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An analysis of hunter kill data Tait, David E. N. 1983

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AN ANALYSIS OF HUNTER KILL DATA by David E.N. Ta i t B.Sc., The University of B r i t i s h Columbia, 1968 M.Sc. , The University of B r i t i s h Columbia, 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the FACULTY OF GRADUATE STUDIES Department of Forestry We accept t h i s thesis as conforming to the required standard ©THE UNIVERSITY OF BRITISH COLUMBIA May, 1983 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6 (3/81) i i ABSTRACT A s t o c h a s t i c model i s developed that can be used to compute the l i k e l i h o o d of observing a s p e c i f i c time stream of harvests of a w i l d -l i f e p o pulation. The harvest i s assumed to be a count of the t o t a l number of animals i n each age i n each sex i n each year removed from the p o p u l a t i o n f o r a sequence of consecutive years. In the s t o c h a s t i c model i t i s assumed that the harvest process and the n a t u r a l s u r v i v o r -ship process can both be t r e a t e d as binomial processes. The r e c r u i t -ment process i s approximated as a product of normal processes. This f o r m u l a t i o n allows f o r the development of an i t e r a t i v e numerical scheme that w i l l r e c onstruct the most probable underlying unknown population given a set of harvest data and a set of l i f e h i s t o r y parameters. A h e u r i s t i c procedure that checks f o r i n t e r n a l consistency between the reconstructed p o p u l a t i o n , the set of harvest data, and the l i f e h i s t o r y parameters may be used to estimate a number of unknown population parameters together w i t h the unknown population. The scheme has been t e s t e d w i t h Monte Ca r l o simulations and, using only harvest data, has simultaneously estimated the s u r v i v o r s h i p r a t e , the r e c r u i t -ment r a t e , the male harvest r a t e , the female harvest r a t e and the y e a r l y harvest e f f o r t together w i t h the unknown population. The scheme has been a p p l i e d t o Alaskan brown bear harvest data demonstrating i t s p o t e n t i a l value as a management t o o l . i i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES v i LIST OF FIGURES v i i ACKNOWLEDGEMENTS x CHAPTER 1 AN OVERVIEW 1 1.1 Introduction 1 1.2 The Population 2 1.3 Data Requirements 3 1.4 Population Requirements 4 1.5 An I l l u s t r a t i v e Example 6 1.6 P o t e n t i a l Value 8 CHAPTER 2 ALTERNATIVE MODELS FOR THE ANALYSIS OF AGE AND SEX DATA 11 2.1 Change i n Ratio Methods 11 2.2 Sex Ratio Estimates 13 2.3 Catch at Age Analysis 14 2.4 Cohort Analysis 22 2.5 Summary 22 CHAPTER 3 THE STRUCTURAL MODEL 23 3.1 Notation 23 3.2 The Harvest Data Set 24 i v Page 3.3 The Harvest Process 25 3.4 The Survivorship Process 26 3.5 The Recruitment Process 27 3.6 The Stochastic Model 29 CHAPTER 4 ESTIMATION 31 4.1 Population Reconstruction 32 4.2 Parameter Estimation 35 4.3 Interpretation 36 CHAPTER 5 NUMERICAL CONSIDERATIONS 38 5.1 Numerical Reconstruction 39 5.2 Numerical Estimation 40 5.3 The MM Algorithm 41 5.4 Al t e r n a t i v e Algorithm's 43 5.5 Summary 44 CHAPTER 6 EXPLORATORY RECONSTRUCTION 46 6.1 Monte Carlo Simulations 46 6.2 The Monte Carlo Simulator 47 6.3 I n i t i a l Conditions 50 6.4 Experiment 1 - The Base Run 52 6.5 Experiment 2 - A Declining Population 70 6.6 Summary 77 CHAPTER 7 EXPLORATORY ESTIMATION 78 7.1 A Consistent Estimate of Harvest E f f o r t 78 7.2 A Consistent Estimate with Unknown Natural History Parameters 79 V Page 7.3 A Consistent Estimate with Unknown Natural History Parameters and a Variable Harvest E f f o r t 85 7.4 Summary 89 CHAPTER 8 SENSITIVITY ANALYSIS 91 8.1 Parameter Errors 92 8.2 Sexing Errors 96 8.3 Ageing Errors 99 8.4 Incomplete Harvest Data 103 8.5 Summary 106 CHAPTER 9 EXPLORATORY DATA ANALYSIS 107 9.1 Brown Bear Harvest Data 107 9.2 The Alaskan Peninsula Data Set 109 9.3 The Alaskan Unit 13 Data Set 119 9.4 F e a s i b i l i t y 122 CHAPTER 10 CONCLUSIONS 123 LITERATURE CITED 125 APPENDIX A: Development of a t r i - d i a g o n a l matrix representation for the numerical implementation of the reconstruction component of the MM algorithm 127 v i LIST OF TABLES Page Table I. I n i t i a l conditions for the base population 51 Table I I . A t y p i c a l harvest data set. Ten years of harvest s t r a t i f i e d by age, sex, and year 53 Table I I I . A t y p i c a l simulated population. The number of animals i n the population, for ten years, s t r a t i f i e d by age, sex, and year 54 Table IV. Population reconstruction from Table I I harvest data 56 v i i LIST OF FIGURES Page Figure 1. Contours representing the average difference i n the number of animals by age, sex, and year for nine simulated populations and the corresponding reconstructed populations based on the simulated harvest data 57 Figure 2. Contours representing the standard errors of the average diffe r e n c e i n the number of animals by age, sex, and year for nine simulated populations and the corresponding reconstructed populations based on the simulated harvest data 60 Figure 3. The simulated and reconstructed male, female, and t o t a l population l e v e l s versus time for a population with a stable age d i s t r i b u t i o n 63 Figure 4. The average difference i n the number of male, female, and t o t a l populations between ten simulated populations and the corresponding ten reconstructed populations based on the simulated harvest data .... 66 Figure 5. The standard errors of the average difference i n the number of male, female, and t o t a l populations between ten simulated populations and the corresponding ten reconstructed populations based on the simulated harvest data 68 Figure 6. The simulated and reconstructed male, female, and t o t a l population l e v e l s versus time for a declining population 71 Figure 7. The average difference i n the number of male, female, and t o t a l populations between ten simulated populations and the corresponding ten reconstructed populations based on the simulated harvest data of a declining population 73 Figure 8. The standard errors of the average difference i n the number of male, female, and t o t a l populations between ten simulated populations and the corresponding ten reconstructed populations based on the simulated harvest data for a declining population 75 v i i i Page Figure 9. The e f f e c t of i t e r a t i n g the MM algorithm on the reconstructed p o p u l a t i o n . Harvest data are generated from a simulated d e c l i n i n g population. The r e c o n s t r u c t i o n s i n (a) are based on the i n i t i a l guesses f o r the harvest r a t e s . Reconstructions i n (b) are based on the f i n a l set of harvest parameters a f t e r 16 i t e r a t i o n s of the MM algo r i t h m 80 Figure 10. The e f f e c t of i t e r a t i n g the MM algorithm when no underlying population parameter values (except variance on recruitment) are known. I n i t i a l estimates produce too l a r g e a population ( a ) , whereas a c l o s e agreement between simulated and reconstructed populations i s a t t a i n e d a f t e r i t e r a t i n g the MM algorithm (b) 83 Figure 11. The simulated population experiencing a v a r i a b l e harvest e f f o r t used to generate a set of harvest data. The i t e r a t i v e scheme i s used to estimate the c o n s i s t e n t s u r v i v o r s h i p r a t e , male and female harvest r a t e s , recruitment r a t e , female vu l n e r a -b i l i t y , and y e a r l y harvest e f f o r t s . In (a) the r e c o n s t r u c t i o n i s based on the i n i t i a l values of the parameters and i n (b) the r e c o n s t r u c t i o n i s based on the f i n a l parameter values a f t e r 200 i t e r a t i o n s of the MM alg o r i t h m 87 Figure 12. A simulated d e c l i n i n g population and a s e r i e s of t o t a l p o p u l a t i o n r e c o n s t r u c t i o n s based on the simulated harvest data f o r a range of a l t e r n a t i v e harvest r a t e s . The r e c o n s t r u c t i o n s are based on harvest r a t e s ranging from 0.25 to 0.50 93 Figure 13. The simulated and reconstructed male, female, and t o t a l p opulation l e v e l s f o r a d e c l i n i n g population i n which the harvest data has been mis-sexed f o r 10% of the i n d i v i d u a l s 97 Fig u r e 14. The simulated and reconstructed male, female, and t o t a l p opulation l e v e l s f o r a d e c l i n i n g population i n which the harvest data has been mis-aged. Ageing e r r o r p r o b a b i l i t i e s range from 0.02 f o r animals i n the f i r s t age c l a s s to 0.10 f o r animals i n the 10th and older age c l a s s e s 101 Figure 15. The simulated and reconstructed male, female, and t o t a l p o p u l a t i o n l e v e l s f o r a d e c l i n i n g population i n which 20% of the harvest data has been ignored . 104 i x Page Figure 16. Reconstructed population l e v e l s based on Unit 9 Alaskan brown bear harvest data and a consistent set of estimated population parameters and estimated harvest e f f o r t s 110 Figure 17. Reconstructed population l e v e l s based on Unit 9 Alaskan brown bear harvest data and a consistent set of estimated population parameters and estimated harvest e f f o r t s with the recruitment rate fixed at 0.24 114 Figure 18. Reconstructed t o t a l harvestable male age classes i n the population based on Unit 9 Alaskan brown bear harvest data and a consistent set of estimated population parameters and estimated harvest e f f o r t s f o r a range of fixed recruitment rates, 0.18, 0.24 and 0.34 117 Figure 19. Reconstructed population le v e l s based on Unit 13 Alaskan brown bear harvest data and a consistent set of estimated population parameters and estimated harvest e f f o r t s 120 X ACKNOWLEDGEMENTS I wish to acknowledge the support that I have received from a number of in d i v i d u a l s and th e i r i n s t i t u t i o n s i n the development of t h i s t h e s i s . The i n i t i a l year of t h i s study was f i n a n c i a l l y supported by a GREAT fellowship that was sponsored by Dr. D. Eastman of the B.C. Fish and W i l d l i f e . The Alaskan Department of Fis h and Game (ADF&G) provided me with access to the i r brown bear harvest data, t r a v e l , and the time of t h e i r s t a f f . Dr. S. M i l l e r of the ADF&G was invaluable i n f a c i l i t a t i n g t h i s i n t e r a c t i o n . L. Glen and R. Modefari of the ADF&G contributed t h e i r time and ideas i n discussions of t h e i r Alaskan brown bear data. I also wish to thank the Faculty of Forestry for their generous support including the D.S. McPhee Fellowship and a teaching a s s i s t a n t -ship. In p a r t i c u l a r I would l i k e to thank Dr. A. Kozak who has served as my 'patron' during these past few years. Dr. F. Bunnell's c o n f i -dence i n my a b i l i t i e s and i n i t i a l encouragement were instrumental i n i n i t i a t i n g t h i s study. Dr. R. Jones and Dr. D. Fournier provided very valuable c r i t i c a l ears i n the formative stages. Dr. D. Williams, Dr. D. Ludwig, and Dr. C. Walters served on my committee and encouraged me to s t r i v e for excellence. Dr. Petkau's c r i t i c a l review and suggestions were a valuable contribution i n the f i n a l stages of t h i s t h e s i s . I appreciate the p r a c t i c a l assistance i n drafting provided by Douglas T a i t . F i n a l l y , I wish to thank my wife P i l l e Bunnell, for her constant support, encouragement, and contribution. 1 CHAPTER 1 AN OVERVIEW 1.1 I n t r o d u c t i o n This t h e s i s i s an a n a l y s i s of b i g game hunter harvest data. I t represents a general methodology that may be used i n the i n t e r p r e t a t i o n of the data representing the age and sex of animals k i l l e d by hunters. Although the approach i s g e n e r a l , the design of the methodology has been motivated by a p a r t i c u l a r s p ecies, the brown bear (Ursus a r c t o s ) . Thus, the brown bear w i l l be used as an example to i l l u s t r a t e concepts and provide a data base. The i n t e n t of the t h e s i s i s to provide the manager of a b i g game population w i t h a t o o l that w i l l a l low him to develop a dynamic popula-t i o n d e s c r i p t i o n . This d e s c r i p t i o n i s to be both c o n s i s t e n t w i t h the observed harvest and w i t h the manager's understanding of the population dynamics of the species. I t i s expected that the d e s c r i p t i o n of the number of animals i n the population over time w i l l r e v e a l both popula-t i o n trends and the impact of the harvest on the status of the popula-t i o n . Information on status and trends i s d i r e c t l y a p p l i c a b l e to the development of f u t u r e harvest p o l i c y . Designers of p o l i c y have been concerned with the question of "what can be learned from hunter harvest data?" I n t h i s t h e s i s the answer i s approached through the development of a s t a t i s t i c a l model that describes the processes which generate observed harvest data. This s t o c h a s t i c model i s simultaneously f i t t e d to harvest data and to the manager's understanding of the population. 2 The remainder of t h i s chapter w i l l i d e n t i f y some of the i m p l i c i t assumptions i n the model, r e f i n e the scope of the t h e s i s , d e fine the type of populations to which the methodology i s a p p l i c a b l e , and i l l u s t r a t e the conceptual framework used f o r determining p o p u l a t i o n s t a t u s from hunter harvest data. 1.2 The Po p u l a t i o n Harvest data can be s t r a t i f i e d i n t o the number of animals i n each age, sex, and year c l a s s . The scope of the t h e s i s i s thus r e s t r i c t e d t o populations f o r which t h i s l e v e l of s t r a t i f i c a t i o n i s appropriate. The r e s t r i c t i o n means that i n d i v i d u a l s belonging to the same age, year, and sex c l a s s should have the same p r o b a b i l i t y of being harvested. I n d i v i d u a l s i n d i f f e r e n t age, sex, and year c l a s s e s can have d i f f e r e n t p r o b a b i l i t i e s of being harvested. The harvested population needs to be clo s e d ; that i s , there i s no net immigration or emmigration. Immigrants from a surrounding population may not have been exposed to the same hunting pressure. The treatment of an open population would r e q u i r e f u r t h e r assumptions r e l a t i n g d i s p e r s a l rates by age and sex to popul a t i o n d e n s i t y . Hunting pressure w i t h i n the population must be d i s t r i b u t e d throughout the population. This r e s t r i c t i o n excludes 'mined' popula-t i o n s . Mined populations are p r o g r e s s i v e l y depleted as hunters generate a f r o n t of encroaching hunting pressure. As a r e s u l t , i n a mined p o p u l a t i o n , the harvest p r o b a b i l i t y i s a f u n c t i o n of the animals l o c a t i o n r a t h e r than e i t h e r the animals age c l a s s or sex. 3 1.3 Data Requirements The intent in this thesis is to provide the manager with a tool that can be applied to currently existing data sets. That objective requires the use of hunter k i l l data rather than hunter effort data. Hunter k i l l data or harvest data do not represent an unbiased sample of the hunted population. The age and sex distribution in the harvest does not reflect the age and sex distribution in the popula-tion. This disparity has severely curtailed the usefulness of harvest data as an estimator of population size. However harvest, survivor-ship, and recruitment processes a l l interact to determine the age and sex distribution of a population and consequently the age and sex distribution of the harvest. These interactions, which produce a temporal pattern in the population structure, are exploited in this thesis to develop a population estimate. A particular feature of the approach in this thesis is the exclu-sion of any explicit, independent estimate of harvest effort. Tradi-tional work developed in fisheries relates population estimates to fishing effort through assumptions about the catch per unit of effort. Unfortunately, l i t t l e confidence can be placed on the current estimates of hunting effort. In British Columbia, effort is measured at the resolution of "hunter days". Hunter days are simply a record of the number of days that a hunter claims to have been hunting for a specific species. They do not measure the actual time spent in the field,, the area hunted, or the technology exploited by the hunter. A day spent at a base camp in the rain, a day spent hiking through the forest, and a 4 day spent on a guided hunt with auxiliary "spotting" aircraft are a l l counted as a single hunter day. A further problem in the traditional interpretation of effort data lies in the calibration of the parameter representing "catcha-b i l i t y " of the animal. The same effort (calibrated in terms of time, area, and technology) may s t i l l result in different harvest levels with no change in the population size or structure. Catchability is a function of many factors. For example, the effectiveness of a hunter day depends on the s k i l l of the hunter, weather conditions, and vegetation cover. The estimate of catchability is further confounded by temporal changes in the distribution of the animals. Variation i n weather from year to year may concentrate the animals in a relatively small area in one year, and in the next year allow a diffuse distribution. The resulting distribution of animals w i l l affect the catchability of the population. 