberately perturbing the system. Eventually, i f w i l d l i f e management agencies can be persuaded, we might be able to employ active adaptive control processes to estimate optimal feedback harvesting strategies for ungulate populations. SUGGESTIONS FOR FURTHER WORK 161 E s s e n t i a l l y f o u r g e n e r a l c a t e g o r i e s of f u t u r e r e s e a r c h should be considered i n connection with e s t i m a t i n g optimal h a r v e s t i n g s t r a t e g i e s f o r ungulate p o p u l a t i o n s : 1. Models developed f o r t h i s study are merely approximate d e s c r i p t i o n s of the p o p u l a t i o n dynamics of ungulate p o p u l a t i o n s . The e f f e c t s of some b i o l o g i c a l p r ocesses, such as d i s p e r s a l , p a r a s i t i s m , and other f a c t o r s were omitted. The obvious next step i s to expand on these models, by not o n l y making them b i o l o g i c a l l y r i c h e r , but a l s o by i n c r e a s i n g the number of s t a t e v a r i a b l e s (4-5) t h a t c o u l d s t i l l be handled by the d e s c r i b e d o p t i m i z a t i o n procedure. 2. To c o l l e c t p r e c i s e l y the type and amount of data r e q u i r e d as input by more complex ungulate p o p u l a t i o n models should be an ongoing and necessary concern. For example, only by having s u f f i c i e n t data about f u n c t i o n a l r e l a t i o n s h i p s such as d e n s i t y dependent n a t a l i t y a c r o s s the range of p o s s i b l e d e n s i t y v a l u e s , can one make p r e c i s e p r e d i c t i o n s about the p r o d u c t i v i t y of a p o p u l a t i o n . 3. The b a s i c concept of maximum sus t a i n e d y i e l d has r e c e n t l y come under severe c r i t i c i s m . The i n t e r e s t s of many groups, some with m u l t i p l e o b j e c t i v e s , are important to i n c l u d e i n developing harvest s t r a t e g i e s f o r ungulate p o p u l a t i o n s . Formal d e f i n i t i o n s of management go a l s have to be made by d e s i g n i n g m u l t i a t t r i b u t e u t i l i t y models and 162. i n c l u d i n g them i n the o p t i m i z a t i o n procedure. The computational e f f o r t r e q u i r e d to estimate o p t i m a l h a r v e s t s t r a t e g i e s using dynamic programming i s s u b s t a n t i a l . For models that are very complex and having more than about f i v e s t a t e v a r i a b l e s , o b t a i n i n g numerical s o l u t i o n s becomes very d i f f i c u l t . I t w i l l be necessary to develop algorithms that produce suboptimal s o l u t i o n s , but are able to d e a l with the computational l i m i t a t i o n s . CHAPTER 6 C o n c l u s i o n s 164 The main c o n c l u s i o n s that emerge from t h i s study are: 1. Developing ungulate p o p u l a t i o n models through which i n s i g h t about the dynamic behaviour of the systems i s gained, proved to be a u s e f u l and necessary f i r s t step i n s o l v i n g the posed o p t i m i z a t i o n problems. With the i n c l u s i o n of resource c o n s t r a i n t s , through d e n s i t y dependent mechanisms, and numerous other b i o l o g i c a l l y important f a c t o r s , the dynamics of ungulate p o p u l a t i o n s were captured r e a l i s t i c a l l y and with some p r e c i s i o n . 2. The r e c u r s i v e o p t i m i z a t i o n procedure of s t o c h a s t i c dynamic programming seems an a p p r o p r i a t e method f o r e s t i m a t i n g o p t i m a l feedback h a r v e s t s t r a t e g i e s f o r ungulate p o p u l a t i o n s , provided the p o p u l a t i o n dynamics can be adequately d e f i n e d with r e l a t i v e l y simple models. While t h i s study was centered around three s p e c i f i c ungulate p o p u l a t i o n systems, the o u t l i n e d o p t i m i z a t i o n procedure i s a p p l i c a b l e i n many other s i t u a t i o n s f o r which the necessary parameter val u e s can be estimated. 