ESTIMATION OF VITAL CHARACTERISTICS OF MICHIGAN DEER HERDS Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY LESTER LEE EBERHARDT 1960- " IIIHHII 3 1293 00998 5627 III \I I I. WWII \g/ This is to certify that the thesis entitled Estimation of Vital Characteristics of Michigan Deer Herds presented by Lester Lee Eberhardt has been accepted towards fulfillment of the requirements for Ph. D degree mFisheries and Wildlife Georg A. Petrides Major professor Date May 16 , 1960 0-169 ‘ none & sons 1; am mum} _ I ,\'I {[1 “\\\\ L .V‘P 2‘ I ' FBOSZOO .313 02 we OVERDUE EINES: 25¢ per day per in. RETU__RN' MG LIBRARY MTERIALS: Place in book retu mto remove charge from circuhtton records ESTIMATION OF VITAL CHARACTERISTICS OF MICHIGAN DEER HERDS By Lester Lee Eberhardt A THESIS Submitted to the School for Advanced Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Fisheries and Wildlife 1960 ACKNOWLEDGMENTS I am indebted to the Game Division, and particularly to Mr. H. D. Ruhl, Chief, for the opportunity to conduct the research reported here. I also wish to express my appreciation to the members of my graduate committee, Professors G. A. Petrides (chairman), J. E. Cantlon, D. W. Hayne, and G. J. Wallace for their advice and counsel. Literally hundreds of the employees of the Michigan Department of Conservation have participated in collecting data used in this study, and I should judge that a majority of the ninety-odd biologists and technical specialists in the Game Division have contributed to this report in one way or another. Especial mention should be made of the participation of the following Game Division personnel: I. H. Bartlett, D. W. Douglass, L. D. Fay, R. A. MacMullan, Mrs. Robert Murray, L. A. Ryel, and S. C. Whitlock. I am most grateful to Dean Armstrong (draftsman) and Mrs. Rex Caster (typist) for preparation of the report. Many of the data used in this report were Obtained in the course of investigations under Federal Aid in Wildlife Restoration Project Michigan 96-R. ***#** ii ESTIMATION OF VITAL CHARACTERISTICS OF MICHIGAN DEER HERDS By Lester Lee Eberhardt AN ABSTRACT Submitted to the School for Advanced Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Fisheries and WiLdlife Year 1960 Approved Z71 W ABSTRACT This study was conducted to analyze certain methods of estimating the relative abundance and vital characteristics of the white-tailed deer (Odocoileus virginianus) in Michigan. The study covers the years 1952 to 1958, with major emphasis on northern Lower Peninsula deer herds. Deer population estimates from the pellet-group count method were used in the study. Extensive field experience with the method, as well as tests on areas of known deer populations, however, show that it can- not as yet be accepted as a wholly reliable standard by which to judge other methods of estimating deer population size. Population estimates based on sex, age, and kill data from the hunting harvest were found to be feasible on the assumption of a low rate of non~hunting mortality in adult male deer. Precise population estimates could be made only for the earlier years of the study (1952 and 1953). The assumption of an exponential kill-effort relationship was necessary to compute estimates for subsequent years. The possi- bility of differential harvest rates was considered, and an under- representation of fawns in samples of the kill was found to be the most important effect in Michigan. Evaluation of adult buck population estimates derived from data on legal kill and hunting effort demonstrated that vulnerability to hunting is not constant. Positive identification of the underlying iv causes could not be established from the available data, but the evi- dence suggested two major aspects: a sharp decline in vulnerability during the first week of the hunting season, and an inverse relation- ship between number of hunters per unit area and hunter-efficiency. Estimation of the proportion of the deer population taken per unit of effort (hunter—day) was further complicated by the necessity of using a biased method of estimation, and by a marked decline in hunting ef- fort during the season. Results of the study show that kill-effort data probably cannot be used to produce direct and trustworthy esti- mates of deer population densities until more information is available on the behavior of deer and of hunters. A method was demonstrated for combining different indices of deer population levels through linear transformations. Several possible criteria for evaluating indices wereinvestigated. It was shown that the lack of absolute measures of deer population levels precludes a completely objective choice of methods for weighting different indices in combining them into a single measure. The chief disadvantage of a combined index was considered to lie in its not providing direct es- timates of numbers of deer, while the major advantages were found to be ease in maintaining a continuity of records and low cost of basic data. Comparisons were made of three independent methods of estimating deer population levels, pellet-group counts. the sex-age-kill method, and the combined index. A high degree of correlation among the methods was demonstrated. Useful estimates of deer survival rates were shown to require a knowledge of the age and sex structure of the herd as well as of the population level in at least two successive years. Rates estimated from the age structure alone were found to be unsatisfactory under Michigan conditions except possibly as representing an average sur- vival value over a span of years. Under-representation of fawns in samples obtained during the hunting season caused considerable dif- ficulty in estimating survival rates for this class. Results of sample surveys for overawinter herd losses were appraised. Illegal kill was considered to be a major mortality factor for antlerless deer. The dynamics of Michigan deer populations were studied by com- paring rates of change calculated directly from annual measures of population level with rates synthesized from data on reproductive and survival rates. Close agreement was noted in an area where the deer population had apparently reached a stable age distribution. Repro- ductive rates were shown to vary inversely with deer population densities. A maximum possible sustained annual mortality rate for adult female deer in northern Michigan was estimated to be about .30, while the much higher reproductive rates observed in the two youngest adult female age-classes in southern Michigan apparently would sustain a mortality coefficient approaching .40. vi TABLE OF CONTENTS I. INTRODIJCTION 000.0 00000000 00000000000000.00000000000000.0900... scope 0.0000000000000000.000.000.0000000000DOOOOOOOOOOOOQQQ Areas and years ........................................... Auspices of the study .............o....................... Background ................................................ Biology of the white-tailed deer .........o................ Definition of terms ....................................... Study areas .......................... .......... ........... II. METHODS OF MEASURING POPULATION IEVELS ........................ Introduction .............................................. Pellet-Group Counts ......................................... Themdmmi.u.u.u.uon.u.u.n.u.n.n.u.n.u.u.u Sampling acoooooooooocacaoooooooooooooooooooooooooooOoooooo Combined surveys .......................................... Errors in pellet—group counting ....o...................... Effect of over-winter mortality ........................... TeStS Of the methOd ooooooooooooooooooooooooooooooooooooooo Population Estimates from Sex, Age, and Kill Data ...........- Buck Population Estimates ................................... Basis of the method ....................................... Natural mortality ......................................... Chronology ................................................ Essential data and assumptions ...................o........ The methOd oooooooooooooooooooooooooooooooooooooooooooooooo The kill-effort relationship .............................. Comments .................................................. Estimates of Total Populations from Sex, Age, and Kill Data . The methOd oooooooooaoooooooooooooooooooooooooooooooooooooo Under-representation of fawns ............................. Herd composition from roadside counts ..................... Population Estimates from Kill and Effort Data .............. Introduction .............................................. Estimating buck populations ............................... Opening day hunting success ............................... Estimates of k from different sources ..................... Changes in vulnerability .................................. Appraisal of changes in vulnerability ..................... Maximum harvest rates ..................................... Effects of errors in estimating population size ........... Vulnerability as a function of time ....................... vii Page H to to MDRDADADAJAJthJPJFJthJFJFI V33g£33gDOCDLnLJLJCD<3<3\O\0\0~d ~a srzruonakaapa TABLE OF CONTENTS - Continued Page Estimating antlerless populations from kill and effort data ............................................ 72 Comparison with other population estimates ............... 75 Population estimates from concurrent special season data . 79 Indices to Deer Population Levels .......................... 81 Introduction ............................................. 81 Indices used in this report .............................. 82 Areas used for comparison ......o......................... 84 The question of criteria ..........}...................... 84 Some possible criteria for combining indices ............. 87 "Attenuation" of coefficients ............................ 92 Transformation to a common scale ......................... 94 Appraisal of results ..................................... 96 III. COMPARISON OF THREE METHODS OF ESTIMATING POPULATION LEVELS .. 104 Introduction ............................................. 104 Independence ........................ ....... ..... . 104 Sampling errors .......................................... 105 Comparison by correlation analysis ....................... 105 Comparison by regression analysis .............. ....... ... 109 Comparison by sequence in time ........................... 112 Effect of illegal kill during the hunting season ......... 114 Discussion ...........o................................... 116 IV. SURVIVAL AND MORTALITY ....................................... 118 The problem .............................................. 118 Herd composition and sampling problems ................... 118 Age determination ...o.................................... 123 Methods of estimating survival and mortality ............. 124 Survival estimates based on index population data ........ 125 Survival estimates from the sex, age. and kill method .... 129 Survival estimates from age distributions alone .......... 129 Adult buck survival rates ................................ 133 Components of mortality °.................................. 136 Survival rates and hunting effort ........................ 136 legal harvest ............................................ 137 Mortality surveys ........................................ 137 Known vs unknown losses .................................. 144 Illegal kill in the hunting season ....................... 