1.4 Population Requirements The effect of the sex differences in the harvested population can be described briefly. The population is regarded as two almost independent dynamic systems; the male subsystem and the female sub-system. The only link between these two subsystems is common recruit-ment, which is a function of the total adult female population. The harvest is a biased sample from the two subsystems in that i t does not accurately reflect the population's age and sex distribution. However, the harvest is also a known perturbation of these two dynamic systems. The difference in the number of male and female animals removed from 5 the population induces a change in the age and sex structure of the population. This difference in the population structure w i l l manifest i t s e l f in the age and sex structure of future harvests. Comparison of the pattern of change through time between the male and female sub-systems allows for an implicit correction or estimate of the bias in the harvest sample. The above procedure places a number of constraints on the type of data that can be analyzed. The data w i l l have to comprise a series of consecutive years. The time sequence should be sufficiently long so that changes in the adult female population can be observed as changes in the recruitment rates to the harvestable age classes. This period allows the analysis to exploit the relationship between the common recruitment of males and females and the status of the adult female population. Because the intention is to examine the perturbations i n the age structure, the data w i l l have to include the age and sex of the animals harvested. The harvest w i l l be used as a known perturbation to the population. Therefore, the harvest data should be a total measure of the harvest, and i t should also represent a significant mortality factor in the population. Since the approach exploits differences between the two sexes, there must be a difference in the harvest rate of males and females. Since the effect of the harvest is measured as changes in the age distributions of future harvests, the harvest must also affect the age distribution. 6 1.5 An Illustrative Example In order to i l l u s t r a t e how the procedure developed in this thesis operates, two contrasting hypothetical populations w i l l be presented. Let us assume that both populations have a stable age distribution; that i s , the proportion of animals in each age and sex class i s the same from year to year. This condition implies that the populations have been subjected to a constant harvest for a number of years and have reached an equilibrium state. For the sake of the argument, the birth rates and survivorship rates are constant, but may differ between the two populations. In both populations, males are assumed to be three times as vulnerable as females. That i s , an individual male has three times the probability of being harvested as an individual female. In the f i r s t population, the harvest mortality is small relative to the natural mortality. Most of the animals die of natural causes. In this population, the harvest would be predominantly male. If the male and female natural mortalities are equivalent and the harvest probability within a sex is independent of age, then the age distribu-tions of the male and female harvests would reflect the age distribu-tions by sex in the population. However, the male subsample would be approximately three times the size of the female subsample. Because the harvest mortality is assumed to be small relative to the natural mortality, the harvest would not be expected to significantly perturb the age or sex distribution of the population. 7 In contrast, the second population supports a high harvest effort. As discussed in Bunnell and Tait (1980), the sex ratio in the harvest would approximate one to one. This conclusion becomes obvious when one considers the extreme case in which natural mortality i s negligible relative to hunter mortality. Almost a l l animals born are harvested. If the sex ratio at birth is even, that is equal numbers of males and females are born, then the sex ratio of the harvest would also have to be even, as an equal number of males and females die. In our example, the harvest probability within a sex is assumed to be independent of the age of the harvestable animals. Because the hunting mortality of males is higher than females, not many males would survive to the older age classes. Both the male harvest and the male population would consist of young animals. The females, in contrast, would be experiencing a lower mortality rate and so would tend to live to an older age. In the younger age classes, the harvest would consist of three times as many males as females. The i n i t i a l high loss of males would leave a low male:female sex ratio in the older animals. At some age, the females would outnumber the males three to one. The sex ratio for the harvest of this age class would be one to one. For older age classes, the harvested females would outnumber the males. The differences in the age and sex distribution in the harvests of the populations described above were generated solely by a dif f e r -ence in the hunting effort. Thus, for a given population with a differential vulnerability between the sexes, one can interpret the age and sex distribution in the harvest as an indicator of the harvest effort in the population. The f i r s t harvest is recognized as being a 8 negligible contribution to the total mortality because the age structure in the harvest of males is identical to the age structure in the harvest of the females even though the harvest of males is three times the harvest of females. In the second example, the harvest is recognized to be a significant mortality factor because the rate of decline in the number of males by age i s observed to be three times the rate of decline in the number of females by age in the harvest; the ratio of males to females in the harvest in the f i r s t harvestable age class is three to one; and the ratio of the total male harvest to the total female harvest is one to one. The harvest data can supply adequate information for estimating population size. As discussed above, an estimate of the harvest effort can be obtained from the age and sex distribution in the harvest. The number harvested is also known. Because the effort indicates what fraction of the population is being harvested, the total population size can be estimated. 1.6 Potential Value The central question, "what can be learned from harvest data?", could be rephrased as "how can the manager modify the harvest policy i n order to increase his understanding of the harvested population?". The approach developed i n this thesis provides a population estimate as a function of harvest data. Successfully applied, the approach means that the manager can use the hunter to create a population inventory. In this situation, the harvest plays a dual role. On the one hand, i t represents a valuable resource that is sought after by big game 9 hunters. It supports a small industry that ranges from guides through to taxidermists. On the other hand, i t represents one of the few data rich sources of information on the population. (Alternative inventory approaches, such as survey counts or mark and recapture estimates, are often very expensive and appear to have limited u t i l i t y for some species.) As a result, the manager is also seen to play a dual role. On the one hand, he manages the resource to supply "optimum"* benefits to the resource users. On the other hand, the manager can manipulate the harvest to provide information on the population status. An experimental design of this manipulation may enrich the manager's understanding of the population and allow for better future management. The manager must solve a dual control problem. He must control the harvest for some social objective function, and he must control the harvest for information so that he may improve the future management of the system. There w i l l almost always be a trade off between these two control features. The manager who is manipulating the harvest to extract information w i l l almost always appear to be compromising short term social values. This thesis w i l l not consider the dual control problem explicitly. It w i l l provide a tool that allows the manager to use harvest data i n A question of what is optimum w i l l not be pursued here. The manager could be providing benefits to hunters, or to public at large. In one situation, he needs to supply a harvest, in the other, a large number of visible animals, and sometimes both. For any situation, the manager is charged with meeting some social objective with respect to the population. 10 the deve lopment o f h i s management p o l i c y . I t w i l l n o t e v a l u a t e p o l i c y t r a d e o f f s between the v a l u e of e x p l o r a t o r y management today and t h e p o s s i b i l i t y o f good management i n the f u t u r e . 11 CHAPTER 2 ALTERNATIVE MODELS FOR THE ANALYSIS OF AGE AND SEX DATA A number of d i f f e r e n t approaches have been used i n the analysis of k i l l data. The change i n r a t i o methods used i n w i l d l i f e management and the catch at age analyses developed i n f i s h e r i e s represent two approaches that have been consolidated i n the development of t h i s t h e s i s . 2.1 Change i n Ratio Methods Change i n r a t i o methods developed for w i l d l i f e management were based on c a l c u l a t i o n s involving the sex and age r a t i o s of the population and the harvest. Hanson's monograph (1963) reviews the approaches developed p r i o r to 1963. These e a r l i e r approaches did not have accurately aged animals, but rather r e l i e d on aggregate age classes such as adults, sub-adults, and j u v e n i l e s . The approaches were a l l based on changes i n r a t i o . A change i n r a t i o , such as the change i n the sex r a t i o i n the population a f t e r a d i f f e r e n t i a l removal of males and females during a hunt, was used i n conjunction with an estimate of the magnitude of the harvest to estimate the harvest population (Kelker, 1940; i n Hanson 1963). These change i n r a t i o c a l c u l a t i o n s require only that the r e l a t i v e proportions i n the population be known before and a f t e r a known change occurs. These subgroups could be age classes, or sex classes, or sex and age classes. The methods do not apply to time sequences of data, and are not r e s t r i c t e d to the harvest data. However, they a l l require a sample of the l i v e population to 12 estimate r a t i o s . The approach i n t h i s thesis could be reduced to a change i n r a t i o method i f i t were r e s t r i c t e d to two years of data. Because, i n t h i s t h e s i s , no d i r e c t measure of the population i s required, the harvest would have to be treated as a biased population sample. The f i r s t age c l a s s would be used to estimate the harvest bias given the sex r a t i o i n the population p r i o r to the f i r s t harvestable age. The second age class harvest i n the second year would be corrected for t h i s bias and would serve to estimate the population sex r a t i o . These corrected sex r a t i o s could be used i n conjunction with the magnitudes of the f i r s t year harvest to estimate the population s i z e . Change i n r a t i o methods, however, appear to have l i m i t e d u t i l i t y . As Caughley, 1974, cautioned "... age r a t i o s often provide ambiguous information and t h e i r f a c i l e i n t e r p r e t a t i o n can lead to serious manage-ment blunders" and "... age r a t i o s cannot be interpreted, even i n a general way, without a d d i t i o n a l demographic information, p a r t i c u l a r l y on the population's rate of increase". Caughley was being c r i t i c a l of the t r a d i t i o n a l way that he thought managers were attempting to i n t e r -pret the age structure of the k i l l from a single harvest year. He was not considering the p o s s i b i l i t y of examining the complete time series of harvests with the i d e n t i f i c a t i o n of i n d i v i d u a l cohorts preserved throughout the harvest period. Bunnell and T a i t (1980) extended Caughley's c r i t i c i s m to the careless i n t e r p r e t a t i o n of population age d i s t r i b u t i o n s . The assumption of a stable age d i s t r i b u t i o n applied to a nonstable age d i s t r i b u t i o n can lead to wrong conclusions. This t h e s i s avoids these problems by t r e a t i n g the t o t a l time sequence 13 of the harvest simultaneously and allowing the population to fluctuate and respond i n a dynamic fashion to the harvest. It i s expected that, i n some s i t u a t i o n s , the proposed analysis w i l l benefit from contrasting harvests and changes i n age d i s t r i b u t i o n s . As indicated i n chapter one, the approach i n t h i s thesis examines the differences between the male and female subpopulations as expressed i n the time series of the age d i s t r i b u t i o n of the male and female harvests. 2.2 Sex Ratio Estimates Paloheimo and Fraser (1981) claim to avoid most of the biases that concerned Caughley (1974). Their approach estimates the harvest rate and v u l n e r a b i l i t y from age and sex data. Paloheimo and Fraser (1981) and Bunnell and T a i t (1980) recognized the extra information that i s available i n the d i f f e r e n t i a l v u l n e r a b i l i t y by sex of w i l d l i f e populations. However, Paloheimo and Fraser (1981) analyzed the change i n sex r a t i o i n the harvest within a cohort as a function of the hunting e f f o r t . A number of cohorts could be simultaneously analyzed i n a multiple regression format to pool estimates of r e l a t i v e v u l n e r a b i l i t y and harvest rates. Fraser et a l . (1982) applied the approach of Paloheimo and Fraser (1981) to the estimation of harvest rates for black bears from age and sex data. Paloheimo and Fraser's model w i l l be discussed i n greater d e t a i l i n the context of some si m i l a r f i s h e r i e s models. 14 2.3 Catch At Age Analysis In fisheries, the analysis of the harvest data is limited to an analysis of the catch at age data. Fish are not classified by sex. A typical fisheries data set generally consists of a time sequence of estimated catches with an estimate of the age distribution of the f i s h within each year's catch. Both the estimate of the year's catch and the estimate of the age distribution have potentially large errors. Despite or because of these limitations, the analysis of catch at age data in fisheries i s generally far more sophisticated than the compara-ble analysis of wildlife harvest data sets by wildlife managers. This may reflect the fact that for some fisheries, the only source of infor-mation about the population is the harvest i t s e l f . Earlier fisheries analyses required a large number of assumptions and did not treat a data set as a dynamic time series. For example, the estimate of mortality from a catch curve by Robson and Chapman (1961) assumed that the survival rate was constant for a l l ages and years, that recruitment was constant for a l l years, and that the harvest represented an unbiased sample of the population. Ricker (1975) provided a good review of these earlier approaches. The recent approaches are similar to this thesis in that they examine the harvest data set as a time series with an implicit under-lying s t a t i s t i c a l model. A l l of the methods discussed here, including that of Paloheimo and Fraser (1981), extend the catch equations developed by Beverton and Holt, (1957; i n Paloheimo 1980). These equations assume that an instantaneous natural mortality rate and an 15 instantaneous f i s h i n g mortality can be applied to the cohort of f i s h for the e n t i r e year. The t o t a l mortality can then be pa r t i t i o n e d propor-t i o n a l l y between f i s h i n g losses and natural losses to generate an estimate of the catch within a s p e c i f i c year and age c l a s s . H = _P_ ( l - e ^ P + ^ N (2.1) p+m where, H = the harvest i n a year from a s p e c i f i c cohort p = the instantaneous harvest rate m = the instantaneous natural mortality rate N = the number of animals i n the cohort at the start of the harvest year An a l t e r n a t i v e model, referred to as the seasonal f i s h e r y model, assumes that the f i s h e r y i s operated for a b r i e f period of time and that during t h i s time e s s e n t i a l l y a l l mortality i s f i s h i n g mortality. The catch within a s p e c i f i c year and age i s the product of the age clas s size and the f i s h i n g mortality rate. The size of the age class i n the following year i s generated by a s u r v i v a l rate and the remaining unharvested f i s h . That form of the model has been applied i n t h i s thesis to approximate a w i l d l i f e harvest process. H = ( l - e ~ p ) N (2.2) 16 Paloheimo (1980) examined both models in his estimate of mortality rates in fish populations. The points made in the following discussion relate to either model form but w i l l be phrased within the context of the seasonal fishery model. The fisheries models, whether they are expressed as a seasonal fishery or as a continuous fishery, require a set of assumptions with additional data. The models a l l employ a basic population dynamic paradigm; animals are born or recruited, age, and die of natural causes or are harvested. A l l the models considered here deal with a closed population. A difference between the model in this thesis and the models in the current literature i s that the latter estimate the harvest by an explicitly or implicitly assumed s t a t i s t i c a l model while in this thesis the harvest is assumed to be accurately known. The general approach used in the literature to analyze harvest data consists of developing a predictive model for the harvest as a function of a number of parameters, and possibly some independent variables such as effort: h = f(E,Q) (2.3) where, h = vector representing the estimated harvest for a l l age classes and years; E_ = a vector representing the independent variables such as effort; and Q = a vector representing the parameters to be estimated. 