3. For ungulate p o p u l a t i o n s harvested i n a randomly f l u c t u a t i n g environment, the optimal h a r v e s t i n g d e c i s i o n i n any given year must be based on the s t a t e of the system i n that year. In g e n e r a l , given the in h e r e n t u n p r e d i c t a b i l i t i e s of the r e a l world, i t i s i n d e f e n s i b l e to use open loop c o n t r o l p o l i c i e s , such as f i x e d h arvest r a t e s or quota systems. 165 In g e n e r a l , i n t r i n s i c p o p u l a t i o n f a c t o r s were found to be more important i n making h a r v e s t i n g d e c i s i o n s than e x t r i n s i c f a c t o r s . T h i s i s a c o u n t e r i n t u i t i v e r e s u l t , s i n c e many w i l d l i f e agencies are pursuing r e s e a r c h s t r i c t l y on e x t r i n s i c f a c t o r s . The estimated o p t i m a l feedback h a r v e s t s t r a t e g i e s apply only when the d e f i n e d o b j e c t i v e s are to maximize f u t u r e average r e t u r n s . I f other f a c t o r s , such as hunting q u a l i t y , number of hunters, season l e n g t h , bag l i m i t s were i n c l u d e d i n the o b j e c t i v e f u n c t i o n s , the o p t i m a l feedback s t r a t e g i e s would undoubtedly be d i f f e r e n t . Thus a c r i t i c a l problem now i s to d e f i n e more p r e c i s e l y the management g o a l s , a l l o w i n g r e p r e s e n t a t i o n not only of b i o l o g i c a l but a l s o s o c i a l aspects of hunting. While optimal feedback s t r a t e g i e s estimated i n t h i s study were g e n e r a l l y found to be i n s e n s i t i v e to random f l u c t u a t i o n s of the environment, i t was found that assumptions re g a r d i n g b i o l o g i c a l processes have to be c a r e f u l l y i n v e s t i g a t e d f o r t h e i r e f f e c t on the f u n c t i o n a l form of the s t r a t e g i e s . Returns obtained from a p p l y i n g optimal h a r v e s t i n g s t r a t e g i e s were always higher than r e t u r n s obtained from a p p l y i n g t r a d i t i o n a l h a r v e s t i n g p o l i c i e s . On the other hand, n e a r l y optimal r e t u r n s were o f t e n obtained from a p p l y i n g s i m p l i f i e d s t r a t e g i e s that were d e r i v e d from 166 optimal s t r a t e g i e s . The best s i m p l i f i e d s t r a t e g i e s were based on p o p u l a t i o n d e n s i t y i n f o r m a t i o n . S i m p l i f i e d s t r a t e g i e s based i n s t e a d on a g e - r a t i o or h a b i t a t i n f o r m a t i o n alone were p r e d i c t e d to lead almost i n e v i t a b l y to mismanagement. 8. The major c o n c l u s i o n I can draw from t h i s study i s t h a t , although i n d e f e n s i b l e , a c t u a l w i l d l i f e management p r a c t i c e s have not l e d to more frequent d i s a s t e r s not because the p o p u l a t i o n systems are r e s i l i e n t to e x p l o i t a t i o n , but because a c t i o n taken based on minimum i n f o r m a t i o n i s o f t e n s u f f i c i e n t . T h i s c o n c l u s i o n f o l l o w s from seven (above), which s t a t e s that r e t u r n s obtained from a p p l y i n g s i m p l i f i e d s t r a t e g i e s based on a c o l l a p s e d i n f o r m a t i o n system can be j u s t as high as those obtained from a p p l y i n g a l l the i n f o r m a t i o n that i s a v a i l a b l e i n the d e c i s i o n making p r o c e s s . Thus the l a c k of i n f o r m a t i o n o f t e n observed i s probably not important f o r making w i l d l i f e management d e c i s i o n s . 167 L i t e r a t u r e C i t e d A l l e n , E.O. 1977. A new p e r s p e c t i v e f o r e l k h a b i t a t management. Proc. West. Assoc. Game and F i s h Comm. 57: 195-205. A l l e n , K.R. 1971. R e l a t i o n between p r o d u c t i o n and biomass. J . F i s h . Res. Board Can. 28(10): 1573-1581. A l l e n , R.L. and P. Basasibwaki. 1974. P r o p e r t i e s of age s t r u c -ture models f o r f i s h p o p u l a t i o n s . J . F i s h . Res. 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