145 Fawn survival ... 146 V. POPUI‘ATION DYNMCS 0°00000099000000000.000000000090000.00.... 153 Introduction ............................................. 153 Rate of population change from density measurements ...... 155 Lotka's analysis of population growth .................... 160 Methods of solving Lotka's equations ..................... 161 Reproductive and survival data ........................... 162 Estimates of r from Lotka's equations ............;....... 163 Stable age distributions ................................. 166 Ages of "shot and accidentally killed" female deer ....... 166 viii TABLE OF CONTENTS - Continued Page “68 Of fGMle deer found dead 0000000000 00000 0000000000 0 171 Ages of female deer in legal harvests ................... 172 "Survivorship" curves ..................0................ 175 Reproduction and population density 0.................... 177 V10 SWY00000000000000000000000090000000000000 00000000 0000000 182 APPWDIX .0000000°0°00000000000000°000°0o00000.0000000000000000.00 185 Proposed alternate derivation of a kill-effort relationShi-p 000000000000000090900000000.0000000000000. 185 HTERATIJRE CITED 0°000000000000000000000000000000.0000.00000000000 188 ix 50 6. 7. 8. 160 17. 18. LIST OF TABLES Results of Chi-square Tests of Heterogeneityu-Fraction 1%9Year-Olds by Day Of season oooooooooooooooooOooooooooo Adult Buck Population Estimates by Game Management District Records of Age and Sex Data on Antlerless Deer Examined at Roadside Checking Stations .......o...................... Population Estimates for Study Areas Based on Sex, Age, and Kill Data 00000000000000.00000000.0000009000000000000 Estimates of District Deer Populations Based on Sex, Age, and K111 Data 0000000°00000000000000000000000000000000000 Ratios of Antlerless Deer to Bucks as Computed from Two Sources 00.000000000000000000000.00000000000000.000000000 Buck Hunting Data from the Rifle River Area ............... Summary of Deer Population Estimates Obtained by Three Methods ................................................. Analysis of Variance of Index Data ........................ Highway Kill per 100 Square Nfiles ......................... Camp Kill per 100 Square Miles ............................ July Deer Counts .......................................... Archery Kill per 100 Square Miles ......................... POpulation Data from Three Sources ........................ Difference Between Pellet-group Count and Sex-Age-Kill Population Estimates .................................... Results of Chi-square Tests of Heterogeneity--Composition Of Antlerless K111 by Day Of Season 00.090900000000000... Survival Estimates from Index Data ........................ Adult Doe Survival Estimates from District Population Data X Page 23 29~30 35 36 4o 41 53 78 90 97 98 99 100 106-107 115 121 126-128 130 LIST OF TABLES - Continued 19. 20. 21. 22. 23. 24. 25. 260 27. 28. Adult Doe Survival Estimates from Age Structure .......... Comparison of Adult Buck Survival Estimates from Two sources 00000000000000000000000000000000000O000000000000 Deer Kill in Northern Lower Peninsula Special Seasons .... Summary of Results of Deer Mortality Surveys ............. Estimates of Rate of Population Change (r) from Trend inDeer Population levels 0000000000000.000.00.00000000. Estimates of Rate of Population Change (r) from Reproductive and Mortality Data ........................ Stable Age Distributions as Computed from Reproductive and SurVival Data 00000000000000000000000000000000000.00 Ages of "Shot or Accidentally Killed" Female Deer ........ Ages of Female Deer Found Dead in Extensive Mortality Surveys 00.000000000000000000000000000000.00000000000000 Estimates of Adult Female Survival Rates for Stationary POPUlati-ons 0.000000000000000000000.000000000900000090000 Page 131 135 140 143 157 165 167 170 173 179 Figure l. 5. 6. 7. 8. 13. 14. 15. LIST OF ILLUSTRATIONS Geographic Subdivisions of Michigan Used in This Report .. Comparison of Pellet-group County Estimates with Actual Populations 00000000009009000000000000000000000.00000000 Age Composition Of Legal BUCk Kill ooooooooooooooooooooooo Assumed Relations Between Hunting Effort and Proportion of Buck Population Harvested ........................... A Comparison of Annual Ratios of Antlerless Deer to Bucks as Computed from Two Sources ........................... Comparison of Buck Population Estimates as Obtained by No HethOds 0.00000000000000000000.000000000000000000060 Kill-effort Data for Northern Lower Peninsula Showing Values for Each Day of the Hunting Season .............. Kill per Unit Effort on Opening Day (November 15) Vs. Buck Population Level (1953-1958) ...................... Comparison of Average values of Proportion (k) of the Buck Population Shot per Hunter-day as Estimated through NO MethOds .00000000000000000000000000000000... Daily Estimates of PrOportion (k) of the Buck Population $°t pr H‘Jnter-day 000000000000000000000000000000.0000. Combined Daily Estimates of Proportion (k) of the Buck Population Shot per Hunter-day ......................... Estimates of the Proportion (k) of the Buck Papulation Shot per Hunter-day on November 15 as Compared to mntim hesmre 000000000000000000000000000000000...000 Joint Effect of Day of Season and Hunter Density on k .... Proportion (k'E) of Population Harvested on Opening Day (November 15) as Compared to Hunting Pressure .......... District Buck Population Estimates from Kill—effort and Age-kill HethOdS 000.000.00.00...00000000000000.00000000 xii Page 17 2a 31 42 1+7 49 50 56 57 59 60 62—63 67 LIST OF ILLUSTRATIONS - Continued 16. 17. 18. 19. 20. 21. 22. 230 24. 25. 26. 27. 28. 29. 30. 31. 32. Cumulative Daily Buck Kill per Square Mile, Northern IOVBI‘ Penin8u189 1953-1958 ooooooooooooooooooooooooooooo Buck Population Estimates for Northern Lower Peninsula as Estimated from Age-kill Mbthod and Two Forms of Kill‘effmt MethOd 000.0 000000 00..00000000000000.0000000 Estimates of Proportion (k) of Buck Population Shot per Hunter-day for Northern Lower Peninsula 0............... Areas in Which "Subsequent" Special Seasons were Held, 1952‘1957 00009OOOOODOOOOOOOOOOOOOOQOOO00000000000000... A Comparison of Study Area Deer Population Estimates from Kill-effort and SexaAge-Kill Methods ................... A Comparison of Deer Population Estimates from Kill-effort and Pellet-group Count Methods ......................... A Comparison of Deer Population Estimates from Kill-effort (Concurrent Season Data) and Sex~Age~Kill Method ....... A Comparison of Two Measures of Variation in Deer Papula- tion Indices coocooooooooooooooooooooooooooooooooooooooo Frequency Distribution of Transformed variables .......... Combined Deer Population Indices by Game Management DiStriCt arld Year 00090000000000000000000000000000000000 A Comparison of Measures of Deer POpulation Levels Obtained by Three Methods on Six Areas ................. . A Comparison of Measures of Deer Population Levels Obtained by Three Methods on Four Major Study Areas .... Deer Population Trends on Six Areas as Shown by Three Measures of Population Level ........................... AdultzFawn Ratios in 1952 Special Deer Season Compared to mmlative Hunting Effort 0.0000000000000000000000.00 A Comparison of Estimates of Adult Doe Survival Rates as Obtained by Three HethOdS oooooooooooooooooooooooooooooo A Comparison of Estimates of Adult Buck Survival Rates as Obtainw. by NO yethds 000000.00.00000000000000.0000000 Negative Natural Logarithms of Adult Doe Mortality Estimates Compared to Hunting Effort ................... xiii Page 70 71 7n 76 77 80 91 101 102 108 110 113 122 132 134 138 LIST OF ILLUSTRATIONS - Continued 33- 3“. 35- 36. 37. 38. 39. 41. 42. H3. 45. A Comparison by Sex of Estimated Survival.Rates for Adult mar 0..00......OOOOOOOOOOOOOOOOOOOO0000......00.. The “Northeast Area" and Game Management Districts 6 am 7 .OOOOOOOOIOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO Three Méasures Of Fawn PrOdUCtion 00000000000000.000000000 A Comparison of Two Measures of Reproduction Based on Hunting Season Ratios oooooeoeoooeooooooooooooooococo... POPUlation Trends on StUdy Areas cooooooooooooooeooooOoooo Trend of Two Deer Population Indices for Southern.hower Peninsula .OOOOOOOOOOOOOGOOOOOOOOOOCOOOOOOOOOOOOOCOOOOOO Deer Population Size and Removal Rates for the Edwin S. George Reserve OOOOCOOOOOOOCOOOOOOOOOOOOOOOOOOOOOOOOOOOO Embryo Production Rates as Determined from Autopsy of ACCidentally Killed Deer 00oeoeooo00000000000000.0000... Some Stable Age Distributions as Calculated in This Report Adult Doe Age Distributions as Determined from Deer Examined during Hunting Seasons ........................ Survival Rates from Present Study Compared to Three I'Survivorship" Curves Reported by Taber and Dasmann (1957) ................................................. A Comparison of Age-specific Reproductive Rates fer Four Areas Of MiChigan 00.000.000.000.000000000000000000000QQ Calculated Mortality Rates for Stationary Populations .... Page 139 148 1H9 150 156 158 159 164 168 17“ 176 178 180 I. INTRODUCTION Scope. This report is chiefly concerned with methods of estimating the relative abundance and certain vital characteristics (in the demographic sense) of the white—tailed deer (Odocoileus virginianus) in Michigan. The basic information used here is derived principally from the following sources: (1) Records of the age and sex of deer shot by hunters and subsequently examined by biologists employed by the Michigan Department of Conw servation. (2) Estimates of numbers of deer killed in hunting seasons and of hunting effort, as obtained through use of mailed questionnaires. (3) Field surveys designed to estimate deer population levels and overwinter mortality. (h) Records of the number of embryos borne by female deer in the spring of the year (obtained mainly through autopsy of deer accidentally killed on highways). (5) A variety of sources which may serve as indices to deer abundance. such as roadside counts. Areas and years. Data used here have been obtained throughout Michigano but the greatest volume and variety of information pertain to the north- ern lower Peninsula over the years 1952 to 1958. Auspices of the study. Virtually all of the data used here have been collected by employees of the Michigan Department of Conservation in the course of research and management activities conducted by the Game Division. I have participated in the collection and tabulation of field data in the capacity of biometrician for the Game Division, but my chief responsibility has been for statistical design of the sample surveys and particularly for analysis of the data as presented here. Background. Michigan's deer herd usually numbers in the neighborhood of 700,000 animals in the fall of the year (Jenkins and Bartlett, 1959, Ryel, 1959b). In recent years from 70,000 to 100,000 deer have been legally harvested annually by about h50,000 hunters (Eberhardt and Jen: kins, 1959). Precise figures are not available, but deer hunting is certainly a multimillion dollar business in Michigan, and deer are also an important tourist attraction in many areas. A major part of the deer range is now, and has been for many years, overpopulated'with deer. In much of the area, available food supplies are inadequate during the winter, resulting in both starvation and raw duced reproductive success (Bartlett, 1938, 1950, Jenkins and Bartlett, 1959). From 1921 through 1951 Michigan deer