17 It i s important to note that Equation (2.3) i s not a function of h, the observed harvest. The estimation procedure consists of minimizing some objective function which relates the observed harvest to the estimated harvest. A simple objective function i s the sum of squared differences between the observed harvest h and the estimated harvest h. In these models, the objective function of the observed and estimated harvest represents e i t h e r an i m p l i c i t or e x p l i c i t proba-b i l i t y statement related to the l i k e l i h o o d of the observed harvest. This thesis also develops a l i k e l i h o o d statement for the observed harvest. The c r i t i c a l difference i s i n the development of the recon-structed underlying population. In a l l the f i s h e r i e s models discussed the reconstruction i s based on an estimated harvest whereas i n t h i s thesis the reconstruction i s based on the actual harvest. Doubleday's (1976) l e a s t squares approach to the analyses of catch at age data minimizes the sum of squared differences between the logarithm of the observed catch at age and the logarithm of the predicted catch at age. He assumes that the natural mortality rate i s known f o r a l l ages and years. No assumptions about the rel a t i o n s h i p between recruitment and the adult population are incorporated. The model estimates the yearly recruitment rather than a parameter repre-senting a recruitment rate as i s suggested i n t h i s t h e s i s . Doubleday reduces the number of parameters to be estimated by assuming that the harvest rate may be represented as a product of two fa c t o r s , yearly harvest e f f o r t and an age s p e c i f i c but year independent harvest v u l n e r a b i l i t y . The same approach i s followed i n th i s t h e s i s . The 18 major distinction between Doubleday's method and the estimator in this thesis is in the reconstructed population. Doubleday assumes that the population model is correct. The difference between the observed catch and the estimated catch is assumed to be observation error. The reconstructed population u t i l i z e s the estimated catch and a constant mortality to deterministically calculate the surviving members of an aging cohort. In contrast, in this thesis both the catch and the surviving members of a cohort are assumed to be stochastic functions of the population. The observed harvest is assumed to be the actual harvest. The consequences of this distinction can be identified by considering the following situation. Imagine an abnormally high observed harvest from a specific age class for a specific year. In Doubleday's approach, this high observed harvest w i l l affect the estimated harvest effort for that year and the estimated vulnerability for that age class. However, since the estimated harvest effort and the estimated vulnerability are pooled estimates based on a combination of years and age classes, the estimated harvest w i l l be lower than the observed harvest and probably lower than the actual harvest. As a result, the reconstructed cohort for a l l years following the high harvest w i l l be overestimated. On the other hand, the estimated harvest rates following the high harvest w i l l tend to be underestimated. In contrast, the approach in this thesis reconstructs the age and year class based on a weighted average of estimates based on the harvest, on the estimated number in the cohort in the previous year, 19 and on the estimated number i n the cohort i n the following year, as w e l l as the observed harvest. This approach avoids the problem of introducing a bias with repercussions through a l l following years i n the cohort. For example, i n the approach i n t h i s thesis, the d i f f e r -ence between two years of a reconstructed cohort w i l l always be greater than the observed harvest; a harvest greater than the cohort size i s not possible. The l o g i c a l structure of the Doubleday approach does allow the difference i n the number of animals between successive years i n a cohort to be smaller than the observed harvest from the cohort. Paloheimo's (1980) estimation of mortality rates i n f i s h popula-tions i s s i m i l a r to Doubleday's least squares approach. Paloheimo estimates the natural mortality rate, the recruitment rate, and the harvest rate but requires an estimate of the catch per unit of e f f o r t . Paloheimo s t a r t s with the Beverton and Holt (1957) catch equations for both a continuous f i s h e r y and a seasonal f i s h e r y . The i m p l i c i t reconstructed population i n the Paloheimo model represents the number of f i s h i n a cohort t h i s year as those f i s h that were not removed i n the "estimated" harvest l a s t year and that survived to t h i s year. Paloheimo's model, used to estimate a constant mortality rate repre-sents a deterministic population sampled with error. In terms of the reconstructed population, there i s no difference between the Doubleday approach and the Paloheimo approach. They both search for the best set of parameter values that can be used to generate a deterministic model which i s used to "predict" the catch at age data, and then minimize the error between the observed catch at age and the 20 predicted catch at age. The predictions are based on a deterministic reconstruction that i s i n d i r e c t l y a function of the observed harvest i n that the observed harvest a f f e c t s the estimated parameters. As with the Doubleday model, the consequences of an unusually large harvest f o r a s p e c i f i c age/year class are inadequately incorporated. The w i l d l i f e model of Paloheimo and Fraser (1981) i s s i m i l a r to the models of Doubleday and Paloheimo i n that i t also creates a harvest estimate and uses t h i s i n the deterministic reconstruction of the population. The objective function i n the Paloheimo and Fraser model i s a weighted l e a s t squares difference between the logarithm of e s t i -mated sex r a t i o by age and the logarithm of observed sex r a t i o by age for the sequence of ages i n a cohort. Pope and Shepherd (1981) provided a simple method for the consis-tent i n t e r p r e t a t i o n of catch-at-age data. Their analysis draws an analogy between a two way ANOVA and the basic model that underlies the models of Paloheimo (1980) and Doubleday (1976). Pope and Shepherd's approach s p l i t s the harvest rate i n t o a year s p e c i f i c harvest e f f o r t and an age s p e c i f i c v u l n e r a b i l i t y . The two way ANOVA c l e a r l y i d e n t i -f i e s what can be expected i n any estimate of these two factors and allows for a reasonable i n t e r p r e t a t i o n of confidence i n t e r v a l s or errors i n these estimates. Fournier and Archibald (1982), i n t h e i r development of a general theory for analyzing catch at age data, come closest to developing a f i s h e r i e s model comparable to the model presented i n t h i s thesis. Their approach i s also a maximum l i k e l i h o o d model. Since i t i s a 21 fisheries model, i t does not exploit the distinction between the sexes. Fournier and Archibald recognized that the harvest rate from a specific age class in a specific harvest year can be distinct from the harvest rate for any other specific age class and harvest year. Their model would be similar to this thesis i f the harvest effort were represented by a year effect, an age effect, and an interaction or error term. However, their implementation assumes that most of the "errors" are associated with the estimate of the total catch and the estimate of the age distribution in the catch. In other words, they reduce their general model to a model that has the same form and the same criticisms as those discussed above. A l l of these fisheries models presumably were designed for large populations. It is thus reasonable to expect the observed harvest to be only an estimate of the actual harvest. For small populations, the variation i n abundance between age classes and years i s much more lik e l y to be due to the stochastic nature of the harvest and survivor-ship processes. A consequence of this distinction is that the proce-dure in this thesis can treat small populations containing empty age classes. Many of the older age classes in a wildlife population w i l l not be represented in the harvest. One could even expect that some of these older age classes w i l l not exist in some years in the wild population. These empty age classes, both in the harvest and in the population, can be included in the population reconstruction used i n this thesis. The fisheries models, particularly those that use some form of the logarithm of the catch, cannot tolerate empty age classes. 22 2.4 Cohort Analysis An a l t e r n a t i v e f i s h e r i e s approach, cohort analysis (Pope, 1972), creates a population reconstruction that assumes the actual harvest i s the observed harvest. In the cohort analysis, the natural mortality i s assumed and i s included i n the reconstruction as a deterministic reduction i n a cohort with age. In t h i s t h e s i s , a natural mortality rate i s used to determine the d i s t r i b u t i o n of possible population s i z e s within an age c l a s s . The population size incorporated i n the recon-s t r u c t i o n i s the most probable one given the d i s t r i b u t i o n of possible population s i z e s . It i s worth noting that Paloheimo (1980) recommends mixing cohort analysis with h i s mortality rate a n a l y s i s , going back and f o r t h from one to the other. 2.5 Summary In summary, the model i n t h i s thesis d i f f e r s from most of the a v a i l a b l e techniques for analyzing harvest data, i n that i t includes the two sexes, excludes e f f o r t data, and includes a recruitment process. It also characterizes the stochastic nature of the data as r e s u l t i n g from a sequence of stochastic processes rather than a stochastic sample from an underlying deterministic population model. It i s expected that the procedure to reconstruct the population and the r e s u l t i n g analysis could be combined with e f f o r t estimates and used e f f e c t i v e l y to reconstruct asexual f i s h populations. That, however, represents an extension beyond the scope of t h i s t h e s i s . 23 CHAPTER 3 THE STRUCTURAL MODEL At the heart of any s t a t i s t i c a l estimate i s a s t r u c t u r a l model. The s t r u c t u r a l model embodies our understanding of the processes and int e r a c t i o n s that we believe are s u f f i c i e n t to generate a set of observations. This chapter w i l l develop a s t r u c t u r a l model that i s assumed to be adequate to characterize a set of hunter harvest data. As Bard (1974) noted, a model i s not complete unless i t also describes, i n an appropriate manner, the random elements of a s i t u a t i o n . Thus, t h i s model w i l l be p r o b a b i l i s t i c ; i t w i l l compute the l i k e l i h o o d of observing a s p e c i f i c time stream of harvests. 3.1 Notation As i n most population dynamics models, a dis c r e t e age structure model i s used for the reconstructed population. In an age structure representation, the population i s p a r t i t i o n e d into the number of animals i n each age and sex class. Thus, x ^ j ^ , the number of animals i n the i t h year, i n the j t h age c l a s s , belonging to sex k (1 = males, 2 = females), i s assumed to represent the number of animals i n the population at a s p e c i f i e d point i n time. For t h i s discussion, i t i s convenient to consider X-JJ^ to be the number immediately p r i o r to the hunting season. In t h i s paradigm, the hunting season i s considered to be a short period of time during which there i s n e g l i g i b l e natural mortality. 24 Each animal in each age and sex class is exposed to a hunting pressure or hunting mortality represented by a probability of being harvested, P i j k * T M S parameter represents the probability that an animal in the i t h year, in the jth age class, and kth sex w i l l be harvested. The number of animals harvested in year i that are age class j , and sex k is given by h ^ ' A fraction of the remaining unharvested animals are assumed to survive to the following year. The probability that an individual in the i t h year, the jth age class, and the kth sex w i l l survive to the following year is represented by s ^ j ^ ' 3.2 The Harvest Data Set This thesis i s limited to an analysis of complete harvest data sets. A complete data set consists of a time series record of a l l the animals harvested from a closed population for a contiguous sequence of years. The data are to be stratified by age and sex for a l l harvest years. A typical data set would consist of the total number of animals harvested by sex in each age class over a number of years. In the previous section, the harvest was described as the set of elements h , • This set w i l l be considered the complete set and w i l l , at times, i j k be referred to as a vector H. The subscript i ranges from 1 to iiy, the number of recorded harvest years, and j ranges from 1 to na> the number of harvestable age classes. It should be noted that the harvestable age classes do not refer to the age of the animal. For some species, for example bears, young animals such as cubs or 25 yearlings are not hunted. In this case, age class 1 would correspond to the youngest aged animal represented in the harvest, and age class n a to the age of the oldest animal. It i s assumed in the following development of the estimator that the harvested animals have been accurately aged and sexed and that a l l harvested animals are included in the data set. 3.3 The Harvest Process The harvest is assumed to be a random sample of the population at the time of the harvest. That assumption implies that the harvest period i s a relatively short time with negligible natural mortality. If x. i s the number of animals in the population (in year i , age class j , and sex k) immediately prior to the harvest, then the harvest . N.jk' *"S a s s u m e < * t 0 k e a binomially distributed random variable. The parameters of the distribution are x ^ j k an^ ?±fo' ^ st r i c t interpretation of this process would correspond to the assumption that each of the x ^ j ^ animals have an equal probability of being harvested. The harvest process would be analogous to flipping a biased coin for each animal in the age, sex, and year class and harvesting the animal i f the coin lands heads and allowing the animal to stay in the popula-tion i f the coin lands t a i l s . The probability of the coin landing heads is p . • Because the estimator w i l l not be used to estimate the * i j k population for those age classes that are not represented in the harvest, the subscripts for the elements x.,..t are assumed to follow 26 the same convention as used to characterize the harvest. In this text, the underlying population, the set of elements xijk> w i l l sometimes be represented by a single vector X . If each animal harvested from the population is treated as an independent event, then the likelihood or probability of perceiving a specific instance of the harvest H , given values for the underlying population and harvest probabilities P , i s : » P = a vector of harvest probabilities. 3.4 The Survivorship Process The second process, the survivorship process, i s also assumed to be a sampling process. Within a year, each animal has a certain chance, or probability, of dying. As with the harvest process, i t is assumed that animals belonging to the same age, sex, and year class have the same average probability of surviving, s^.^' Thus, the survivorship of the unharvested animals is also analogous to flipping a biased coin. If the survivorship of the individual animals is treated as a set of independent events, then the likelihood of a specific L ( H | X , P ) = n (3.1) where, H X a vector representing the observed harvest; a vector representing the population prior to the harvest, and 27 representation for the underlying population can be recursively generated from an i n i t i a l population and r e c r u i t s to the population as a product of binomial d i s t r i b u t i o n s : L ( X e I H,X ,X S) (3.2) S 1 J. • • • J. • j = I T 7 ' 1 n a _ 1 I f x i j k " h i j k N s x i + i , j + i , k a N x i j k - h i j k - x i + i , j + i , k i= l j - l k-1 V * i + l , j + l , k ' where, X g = the "surviving" population age and sex classes (age and sex classes that are neither r e c r u i t s to the harvestable age classes nor members of the i n i t i a l population); H = the harvest; Xn = the state of the population i n the f i r s t harvest year; the number of animals i n each of the age and sex classes i n the f i r s t year; X ^ = the set of r e c r u i t s to the f i r s t harvestable age class for a l l harvest years and both sexes; and S = the set of survivorship rates s ^ j ^ -3.5 The Recruitment Process It i s assumed that recruitment may be represented as the sum of independent, d i s c r e t e , random processes. Each adult female of age class j i n the population i n year i has the pot e n t i a l of r e c r u i t i n g 0, 1, 2, 3, ... i n d i v i d u a l s to the population i n year i+r where r i s the age of 28 the f i r s t harvestable age class. In this situation one would expect that the distribution of recruits within a sex would be approximately normal with the mean equal to the sum of the individual mean recruit-ment rates and the variance equal to the sum of the individual recruit-ment variances. The likelihood of recruiting X .i. individuals to the population is then approximated by: - % ( x i l k - E R l k ) 2 L C x J X - . B ) = ^ n , 1 e E V i k ( 3 . 3 ) 1 i=l k=l • 2 TT EV i k where, X -, = the set of recruits to the population; X^ = the set of adult females in the population; B = the set of recruitment parameters, b^ .=k and o^tjk used to generate ER i k and EV i k; b ^ j k i s the average birth rate and o^ijk is the variance in the number of births; n a E R i k = jSi *>i-r j k x i - r j 2 the assumed form of the expected number of recruits to the f i r s t harvestable age class for year i and sex k; ( x i _ r j 2 ^ t n e number of age class j females in the population i n year i - r ; n EV ± k = a 2 i - r j k x i - r j 2 the assumed form of the estimated variability in the number of recruits to the f i r s t harvestable age class for year i and sex k; and r = the age of the f i r s t harvestable age class. 29 The estimated recruits (ER) and the estimated variance in the number of recruits (EV) are computed from the number of females in each age class r years previous, as well as age and sex specific parameters representing an average recruitment rate and the variance in the recruitment rate. In the above presentation, ER and EV have not been defined for the f i r s t r harvest years. For these f i r s t few years, ER and EV can be considered as extra parameters which are members of the parameter set B. 3.6 The Stochastic Model The joint distribution for X and H i s given by: L ( x , H | 0 ) = n y n a n ( * W ) P ^ H i - v , ) x W h i * 1 i=l j=l k=l v h i j k ' i J k i j k X X X ( x l j k " h i d k 1 + 1 j + 1 k < i - i j k > X i j k " h i j k " X i + 1 j + 1 k i - 1 j - 1 k - 1 x i + 1 j + 1 k 2 - ^ ( x i l k - E R i k > 2 n y n 1 e E V i k (3.4) i=l k=l / 2irEV i k where, 0 = a composite set of the parameters P, S, B and X^^ Intuitively, Equation (3.4) appears to be simply the product of the likelihood representations of the three processes. However, to be 30 s t r i c t l y c orrect, Equation (3.4) must be b u i l t up r e c u r s i v e l y , a year cla s s at a time. The j o i n t p r o b a b i l i t y of the harvest i n the i t h year can be computed given the i t h year's population. The p r o b a b i l i t y of the i t h year's population can be computed given the previous year's population and an estimate of the expected number of r e c r u i t s from r years previous. The expected number of r e c r u i t s can be computed given the status of the female population r years previous. This recursive sequence of conditional p r o b a b i l i t y statements extends back to the f i r s t harvest year and can be m u l t i p l i c a t i v e l y combined to y i e l d Equation (3.4). 