hunters were legally restricted to one adult male deer per year, except that beginning in 1941 a limited number of antlerless deer were taken, under permit, to alleviate crop damage by deer in certain areas. Legislation in 1952 gave the Conservau tion Commission (governing body for the Department of Conservation) much broader authority, and significant numbers of antlerless deer have been taken in most of the succeeding years. The harvesting of antlerless deer is still a controversial issue, with many people sincerely-contenda ing that there are not too many deer, and that shooting does and fawns is improper and will result in destruction of the herd. Biology of the white-tailed deer. A brief description of pertinent biological aspects may be useful to readers who are not familiar with the species. My reference for the following paragraphs has been The Deer of North America, edited by Taylor (1956), which provides much more detail than can be summarized here. The white-tailed deer (Odocoileus virginianus) is a hoofed mammal having average adult weights of 100 to 200 pounds, depending on age and plane of nutrition. The species is found in a wide range of habitats, but major populations of the northeastern United States and Canada are found in areas of immature or second-growth forests interspersed with openings and coniferous swamps. A great variety of plant foods are eaten by deer, but the basic winter diet is composed of twigs of trees and shrubs. In northern areas of deep snow, deer congregate in shel— tered areas (frequently coniferous swamps) during the winter months. Such concentrations may result in rapid depletion of local food supplies, followed by many deaths from starvation. USually the youngest animals (fawns) are most affected. Sexual capability is ordinarily reached at 18 months. Under the very best of food conditions, however, some females may breed and give birth to a single fawn during their first year of life. Females might be classed as fully mature at 3 or 4 years of age, but maximum repro- ductive rates may not be achieved until about the sixth year of life (Part V of this report). In northern areas breeding is restricted largely to the fall of the year, reaching a peak in November. A gesta- tion period of a little over 200 days results in the birth of most fawns in early June. Reproductive rates vary considerably with age and nutritive conditions, but the maximum average rate seems to be about two fawns per mature doe. Adult male deer grow antlers each summer and shed them in late winter, so that sexes of adult animals may readily be distinguished during fall hunting seasons. This sex differentiation combined with the fact that males will mate with more than one female permits sex-discrim- inate hunting regulations. Definition of terms. For clarity and convenience, certain terms and areas are defined here. "Buck" and "doe" refer to male and female deer in at least their second year of life, and "fawn" refers to the age class of animals between birth and one year. The word "population," unless otherwide identified, is used to identify aggregates of the white-tailed deer in Michigan. The "regular season" or "buck season" pertains to the statutory deer hunting season wherein male deer having antlers at least three inches long may be taken during the period November 15 to 30, inclusive, in any part of Michigan. "Special" seasons are those in which any deer may be taken, regardless of sex or age, either by any licensed hunter, or only by those specifically authorized to do so under permits issued by the Michigan Department of Conservation. Since dates and regulations for such seasons have varied during the period covered by this study, descriptions are provided at the appropriate places in the text. The word ”we" is used, in cases where specific references are not available, to identify practices, procedures, and policies employed by the Game Division of the Michigan Department of Conservation. Study areas. Two sets of geographic subdivisions of Muchigan (Figure 1) are used throughout this report. The Game Management Districts (usually referred to here simply as "Districts') are groups of counties serving as administrative units of the Nfichigan Department of Conservation. The "Study Areas" are areas which I have delimited for use here, both on the basis of deer population densities, and as a consequence of sampling in- tensity in some of the surveys. Upper Peninsula Cusino Wildlife Ebcperianent Station Northern Lower Peninsula 6 7 Rifle River Area Southern Lower Peninsula Game Management Districts, Michigan Department of Conservation Mio Ranger District of ‘ Huron National Lake Forest 9 County a Major Study Areas Areas of special studies Figure 1. Geographic subdivisions of Michigan used in this report. I. II. METHODS OF MEASURING POPULATION LEVELS Introduction. Four different types of data for measuring deer population levels are considered in this report: (1) fecal pellet-group counts, (2) sex, age, and kill data. (3) kill and hunting-effort records, and (h) var— ious indices (e.g., roadside counts). The first three sources are used for direct estimates of total population, and the fourth yields a composite index. Data for large areas can be obtained from these sources at a rea- sonable cost. Many other possibilities exist (Hazzard, 1958), which have not been extensively tested in Michigan, or which are not well suited for use on a continuing basis over large areas. Two principal problems turn up repeatedly here: (1) Bias. Practi- cally all of the estimates used in this report depend on various assumptions of uncertain validity. Since there is no way of directly testing these as- sumptions, I have relied on comparisons of different estimates of the same quantity as a measure of bias, and have therefore attempted to make indi- vidual estimates as nearly independent of each other as possible. (2) Ef- ficient use of the available information. 'When estimates can be formed in different ways, the question of the "best" estimate includes not only whether or not it is unbiased, but also whether a particular form of the estimate will have a smaller sampling error than others, or whether several different estimates can be combined to yield a single value more precise than the individual estimates. In attempting to keep different estimates of the same quantity independent, I have necessarily lost in efficiency in order to appraise the possibilities of bias. The four major methods of estimating population level are described in this part of the report, and Part III deals with comparison of the in- dividual methods. PELLET-GROUP COUNTS The method. The method depends basically on measurement, by sampling, of the accumulation of fecal pellet~groups over some protracted period of time. various features of pellet-group count investigations in Michigan were reported by Eberhardt and Van Etten (1956) and Ryel (1959a). Some general aspects are reviewed here. Results of sample counts may be reported simply as the average number of pellet-groups found per unit of area, and thus serve only as an index to deer abundance. Use of the method in Michigan has, however, depended on conversion of pellet-group counts to estimates of actual numbers of deer present on the area sampled. The discussion (and use) here is there- fore in terms of estimates of deer numbers. While survey results have not always been satisfactory, we have found the method sufficiently useful to warrant using it for annual surveys of all the major Michigan deer range. Some remarks on the several assumptions basic to the method follow: (1) A knowledge of the average daily defecation rate is essential for conversion of counts to deeredays of use for a particular area. Average daily defecation rates vary somewhat both with diet and size of deer, but are remarkably constant from day to day. Both our earlier experience (Eberhardt and Van Etten, 1956) and more recent unpublished Nfichigan studies confirm.this, but Rogers 23 al. (1958) have reported higher defecation rates for mule deer (Odocoileus hemigggs hemionus). (2) length of the deposition period represented in the samples must be determined, either by advance clearing of pellet-groups from the plots, or by depending on the autumnal fall of leaves as a reference point. Nearly all Michigan work with the method has been based on a 10 fall-to—spring accumulation of pellets. Summer use of the method does not seem feasible because of the short deposition period, and the rapid deterioration of pellets deposited in late spring and early summer. Leaf~fall dates are recorded each year by biologists living in the various parts of the state. In northern Michigan, leaves of most deciduous trees and shrubs fall during a fairly short period, but those of some of the oaks may persist on the trees until late in the winter or into early spring. Under these oak stands it seems certain that some pelletmgroups are covered by leaves after the date used as the starting point for the winter pelletwgroup accumulation. 'we find, however, that even in an area of rather extensive oak cover in southern Michigan (The Edwin 5. George Reserve, of the University of Michigan), reasonable care in searching the plots will apparently turn up most of the groups present, although often only a few pele lets are visible, and the risk of missing groups is thereby increased. In open areas, or under coniferous cover, it becomes necessary to estimate the age of pelletegroups. A few criteria have been des- cribed that help in this process (Eberhardt and Van Etten, 1956), but actual field experience, including the examination of groups of known age under various conditions, provides the best basis for such determinations. (3) Additional important assumptions are that sampling is representative and adequate, and that all winter-deposited groups on sample plots are tallied. These points are covered in more detail beyond. Sampling. Sampling methods for pellet-group surveys necessarily depend on the particular situation and on the kind and quality of results needed. 11 For very small areas systematic or 'grid' samples may be preferable, for ease in locating the plots. 0n larger areas, however, which usually show considerable variation in deer populations, considerations of time and cost dictate the use of stratified sampling methods to increase sampling efficiency (Eberhardt, 1957b). A rectangular plot 12 feet by 72.6 feet (1/50 acre) has been used as the basic unit in Michigan, but it is necessary to locate several such plots fairly close‘together in order to reduce travel time and costs which are a major item in surveys covering several thousand square miles. The section (square mile) is used as a sampling unit, and eight plots are located on a half-mile line penetrating the section from a random starting point on the periphery.) There is some evidence (Ryel, 1958) that fewer plots at a location might be desirable. The Michigan surveys are based on four or five strata of estimated overawinter deer population levels. The actual classification of each section of a given area into one of the several strata is done by Game Division field men, and has proved to be fairly accurate. An initial difficulty of some importance has been that of getting the several people involved to think in roughly the same terms in defining strata. 