31 CHAPTER 4 ESTIMATION The stochastic model, Equation (3.4), developed i n the previous chapter represents a statement of the l i k e l i h o o d or p r o b a b i l i t y of observing a s p e c i f i c harvest set H and an underlying population X given a set of population dynamic parameters. If X and H were known, Equation (3.4) could be used to generate maximum-likelihood estimates fo r the population dynamic parameters. However, X i s not known. I t represents missing data. Dempster et a l . (1977) describe a procedure that i t e r a t i v e l y computes maximum-likelihood estimates when the observations can be viewed as incomplete data. Their approach, known as the Estimation Maximization or EM algorithm, f i r s t ' f i l l s ' i n values for the missing data with the expected values of the missing data. These expected values are computed from the current estimate of the parameters. They then use the f i l l e d i n complete data de s c r i p t i o n to generate maximum-l i k e l i h o o d estimates for the parameters. These new parameter estimates are i n turn used to recompute the expected values of the missing data. These two steps, the expected value step and the maximum-likelihood step are i t e r a t i v e l y applied with the r e s u l t i n g parameter estimates converging to the maximum-likelihood estimates appropriate for the incomplete data set. The algorithm developed i n t h i s thesis i s s i m i l a r to the EM algorithm i n that i t i s a two step algorithm that may be used to generate estimates from incomplete data. The f i r s t step i s used to 32 f i l l i n the missing data and the second step i s used to generate para-meter estimates from the f i l l e d i n complete data s p e c i f i c a t i o n . The diffe r e n c e between the EM algorithm and the algorithm i n t h i s thesis i s that the missing data are computed as the most probable values, a maxi-mization procedure, rather than the expected values. The f i r s t step i n t h i s Maximization/Maximization or MM algorithm represents a population reconstruction. The reconstructed population i s simply the most probable set of values for the underlying population given a harvest H and an assumed set of parameter values. It i s com-puted by maximizing Equation (3.4). The second step i s a parameter estimation using the maximum-likelihood estimate of the population parameters given the computed values for the underlying population and the harvest data set. 4.1 Population Reconstruction Equation (3.4) together with a harvest data set may be used to reconstruct the underlying population X. The d e r i v a t i v e of the logarithm of Equation (3.4) with respect to each X-JJ^ i s set equal to zero to y i e l d a set of iiy * n a * 2 simultaneous equations i n X. Using ln(x) to approximate the d e r i v a t i v e of In (x!) (6.3.18 i n Abramowitz and Stegun, 1964), the exponential forms of the r e s u l t i n g equations are: X i j k h i j k X i + l , j + l , k ) i j k l BT (X) = x for i = 1, j 4 n a or j = 1, i 4 n y (4.1) 33 ( x i - l , J - l , k - h i - l , J - l , k - x i J k > ( 1 ~ p i j k > s i - l t j - 1 , k B T i j k < X ) m . ^ I j k ^ i j k X ^ i - l . J - l . k * (4.2) f o r i = n y , j ^ 1 or j = n , i 4 1) x -h i j k i j k = 1 for i = 1, j = n a or j = 1, i = n y (4.3) -h . ( l ~ s )BT „ (X) I j k " " l - l , j - l , k i - l , j - l , k ' i j k ^ i - l . j - l . k ^ i j k ^ i j k ( 1 ~ S i - l , j - l , k } ( x i j k " h i j k " x i + l , j + 1 , k ) = 1 (4.4) where, B T i j k i s defined by: for a l l other i and j M = e x p { - ( X i J k - E R i k > i J k J 1 E V i k (4-5) .2, 4. 2 c - b ^ i + r . l . m ^ i+r,m ; n„-r-i)f 1 \ <y -on ) i_im u,, i+r 6 k l H (  - i ) [ Z V 2 — " > E V m=l I+r ,m + o 2 i jm^ xi+r, 1 .m'^i+r 2 j ^  2(EV ) 2 i+r ,m 34 and 6J;L 1 i f j = 1 0 i f j i 1 H ( n v - r - i ) 0 i f n y - r - i < 0 1 i f n y - r - i > 0 Equation (4.1) represents those age and year classes that have no predecessors i n the population, either the f i r s t year of harvest data or the f i r s t harvestable age c l a s s . Equation (4.2) represents those age classes that have no survivors represented i n the population, the oldest age cla s s and the f i n a l harvest year. Equation (4.3) represents those age classes that have neither predecessors nor survivors. Equation (4.4) represents the majority of the age classes, those age classes that have both predecessors and survivors represented i n the population. The terms B T i : j k , Equation (4.5), represent b i r t h terms that have been generated by the recruitment component of Equation (3.4). 35 4.2 Parameter Estimation The second step in the two part maximization algorithm consists in developing maximum-likelihood estimates for the underlying popula-tion parameters, given values for the underlying population and the harvest data. In this situation i t i s as i f the manager knows the number of animals in each age class for each year and knows the harvest removed from each age class each year. The problem Is to compute the birth rates, survivorship rates, and harvest rates. The actual com-putations depend on the assumptions that the manager is willing to make in order to pool some of his observations. For example, the manager may be willing to assume that the adult harvest rate is constant over a l l years and age classes within a sex. The pooled estimate would be different i f the manager assumed that there was a constant increase in harvest rate over time and estimated the best linear relationship to represent this harvest effort. Therefore, the particular estimation procedure used by a manager who knows his underlying population would have to be specifically created to correspond to his assumptions and the known l i f e history. Even the type of estimator could be tailor -made to f i t a specific problem. In this thesis, the emphasis has been on the use of maximum likelihood estimators. (Other types of estimators, such as a "least squares" estimate, may, in some situations, be easier to use and more appropriate to the problem). As an example of the types of computations required, the maximum-likelihood estimates of the assumed constant recruitment rate b, the male and female constant harvest probabilities pm and pf and the constant natural survivorship rates are given by: 36 »y b = i - r J l X i l - / ( 2 ^ Ft) Pf h . . l (4.9) (4.10) h , p = _ii£ m x..2 (4.11) 2 n a n y 2 n a - 1 n y _ 1 s = I S IT x ± 1 k / ( Z E f ( x i i k - h i i k ) ) ( 4 " 1 2 > k=l j=2 1=2 1 J R Vk=l j=l 1=1 1 J k i J k y where, x^ -^  = the t o t a l r e c r u i t s i n year i ; F^ = the t o t a l number of adult females i n year i ; h i = the t o t a l female harvest; h o = t h e t o t a l male harvest; x ^ = the t o t a l harvestable females over a l l years and ages; and x..2 = t n e t o t a l harvestable males over a l l years and ages. 4.3 Interpretation There are two suggested interpretations for the r e s u l t i n g set of parameter values generated by a converging MM algorithm. The f i r s t i n t e r p r e t a t i o n i s expressed i n terms of the r e s u l t i n g reconstructed population based on the f i n a l estimated parameter values. This f i n a l population reconstruction represents the most probable set of values for the underlying population that i s I n t e r n a l l y consistent, given the harvest data set. This i n t e r n a l consistency r e f l e c t s the fact that the 37 reconstructed underlying population generates, as parameter values, the same set of parameter values that were used in developing the popula-tion reconstruction. The second and intuitive interpretation is to consider the most probable reconstruction as an estimate for the expected value of the underlying population and the MM algorithm as an approximation to the EM algorithm of Dempster eit a l . (1977). The resulting parameter values are interpreted as the maximum-likelihood values given the harvest data set. The d i f f i c u l t y with the second interpretation is that i t can not be verified without directly estimating the expected value of the underlying population and comparing i t to the most probable reconstruc-tion. The alternative approach, followed in this thesis, treats the algorithm as a heuristic procedure and tests i t s u t i l i t y directly on a set of Monte Carlo simulations. CHAPTER 5 NUMERICAL CONSIDERATIONS The population reconstruction and the maximum-likelihood parameter estimates represent the simultaneous s o l u t i o n to the two sets of equations: 2*&alii.- o i j k ^ % p - > - 0 (5.2) where, L(X,H |0 ) = the l i k e l i h o o d of X and H given 9, equation (3.4) X = the underlying population H = the observed harvest 6 = the set of population dynamic parameters to be estimated x^jk = the number of animals i n the population i n year i age j and sex k 0 ^  = a s p e c i f i c population dynamic parameter. The MM algorithm consists of i t e r a t i v e l y solving Equation (5.1), using the so l u t i o n to Equation (5.1) i n Equation (5.2), solving Equation (5.2), r e s o l v i n g Equation (5.1) and so on. 39 5.1 Numerical Reconstruction For a problem with 10 years of harvest data and 2 5 distinguish-able age classes, Equation 5.1 represents a set of 5 0 0 nonlinear simultaneous equations in X. These equations have been explicitly represented by Equations 4.1 through 4.4. Using matrix notation, Equations 4.1 through 4.4 may be rewritten (Appendix A) as: D Q H(X)X = R 0 H(X) ( 5 . 3 ) where, D Q H(X) = a matrix that is a function of X and is dependent on the values of H and 0 X = a column vector representing the underlying population Rgjj(X) = a column vector that is a function of X and is dependent on the parameters 0 and the harvest H. Equation 5.3 may be numerically solved using an iterative scheme based on the Newton method: I + 1 X = *X - ^ G - J ^ X ) g 0 H ( I X ) ( 5 . 4 ) where, SQH(^X) = a vector of the error or difference between the l e f t and right sides of equation ( 5 . 3 ) = D Q H ( I X ) I X - R9R( IX) G . „ ( I X ) = matrix of partial derivatives 3 g i , OH n 1 g^ i s the i t n element of g ^ X j i s the j t h element of X 40 *A = a scaler used to adjust the step size of the i t n i t e r a t i o n . In the above example, with 10 years of harvest data and 25 age classes, G, the matrix of p a r t i a l s , t h e o r e t i c a l l y represents a 500 by 500 square matrix. As described i n Appendix A, G has been approxi-mated by a t r i d i a g o n a l matrix that allows for e f f i c i e n t inversion. The step size ^ has been chosen as that value of X (from 0.35, 0.7, 1.05 and 1.4) that minimizes g. In t h i s thesis the population reconstruction component of the MM algorithm inadvertently contained an extra product term i n the computation of B T i j ^ i equation 4.5: e x p f 6 k l H ( n - ^ l ) [ Z - ^ ]} 3 m=l i+r ,m The Inclusion of t h i s term appears to marginally increase the rate of convergence and to s t a b i l i z e the i t e r a t i v e scheme for reconstructions based on poor i n i t i a l parameter estimates. The f i n a l population reconstructions, based on computations that have included t h i s term, do not appear to q u a l i t a t i v e l y d i f f e r from reconstructions based on computations that have excluded t h i s extra erroneous term. 5.2 Numerical Estimation Equation 5.2 represents a set of simultaneous equations i n 0 . The number of equations depends on the number of parameters that are to be estimated. This can not be a l l of the population dynamic para-41 meters that are i m p l i c i t l y represented i n Equation (3.4). Equation (3.4), the s t r u c t u r a l model, allows each age year and sex class to have i t s own independent harvest rate and survivorship rate. For females, each age and year class could t h e o r e t i c a l l y have i t s own independent b i r t h rate and i t s own independent variance for the number of b i r t h s per female. For the 500 d i f f e r e n t age sex and year classes, t h i s would represent approximately 2000 parameters. In general the manager must use h i s understanding of the system to pool and reduce the number of parameters to be estimated. The so l u t i o n to Equation (5.2) depends on the exact nature of the set of equations. For the example i l l u s t r a t e d i n Chapter 4, a constant survivorship rate, a constant b i r t h rate, a constant male harvest rate, a constant female harvest rate, and an assumed known b i r t h variance, the so l u t i o n represents a set of e x p l i c i t a n a l y t i c equations for each of the parameters. The numerical s o l u t i o n i s computed d i r e c t l y from these equations. For more complicated problems numerical algorithms may have to be employed to i t e r a t i v e l y converge to a so l u t i o n set. This component of the MM algorithm must be t a i l o r e d i n d i v i d u a l l y for the s p e c i f i c problem and depends on the managers understanding of the population dynamics and harvest dynamics of the species being exploited. 5.3 The MM Algorithm The d e s c r i p t i o n of the MM algorithm, as an i t e r a t i v e sequence of solutions to Equations 5.1 and 5.2 i s incomplete. As with any other i t e r a t i v e scheme a set of s t a r t i n g values must be s p e c i f i e d . The 42 approach that has been followed i n th i s thesis has been to u t i l i z e an i n i t i a l guess for the parameter set 8. This i n i t i a l guess would be provided by the manager or be based on the current understanding of the population dynamics of the species. This i n i t i a l set for 0 may be used to generate an i n i t i a l reconstruction X. Equation (5.3) i s a representation of Equations (4.1) though (4.4). The dependencies of D and R on X are only through the b i r t h term adjustment factor BT, Equation (4.5). If BT i s assumed to be 1, which implies no b i r t h term adjustment, Equation (5.3) represents a system of l i n e a r equations i n X and may be solved d i r e c t l y . This d i r e c t solution has been used as the st a r t i n g value for the f i r s t i t e r a t i o n of Equation (5.3) for the MM algorithm. The stopping rule that has been used i n t h i s thesis for the MM algorithm has been based on the maximum observed change i n the elements of X within an i t e r a t i o n . In general, the algorithm has been allowed to proceed u n t i l the maximum change i n X has been less than 0.1. The MM algorithm, as i t has been presented, i s not a s o p h i s t i -cated numerical scheme. It has the p o t e n t i a l , as do most i t e r a t i v e schemes, of not converging, converging very slowly, or possibly converging to one of a number of l i m i t points. These p a r t i c u l a r problems have not been e x p l i c i t l y examined i n t h i s thesis. The j u s t i f i c a t i o n for i t s use has been based on the empirical, Monte Carlo simulations to be described i n Chapter 6. However, from the perspective of numerical considerations i t should be noted that the scheme converged, i n a r e l a t i v e l y small number of i t e r a t i o n s , to a 43 set of parameter estimates that were appropriate or close to the known parameter values. The number of i t e r a t i o n s depended on the number of parameters i n the parameter set 6• For a problem that was assumed to have only two unknown parameters, the MM algorithm converged a f t e r 16 i t e r a t i o n s . For a problem with 13 parameters the MM algorithm converged with just under 200 i t e r a t i o n s . 5.4 A l t e r n a t i v e Algorithm's The simultaneous so l u t i o n to Equations (5.1) and (5.2) may be found by s u b s t i t u t i n g the so l u t i o n to Equation (5.1) into Equation (5.2) and treating Equation (5.2) as a system of equations i n 9 alone. From t h i s perspective the MM algorithm i s represented by the i t e r a t i v e scheme: 1 + 1 e = k H ( x ( i e ) ) (5.5) a vector function representing the maximum-likelihood estimates for 8 the vector of solutions to the system of Equations 5.1. This represents a very s i m p l i s t i c approach to finding the solu-t i o n to a system of non-linear equations. However, an a l t e r n a t i v e algorithm, such as the Newton method requires the p a r t i a l derivatives of k with respect to 0 j_. where, k H(X) = X( i 9) = 44 For the problems considered in this thesis, the Newton method is not expected to offer substantial computational savings. A second consideration in applying a Newton style algorithm is the d i f f i c u l t y in numerically determining the values of the partial derivatives. A numerical differential scheme requires accurate function evaluations. The derivative i s based on minor differences in values of the function resulting from minor changes in the values of 8. However, k is a function of X which is i t s e l f a function of a numerical scheme. As a result k is evaluated only approximately. The application of the Newton method would require analytic derivatives. 5.5 Summary The MM algorithm represents a two step iterative process. The f i r s t step, the population reconstruction, could be regarded as a population description that represents a compromise between the current estimate of the underlying parameter values and the harvest data. As such, the reconstructed population may be considered as not entirely consistent with the parameters used to generate i t . The second step of the iterative scheme re-estimates the parameter values. The new set of parameter values approaches a set of para-meters that is consistent with the harvest data. The next population reconstruction again represents a compromise solution between the parameter estimates and the harvest data. In a system with a large number of parameters one would expect a number of different parameter combinations that would produce result almost consistent with the harvest data. In that case the MM 45 algorithm would be expected to converge very slowly and more sophis-t i c a t e d search a l g o r i t h m s , such as the Newton s t y l e algorithms may be required to develop an optimum s o l u t i o n i n a p r a c t i c a l number of i t e r a t i o n s . 4 6 CHAPTER 6 EXPLORATORY RECONSTRUCTION The population reconstruction component of the MM algorithm discussed i n section 4.1 generates a single vector estimate or reconstruction of the population for any complete set of harvest data and population dynamic parameters. In t h i s chapter the s t a t i s t i c a l properties of the reconstructions, i n p a r t i c u l a r the biases and the variances, are explored through Monte Carlo simulations. 