'we find that general terms ('high,‘ "medium," and the like) are not satisfactory. for this purpose, and that actual estimates of numbers of deer per square mile are best. In nearly every case it has been initially necessary to hold two meetings, one to outline the basic plan of stratification, and the other to compare maps prepared by individual biologists. The ensuing discussions usually result in revision of maps and fair agreement in strata at the joint boundaries of the various districts. Allocation of sample plots among strata has been based on past survey results, and is described in detail in several reports (Eberhardt, 12 1957a, Ryel, 1958, 1959b). Assumption of the negative binomial as the theoretical frequency distribution most closely fitting pellet-group tallies seems to provide a satisfactory basis for sample allocation (Eberhardt , 1957b) . Some improvement of the stratification seems possible on the basis of our accumulated results, and possibly aerial photographs may aid the process. It may, however, be difficult to improve on intimate field familiarity with the areas as a basis for stratification. Permanently located plots may have several advantages.. In most past years, we have changed the areas to be sampled from year to year, ranging from rather small (about county size) areas of particular in- terest in some years, to half or all of the 30,000 square miles of major deer range in others. 'Hhile the sampling flexibility of the pellet-group count method does make it particularly suitable for such changes in scope, a continuity of records from year to year on the same area is also desir- able. For one thing, counts on permanently located plots should give better estimates of change in population level from year to year, and thus of trends in population, than do new samples taken each year. (Cochran, 1953). A further important point is that it is often difficult to get the necessary arrangements, plot locations, and so on, set up in the spring sufficiently far ahead of the time when counts can be started. When the plots are permanently located, field biologists can begin visit» ing them as soon as the snow melts. A disadvantage to permanently located plots, since the stratifica- tion is fixed for a number of years, is losing the chance to take into account over-winter conditions each year. Stratification in a year of deep snows may be considerably different from that in a year of relatively 13 mild conditions when deer can range far out from heavy cover. It is likely, though, that an intermediate system may prove practicable, where part of the plots are changed annually; various possibilities are dise cussed in the standard references for sampling (e.g., Cochran, 1953). Combined surveys. If pellet-group surveys can be conducted simultan- eously with those for other purposes, an appreciable reduction of overall effort may be possible. No veny explicit studies have been made in Miche igan of the advantages of combining other investigations with the pellet- group counts, but I believe that much time and effort may be saved pro- viding the stratification is approximately the same, as it may well be in the case of deer mortality and range surveys, or where it is possible to add an additional feature with little change in the total effort in- volved. Errors in pellet-group counting. An obvious and probably the most common source of error in the pellet-group survey is that of missing groups on the sample plots. I suspect that this type of error increases in impor~ tance in direct ratio to the amount of area any individual worker tries to cover in a given period of time. A good deal of experience has shown that it does not pay to try to hurry on such a survey. This particular source of error is often insidious, in that the worker can easily decide that things are really going pretty well and that no groups are being missed. There is also evidence that individuals differ appreciably in their ability to detect pellet-groups (Ryel, 1959a). In 1955, a few biologists attempted to survey the northern lower Peninsula in a very short period of time. As a result, the estimated population was approximately half of that believed present in the area. 14 The same degree of error also occurred that year in an area of known population (the George Reserve), but here a recheck of a number of plots (Ryel, 1959a) showed that the original counters had missed a sizable pro- portion of the groups present. After 1955, we planned to recheck 20 per cent of the sample plots. Such a system requires that all of the original plots be marked accurately, inasmuch as the counter cannot be permitted to know that only certain plots will be rechecked. Some difficulty was encountered in finding the plots and in being sure that the exact plot outline was used in the re- check. The use of aluminum (or steel) disks to mark all groups found on the first check helped on this score, as did the use of two colors of disks, whereby the "old" (pre-leaf fall) and "new“ group classification by the first man was available to the rechecker. In most cases, the re: check results increased the survey estimates by 15 to 20 per cent, but in a few instances apparent mistakes in classification of the age of pellet—groups resulted in decreasing the estimates. In any case, the addition of the recheck not only increased the survey effort by 20 per cent or more, but the added estimation procedure increased the computed sampling error appreciably, inasmuch as the final estimates must include not only the number of pellet-groups present, but also the proportion missed on the first count (Eberhardt, 1957a). Since the 1959 survey also included searches for dead deer, we used a two-man team, and had the individuals of the team check up on each other, and so did not use a subsequent subsample recheck. So far as we can tell, this procedure worked satisfactorily; at least on the George Reserve it has proved adequate (Ryel, 1959a). The difficulty in sorting out groups which have been deposited 15 before the time of leaf—fall might possibly be avoided by clearing the plots of groups just after leaf-fall. This was attempted in two Districts in the fall of 1958, but we found altogether too many groups in the spring of 1959 that were clearly deposited before leafefall, and consequently discarded the system. The only sure way of clearing plots is to remove each individual pellet, and we find that this is a great deal more work than the actual survey itself, and must be done with even more care than the final counting, inasmuch as a few missed pellets may well show up in the Spring. [1 do not believe that this doubling of the effort for the survey is justified except perhaps as a special check on overall results. Effect of overawinter mortality. If any deer die on the survey area between fall and spring, the pellets they dropped will make the estimates represent some sort of an "average over-winter population" rather than a true measure of the spring or fall population. Fortunately, the two major sources of mortality seem to be concentrated in such a manner as to make for useful estimates of fall populations. The effects may be listed as follows: (1) Legal harvest occurs not long after leaf~fall, and is measured rather accurately; so suitable corrections may be made. (2) Illegal kills in the hunting season appear to represent a sizable portion of the annual herd mortality, and are not well accounted for. These deer will be present for only about 1/6 of the period covered and will thus not contribute very many pellet-groups to the spring total. (3) Deer removed by poaching are quite likely taken in greatest numbers before the hunting season and will thus have little effect on the survey. 16 (14) losses through starvation occur mainly early in the spring, and when mortality survey data are available, corrections may be used for these losses. We frequently do not have such data, however, and will thus tend to overestimate the size of the spring population on this score, but will come reasonably close to estimating the numbers pres- ent in the fall. Quite likely many losses from other causes, includ- ing dog-kills, are greatest in the late winter and early spring and thus have an effect similar to that of starvation losses. In general, it seems that estimates of the fall populations are to be preferred, and the effect of most losses will be in the direction of under-estimation of the fall population level. While this is undesirable, if we must err we would rather err on the conservative side. When results of winter-loss surveys are available, approximate adjustments for winter losses can be made. Tests of the method. Data on accuracy of the pellet-group count method on I'fi.chigan areas of known populations (the Cusino enclosure in the Upper Peninsula, and the George Reserve in southern Michigan) are given in papers by Eberhardt and Van Etten (1956) and Ryel (1959a). A summary of the re- sults is depicted in Figure 2. Confidence limits on the estimtesindicate that several of the survey errors have been too large to be due to chance cauSes alone. Some possible reasons for this are given in the aboveu mentioned papers. I believe that mistakes in aging pellet-groups and failure to count all groups on the plots are the chief errors. Comparisons between pellet-group counts on large areas and other Population estimates for the same areas are given in Part III of this report, and provide a basis for further appraisal of the method. The results on areas of known populations indicate that the method Deer per square mile by pellet-group count estimate 5Q-—- 30 20 10 17 T l I I I56 T T l I ' l | y=x 0 Cusino Enclosure | l I 0 George Reserve ! if : l _ 1'53 | ' I 'T o + 58 I ll '55l I I '5' ' ‘ T | ' I _ 1" H53 : ._ II '5 I 56f e' '1 I i '58 ' 1 I I T55 1 h- l l l j —- {2?2 standard errors on pellet count estimate J 1 I l l x 10 20 30 1+0 50 Known deer population per square mile Figure 2. Comparison of pellet-group count estimates with actual populations. does work, and may provide satisfactory estimates of the actual deer population level. However, we cannot yet regard pellet-group count population estimates as a wholly reliable standard by which to judge the validity of another method. l9 POPULATION ESTIMATES FROM SEX, AGE, AND KILL DATA ‘When all sex and age classes are harvested, the ready availability of samples of age and sex structure may provide a means of estimating population size. The hunting regulations prevailing in MiChigan during the period covered by this report and a marked difference in the causes of mortality in adult males and antlerless deer make it logical to divide the treatment of such population data into two classes, one dealing with adult bucks only and the other with all deer. Since in many of the areas and in some of the years covered in this report there have been no seasons on antlerless deer, the data on antler- less deer are less complete than the continuous set available for adult bucks. This section will thus be split into two parts: (1) Buck population estimates. (2) Total population estimates. 20 BUCK POPULATION ESTIMATES Bgsis of the method. A number of attempts have been made (Ricker, 1958, Beverton, 1954) to estimate animal populations on the basis of a knowledge of the harvest and age structure. One of the principal problems in forming such estimates is the usually unknown, but often large, loss from causes other than legal harvest. In the case of adult bucks, it seems that such losses are not particularly large and may, in.fact, be only a very small proportion of the fall popu- lation. In our extensive mortality surveys (Part IV of this report), we found very few adult male deer. One of the commonest deer hunting stories is that of wounding a buck and tracking it to the point where some other hunter has killed it. However, adult bucks are probably taken out of season by poachers, and we have no adequate means of measuring such losses. The methods used here to estimate the population of adult sales are essentially similar to those known in fisheries work as estimates of "virtual“ populations (Ricker, 1958), in which the annual catches of a given year-class are summed until the class disappears from the catch. An important difference in my use of the method is that the natural mor- tality is unquestionably much smaller than in fish populations. Further- more, I have used a rough estimate of the natural mortality rate to obtain estimates of the actual total population, and not the minimal values given by adding up the known harvest. Natural mortality. Mortality from causes other than legal harvest is used here as a constant rate over all years and areas. This is done simply for lack of any better information beyond the mortality-survey evidence that 21 such losses are small. The rate is applied from the close of one hunting season to the beginning of the next, and is based roughly on the losses recorded in the mortality surveys (Part IV), which cover a period from November to April. Crippling losses from hunting are probably a major factor in such so-called natural mortality. The estimates of natural mortality of bucks may well be a weak point in the whole procedure. If the natural mortality rate is assumed to be constant, any error in its estimation.will result in a proportionate constant change in the calculated population level. Here, as in most of the rest of this report, I do not expect to obtain a precise direct measure of the degree of biases, but can only depend on the comparison of independ- ent estimates to determine overall validity of the population figures. These buck population estimates are used in the next section of this report as a basis for computing total deer populations. In a later section (Part III) comparing the end results with pelletsgroup count estimates provides a measure of the reliability of the results obtained here. Chronology. Since hunting is the major mortality factor for adult bucks, it is convenient to start population calculations as of the opening of the deer hunting season. The present mail survey system of determining legal harvest was initiated in 1952. Inasmuch as that year was also the first in which we obtained extensive age samples, the chain of figures used here to obtain buck population estimates begins in 1952. Essential dgta and assumptions. The necessary data include age ratios, number of deer legally killed, and hunting effort records, as well as the assumption of a specific rate of natural mortality. Aging must be accur- ate in at least the separation of l%-year-olds from older deer, since the 22 l%-year-olds are here regarded as I'recruits." A further essential assumption is that the 1%wyear-olds are neither more nor less vulnerable than older deer. The possibility of greater vulnerability of the'l%-year~old class does not seem to be a matter for major concern in Michigan (Table l), although.Naguire and Severinghaus (l95h) have reported a higher vulnerability of l%-year-olds in New York. The converse situation, however, is important here inasmuch as a sizable proportion of the Upper Peninsula l%—year-old class evidently is not as vulnerable as are older deer. Evidently these 1%eyear-olds either have antlers less than the 3 inches required by law, or else their antlers are short enough to prevent many hunters from taking a chance on shooting such deer. The existence of this situation is evident in graphs of age distributions of Upper Peninsula bucks (Districts 1 to h in Figure 3). In Figure 3, each line represents the age sample obtained in one hunting season, with the ages arranged in sequence from left to right (1% to 4%-year-old classes are shown). The dates given are those of the "year-class," 1.6., the points plotted above, say, 1955, represent the bucks born in that year, but examined as l%-, 2%—, and 3%-year-olds in the hunting seasons of 1956, 1957, and 1958. This arrangement makes it possible to trace the history of one particular year-class through the years of its major contribution to the harvest. The vertical scale is the logarithm of the proportion of each class in each year's sample. Further details of the use of the "catch-curves" are described for fisheries studies by Ricker (1958), and for deer investigations by Hayne and Eberhardt (1952). The age distributions of Lower Peninsula bucks do not suggest pro- nounced shortages of l%—year-olds, and the effects are not apparent in population estimates, while in the Upper Peninsula such estimates simply TABLE 1 23 RESULTS OF CHI-SQUARE TESTS OF HETEROGENEITYnFRACTION 1%-YEAR OLDS BY DAY OF SEASON D . a .. 0 a 0 :3 n :1? 8 s ”8 s I. 'o z .—I m ta I-I a as a. a a a c: o I o b. .a at a. 2 a at Area 0 Cu 0 4-! “O m m [:1 to a 8 I: 7’ 'E'd 3' o In 8' d) (calm O I (D 0 a: U) a ag EH a5 é s 8% :3 g o 5 ''6' IE: >. o 8 6: 0 Upper Peninsula 1955 1,909 .456 .480 6.96 5 .20 ‘ C r F'Upper Peninsula District - Tr— 3" 7’ F-Northern Lower Peninsula District y:x '5 <36 IA? A8 )6? I I I 111 5.640 *10,000 15,000 20,000 25,000 Age-kill method Figure 6. Comparison of buck population estimates (1953-1958) as obtained by two methods. 47 Kill per hunter-day, C(t) C(t) C(t) “I 48 1953 005 r- e e e e» e .. .... I I ‘ .J- O 1.0 2.0 3.0 .10 1955 .05 " . e e e e’ 9.. a" a I I - l O 1.0 2.0 3.0 .10 " A 1957 005 '- e . e 0.00, I .7 I ‘1? I 0 1.0 2.0 3.0 Cumulative kill per square mile, K(t) Figure 7. Kill-effort data for northern Lower Peninsula showing values for each day of the hunting season. f thflt .laIQ-I Il‘ tutu..- 49 '12 Upper Peninsula 0 .10 o o A3 008 . ‘ . .‘. IA ‘ A an 0 A 3.06- :3 A .04— District 9 l , o .02— ‘g At4 I <4 I I I 1 I ‘J 0 ‘1 2 3 4 5 6 7 .8 Bucks per square mile (age-kill method) ’ Northern Lewer Peninsula “ 3 .10 X X X C (i X. ' A >> 0 CU ' .0 . 1.3 o 8 “A C ‘ ‘ a x A o o A]! 15 +5 .06— . A .5 9 ‘ 3 A» District o.’ “.5 H o 6 a 0 A7 ' IA8 f . )(9 I I I 4 ‘ I I L I I I 0 1 2 3 ‘4 5 6 7 8 9 10 Bucks per square mile (age-kill method) Figure 8. Kill per unit effort on opening day (November 15) vs. buck population level (1953—1958). Regression method 50 .2045) .030 r—' 03(50) 04(52) .025 ‘- .1(40) 0020 t—- ”(95) 06(95) 08(95) .015 — 0 mm .010 p— 97(155) ..C)<25 "' I l I I I C) .005 .010 .015 .020 .025 Age-kill method IFigure 9. Comparison of average (1953-1958) values of proportion (k) of the buck population shot per hunter-day, as estimated through two methods. The first number at each point represents the Game Management District, while the number in parenthesis gives the median number (1953-1958) of hunter-days per square mile for the District. 51 Other factors possibly influencing the value of k are those associ- ated with the difference in type of cover and in the types of hunting employed in the several areas. Many Lower Peninsula hunters depend on the presence of other hunters to keep the deer moving, but in most Upper Peninsula areas, only organized “drives" can be counted on for this pur- pose, and techniques of still-hunting and waiting by runways tend to be more important. One might go to great lengths to ascribe the observed differences in k to variations in hunter ability, relative areas of dense swamp, and so on, but any really objective analysis will probably have to wait for careful field studies or controlled experiments. Such studies will be difficult at best, but will also be important as a check on the validity of hunter reports. Some data is now available on two areas where trained observers keep daily hunting records.” These are the fenced square-mile experimental area at the Cusino Wildlife Experiment Station, and the 7.5—square-mile Rifle River Area. locations of these areas are shown in Figure l; the Cusino area has been described by Van Etten (1957) and the Rifle River Area by Howe (1954). The Cusino area is maintained as an experimental area, where the herd is censused repeatedly to provide accurate popula- tion figures. Experimental hunts have been used to maintain the desired population levels since 1954, and three year's data on the results are given by Van Etten (1957). The small size of the area permits harvest of only a few deer each year, so precise measures of kill-effort relation- Ships cannot be obtained there. Estimates of k computed from Van Etten's data are: 52 Initial Length of Hunter- Popula- Type of Season days tion Harvest Estimate Year Season in Days E(t) N(o) K(t) of k 1954 Any adult deer 25 21'”I 4“ .008 54 10 7 .022 I. 19 5 5 Bucks only 7 19 5 5 Any-deer" 1 7 29 3 .016 1956 Bucks only 2 l7 5 l .013 1956 Any—deer" 4 25 25 9 .018 *Following buck hunting. *‘Adult deer only; shooting of fawns stopped after lst day. The Rifle River Area is a lower Peninsula area with extremely high hunting pressure. but exact census data are not available. Hunters must check in and out of the area. and since only one access road is for the general public, precise records on hunting pressure are obtained. Estimates of k from eye-fitted regression lines are given for the Rifle River Area in Table 7. Hunting pressure figures used here are based on the number of daily permits issued. but the average daily length of hunt is about that observed at Cusino. The average value of R (.0078) for the Rifle River Area falls between those already estimated (Figure 9) for District 7, in which the area is located. This seems reasonable ac- cording to the relationship exhibited in Figure 9. since hunting pressure at the Rifle River Area is higher than in District 7 generally. Changes in vulnerability. Results given above Show that the regression method apparently seriously overestimates the proportion (k) of the popu- lation taken per unit of effort (hunter-day). The relationship (Figure 7) of the kill per unit effort, C(t). to the cumulative kill, K(t), seems to be more nearly curvilinear than straight, with a rapid drop early in the Season and a tendency to level off as the season progresses. Since k is the slope of a line through the plotted points, it seems that k decreases as the hunting season advances. 53 TABLE 7 BUCK HUNTING DATA FROM THE RIFLE RIVER AREA Hunter-days Per Estimate Hours Per Year Square Mile of k Hunter Day 1945 167.6 .0086 5.4 1946 205.9 .0122 5.1 1947 193.8 .0080 4.8 1948 146.1 .0086 5.1 1949 159.3 .0077 4.8 1950 173.4 .0080 5.0 1951 119.0 .0098 4.4 1952 112.5 .0060 4.8 1953 282.7 .0060 5.8 1954 231.2 .0047 5.0 1955 196.9 .0066 4.9 Average 180.8 '.0078 54 Behavior of k as the season progresses may be shown more precisely by rewriting the first equation given in this section as: k 2 C(t) N(0) - K(t) and using the buck population estimates from age-kill data for N(O), and the killmeffort records for C(t). A difficulty here is that since C(t) is actually a mean value for each day. the above equation tends to underestimate k. This may be demonstrated by obtaining a mean value of c(t) from the eXponential relationship (DeLury. 