6.1 Monte Carlo Simulations For a manager, the important question may not be the value of a p a r t i c u l a r parameter, such as the estimate of the f i n a l population s i z e , but rather whether the reconstructed population c o r r e c t l y indicates population trends. The manager i s also interested i n the robustness of the estimator. He would l i k e to know i f the estimator can indicate trends even i f a l l of the assumptions i n the model are not met. Correct trend information provides a signal that can be used to modify future harvest p o l i c y and allow the manager to manage. These, and other questions, can a l l be explored with Monte Carlo simulations. As Bard (1974) indicated, Monte Carlo simulations can be used to generate sampling d i s t r i b u t i o n s from s t a t i s t i c a l models, and to examine the e f f e c t s of errors i n model formulation. For the manager, these studies provide some measure of confidence i n the u t i l i t y of the s t a t i s t i c a l model and i n the robustness of the estimator. 47 Repeated stochastic simulations, using duplicate i n i t i a l condi-tions and parameters, provide a set of r e p l i c a t e d experiments. These generated data sets may be used to develop a set of reconstructions which may be compared to the simulated populations. S u f f i c i e n t r e p l i -c a tion allows one to examine the bias and p r e c i s i o n of the estimator. 6.2 The Monte Carlo Simulator The computer simulation i s a d i s c r e t e , stochastic, state-oriented simulation model with an annual i t e r a t i o n . That means the population i s represented by the number of males and females i n each of 30 age classes for the current i t e r a t i o n year. A random harvest i s removed from each age and sex class. This harvest i s chosen as a binomially d i s t r i b u t e d random variable for each of the age and sex classes. The parameters for each of these d i s t r i b u t i o n s are the number i n the age sex^class and a harvest p r o b a b i l i t y . The harvest p r o b a b i l i t y i s assumed to be composed of a year e f f e c t representing harvest e f f o r t and an age-sex e f f e c t representing r e l a t i v e v u l n e r a b i l i t y . If the r e l a t i v e v u l n e r a b i l i t i e s are scaled so that the v u l n e r a b i l i t y of an age 6 male i s a r b i t r a r i l y set to 1, then the yearly e f f o r t component of the harvest p r o b a b i l i t y represents the p r o b a b i l i t y of an i n d i v i d u a l s i x -year-old male being harvested. The p r o b a b i l i t y of an i n d i v i d u a l of age j and sex k i n year i being harvested i s thus given by: P i j k " V j k 6 i (6.1) 4a the probability of a six-year-old male in year i being harvested; the ratio of the probability of a j year old animal of sex k to the probability of a six-year-old male being harvested; and the probability of an animal of age j and sex k in year i being harvested. It should be noted that i s n o t t n e probability that a harvested animal w i l l be age j and sex k in the i t h harvest. The actual distribution i n the harvest depends on the age and sex distribu-tion in the population. p . r e p r e s e n t s the probability from the animal's perspective. It is equivalent to flipping a biased coin for each animal in the population. If the coin lands with the head up, the animal is harvested. If the coin lands with the t a i l up, the animal remains in the population. The probability of the coin landing heads up is p ^ . Although the harvest probability has been described as a product of an age and a year factor, the simulations can include other factors. In a simulation, the vulnerability may be changed for each simulated year. That means the experimenter could specify independent harvest probabilities for each age sex and year class. From the remaining nonharvested animals a random number are allowed to survive to the next age and year class. The number surviving i s again a binomially distributed random variable for each of the age and sex classes. The parameters are the number of nonharvested animals where, e i ' V j k = P i j k = 49 in each age and sex class and an age, sex, and year specific survivor-ship rate. The third simulated process, recruitment, is treated in an ad hoc fashion. In the simulation model, the number of male and female recruits in any specific year is assumed to be a sum of binomially distributed random variables. The binomial parameters are equal to the number of reproductive females in each age class and the corresponding age-specific male and female birth rates for the current simulated year. Birth rate may not be the most accurate term. The simulation represents the status of the population at a specific time of the year. This "census" point in time may or may not correspond to the time of birth for a seasonally reproducing population. To be accurate, this rate is the average number of offspring that accompany a female at the time of the annual census. In the simulations, b-jj^ was assumed to be the average number of offspring of sex k for a female age j in year i . For the general simulation model, these age specific birth rates may be changed for each simulated year. It should be noted that the age specific birth rates may be separately specified for each sex of offspring. For a species with an equal sex ratio at birth and for the following simulation experiments, these sex specific rates w i l l be equal. In these simulations, the birth rate does not directly correspond to the recruitment rate discussed in Chapter 3. The recruitment rate in Chapter 3 represents the product of the birth rate and the survivor-ship rates up to the age of the f i r s t harvestable age class. 50 6 • 3 I n i t i a l Conditions For convenience, the i n i t i a l conditions for the following set of experiments were established by s t a r t i n g with a population of 3000 animals; 50 males and females i n each of 30 d i s t i n c t age classes. Survivorship rates for a l l age, sex and year classes were set at 0.93. On average, 93% of the unharvested animals survive to the following year. Harvest rate was established as 0.30 for a l l adult males (fourth age c l a s s and o l d e r ) . The r e l a t i v e v u l n e r a b i l i t y of adult females was assumed to be 0.3 that of adult males and the r e l a t i v e v u l n e r a b i l i t y for juvenile males and females was zero. The r e s u l t was that, on average, 30% of the adult males and 9% of the adult females were harvested per year. The average male and female b i r t h rate for a l l reproductively active females was established as 0.22. The f i r s t age of reproduction was assumed to be age 4, the 5th age c l a s s . An i n i t i a l population of 50 animals i n each age and sex class was then simulated with the above parameter values for 70 years to develop a r e a l i s t i c age d i s t r i b u t i o n . It i s t h i s population that i s used as the base population for the following experiments. This i n i t i a l population and the above parameters are summarized i n Table I. 51 TABLE I. I n i t i a l conditions for the base population. Male survivorship rate: 0.93 Female survivorship rate: 0.93 Average b i r t h rate age 0 to 3: 0.0 Average b i r t h rate age 4 to 29: 0.22 yearly harvest p r o b a b i l i t y : 0.3 Male harvest v u l n e r a b i l i t y - age 0 to 2: 0.0 - age 3 to 29: 1.0 Female harvest v u l n e r a b i l i t y - age 0 to 2: 0.0 - age 3 to 29: 0.3 BASE POPULATION AND HARVEST Males Females Age Preharvest Harvest Preharvest Harvest 0 97 0 116 0 1 108 0 121 0 2 89 0 101 0 3 93 18 94 8 4 69 17 81 5 5 34 5 57 5 6 24 6 51 3 7 14 2 52 6 8 10 4 33 3 9 8 1 37 5 10 3 1 28 1 11 5 1 31 3 12 0 0 21 0 13 1 0 13 0 14 1 0 9 1 15 1 0 9 1 16 0 0 6 0 17 0 0 14 2 18 0 0 11 0 19 0 0 7 2 20 0 0 8 1 21 0 0 3 0 22 0 0 4 0 23 0 0 3 0 24 0 0 3 0 25 0 0 2 0 26 0 0 3 0 27 0 0 0 0 28 0 0 4 0 29 0 0 2 0 52 An experiment consists of removing the base harvest from the base population, allowing the remaining animals to survive and reproduce and then recording the harvest and population l e v e l s . A l l r e p l i c a t e s use exactly the same base population p r i o r to the f i r s t harvest. However, since the survivorship to the f i r s t recorded year i s a random var i a b l e , the r e p l i c a t e d populations w i l l a l l show minor differences by the time of the f i r s t harvest. 6.4 Experiment 1 - The Base Run In t h i s f i r s t experiment, the i n i t i a l population and the para-meters were l e f t exactly as described i n Table I and simulated for a further ten years. The harvest was recorded and used by the estimator to develop a population reconstruction. The base population was reestablished and the process was repeated to generate 9 r e p l i c a t e harvest data sets each of which contained ten years of harvest data. The actual simulated population was recorded for each of these simula-tions for comparison to the reconstructed population. Table II i s a t y p i c a l simulated harvest data set. Table I I I i s a record of the simulated population that gave r i s e to the harvest data i n Table I I . The base run experiment was used to examine the properties of the population reconstruction component of the estimator. The reconstruc-t i o n requires an average and variance i n the recruitment rate to the f i r s t harvestable age c l a s s . The recruitment rate to the f i r s t harvestable age class was defined as the average number of r e c r u i t s per female that would survive for 3 years: 0.22 x 0.93 x 0.93 x 0.93 = 0.177. TABLE I I . A t y p i c a l harvest set. (*The population simulator used 29 age classes but the harvest was recorded to age 27). Age 1 2 3 4 5 6 7 8 9 10 MALE HARVEST 3 28 28 31 23 23 32 33 34 28 26 4 17 14 18 13 24 14 18 13 14 24 5 11 11 IS 16 6 11 11 7 11 6 6 11 11 8 4 8 4 8 8 5 8 7 4 3 10 5 4 6 2 4 5 3 8 3 3 5 4 3 3 2 3 2 4 9 3 3 2 2 1 2 0 1 1 3 10 5 0 0 0 3 0 3 1 2 1 11 0 1 0 0 1 0 1 0 0 0 12 0 0 1 1 0 1 1 0 1 1 13 0 1 0 0 0 0 0 1 0 0 14 0 0 2 0 0 0 0 1 0 0 15 0 1 0 0 0 0 0 1 2 0 16 0 0 0 0 0 1 0 0 0 0 17 0 0 1 0 0 0 0 0 0 0 18 0 0 1 0 0 0 0 0 0 0 19 0 0 0 0 0 0 0 0 0 0 20 0 0 0 0 0 0 0 0 0 0 21 0 0 0 0 0 0 0 0 0 0 22 0 0 0 0 0 0 0 0 0 0 23 0 0 0 0 0 0 0 0 0 0 24 0 0 0 0 0 0 0 0 0 0 25 0 0 0 0 0 0 0 0 0 0 25 0 0 0 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 0 0 0 27* 0 0 0 0 0 0 0 0 0 0 FEMALE HARVEST 4 5 5 6 8 5 11 6 2 11 8 5 5 4 5 4 8 4 4 4 6 6 6 5 6 5 5 6 3 3 2 5 11 7 6 4 6 4 1 6 3 2 2 4 8 5 2 4 3 5 4 4 1 4 3 9 1 5 2 1 2 2 1 1 2 0 10 1 0 4 4 2 5 1 0 2 6 11 2 2 3 6 0 0 2 2 7 3 12 2 3 4 0 2 0 0 4 4 1 13 4 0 0 2 2 5 0 3 0 2 14 1 2 1 1 1 0 0 2 2 2 15 0 0 2 2 3 2 3 0 0 2 16 0 0 1 0 3 0 0 2 0 0 17 0 1 0 0 0 2 3 1 0 0 18 3 0 0 2 0 1 0 0 0 2 19 2 0 1 0 0 0 0 0 0 1 20 1 1 0 0 0 1 1 0 1 0 21 0 0 0 0 0 0 1 1 1 0 22 0 0 0 0 1 1 0 1 1 1 23 0 0 2 0 0 1 0 0 0 0 24 0 1 0 0 1 0 0 0 0 0 25 0 0 0 0 1 0 0 0 0 0 26 1 0 0 1 0 0 0 0 0 0 27* 0 0 0 0 0 0 0 0 0 0 TABLE I I I . A t y p i c a l simulated population. Year Age 1 2 3 4 5 6 7 8 9 10 SIMULATED MALE POPULATION 0 114 98 94 118 106 118 104 90 97 197 1 94 108 94 84 109 93 108 101 84 93 2 100 86 102 90 78 101 92 104 89 80 3 84 92 80 95 82 75 91 85 91 85 4 64 52 61 42 66 55 42 53 44 56 5 50 44 34 39 23 41 41 22 38 28 6 28 38 30 17 23 16 28 29 14 26 7 15 17 22 19 9 14 12 18 18 9 8 12 11 13 7 12 5 7 10 11 11 9 5 6 7 8 2 9 2 5 6 9 10 7 2 3 5 5 1 7 2 4 5 11 2 2 2 3 5 2 1 4 1 2 12 4 2 1 2 3 4 2 0 4 1 13 0 4 2 0 0 2 3 1 0 3 14 1 0 3 2 0 0 2 3 0 0 15 1 1 0 0 2 0 0 2 2 0 16 1 1 0 0 0 2 0 0 1 0 17 0 1 1 0 0 0 1 0 0 1 18 0 0 1 0 0 0 0 1 0 0 19 0 0 0 0 0 0 0 0 0 0 20 0 0 0 0 0 0 0 0 0 0 21 0 0 0 0 0 0 0 0 0 0 22 0 0 0 0 0 0 0 0 0 0 23 0 0 0 0 0 0 0 0 0 0 24 0 0 0 0 0 0 0 0 0 0 25 0 0 0 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 29 0 0 0 0 0 0 0 0 0 0 SIMULATED FEMALE POPULATION 0 101 96 115 98 96 124 115 103 103 110 1 111 97 89 106 91 90 115 106 90 99 2 no 105 90 85 95 87 85 106 96 78 3 92 100 98 85 78 91 82 81 102 86 4 79 81 82 84 67 70 73 71 68 92 5 68 71 70 71 71 60 56 57 68 53 6 48 60 61 59 64 58 51 50 52 59 7 43 41 51 52 51 50 53 48 43 43 8 41 32 37 39 43 45 39 50 43 37 9 28 35 26 30 35 34 36 33 45 37 10 30 26 28 20 28 32 26 32 30 41 11 25 26 25 22 16 24 25 25 31 28 12 27 21 21 20 16 16 22 20 23 24 13 20 25 16 17 19 14 16 22 16 18 14 12 14 23 16 13 17 8 14 19 15 15 8 10 12 21 14 12 15 8 11 17 16 7 8 10 10 19 11 9 10 8 9 17 5 7 8 9 9 15 10 9 8 8 18 11 4 5 7 9 9 11 7 6 5 19 10 5 4 5 3 7 8 9 5 5 20 5 7 5 3 3 3 5 8 7 5 21 7 4 6 5 3 3 2 4 8 6 22 3 6 3 6 5 3 3 1 3 7 23 3 3 5 3 4 2 2 3 0 2 24 2 3 3 3 2 3 0 2 3 0 25 0 0 0 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 29 0 0 0 0 0 0 0 0 0 0 55 Recruitment was treated as a B e r n o u l l i process y i e l d i n g a variance estimate of 0.15. Given t h i s information, and the harvest data of Table I, the estimator generates a reconstruction of the population, Table IV. These reconstructions are not u t i l i z i n g the complete MM algorithm but rather are only examining the population reconstruction component. As a r e s u l t the numerical scheme used to search for the most probable population has not been l i m i t e d to 2 i t e r a t i o n s . The algorithm has been allowed to i t e r a t e u n t i l the maximum change i n the reconstructed population was less than 0.1. The estimator can only estimate harvest-able age classes so the reconstruction s t a r t s at age 3. Tables I I through IV are the r e s u l t s of a s i n g l e simulation and reconstruction. By themselves, they provide l i t t l e information on the accuracy or bias of the estimator. Figure 1 i s a composite view that has been generated from the 9 r e p l i c a t e simulations of Experiment 1. For each simulation, for each harvest year, for each sex, and for each age c l a s s , the reconstructed population s i z e was subtracted from the "known" simulated value. Figure 1 i s a contour plot of the average of these differences or errors for the 9 simulations. The elevation of the contour represents the magnitude of the average error. For the males, the average errors range between -5.1 and 2.1; for the females between -5.7 and 7.8. These extreme values are not represented by contour l i n e s . The con-tours represent a series of ridges and v a l l e y s that follow i n d i v i d u a l cohorts. These cohorts approach the zero contour with increasing age. TABLE I V . P o p u l a t i o n r e c o n s t r u c t i o n from Table I I h a r v e s t d a t a . Age 1 2 3 4 5 6 7 8 9 10 RECONSTRUCTED MALE POPULATION 3 83 96 76 91 83 86 90 86 93 84 4 59 51 64 40 63 55 50 53 48 61 5 45 39 34 42 25 37 39 29 37 31 6 31 32 26 17 24 17 24 26 20 25 7 17 18 19 17 12 15 12 15 17 14 8 12 12 14 12 11 7 12 9 10 11 9 5 5 12 13 4 8 4 6 5 8 10 12 1 2 6 6 2 5 4 4 4 11 1 3 1 1 6 3 2 2 2 2 12 4 1 2 1 1 4 3 1 2 2 13 0 3 1 0 0 1 3 2 1 1 14 1 0 2 1 0 0 1 3 1 1 15 1 1 0 0 1 0 0 1 2 0 16 1 1 0 0 0 1 0 0 0 0 17 0 1 1 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 19 0 0 0 0 0 0 0 0 0 0 20 0 0 0 0 0 0 0 0 0 0 21 0 0 0 0 0 0 0 0 0 0 22 0 0 0 0 0 0 0 0 0 0 23 0 0 0 0 0 0 0 0 0 0 24 0 0 0 0 0 0 0 0 0 0 25 0 0 0 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 0 0 5NSTRUCTED FEMALE POPULATION 3 81 98 90 82 82 86 85 88 83 83 4 72 68 80 76 66 71 71 75 77 73 5 74 63 58 68 63 56 56 61 69 61 6 49 65 55 50 60 51 48 48 53 59 7 34 41 55 47 41 50 45 41 42 44 8 57 26 34 45 40 38 41 39 36 37 9 28 49 22 28 40 32 31 34 35 30 10 30 25 41 19 25 35 28 28 31 31 11 26 27 23 34 13 22 28 26 27 27 12 23 22 23 19 26 12 20 24 22 18 13 23 20 18 18 18 18 22 11 19 17 14 11 18 19 16 15 15 15 14 15 17 15 11 9 15 17 14 13 14 14 8 12 16 2 10 12 12 14 15 10 11 13 7 17 4 2 13 7 11 10 10 13 12 12 18 11 4 1 9 7 10 7 6 8 8 19 7 7 4 1 7 6 12 7 6 7 20 5 4 6 3 1 6 6 8 6 5 21 8 3 3 6 2 1 5 5 7 5 22 0 7 3 3 6 2 1 4 3 6 23 6 0 7 3 2 4 1 1 2 2 24 0 6 0 4 3 2 3 1 0 2 25 0 0 4 0 4 2 2 3 1 0 26 6 0 0 4 0 3 1 2 2 1 27 0 4 0 0 3 0 3 1 2 1 57 Figure 1. Contours representing the average d i f f e r e n c e i n the number of animals by age, sex, and year f o r nine simulated populations and the corresponding reconstructed populations based on the simulated harvest data. 59 Figure 2 i s a contour surface f o r the standard e r r o r of these estimates. I t represents a p l o t of the sd/^9~, where i s the estimated standard d e v i a t i o n around the average e r r o r s of Figure 1. In Figure 2, the male e r r o r s range between 0.1 and 2.4 and the female e r r o r s between 0.6 and 3.6. Again the contours g e n e r a l l y represent ridges and v a l l e y s f o l l o w i n g ageing cohorts. The highest peaks, or hiehest contours tend to be f o r the younger age c l a s s e s and l a t e r harvest years. Together, Figures 1 and 2 suggest that the population reconstruc-t i o n i s unbiased. The l a r g e s t absolute average male e r r o r i n t h i s experiment was f o r the second harvestable age c l a s s (age 4) i n the tenth year of harvest. The average e r r o r was -5.1. The standard e r r o r f o r t h i s same age and year c l a s s was 2.0. For females, the l a r g e s t absolute average e r r o r was f o r the f i r s t harvestable age c l a s s i n the f i r s t year of harvest. The average e r r o r was 7.7. The standard e r r o r f o r t h i s age and year c l a s s was 1.6. Both of these extreme male and female e r r o r s represent a s l i g h t b i a s f o r these s p e c i f i c age, year, and sex c l a s s e s . However, these biases and standard e r r o r s are small r e l a t i v e to the number of animals i n the r e c o n s t r u c t i o n . The l a r g e s t average e r r o r , -5.1 f o r the age 4 males represents only a 10% e r r o r i f compared to the number of age 4 males, Table I I I . Most of the average e r r o r surface f o r males (Figure 1) i s at a l e v e l which corresponds to an average e r r o r of 0.4. For females, most of the surface corresponds to an average e r r o r of 2.3. Figure 2 i n d i c a t e s a c o n s i s t e n t and expected p a t t e r n i n the variance of the e r r o r f o r the reconstructed population. The variance •60 Figure 2. Contours representing the standard e r r o r s of the average d i f f e r e n c e i n the number of animals by age, sex, and year f o r nine simulated populations and the corresponding reconstructed populations based on the simulated harvest data. YEARS 62 shows a general increase from the older classes to the younger and from the e a r l i e r harvest periods to the l a t e r harvest periods. A l l of the processes have been modelled as binomial processes which have variances i n proportion to the si z e of the population being sampled. As a r e s u l t , one would expect the standard errors to increase with increasing age class s i z e s (decreasing age). One would also expect the error variance to increase as the amount of information bearing on a s p e c i f i c age and year class decreases. For the f i r s t harvestable age class of the f i n a l harvest year, the only information used i n the estimate i s the harvest for that s p e c i f i c year and age cl a s s and the estimated female population that i s r e c r u i t i n g to that year c l a s s . In contrast, older age classes or e a r l i e r harvest years u t i l i z e information from the e a r l i e r younger and l a t e r older age classes as well as the estimated recruitment values. The information i n Figures 1 and 2 can be further consolidated by considering a number of i n d i c a t o r variables that can represent the status of the population. The three that w i l l be used here are the t o t a l harvestable male population by year, the t o t a l harvestable female population by year, and the t o t a l harvestable population by year. Figure 3 i s a plot of these three t o t a l s for both the simulated popula-t i o n and the reconstructed population for the experimental data set of Tables I I to IV. The t o t a l population i s the sum of the male and female populations. The male population, experiencing a higher harvest rate, i s s l i g h t l y less than h a l f of the female population. The popula-t i o n i s stable, with no s i g n i f i c a n t increase or decrease. The recon-structions are e s s e n t i a l l y i ndistinguishable from the simulated 63 Figure 3. The simulated and reconstructed male, female, and t o t a l population l e v e l s versus time f o r a population w i t h a s t a b l e age d i s t r i b u t i o n . 64 1 0 0 C H T O T A L CO < u_ O CC 111 m F E M A L E S 5 0 0 -M A L E S simulated reconstructed 4 6 YEARS 8 1 0 65 populations. The difference between these simulated and reconstructed t o t a l s may be used to generate an average "error" and an estimate of the standard error from a series of r e p l i c a t i o n s . Figure 4 has been constructed from the average differences between the simulated and reconstructed male, female, and t o t a l population l e v e l s i n the harvest-able age classes for 10 simulation runs of experiment 1. Figure 4 i s scaled from -5 to 15 i n contrast to the scale of 0 to 1000 of Figure 3. The bias, or the average error i n these estimates i s only one percent. However, none of the biases are s i g n i f i c a n t . Figure 5, represents the standard errors associated with the average errors i n Figure 4. The estimated standard errors i n Figure 5 are of the same order of magni-tude as the average errors i n Figure 4. The conclusion to be drawn from Figures 4 and 5 i s s i m i l a r to the conclusion from Figures 1 and 2. The bias i n the reconstruction of the populations l e v e l s i s , at most, small and the p r e c i s i o n of the estimate i s lowest for the most recent population estimates. A further i n t e r p r e t a t i o n that i s suggested by Figures 4 and 5 i s that the accuracy and pr e c i s i o n of the t o t a l population estimate i s highest for the intermediate years. The estimates for the intermediate years exploit a larger propor-t i o n of the data than do the estimates for the i n i t i a l or f i n a l years. The estimates for the intermediate years can be considered to pool the harvest data from both the preceding and following years. 66 Figure 4. The average difference in the number of male, female, and total populations between ten simulated populations and the corresponding ten reconstructed populations based on the simulated harvest data. 68 Figure 5. The standard errors of the average difference in the number of male, female, and total populations between ten simulated populations and the corresponding ten reconstructed populations based on the simulated harvest data. 69 70 6.5 Experiment 2 - A Declining Population Experiment 2 continues to examine the reconstruction component of the population estimator. The purpose of t h i s experiment was to examine the e f f e c t s of a nonstationary age d i s t r i b u t i o n on the popula-t i o n reconstruction. In experiment 1, the harvest, survivorship, and n a t a l i t y rates had been constant for 70 simulated years. The harvest was a sample of a further 10 years of simulation with these same rates. In contrast, i n experiment 2, the harvest e f f o r t was increased from 0.3 to 0.35 for the ten years of recorded harvest. (The base population i s the same as i n experiment 1.) Adult males were thus harvested at a rate of 35% per year and adult females were harvested at a rate of 10.5% per year. This increase i n the harvest rate generated a 7% per year decline i n the population. Figure 6 i s a t y p i c a l plot of the simulated and reconstructed t o t a l s for the harvestable population, harvestable males, and harvestable females. It shows a slowly d e c l i n i n g male female and t o t a l population. The simulated and recon-structed populations, i n t h i s p a r t i c u l a r example, are e s s e n t i a l l y i n d i s t i n g u i s h a b l e . Figures 7 and 8, the average errors and the estimated standard errors for ten simulations of experiment 2, are i n agreement to the r e s u l t s of experiment 1. The most accurate and precise population estimates appear to be for the intermediate harvest years. Given the correct values for the population and harvest parameters, the population reconstruction i s not s i g n i f i c a n t l y d i f f e r e n t from the simulated population. 71 Figure 6. The simulated and reconstructed male, female, and t o t a l p opulation l e v e l s versus time f o r a d e c l i n i n g p o p u l a t i o n . 72 YEARS 73 Figure 7. The average difference in the number of male, female, and total populations between ten simulated populations and the corresponding ten reconstructed populations based on the simulated harvest data of a declining population. 15.0n 75 F i g u r e 8. T h e s t a n d a r d e r r o r s o f t h e a v e r a g e d i f f e r e n c e i n t h e n u m b e r o f m a l e , f e m a l e , a n d t o t a l p o p u l a t i o n s b e t w e e n t e n s i m u l a t e d p o p u l a t i o n s a n d t h e c o r r e s p o n d i n g t e n r e c o n s t r u c t e d p o p u l a t i o n s b a s e d o n t h e s i m u l a t e d h a r v e s t d a t a f o r a d e c l i n i n g p o p u l a t i o n . 77 6.6 Summary The results of the Monte Carlo experiments suggest that the most probable population reconstruction based on the harvest data is i n close agreement to the actual harvested population. The results are not a function of the state of the population. The age distribution can be stable or nonstable. The population can be stationary or declining. The most probable reconstruction retains these features. As expected, the reconstruction has the highest accuracy associated with the middle years and the lowest accuracy for the youngest age classes of the i n i t i a l and f i n a l years. The cohorts represented by these age classes have been exposed to the least amount of recorded harvest; they have been sampled the smallest number of times. In this thesis, the population reconstructions represent a component of the MM algorithm, and as such are simply a means to generate population parameter estimates. However, the population reconstructions may be interpreted independently. The reconstructed, most probable population can be considered to be the result of a specific stochastic simulation. This simulation would have used as parameters the population dynamic parameters used in the reconstruc-tion. The reconstruction represents the most likely simulated population that would have had the observed harvest. 78 CHAPTER 7 EXPLORATORY ESTIMATION The Monte Carlo simulations discussed i n Chapter 6 were used to evaluate the reconstruction component of the MM algorithm. In these evaluations a l l of the underlying population parameters were assumed to be accurately known. In t h i s chapter the MM algorithm, as an estima-t i o n procedure i s examined. The approach again uses Monte Carlo simulations to produce harvest data sets with 'known' underlying populations. The comparisons are done g r a p h i c a l l y . The reconstructed population t o t a l s based on the estimated parameter values are con-trasted to the known simulated population t o t a l s . As an indicator of the performance of the MM algorithm the reconstructed population t o t a l s based on the i n i t i a l guess of the parameters i s also presented. These before and a f t e r contrasts represent the degree to which the MM algorithm has improved the f i t of the best population reconstruction as a r e s u l t of the improved parameter estimates. 7.1 A Consistent Estimate of Harvest E f f o r t The simulated population (Figure 6) had a male harvest rate of 0.35 and a female harvest rate of 0.105. In the following i t e r a t i v e scheme these rates were assumed to be unknown and were i n i t i a l l y , a r b i t r a r i l y assigned values of 0.14 and 0.07 respectively. The remaining parameters were fixed at the values used i n the simulation. The r e s u l t i n g most probable population reconstruction, using the values used i n the simulation f o r the survivorship and recruitment rates i s 79 i l l u s t r a t e d i n Figure 9a. The reconstructed population based on these i n i t i a l values bears l i t t l e resemblance ot the simulated population. The reconstructed population i s 30% larger and growing rather than d e c l i n i n g . A f t e r 16 i t e r a t i o n s of the MM algorithm, the harvest rates had converged to values of 0.355 and 0.125 re s p e c t i v e l y . The r e s u l t i n g reconstructed population i s represented i n Figure 9b. The recon-structed male component of the population i s e s s e n t i a l l y i n d i s t i n g u i s h -able from the simulated population. The female, and thus t o t a l popula-t i o n reconstructions are both le s s than t h e i r simulated counterparts but have the same rates of decline. The MM algorithm applied to the estimate of the male and female harvest rates for a d e c l i n i n g population with a nonstable age d i s t r i b u -t i o n generates a population reconstruction that i s i n close agreement with the o r i g i n a l simulated population. 7 »2 A Consistent Estimate with Unknown Natural History Parameters In the previous i l l u s t r a t i o n of the i t e r a t i v e scheme i t was assumed that the manager not only knew the functional form corres-ponding to the parameter values ( for example, a constant recruitment rate) but also knew the exact values for some of these parameters. The i t e r a t i v e scheme was only required to estimate the male and female harvest rates and the underlying population. In t h i s i l l u s t r a t i o n i t was assumed that the manager s t i l l knew the functional form but did not know any of the underlying parameter values (except the variance associated with average female recruitment). 80 Figure 9. The effect of iterating the MM algorithm on the reconstructed population. Harvest data are generated from a simulated declining population. The recon-structions in (a) are based on the i n i t i a l guesses for the harvest rates. Reconstructions in (b) are based on the f i n a l set of harvest parameters after 16 iterations of the MM algorithm. 81 Y E A R S 82 The simulated population was the slowly d e c l i n i n g population i l l u s t r a t e d i n Figure 6. The survivorship rate, the male harvest p r o b a b i l i t y , the female harvest p r o b a b i l i t y , and the recruitment rate were 0.93, 0.35, 0.105 and 0.176 respectively. The i n i t i a l values i n the i t e r a t i v e scheme were assumed to be a survivorship rate of 0.8, a male harvest rate of 0.15, a female harvest rate of 0.07, and a recruitment rate of 0.25. As i n a l l of the other examples, the population si z e was also assumed to be unknown. The most probable reconstruction f o r these s t a r t i n g values i s summarized i n Figure 10a. The reconstructed population i s almost twice as large as the simulated population and the reconstructed male component shows no net decline. The MM algorithm converged to an estimate of 0.95 for the s u r v i -vorship rate, 0.364 for the male harvest r a t e , 0.122 for the female harvest rate, and 0.174 f o r the recruitment rate. The reconstructed population, Figure 10b, has the same t o t a l number of males over time as the simulated population that was used to generate the harvest data. The larger difference between the simulated and reconstructed female population, and thus, t o t a l population, r e f l e c t s the smaller sample size of harvested females. 83 Figure 10. The effect of iterating the MM algorithm when no underlying population parameter values (except variance on recruitment) are known. I n i t i a l estimates produce too large a population (a), whereas a close agreement between simulated and reconstructed populations is attained after iterating the MM algorithm (b). 1500-1, TOTAL TOTAL FEMALES FEMALES MALES simulated reconstructed 0-1000-1 ~i r - 1 -4 ~i r 8 MALES T > 10 TOTAL 500-^ FEMALES MALES simulated reconstructed 0- - T -2 —r -4 6 —r 8 To YEARS .85 7.3 A Consistent Estimate with Unknown Natural History Parameters and  a Variable Harvest E f f o r t In t h i s f i n a l experiment, the population was simulated using a natural survivorship rate of 0.93, a recruitment rate of 0.176, a recruitment variance parameter of 0.15, and a female to male harvest v u l n e r a b i l i t y r a t i o of 0.30. Females were assumed to be only 30% as vulnerable as males. The i n i t i a l conditions are given i n Table I, the base population. In t h i s simulation, the harvest e f f o r t or male harvest p r o b a b i l i t y varied from year to year and was 0.35, 0.4, 0.4, 0.2, 0.3, 0.2, 0.4, 0.35, 0.3 and 0.4 for the ten simulated harvest years. In the following population reconstruction and estimation, i t was assumed that the manager had no prior knowledge of any of the parameter values or of the size of the population. However, i t was assumed that the natural survivorship rate was constant for a l l years, age classes, and both sexes; the recruitment rate was a constant proportion of the number of active females; the r e l a t i v e v u l n e r a b i l i t y of females was constant for a l l harvestable ages and years; and that the harvest rate of males within a year was constant for a l l ages. The recruitment variance parameter was s t i l l fixed at 0.15. Equations 4.9 and 4.12 were used to generate the maximum-likeli-hood estimates of the recruitment and survivorship rates from the reconstructed population. Male and female harvest rates within any s p e c i f i c year were computed from the r a t i o of the t o t a l male and female harvest and the t o t a l male and female reconstructed populations for each year. These would represent maximum-likelihood estimates for a 86 known underlying population i f the male and female harvest rates were independent within each year. The relative vulnerability may be estimated as the average ratio of the above estimated female and male harvest rates over the ten harvest years. (This estimate of the relative vulnerability rate i s not a maximum likelihood estimate.) Given this pooled estimate of the relative vulnerability that is based on a l l of the harvest years, one can generate a male and female pooled estimate of the harvest effort within any year. In the following procedure, the pooled harvest effort estimate was the average harvest effort based on the male harvest and the harvest effort based on the female harvest corrected by the difference in vulnerability. The i n i t i a l parameter values used in the following iterative scheme were a survivorship rate of 0.80 and a recruitment rate of 0.22. The i n i t i a l efforts for a l l ten years were assumed to be 0.23. The relative vulnerability was assumed to be 0.30. (The relative vulnera-b i l i t y can be estimated from the total ratio of the male and female harvests for the f i r s t harvestable age classes. That estimate assumes that the recruitment into these age classes is equal for the two sexes for any individual year). The simulated population and the reconstruc-tion based on these i n i t i a l assumed parameter values is portrayed in Figure 11a. The i n i t i a l parameter values generate a reconstruction that i s substantially different from the simulated population. The simulated population i s slightly decreasing. However the reconstructed male population shows no net decline while the reconstructed female and total populations have high rates of decline. 87 Figure 11. The simulated population experiencing a variable harvest effort used to generate a set of harvest data. The iterative scheme is used to estimate the consistent survivorship rate, male and female harvest rates, recruitment rate, female vulnera-b i l i t y , and yearly harvest efforts. In (a) the reconstruction is based on the i n i t i a l values of the parameters and i n (b) the reconstruction is based on the f i n a l parameter values after 200 iterations of the MM algorithm. 1400-1 ^ — — — " (fi _l < Z < LL O cc L U CO Z* 3 Z TOTAL TOTAL FEMALES FEMALES MALES MALES simulated reconstructed 0 1000-1 —j— 2 — i 4 10 CO _l < z < LL O 0C L U CD 3 Z 500 H TOTAL MALES simulated reconstructed —r-2 —r— 6 8 T • 10 Y E A R S 89 The MM algorithm after approximately 200 iterations generated final- estimates of 0.933 for the survivorship rate and 0.189 for the recruitment rate. Final estimated harvest values were 0.33, 0.38, 0.39, 0.20, 0.28, 0.24, 0.39, 0.36, 0.34 and 0.44 for the ten years. The f i n a l estimated relative vulnerability was 0.38. The reconstructed population based on these estimates i s portrayed in Figure l i b . Again the reconstructed male component of the population i s indistinguishable from the simulated counterpart. The female and total reconstructions are slightly less than the simulated populations but have the same pattern of increase and decrease. The MM algorithm successfully reconstructed an unknown underlying population, and simulaneously developed estimates for an unknown survi-vorship rate, an unknown recruitment rate, an unknown relative male/ female harvest vulnerability, and a sequence of unknown yearly harvest rates. This estimate was done with a declining population with a non-stable age distribution. The reconstruction of the underlying male component of the population is in close agreement with the simulated values, Figure l i b . 