1947): C(t) = kNe“kE(t) Where N now represents the population at the beginning of the day, and E(t) represents the effort cumulated from the beginning of the day to some time, t. during the day. The mean value of C(t) is: C(t) = FHF’ q E kN e”kE(t)dE(t) = %{1—e"kEj 0 so that the estimate of k (now represented by k' to distinguish it as an estimate) is: . - C(t) = 1 _. -kE 1‘ " Nara-K(t) EILS .I Since N(O) _ K(t) = N. the p0pulation at the beginning of the particular day of interest. When kE is small, k' is very close to k, as shown by 2 3 “kE - kE‘ £E§1-’+ ££§%-’“ see 3 for small the series expansion of 1 u e - e 2’ kE the power terms are negligible and k' is practically identical to R, but if kE is on the order of .1 to .3 (as it is in the first few days of the season), than k' becomes an underestimate of k. This means that values of k' obtained for the first several days of the season are apprea ciably lower than the true values. but later in the season kE becomes 55 small and k' approaches the true value. Values of k' are shown in Figure 10 for the Upper and northern lower Peninsulas. There is evidently a marked drop in vulnerability from opening day (November 15) to at least November 20th. Daily values of k' for the years 1953 to 1958 (Figure 11) Show a definite tendency for k' to level off later in the season. Unfortunately. so little hunting occurs late in the season that the data are highly variable and offer little promise for population estimation. The leveling-off of values of k' provides further evidence that the initial decline in k' is not simply a consequence of overestimation of the population from the ageukill data. If the population were overestimated. k' would decline throughout the season. and conversely. if the population were underesm timated. there would be a tendency for k to increase throughout the season (see beyond). Appraisal of changes in vulnerability. The drop in k' evidently repre~ sents a decreasing vulnerability of surviving deer as the season advances. One possibility is that the youngest age group (1%myearmolds) may be more vulnerable than older deer. so that as these animals are shot off, over- all vulnerability decreases. Such a situation would necessarily be ac- companied by a decrease in the proportion of 1%-year-olds in the harvest. but it has been shown (Table 1) that this evidently does not occur in any great degree. Another possibility is that deer in relatively open areas are more vulnerable to hunting and are taken early in the season. and that late season hunting is largely confined to heavy cover. where deer are harder ‘to shoot. Also, survivors of the first few days of hunting may tend to ermer heavy cover and stay there during the daylight hours. There are 56 ‘03 ' 1953 F 1954 .02 k! .01 oIlLJIIfiI IlLllIl IIIIILI 15 20 15 20 15 20 — -' — Upper Peninsula Northern Lower Peninsula .0 — A. 3 1956 I 1957 1958 I I I \ I \ .02 -\ .\ I \ \ /\ k' \l/ \ " ’\ L4 \ .01 _ \ o I I l I l l l_L l I I I J l I I 15 15 20 15 20 Day of season Figure 10. Daily estimates of proportion (k) of the buck population shot per hunter-day. .02 57 Northern Lower Peninsula 15 November .03 Upper Peninsula I I L I .1 I I L I I I I I,J 15 20 25 30 November Figure 11. Combined (1953-1958) daily estimates of proportion (k) of the buck population shot per hunter-day. The curves were plotted from median values of k'. 58 few concrete data on deer behavior during the hunting season, beyond the general knowledge that most deer seem to be in the uplands prior to the season. On several occasions, Conservation Department personnel have attempted to remove deer from small areas (5~10 acres) of cedar swamp which have just been fenced, and have found it virtually impossible to drive such animals out, and very difficult to shoot thenh Further information on the behavior of k' is available through com- putation of daily values for each District. and by comparing these values with hunting effort. Figure 12 shows values for November 15th (opening day). Graphs for succeeding days (not shown) are of essentially the same form. but with k' steadily decreasing up to the sixth or seventh day. Ev- idently k' not only decreases as the season advances, but is also lower in areas of high hunting pressure. A three-dimensional graph (Figure 13) is needed to Show both behavior of k' as the season advances and its relation to hunter density. More precise results than those of Figure 13 might be obtained through use of some of the methods presented by Ezekial (1941). but would not be partic~ ularly useful without more detailed knowledge of the factors responsible for fluctuations in the proportion (k) of the population shot per unit of effort. It seems entirely reasonable to assume that there will be a tendency .for hunters to interfere with each other and reduce individual efficiency ‘with increasing numbers of hunters per unit area, as suggested in Figure 13. but there is also a possibility that some of the observed differences :Ln k' may be associated with relative amounts of dense cover in the various JDistJdcts, or be due to errors in estimating population level. MaXimum harvest rates. An important further consideration here is the 59 Upper Peninsula 0 District 5 0 A 6 o .037.— 7 A .0 8 A k' a 0A): x 9 X IIa 0A 0 o .02L— 8 o A A x A 0 x X X .‘I e e ‘I .A A. t I If I J I 51 1 20 0 Hunters per square mile Figure 12. Estimates of the proportion (k) of the buck population shot per hunter-day on opening day of deer season (November 15) as compared to hunting pressure (1953-1958). Figure 13. Joint effect of day of season and hunter density on k'. 61 point at which increasing hunter densities may cease to yield an increase in deer kill. Using the equation given previously (age-kill section): 1 - p = emkB where p is the prOportion of the population harvested, and plotting the product k'E against hunting effort should indicate whether increasing effort will take an increasing fraction of the population. Since k' is a biased estimate of k (due to the use of an average value of C(t) ob— tained by dividing the kill for a given day by the effort expended in that day), the values plotted for k'E (Figure 14) are, in fact, actually the proportion (p) of the available population harvested in that day rather than the exponential index (kE) of the equation above. In any case hunting densities beyond about 15 hunters per square mile (Figure 14) apparently do not result in much increase in the pro- portion of the population harvested, the maximum proportion seeming to be about .30 on the first day of the season. Effects of errors in estimating pepglation size. The effects of errors in the estimation of population size on estimates of k may be examined rather simply if the bias in the estimator k' is disregarded for this pmrpose. The estimating equation is: g = C(t) N(O) - K(t) and, substituting N' as the estimate of N(O), and the defined value of C(t): k 0 - K k = N' - K(t) = the ratio of the estimated value of k to the true value, p = prOportion of the population shot at time t, and R = fi%é7-= Further, let B = ms» ratio of estimated to actual population. '50F— November 15 Upper Peninsula 0 I’,/” District 5 1’ .40 .30 k'E .A .20 .10 I .I L, iIi I, I I l l O loL 20 Figure 14. Hunters per square mile. Proportion (k'E) of population harvested on opening day of hunting season (November 15) as compared to hunting pressure (1953-1958). The solid line shows the relationship to be expected for a true value of k, while the broken line represents the effect of using the biased estimate, k', of the same quantity (see page 61). Figure continued on next page. 62 63 .30.. November 16 _ O .20-— k'E _ ., ° A A. 0 . . . ‘ .10— "a . . . ‘ 3‘. 9 I I, I L I I I I I O 10 20 Hunters per square mile. Upper Peninsula 0 District 5 0 6 o 7 A 9 X e- November 17 i— .20 0 t _. k E )( I . ‘ 9 A F A36 .' .0 ‘ b. e )( I I, I I I I I I I O 10 20 Hunters per square mile. Figure 14 (continued). Data for November 16 and 17. Then: B= ———P-—1‘ R-p and the derivitive of B with respect to p is: 5113,: l-R dp (R - P)2 If R is less than unity (population underestimated), B will be greater than unity and k is overestimated. Furthermore, as p increases, the bias in estimation of k will increase rapidly. In the converse situation, where R is greater than unity, B must be less than unity and k is underestimated, but the degree of bias tends to decrease with increasing p. (3% is negative). Ryel 93 gl. (1960) found important errors in aging of deer 2%-years- old and older in Michigan. The net result seems to be a tendency to under~ estimate the number of 2%-year~olds. In the Upper Peninsula Districts, where population estimates (age-kill method) must be based on ratios of 2%-year—olds to older deer, such underestimates tend to result in under- estimation of population size. The above discussion of bias suggests overestimates of k in the Upper Peninsula, with the degree of overesti» mation increasing as the season advances. It may thus be possible that the difference (Figures 12 and 13) in k' between 5 and 10 hunters per square mile may be due to underestimation of Upper Peninsula buck pepu1a~ tions. This possibility would also account for the high values of k' late in the season (Figure 13) in the Upper Peninsula, and would make for better agreement in the overall pattern of the relationship between k', hunter density, and day of the season. Vulnerabilityas a function of time. The results given so far in this section show that vulnerability to hunting, represented as k or k', evi- dently'changes drastically during the hunting season, making it impossible 65 to use the two equations given in the introduction to this section for population estimation. However, if the factors responsible for change in vulnerability can be sufficiently identified, it seems to me that a method of Obtaining unbiased population estimates from kill-effort data can be devised. 'Ne do not, at present, know just why vulnerability dew clines rapidly early in the season, so attempts at deriving a method for pepulation estimation must be based on some sort of empirical relation~ ship. Also, since hunter efficiency apparently is greater at lower levels of hunter density, evidently there are two factors operating in opposite directions-~a declining vulnerability as the season advances, but an in- creasing hunter efficiency as fewer and fewer hunters remain afield. At- tempts to consider both of these factors as separate entities in an esu timating equation lead to considerable complications, due in large part to the highly variable nature of the kill-effort data. I have therefore considered only cases where the proportion of the population killed per unit effort, k, is taken as a function of time (t), represented by day of the season, numbering the first day as l, and so on to the 16th day. There are, of course, a very large number of functions that might be used here. Because k apparently drops off very rapidly early in the season, and then seems to level off to a constant value, one might logi- cally consider a function of the form: k = a + Ea ‘where a, b, and c are constants. Such a function will "fit'I the estimates, k‘, very nicely, but direct estimation of the constants from kill-effort data requires a very complex analysis. Two simpler equations for k are: 66 (1) k= "Ea— l. (2) k = bat and the equation: C(t) = k{N(O) - K(t)} may be rewritten with the above subatitutions for k. Fitting such repre- sentations to actual data (using least-squares methods) still leads to equations which are difficult to solve, and I have confined the investigam tions here to use of arbitrarily chosen values of the constant a, so that ordinary multiple regression methods may be used. Using equation (1) and a = % (i.e., vulnerability decreases as the reciprocal of the squarewroot of day of the season), the general equation becomes: p(t) = bN(O)t'% . bt“%h(n) and leastwsquares methods may be used to estimate b and N(O) (the "normal" equations are equivalent to those of multiple regression without correc- tions for the means). Using this equation, population estimates for the six years of avail» able data in the Lower Peninsula may be compared with those from the age- kill method (Figure 15). ‘Hhile averages of these estimates are close to those of the age-kill method, there are evidently some wide deviations, with extreme cases in 1955 and 1958. There was less of a drop in k' early in the season in 1955 and 1958 (Figure 10), and behavior of the cunmlative kill, K(t), (Figure 16) is also different for these years, with a low kill early in the season being made up in the next few days, so that it seems some change in deer- or hunterébehavior may have changed the functional relation between k and t. ‘Weather records show that the open~ ing day (November 15) of 1955 was particularly cold and rainy, and there 8 _ Kill-effort L method Age-kill ‘a/ method 0 6 — :1 E 0 — a (U :3; h 4 & 3 Buck kill 0 / 2 p - I- District 5 District 6 isms 'E+L_'E_I_J3'5 '55 '5 5758 o H -H E o h ‘0 5" U) 56 Q, U) a: c: :3 cm District 8 District 9 0‘5'T§E1§L'T§313§'T§ 3-E£71§§1-%_flér13£ '5 5 8 '5 5 Figure 15. age-kill methods. 