7.4 Summary The results of the Monte Carlo simulations in this chapter suggest that the MM algorithm represents a viable approach to the interpretation of hunter k i l l data. Not only does the MM algorithm generate estimates of parameters such as the natural survivorship rate, the birth rate and yearly harvest rates but the algorithm also generates a description of the most probable underlying population. 90 This historical description represents a population census or indicator of population trends that is derived from a data source that is readily available to most wildlife management agencies. These estimates and reconstructions are based on the k i l l data alone. No additional population census, hunter census, or catch per unit of effort estimate has been employed. The value of this approach lies in the potential of u t i l i z i n g the harvest data as a monitor that w i l l indicate the impact of alternative harvest policy. 91 CHAPTER 8 SENSITIVITY ANALYSIS The population reconstruction and parameter estimation scheme presented in this thesis is intended to be a practical management tool. It has been developed by exploiting relationships incorporated in a specific model or stochastic representation of the processes involved in the population dynamics of a harvested population. In Chapter 7 i t was demonstrated, through the use of Monte Carlo simulations, that the MM algorithm can be effective at estimating population dynamic parameters and reconstructing underlying populations. However, these simulations represented ideal experiments. The Monte Carlo simulations reproduced exactly a l l of the assump-tions that were contained in the idealized population paradigm that was exploited in developing the estimators. The simulated population was a closed population. The simulated harvest was accurately aged and sexed. The simulated harvest represented the complete harvest from the population. Generally, the manager knows that none of these assump-tions or restrictions are totally correct for his specific animal population. Before he can r e a l i s t i c a l l y use the population reconstruc-tions or parameter estimates he needs to know how drastically his conclusions may change as a result of the failures of some or a l l of these basic assumptions. A complete analysis of the value of the current analysis approach would require an understanding of the managers decision making process. For the manager, the question is not how sensitive is the estimate of a 92 p a r t i c u l a r parameter or population reconstruction to v i o l a t i o n s of one or more underlying assumptions but rather how l i k e l y i s the manager to inappropriately change one or more of h i s management decisions. 8.1 Parameter Errors The MM algorithm contains two components, a population recon-s t r u c t i o n component and an estimation component. If a manager has a good understanding of the population dynamics of the harvested species, he may be w i l l i n g to only use the reconstruction component to generate a probable h i s t o r y for the hunted population. In t h i s s i t u a t i o n the manager would assume that he accurately knew the population parameters and was only uncertain about the number of animals i n the population. The s e n s i t i v i t y of the population reconstruction to p o t e n t i a l errors i n the population parameters may be examined by considering a sequence of reconstructions with a range of a l t e r n a t i v e parameter values. For example, Figure 12 represents a sequence of reconstruc-tions based on a set of a l t e r n a t i v e male harvest rates. A l l of these reconstructions used the same harvest data. The harvest data set was generated by a simulation using the base population, the base popula-t i o n parameters, a male harvest rate of 0.35 and a female harvest rate of 0.105 (the same data set used to represent a d e c l i n i n g population with an unstable age d i s t r i b u t i o n i n Section 7.2). In the following reconstructions, i t was assumed that the manager knew a l l of the population parameters accurately except the male and female harvest rates. However the r e l a t i v e harvest rates between males and females was assumed known; i n a l l of the reconstructions the 93 Figure 12. A simulated d e c l i n i n g population and a series of t o t a l population reconstructions based on the simulated harvest data for a range of a l t e r n a t i v e harvest rates. The reconstructions are based on harvest rates ranging from 0.25 to 0.50. 95 female harvest rate was set at 0.3 of the male harvest rate. The sequence of reconstructions depicted i n Figure 12 i s based on male harvest rates that range from 0.25 to 0.50. The reconstructed populations represent a sequence of s i m i l a r curves. The curves based on high harvest rates have fewer numbers of animals and higher rates of decline than the simulated population. The curves based on low harvest rates have more animals and show an increase over time. In these reconstructions, i t was assumed that the survivorship rates, reproduction rates and r e l a t i v e v u l n e r a b i l i t y rates were a l l known. The only change from reconstruction to reconstruction was the harvest p r o b a b i l i t y , which ranged from a p r o b a b i l i t y of 0.25 to 0.50 for adult males; adult female harvest rates thus ranged from 0.075 to 0.15. With an increasing harvest rate the reconstructed populations (Figure 12) show a decline i n the estimated population l e v e l and a decline i n the population growth rate. Both of these r e s u l t s are to be expected. With a high harvest rate, one would expect the population to decline over time. One would also expect only a smaller population to y i e l d the same harvest with a higher harvest rate. For a w i l d l i f e manager, the reconstructions represented i n Figure 12 are r e l a t i v e l y robust. Although the male harvest ranges from 25% of the males being removed per year to 50% of the males per year, the f i r s t f i v e years of the reconstruction a l l show a consistent and appropriate rate of decline i n the population. It i s only i n the l a t e r years that the estimates or reconstructions show q u a l i t a t i v e l y d i f f e r e n t trends. 96 8.2 Sexing Errors Generally, i t i s expected that the MM algorithm w i l l be used to estimate unknown parameter values as well as generate an h i s t o r i c a l reconstruction of the population. In these situations the manager may be concerned about possible errors i n the data. As an example some animals may be mis-sexed. This s i t u a t i o n was simulated by randomly changing the sex class of 10 percent of a l l of the harvested animals generated i n the simulation described i n Section 7.2 and Section 8.1. This harvest represented the base population subjected to a 35% male harvest rate. A reconstruction was generated using the MM algorithm to estimate the constant survivorship rate, the b i r t h rate, the male harvest rate, and the female harvest rate. The i n i t i a l estimates for these parameters were 0.95, 0.13, 0.37, and 0.18. These are close to the correct values for these parameters and represent the values that the MM algorithm converged to when there were no errors i n sexing the harvest, Section 7.2, Figure 10. With a ten percent sexing error, the f i n a l parameter estimates were 0.96, 0.15, 0.36, and 0.204. The f i n a l reconstruction i s sum-marized i n Figure 13. The reconstructed male population i s s t i l l e s s e n t i a l l y i d e n t i c a l to the simulated male population. The recon-structed female and thus t o t a l population underestimates the simulated values. Associated with t h i s smaller female population i s a higher estimated b i r t h rate. The larger number of apparently harvested females and smaller number of apparently harvested males has resulted i n appropriate s h i f t s i n the male and female harvest rate estimates. However, despite these changes i n parameter estimates, the 97 Figure 13. The simulated and reconstructed male, female, and t o t a l p o p u l a t i o n l e v e l s f o r a d e c l i n i n g population i n which the harvest data has been mis-sexed f o r 10% of the i n d i v i d u a l s . 98 1000-1 < 2 500 H tr LU CD T O T A L F E M A L E S 3 Z M A L E S simulated reconstructed 2 T — r -6 8 To YEARS 99 reconstructed t o t a l population retains the general d e c l i n i n g trend that i s exhibited by the simulated population. 8.3 Ageing Errors Ageing errors represent a second example of the type of error that might be associated with actual data. In t h i s example, the harvest data described i n the previous two sections was re-aged i n a way that simulated ageing e r r o r s . For brown bears, the age of the k i l l e d animal i s estimated by counting the annual rings on a stained, sectioned, tooth sample. Young animals have fewer rings and are thus expected to be more accurately aged. To simulate t h i s ageing error process each age class was assigned an ageing error p r o b a b i l i t y . For the f i r s t age class the p r o b a b i l i t y of i n c o r r e c t l y ageing the animal was assumed to be 0.02, a 2 percent chance. For age class ten and over there was an assumed ten percent chance of mis-ageing. For these older age classes the chances of overestimating the age was assumed to be equal to the chances of underestimating the age. For the f i r s t age class i t was assumed to be impossible to underestimate the age. For the f i r s t ten age classes the chances of mis-ageing increased l i n e a r l y from 0.02 to 0.10 and the chances of a mis-aged animal being under-aged increased from 0.0 to 0.5. As i n Section 8.2, the MM algorithm was used to re-estimate the survivorship rate, the male harvest rate, the female harvest rate and the b i r t h rate. The i n i t i a l estimates for these rates were again taken to be 0.95, 0.13, 0.37, and 0.18, the f i n a l estimated values for the e r r o r l e s s harvest data set. The f i n a l reconstruction i s summarized i n 100 Figure 14. With the above described ageing e r r o r s , the f i n a l parameter estimates were, a f t e r 26 i t e r a t i o n s 0.96, 0.13, 0.37, and 0.18. The parameter estimates and the reconstructed population shows no substan-t i a l change from the population parameter estimates and population reconstruction based on the error free harvest data set, Figure 10 and Section 7.2. The male population reconstruction i s i n close agreement with the simulated male population. The female and thus t o t a l recon-structions underestimate but r e t a i n the general pattern of the simulated female and t o t a l populations. It should be noted that the r e s u l t s of t h i s simulation can not be interpreted to mean that ageing errors do not a f f e c t the estimation procedure. Ageing errors would be expected to smooth over any d i f f e r -ences or contrasts between cohorts i n the population. Estimates that were e x p l o i t i n g these contrasts would be affected by ageing errors. In t h i s p a r t i c u l a r example t h i s was not a consideration as there were no major changes i n the r e l a t i v e sizes of successive cohorts. 101 Figure 14. The simulated and reconstructed male, female, and t o t a l population l e v e l s for a declining population i n which the harvest data has been mis-aged. Ageing error p r o b a b i l i t i e s range from 0.02 for animals i n the f i r s t age class to 0.10 for animals i n the 10th and older age classes. 102 100CH co _i < z < LU m 500-T O T A L F E M A L E S • e r M A L E S simulated reconstructed T 6 8 10 YEARS 103 8.4 Incomplete Harvest Data ' A major assumption i n the development of the stochastic model was that the harvested population i s a closed population. This assumption would be v i o l a t e d i f the harvested population represented only a portion of a population. A l t e r n a t i v e l y , not a l l of the harvest may be observed. There may be poaching losses or f a i l u r e s to report a k i l l . This s i t u a t i o n was simulated by randomly ignoring 20 percent of the harvest data from the same harvest data set exploited i n the previous two examples; that i s the base population with a 35 percent male harvest rate. The MM algorithm, s t a r t i n g with i n i t i a l estimates for the survivorship rate, male harvest rate, female harvest rate, and b i r t h rate of 0.95, 0.13, 0.37, and 0.18 converged to f i n a l parameter values of 0.95, 0.13, 0.37, and 0.19. The r e s u l t i n g reconstruction, Figure 15 i s , as expected, e s s e n t i a l l y a scaled down version of the population reconstruction based on the complete harvest data set (Figure 10). The trend information i s s t i l l represented i n the reconstruction but the absolute magnitude of the population i s underestimated. 104. Figure 15. The simulated and reconstructed male, female, and t o t a l p opulation l e v e l s f o r a d e c l i n i n g population i n which 20% of the harvest data has been ignored. 105 1000-1 YEARS 106 8.5 Summary It i s not possible to make a general statement or develop general conclusions about the robustness of the estimator or of the population reconstructions. The sensitivity depends on the assumptions that are being made, the particular assumptions that are being violated, the particular way that the assumptions are being violated, and the particular patterns in the harvest data i t s e l f . The purpose of the preceding analyses was to indicate how one could gain confidence in a set of parameter estimates or in a population reconstruction. The recommended approach to sensitivity analysis is through simulations. This could be done by examining alternative model parameters as was done with the alternative harvest efforts. Alterna-tive assumptions that are expected to affect the data could also be simulated. The approach used in this thesis of resampling the harvest data with the appropriate bias could be applied to any data set, including real data. This would allow the manager to examine the impact of ageing errors, sexing errors, and sampling errors in the context of his particular data set given his particular set of model assumptions. 107 CHAPTER 9 EXPLORATORY DATA ANALYSIS This f i n a l section represents a statement about the future di r e c t i o n s for the process developed i n t h i s t h e s i s . The value i n the analysis system i s not i n i t s a b i l i t y to reconstruct Monte Carlo simulations but i n the p o t e n t i a l of providing the manager with an exploratory t o o l ; a t o o l that allows him to reconstruct populations and remain consistent with h i s b i o l o g i c a l understanding. The framework may be used to guide research harvest p o l i c y development. As an i l l u s t r a t i o n , the analysis system has been applied to two Alaskan brown bear (Ursus arctos) data sets. The a p p l i c a t i o n i s not intended to generate a d e f i n i t i v e statement with regard to the popula-t i o n status of the bear populations. That would require a commitment by the Alaskan b i o l o g i s t s to engage i n an extensive i n t e r a c t i o n with the data sets and the development of s p e c i f i c a l l y t a i l o r e d estimation systems that could c a p i t a l i z e on t h e i r c o l l e c t i v e understanding of the b i o l o g i c a l and harvest systems. Rather, the intent of t h i s i l l u s t r a -t i o n i s to demonstrate the f e a s i b i l i t y of applying the analysis to r e a l data and to i n d i c a t e the types of explorations and d i r e c t i o n s that a manager could take i n using t h i s system. 9.1 Brown Bear Harvest Data Brown bears represent an i d e a l subject animal f o r the form of analysis developed i n t h i s t h e s i s . The species i s a long l i v e d species with i n d i v i d u a l s reaching 30 years of age (Bunnell and T a i t , 1981). 108 For most current bear populations, hunting has long been i m p l i -cated' as the major source of mortality (Cowan, 1972). As a r e s u l t , one would expect the harvest of a bear population to s i g n i f i c a n t l y a f f e c t the age structure of the surviving population. Bunnell and T a i t (1980) also pointed out that for most bear populations, males appear to be more vulnerable to hunting than females. Often female bears accom-panied by cubs are protected by law. Male bears have larger home ranges and may, as a r e s u l t , be more l i k e l y to come i n contact with a hunter. Often hunters show a preference for the large male animal. F i n a l l y , males may be more curious or less cautious. For a v a r i e t y of reasons related to both the bear's behaviour and the hunter's behaviour, male bears are more vulnerable than females. Bears can and have been aged for the past decade through cementum analysis. For some populations, compulsory sealing programs have ensured that a r e l a t i v e l y large proportion of the harvest has been aged and sexed by a game o f f i c e r . Spot checks of guides and taxidermists with an accompanying threat of loss of li c e n s e has helped to enforce these laws. The harvest pressure i n guided hunts w i l l often be d i s t r i -buted over a r e l a t i v e l y large guiding u n i t . The transport of the hunter to the area i s often done by a i r c r a f t creating the p o t e n t i a l f or a hi g h l y dispersed harvest e f f o r t . The bears themselves are highly mobile with large home ranges. These l a s t two factors together suggest that bears do meet the assumptions of the s t a t i s t i c a l model and are not progressively "mined". 