16 _ 14 _ 12 _ District 7 95M District buck population estimates from kill-effort and 67 Buck kill per square mile 68 005 "' I 0.5 h- I, I I II I I I I I I 15 16 17 18 19 20 15 16 17 18 19 20 November Nobember Figure 16. Cumulative daily buck kill per square mile. Northern Lower Peninsula, 1953-1958. 69 were strong winds and heavy snow from the afternoon of the 16th to the morning of the 17th, so that hunter efficiency may well have been low on these days. There was also rain during the opening days of the 1958 season, but temperatures were milder than in 1955, and conditions then apparently were notvless satisfactory than in 1956 or 1957. Using a = 3 in equation (2) yields results similar (Figure 17) to those obtained with equation (1). They conform a little more closely to the age-kill estimates, but are not entirely satisfactory. Values of k determined from the two equations and compared with es~ timates (k') obtained from.the age~kill method do conform in a general way (Figure 18) to the observed change in vulnerability, but it seems likely that no one arbitrary function may accurately represent the actual situation for all areas and years. 'When some better notion of the actual situation becomes available from field studies, further analysis of these relationships may be worth- while. Three items especially need further consideration: (1) I have used arbitrary values for the constant, a, in the equations, but leastusquares equations can be derived which permit the estimation of values of the constant directly from kill—effort data. Solution of such equations requires laborious computation, and use of an elec~ tronic computer may be advisable if a number of years and areas are to be studied. (2) No attempt has been made here to adjust for the bias due to C(t) being actually an average, rather than an instantaneous value. (3) If possible, the decrease in hunter density as the season advances should be taken into account in further studies. Bucks per square mile 0 7O NIP I Equation (1), a A [I Equation (2), a = 3 #Age-kill method /Buck kill v I I L ! I 1953 '51:, '55 '56 '57 '58 Figure 17. Buck population estimates for northern Lower Peninsula as estimated from age-kill method and two forms of kill-effort method. 71 1954 1955 .x ‘8 o E +3 \. a \\ IIIIIIJ IIIIIIII LIIIIIJ 15 ' 20 15 20 15 20 November Day of Season Observed Value --—Equation (1) ——Equation (2) .02 .02 ,ozw- 1956 1958 .x #4 c Q) 001'— 001‘ 00:” 3% .p :2 I I I I L_I J .1 I II I I I I I I I I I AL I 15 20 15' 20 15 . 20 November Day of Season Figure 18. Estimates of proportion (k) of buck population shot per hunter—day for northern Lower Peninsula. The estimates are based on two equations representing decreasing vulnerability to hunting (page 66) and a direct measure of k (Figure 10). 72 Estimatingfiantlerless populations frgggkill and effort data, ‘While the estimation of buck populations from.kill and effort data is subject to shortcomings, there is evidence that the general relationships between kill per unit effort and pOpulation size are real, and that they behave rather consistently within a given area. It seems logical, therefore, to attempt to apply similar procedures to data from antlerless deer sea- sons. Two different sets of data are to be dealt with in this case, since two different types of seasons have been held in Michigan for hunting ant- lerless deer. In recent years special seasons have been concurrent with the regular buck seasons (November 15-30). Results of kill-effort studies on data from these seasons indicate that few Michigan hunters attempt to shoot the first deer seen; many evidently wait for a buck and hunt antler- less deer later in the season. From 1953 to 1956 one or two days of special hunting for antlerless deer were held after the regular buck season (in 1952, the last three days of November were open for shooting any deer). Since the special seasons subsequent to buck seasons have been too short for satisfactory use of regression analysis, I have Obtained popu» lation estimates from the exponential equation, given by DeLury (1951) as: K(t) = N(o) e'kE(t) where N(t) represents the population surviving at time t. The other sym- bols are as previously defined. If mortality from causes other than legal N(O) - K(t), and the equation may be harvest is disregarded, then N(t) rewritten as: K(t) n(o) _e-kE(t) 1 If values of k are available, the equation may be used to estimate the initial population, n(o). 73 To facilitate comparisons with pellet-group counts, population esti- mates from the "subsequent" Special season data have been made only for the Study Areas (Figure 1). Values of k used here are those obtained from the buck population estimates (age-kill method) made on the same areas. Since it was shown earlier in this section that k changes (decreases) duru ing the season, it is worthwhile here to consider what value was actually obtained for k. If the value of k for a particular day of the season is represented as k(t) (assuming that k(t) is constant through the day, or that a mean value for the day is used), and if e(t) represents the effort expended on that day, then the surviving population after the season is: N(t) = n(o)e“k(1)e(l) e~k(2)e(2) ... e~k(16)e(l6) = Mo) 6 «Zk(i)e(i) whereidenotes summation from i = l to i = 16. The estimate of k was actually obtained in this report from the equation: filégg. a: e“kE(t) where E(t) =ze(i) so that k was estimated as: ”loge M k = MO) Z 6(1) but under the assumption of changing k, the estimate becomes: k =kaegi) [e(i) ‘Which is a weighted average of the daily values of k, the weights being the daily values of hunting effort. The ”subsequent” special season areas have varied in size, and regu- lations have changed over the period studied (Figure 19). Hunting regula- tions in 1952 and 1953 did not limit the number of hunters. In later years 7h ;// 7 ea %%//%2 Figure 19. Areas in which "subsequent" special seasons were held, 1952-1957. Dates given are those on which the seasons were held. Broken lines show boundaries of Study Areas (Figure l). 75 hunting was by permit only, and limited numbers of permits were issued for each specific area. The 1952 results are of uncertain value since our mail questionnaire was not designed to give hunting effort data in the same manner as in the years after 1952. Population estimates obtained here are presumably those pertaining to the deer population alive at the end of the regular season. The legal buck kill has been added to give a total pro-season population, which, however, tends to be an underestimate by whatever losses of antlerless deer occur through illegal shooting in the regular season, and by any crippling losses. Comparison with other population estimates. Results are compared graph- ically with population estimates from the sexwage-kill method and with the pellet-group count estimates in Figures 20 and 21, and summarized in Table 8. Close correlations among the several sources are evident, par- ticularly with the sex-ageukill method. Some of the discrepancies are no doubt due to the fact that the special season areas do not exactly fit the Study Areas geographically (Figure 19). Some parts of the Study Areas are not covered by Specialeseason units, and some of the special season units overlap two or more study areas. Most significant, however, is the fact that the kill~effort estimates generally exceed those from the other two sources. The chief exception seems to be Area 8, and this is explicable on the basis that the special season areas used for these estimates (kill-effort) excluded some of the higher population areas. The apparent overestimation of population level by this method sugu gests that the values of k applying to these subsequent special seasons should be higher. This seems entirely reasonable if antlerless deer be- havertowards hunting in the same manner as do bucks, so that k is highest Kill-effort method. 50 5 K.) O N O 10 76 y = 1.2X Study Areas 6 e 7 A. 8| I 10 20 3o Sexuageekill method 5- Figure 20. A comparison of Study Area deer population estimates from kill-effort and sex-age-kill methods (page 75). Figures are deer per square mile. Ratio line "eye- fitted". Kill-effort method. 77 y y = 1.3x Study Areas ‘.53 6 O 7. 8 I [4.0 I— '53. 30 20 10 I l o 10 20 3 cre- 8 \n c: Pellet-group count method. Figure 21. A comparison of deer population estimates from kill- effort and pellet-group count methods. Figures are deer per square mile. Ratio line "eye-fitted." TABLE 8 SUMMARY OF DEER POPULATION ESTIMATES OBTAINED BY THREE METHODS (Estimates in deer per square mile) 78 Sex, Age, Kill and Pellet- Study Area Year and Kill Effort count 6 1952 23.8 31.3 - 1953 33.6 40.1 29.7 1954 22.5 35.2 1955 1956 23.4 26.7 22.7 1957 23.8 27.8 1958 27.1 (19.7)* 36.5 7 1952 34.7 40.2 1953 38.2 47.0 29.6 1954 26.8 33.2 1955 1956 28.6 27.7 30.2 1957 25.2 29.6 1958 28.4 (23.6)“ 43.2 8 1952 9.9 11.8 1953 11.0 12.0 11.2 1954 11.8 10.4 1955 1956 10.5 7.6 15.6 1957 11.3 (6.7” ' 1958 11.7 (8.2)‘ 12.9 *From concurrent season data; all others from subsequent season data. 79 early in the season and drops off thereafter. Since the subsequent sea- sons are of only one to three days duration, it seems that the average value of k applicable here will be higher than that resulting from data collected over the entire buck season. The values of k used here for estimating populations are a little higher than those used for buck populations, inasmuch as these latter values were based on square miles of deer range, in an attempt to compen- sate for areas of farmland with few deer and low hunting pressure in some of the Districts. Since figures on areas of deer range are not available for the individual special season areas, the values of k were recomputed for total land areas involved. Such changes are generally minor, relative to the other differences noted. 