109 9.2 The Alaskan Peninsula Data Set The f i r s t population considered i s from the Alaskan game manage-ment area, unit 9. This data set represents a peninsular population, that i s from the Alaskan peninsula. The brown bear harvest from t h i s e s s e n t i a l l y closed population i s almost excl u s i v e l y through guided hunts. Because the longest road i n t h i s 117,900 knr- area of land i s only 42 km, access i s by a i r c r a f t . The hunter, with an i n t e r e s t i n trophy animals, s e l e c t s f o r males. The harvest data consists of ten consecutive years of r e l i a b l e age and sex data with an annual harvest of between 100 and 200 bears. In the f i r s t analysis of t h i s data set i t was assumed that the natural survivorship f o r a l l harvestable age classes could be approxi-mated by an average survivorship rate. Recruitment was assumed to be a constant proportion of the adult female population. The harvest e f f o r t s were assumed to be p o t e n t i a l l y d i f f e r e n t f or each of the ten harvest years. The r e l a t i v e v u l n e r a b i l i t y between males and females was assumed to be constant. The only information added to the analysis beyond the harvest data was the age of f i r s t reproduction of adult females. This has been estimated i n independent studies to be age 5 (pers. comm., R. Modefari and L. Glen). As i n the reconstructions of the Monte Carlo simulations the parameter f o r the variance of the recruitment process was f i x e d at 0.15. Figure 16 i l l u s t r a t e s the r e s u l t i n g best consistent reconstruc-t i o n based on the above assumptions. The reconstructed t o t a l popula-t i o n i s slowly d e c l i n i n g from 1972 to 1981. The reconstructed female component shows a net increase over t h i s period while the male 110 Figure 16. Reconstructed population le v e l s based on Unit 9 q Alaskan brown bear harvest data and a consistent set of estimated population parameters and estimated harvest e f f o r t s . 112 component shows a net decrease. The r e s u l t i n g sex r a t i o i n the recon-structed population has, as a r e s u l t , increased from approximately 1:1 to a r a t i o of 3:2 i n favour of females from 1972 to 1981. This reconstruction i s not an estimate of the t o t a l population but only of the population that i s two years old and older. An estimate of the t o t a l population would need to be corrected by the proportion of the population composed of cubs and yearlings. The estimate of a natural survivorship of 0.93 i s i n agreement with the mortality pattern for brown bears described by Bunnell and T a i t ( i n prep.). However, t h e i r analysis of Alaskan brown bear populations has been r e s t r i c t e d to harvested populations and has not c r i t i c a l l y separated the mortality estimate into a harvest mortality and a natural m o r t a l i t y . The estimated recruitment rate of 0.34 represents the average number of male or female o f f s p r i n g that an adult female i s expected to r e c r u i t to the f i r s t harvestable age class, age 2. I t represents a combination of the reproductive rate and the survivorship rates for cubs of the year and yearlings. These survivorship rates are generally not known (Bunnell and T a i t , i n prep.). However, i t i s expected that the survivorship rate of cubs must be less than the survivorship rate of the mother. If the mother dies, the cub i s expected to die as well. If we assume that cubs also experience mortality independent of t h e i r mother's m o r t a l i t y , then the survivorship rate from age 0 to age 2 could be lower than 76%. This rate would be expected i f the cub's independent mortality rate was equal to the adult mortality rate of 7% per year. Using a mean l i t t e r s i z e of 2.2, an i n t e r - l i t t e r i n t e r v a l of 113 3.4 years and the above suggested 76% survivorship, the expected recruitment rate would be 0.24 per year for each sex. This recruitment rate can be exploited i n the reconstruction by assuming i t i s known and f i x e d and estimating the consistent values for the remaining parameters and underlying population. Figure 17 i s the r e s u l t i n g reconstruction. The r e s u l t of the decrease i n recruitment rate i s a larger estimated population and a higher rate of population decline. It should be kept i n mind that t h i s reconstructed population contains an inconsistency. The recruitment rate has been fixed at a l e v e l of 0.24 per year. The estimate of the recruitment rate based on the harvest and the reconstructed population, treated as missing data, was 0.25 per year. The change i n the recruitment rate appears to generate substantial changes i n the population reconstruction. Despite the obvious differences i n the reconstruction, the higher population s i z e and the higher rate of decline, there are some s i m i l a r i t i e s . In both reconstructions, the difference i n the number of males and females i n the population increases from 1977 to 1981. This trend may r e f l e c t a " r e a l " change i n the population age structure. "By the mid to l a t e 1970s, open seasons were shortened and t h e i r timing and occurrence were varied to influence the numbers of males k i l l e d and to a l t e r sex r a t i o s i n such a manner to optimize productivity" (Modefari, pers. comm.). Reducing the average recruitment rate further to 0.182 increased the i n i t i a l t o t a l population estimate to 1650 and increased the rate of decline i n the t o t a l population. However, s i m i l a r i t i e s do p e r s i s t i n the male reconstructions that are based on these three 114 Figure 17. Reconstructed population le v e l s based on Unit 9 Alaskan brown bear harvest data and a consistent set of estimated population parameters and estimated harvest e f f o r t s with the recruitment rate f i x e d at 0.24. 115 116 d i f f e r e n t recruitment rates (Figure 18). A l l star t with the same t o t a l male population, 700 harvestable males, a l l show an i n i t i a l high rate of decline, a l l show an intermediate rate of decline i n the mid 1970s, and a l l show a high rate of decline for the l a s t three to four harvest years. The above population reconstructions are intended as an i l l u s t r a -t i o n of the types of analysis that can be done with hunter harvest data. In the above analysis, the v u l n e r a b i l i t y estimate was a pooled estimate of the average r e l a t i v e v u l n e r a b i l i t y based on a l l sample years. A complete analysis of the Alaskan unit 9 brown bear data would have to take into consideration the changes i n r e l a t i v e v u l n e r a b i l i t y that have been generated by modifications of season length and season timing. That would require modifying the model i n th i s thesis to include possibly two harvest seasons r e f l e c t i n g a spring season and a f a l l season, each having t h e i r own sets of harvest parameters. 117 Figure 18. Reconstructed t o t a l harvestable male age classes i n the population based on Unit 9 Alaskan brown bear harvest data and a consistent set of estimated population parameters and estimated harvest e f f o r t s f o r a range of fi x e d recruitment rates, 0.18, 0.24 and 0.34. 119 9.3 The Alaskan Unit 13 Data Set . The second population data set i s from unit 13 i n Alaska. It represents a smaller sample s i z e for each of the ten harvest years. The assumptions are the same as used i n the unconstrained analysis of the unit 9 data, except that the age of f i r s t reproduction i s 4 ( M i l l e r , pers. comm.). The r e s u l t i n g reconstruction i s depicted i n Figure 19. The population shows an increase from 1972 to 1974 with a general decrease back to the 1972 l e v e l by 1981. The estimated natural survivorship rate of 0.92 i s again reasonable for brown bears. The estimated recruitment rate of 0.29 i s i n agreement with the estimated recruitment rate for a population with an average l i t t e r s i z e of 2.2, an i n t e r - l i t t e r i n t e r v a l of 3 ( M i l l e r , pers. comm.) and a cub to two year old survivorship rate of 0.80. In discussions with S. M i l l e r of the Alaska Department of Fis h and Game prior to the ana l y s i s , i t was anticipated that the harvest e f f o r t would be low and r e l a t i v e l y constant from 1970 to 73, high from 1974 to 76 as a r e s u l t of the Alaskan p i p e l i n e construction, with a drop to the pre 1974 e f f o r t l e v e l i n 1977, and a gradual increase to the 1974-76 l e v e l by 1981. These expected e f f o r t estimates are i n substantial agreement with the independently estimated e f f o r t l e v e l s of 1972 (0.07), 1974 (0.04), 1974 (0.21), 1975 (0.29), 1976 (0.21), 1977 (0.13), 1978 (0.23), 1979 (0.25), 1980 (0.29), and 1981 (0.33) generated i n the above reconstruction. Assuming that cubs and yearlings comprise about 40% of the t o t a l population ( M i l l e r , pers. comm.), the reconstructed population estimate of 600 to 700 animals i s about half the current estimate of 940 - 1,430 120 Figure 19. Reconstructed population le v e l s based on Unit 13 Alaskan brown bear harvest data and a consistent set of estimated population parameters and estimated harvest e f f o r t s . 122 ( M i l l e r , pers. comm.) for unit 13 brown bears. The smaller estimate i s not considered s u r p r i s i n g ( M i l l e r , pers. comm.) and the discrepancy w i l l be u t i l i z e d to c l a r i f y which assumptions regarding the population are most l i k e l y to be inaccurate. 9.4 F e a s i b i l i t y Analysis of the two Alaskan data sets i l l u s t r a t e s both the p r a c t i c a l i t y of performing the analyses and the value which can be anticipated from u t i l i z a t i o n of the procedure for management ap p l i c a -t i o n s . The model structure i s s u f f i c i e n t l y general that f i e l d data and parameter values derived from l i t e r a t u r e or from b i o l o g i c a l p r i n c i p l e s can be included as av a i l a b l e . Uncertainties or inconsistencies not only become apparent but are also amenable to examination. The system provides an i n t e r a c t i v e management too l which allows the user to project the consequences of his current assumptions and the ava i l a b l e data. It d i f f e r s from other simulation models i n the way i t enforces consistency between time streams of harvest data and assumptions regarding population dynamics. 123 CHAPTER 10 CONCLUSIONS "What can be learned from hunter harvest data?". The answer depends on the nature of the data base, the known structure of the population, and the structure of the harvest system. However, the approach developed in this thesis now provides the manager with the opportunity to investigate the question. It provides a framework that w i l l exploit the time series of patterns in the harvest and allow the additional sources of information to be added in a way that w i l l remain internally consistent. The particular formulation of the stochastic model used to represent the harvested population has allowed for the development of a numerical scheme that successfully reconstructs an underlying unknown population. This 'most probable' population essentially reproduces Monte Carlo simulated populations. It is not a function of the sta b i l i t y or stationarity of the age distribution in the population or in the harvest. The reconstruction of the underlying population requires estimates of the l i f e history and harvest parameters. As such, the reconstruction could be considered as a simulation of the population that uses the known parameters and at the same time ' f i t s ' the harvest data. This reconstructed population may also be used to test for internal consistency, "Do the parameters used in the reconstruction agree in value with the estimates of the parameters based on the reconstructed population?". This check of internal 124 consistency has been successfully developed into an heuristic procedure that may be used to estimate unknown parameters. In Monte Carlo simulations, the scheme simultaneously estimated natural survivorship rates, recruitment rates, relative harvest vulnerabilities, year dependent harvest rates while at the same time generating a good reconstruction of the unknown underlying population. Applied to hunter k i l l data, the system has generated believable population reconstructions. These reconstructions represent hypotheses to be tested by the wildlife manager. In particular they suggest alternative observations such as the sex ratio in the population that could be included i n the evaluation of the results. The intent of the system was to exploit the information generated by differential vulnerabilities. However, the process can be used to exploit a much wider range of available information. The approach allows alternative sources of information to be included as constraints in the population reconstructions or as additional structure in the population model. It is expected that this approach could be used to include effort estimates and even be constructively applied to unsexed harvest data such as fisheries catch data. o 125 LITERATURE CITED Abramowitz, M. and I.A. Stegun. 1964. Handbook of mathematical • functions. National Bureau of Standards Applied Mathematics Series 55. U.S. Govt. P r i n t i n g O f f i c e . Bard, Y. 1974. Nonlinear parameter estimation. Academic Press, N.Y. 341 pp. Beverton, R.J.H. and S.J. Holt. 1957. On the dynamics of exploited f i s h populations. U.K. Min. Agric. Fi s h . , F i s h . Invest. (Ser.2) 19: 533 pp. Bunnell, F.L. and D.E.N. T a i t . 1980. Bears i n models and i n r e a l i t y - implications to management, pp. 15-23. In: C.J. Martinka and K.L. McArthur (eds.), Bears - Their Biology and Management. U.S. Government P r i n t i n g O f f i c e , Washington, D.C. Cowan, I. McT. 1972. The status and conservation of bears (Ursidae) of the world - 1970. In: Bears - Their Biology and Management (S. Herrero, ed.), IUCN Publ. New Series 23:343-367. Caughley, G. 1974. Interpretation of age r a t i o s . J . W ildl. Manage. 38(3):557-562. Dempster, A.P., N.M. L a i r d and D.B. Rubin. 1977. Maximum l i k e l i h o o d from incomplete data v i a the EM algorithm. J.R. S t a t i s t . S o c , B, 39, 1-22. Doubleday, W.G. 1976. A l e a s t squares approach to analysing catch at age data. ICNAF Res. B u l l . 12:69-81. Fraser, D. , J.F. Gardner, G.B. Kolenosky and S. Strathearn. 1982. Estimation of harvest rate of black bears from age and sex data. W i l d l . Soc. B u l l . 10(1). Fournier, D. and C P . Archibald. 1982. A general theory for analyzing catch at age data. Can. J . F i s h . Aquat. S c i . 39:1195-1207. Hanson, W.R. 1963. C a l c u l a t i o n of productivity, s u r v i v a l and abundance of selected vertebrates from sex and age r a t i o s . W i l d l . Monogr. 9:1-60. Kelker, G.H. 1940. Estimating deer populations by a d i f f e r e n t i a l hunting l o s s i n the sexes. Proc. Utah Acad. S c i . , Arts and Letters 17:65-69. Paloheimo, J.E. 1980. Estimation of mortality rates i n f i s h populations. Trans. Amer. F i s h . Soc. 109:378-386. 126 Paloheimo, J.E. and D. Fraser. 1981. Estimation of harvest rate and v u l n e r a b i l i t y from age and sex data. J. W i l d l . Manage. . 45:948-958. Pope; J.G. 1972. An i n v e s t i g a t i o n of the accuracy of v i r t u a l "population analysis using cohort a n a l y s i s . Res. B u l l . Int. Comm. Northw. A t l . F i s h . 9:65-74. Pope, J.G. and J.G. Shepherd. 1982. A simple method for the consistent i n t e r p r e t a t i o n of catch-at-age data. J . Du Conseil 40(2):176-184. Ricker, W.E. 1975. Computation and i n t e r p r e t a t i o n of b i o l o g i c a l s t a t i s t i c s of f i s h populations. F i s h . Res. Bd. Can. B u l l . 191. 382 pp. Robson, D.D. and D.G. Chapman. 1961. Catch curves and mortality rates. Trans. Am. F i s h . Soc. 90:181-189. 127 APPENDIX A DEVELOPMENT OF A TRI-DIAGONAL MATRIX REPRESENTATION FOR THE NUMERICAL IMPLEMENTATION OF THE RECONSTRUCTION COMPONENT OF THE MM ALGORITHM • Let X be a column vector representing the underlying population, such that successive elements of X represent, where possible, succes-sive members of the same sex of the same cohort. Then equation 4.1 may be written as: Ki-pm)U - S m ) B T m - l ] + = ~ \ Al Equation 4.2 may be written as: (1-Pm> Sm-l B Tm m-1 + [-1 - "•V S m - l B T i ^ m - l ) 1 x m A2 = - h m + h, Lm-r (1-Pm> Sm-l B Tm Equation 4.3 may be written as: t^-Pm) mm-1] *m = " h i A3 m Equation 4.4 may be written as: m m-1 BT X m m-1 ( 1 - p x U - S v Sm . BT Pm m) m-1 «j ^ _ ^ m ( 1 _ S m - l ) ( l - p m ) ( l - S M ) S M _ 1 BT m V ! where the subscripts m-1, m, and m+1 represent successive elements i n the above ordering of the vector X and correspond to the subscript t r i p l e t s i-1 j - l k, i j k , and i+1 j+1 k used i n Chapter 4. From equations Al through A4 i t i s apparent that the system of equations may be represented as: D Q H ( X ) S = R 0 H ( X ) A5 where D Q h = a t r i - d i a g o n a l matrix function, i m p l i c i t l y defined by equations A1-A4 RQJJ ZS a vector function representing the rig h t hand sides of equations A1-A4 D and R are functions of X only i n that they contain elements BT m which are functions of X. BT i s as defined i n Chapter 4. Using E i n s t e i n notation equation A5 may be written as: g m = 0 where Sm - V1 " Rm D M I = mi th element of DQ J J(X) 129 a m th element of R Q H(X) then e _ i = D J l X i + D 1 - R , U 1 B U Sm.k ml ,k mk m,k Both D „ , and R , are proportional to BT , mi,k m,k m,k. In this thesis BT . has been taken as: m, K BT = - zr~ BT i f m represents a subscript of a recruit to m,m EV m .. , ' m the population = 0 for a l l other m where EV m = the expected variance of X m With this simplification fc i s a tri-diagonal matrix. PUBLICATIONS Bu n n e l l , F. and D. Ta1t. 1974. Mathematical s i m u l a t i o n models of decomposition processes. In A.J. Holding, O.W. Heal, S.F. MacLean J r . , and P.W. Flanagan, eds. S o i l organisms and decomposition i n tundra. Tundra Biome S t e e r i n g Committee, Stockholm, Sweden, pp 207-225. Bunnell, F., D. T a i t , P.W. Flanagan, and K. Van Cleve. 1977. M i c r o b i a l r e s p i r a t i o n and s u b s t r a t e weight l o s s . I - A general model of the i n f l u e n c e of a b i o t i c v a r i a b l e s . S o i l B i o l , and Biochem. 9:33-40. Bunnell, F., D . T a i t , and P.W. Flanagan. 1977. M i c r o b i a l r e s p i r a t i o n and substrate weight l o s s . II -A model of the i n f l u e n c e s of chemical composition. S o i l B i o l , and Biochem. 9:41-47. Ta1t, D. and F. Bunnell. 1980. Estimating r a t e of i n c r e a s e from age of death. J . W i l d l . Manage. 44(l):296-297. T a i t , D. 1980. Abondonment as a reproductive t a c t l c -the example of g r i z z l y bears. Am. Nat. 115:800-808. Bunnell, F. and D . T a i t . 1981. Bears i n models and r e a l i t y - i m p l i c a t i o n s to management. In: C. Martinka and K.L. McArthur (eds.), Bears - Their" b i o l o g y and management. U.S. Government P r i n t i n g O f f i c e , Washington, D.C. B u n n e l l , F. and D . T a i t . 1981. Population dynamics of bears and t h e i r i m p l i c a t i o n s . I n T . D . Smith and C. Fowler, eds. Dynamics 1n l a r g e mammal populations. J . Wiley and Sons, Inc. C l a r k , C.W. and D. T a i t . 1982. S e x - s e l e c t i v e h a r v e s t i n g of w i l d l i f e p o p u l a t i o n s . E c o l o g i c a l Modelling, 14:251-260. 9 

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