'Hithout more data on behavior of k, and on illegal kills of antler— less deer in the buck seasons, it does not seem worthwhile at present to attempt further refinement of the population estimates from subsequent special season data. The close correlation with estimates from other sources provides good support for the notion that the estimates are reli- able measures of deer population levels, but the recent change to concur- rent special seasons means that such methods will not be currently useful unless a similar relationship exists in the concurrent seasons. Population estimates from concurrent special season data. Kill-effort population estimates from concurrent special season data are markedly lower than those from the sex-age-kill method (Figure 22), and thus indi- cate a lower vulnerability to hunting. This is a reversal of the apparent situation in the subsequent special seasons. Evidently hunters in the con- current seasons have not been taking antlerless deer as readily as bucks in the years covered here. Kill-effort method. \A.) O N O 10 80 r. Study Area 6 . y=008x 7 A 8 I 958‘ 358 I I I 30 he 50 x 20 Sex~agenkill method. Figure 22. A comparison of deer population estimates from kill- effort method (concurrent special season data) and sexs age-kill method. figures are deer per square mile. Ratio line "eye-fitted." 81 INDICFS T0 DEER POPULATION IEVEIS Our experience has been that no single method is available Matias- The to estimate extensive deer population levels reliably and directly. pellet-group count may well approach direct and reliable estimates (with- in sampling error) when carefully used, but it is also true that this method is expensive, and it is also subject to both known and unknown sources of bias. There are, then, two reasons for attempting to use several indices or direct estimates of deer populations: (a) cost, and, (b) uncertainty about biases. In most usages, an index is expected to measure relative population levels between areas as well as changes in time (trends). If an index is to do these things satisfactorily, it must be directly prOportional to the actual population level and therefore can theoretically be trans- formed into a direct estimate of population density. Since most indices of deer population levels depend on kill figures or sight records, such things as changes in hunter numbers, differences in vegetation, and weather conditions, along with a number of human factors, may cause the index values to vary in their relation to the actual population densities. Nonetheless, it remains true that the basic data are often collected for other purposes, frequently at a relatively low cost. Also, records that may provide index values are usually available for large areas and over long spans of time, while direct estimates of populations may be too ex- pensive to permit annual use on a number of areas. In the present case the problem is to make maximal use of several 3 011I‘ces of data which can be reasonably expected to vary with deer popula- ti °h densities. Probably optimal use of the indices (i.e., getting the t:Ll'rmm amount of information) can be achieved only through a very detailed 82 study, including much field work and dealing with several factors which probably cannot be reliably evaluated without extensive data on population levels. Indices used in this report. The indices considered in this section are listed below along with descriptions of their sources. (1) July deer counts. These counts are essentially roadside counts, made in the course of other duties, and are presumably recorded only dur- ing work in deer areas. The counts have been maintained in all Mich- igan deer range since the early 1930's. Records are kept by nearly all Conservation Department personnel whose duties carry them reg- ularly through the deer country. The counts are actually made for the four months of July through October, and there are evidently real differences between months, with the July records seeming to show the closest correlation with populations as estimated from pellet-group counts. An additional reason for using here only the counts made in July is that they are available just before hunting regulations are set (in August). The chief advantage to the counts other than their timeliness and the fact that they are readily understood by the general public, is the large volume, averaging around 5,000 deer actually tallied in July alone. Possible sources of bias are easily conceived, in- cluding differences in vegetation, roads, observers, weather, and so on. Some field workers keep the records conscientiously, while others undoubtedly keep none at all, but fill out a form at the end of each month and submit it as required. In earlier years these counts served as virtually the only avail— able measure of deer population levels, but in more recent years we 83 have attempted to restrict their use to measures of long-term trends on large areas. (2) (3) Archery kill. Some 30,000 to 40,000 archers annually take about 2,000 deer in Michigan. Their hunting success (about 6 per cent) is low but rather consistent, and archers seem, by and large, to be persistent hunters. Nearly all counties are open to the taking of antlerless deer, as well as bucks, with bow and arrow, so the kill may vary in closer relation to the population than in seasons where only adult males may be taken. The usual problems of weather con- ditions, different hunting conditions, etc., apply here, too, but the long season (October 1 to November 5) may tend to cancel out such effects. For purposes of this report only the total kill per 100 square miles has been used. It seems desirable to make some adjustment for hunter numbers, but my attempts to do so have not significantly increased the correlation with pellet-group counts. Highway kills. In recent years, rather complete records have been kept of deer accidentally killed on Michigan highways. The care 'with which such records are kept varies from area to area, and not all deer killed by cars are reported to the Conservation Department. Also, there are obviously marked differences in the likelihood of deer being struck by automdbiles in the various areas in the major deer range of Michigan, and highway traffic volume has been increas- ing steadily over recent years. In spite of these difficulties, a correlation between deer accidentally killed and deer populations evidently does exist. Again, attempts to include supplementary data, such as traffic volume, have not increased the correlation between deer killed per unit area and pellet-group counts, but I believe a more detailed study may yield a means of improvement. Here, I will 84 use deer killed per 100 square miles as index values. (4) Camp kill. Under Michigan laws, any four hunters occupying a "camp“ (with definition of "camp" uncertain) may obtain a Special license to take one deer for camp use, over and above the one-deer-per hunter regulations. Current regulations restrict hunters to taking only a buck with threewinch antlers or better on camp permits. These lie censes have not been very popular, averaging under 1,000 sold annue ally for the years 1952e56, and drOpping sharply in 1957 with an increase in the fee. Unless the regulations are changed, the camp kill cannot be eXpected to yield much information on deer populae tions; it is included here for its historical value and in the ex- pectation that the regulation may well be changed in the future, quite possibly to make any kind of deer legal for camp licenses. Again, adjustments for hunter numbers seem worthwhile, but prelim— inary investigations have not shown enough improvement to justify corrections for present purposes. Figures used here are camp kill per 100 square miles. Areas used for comparison. A unit of at least county size is indicated for comparisons, inasmuch as some of the records are kept only on a county basis. Sampling variations seem to indicate that a unit of size larger than a county is necessary, and the District (Figure 1) seems useful as such a unit. hhile ecological and herd management criteria may suggest units which do not follow District boundaries exactly, administrative use of such units and the convenient geographic arrangement make them prefer~ able here. The question of criteria. The major problem, here, and throughout this 85 report, is one of finding criteria for judging potential measures of population level. In essence, the only real criterion is the actual population density, either as an average or as an instantaneous value. 'We do not, however, have any immediate prospect of obtaining exact measures of deer populations, and one of the chief reasons for spending much time on these several indices is to use them as independent measures of population level. One obvious criterion for judging the value of index measures is to compare them with the pelletegroup counts, and such comparisons are shown below. ‘We cannot, however, be sure that the pelletmgroup counts are not biased, and this uncertainty argues against using them as a base for come bining the several indices into one measure. That is, we might use re» gression or correlation methods to determine some sort of weights for each index and thus combine the several sources. Such combinations lack the desirable property of independence; biases in the pellet-group counts may be introduced into the combined index and effectively prevent any fair comparison of the end—products. A further difficulty exists in the fact that the various indices are based on different measures--the kills of deer by archers, automobiles, and "gun" hunters cannot be expected to be in the same ratio to the true population level, and roadside counts are in altogether different units (deer seen per 100 hours). If the indices were directly proportional to population level. the problem would be resolved into a search for correction factors which would adjust each index to the density of deer population per unit area (with the further necessity for weighting according to reliability). I suSpect that the relationship between index and population is not that of a simple proportion, and that the true relation is likely to be curvim 86 linear. However, I assume here that portions of the curves can be approx- imated reasonably well by straight lines. If the true relationship is curvilinear, the linear regression of index on population will have a 'y-intercept" which does not pass through the origin. The assumption of a straight-line relationship between index and population permits linear transformations of the indices without changing anything except the scale of measurement. An illustration of the idea basic to transformations is the conversion of temperatures from one scale (Centigrade, Fahrenheit, Kelvin, etc.) to another. Degrees Centigrade may be converted to Fahrenw heit by multiplying by the factor 1.8 and adding 32; this implies no change in the length of the mercury column, but is simply a change of scale. Under the assumption of linearity, the several indices are all preu sumed to be measures of the same quantity (deer populations) but in dife ferent units or on a different scale. A further feature is implied by the idea that the several indices are not equally reliable. This may be taken to mean that the indices are sampling results and that "sample sizes" vary between indices. In actual fact, this is evidently not strictly true, as various sorts of bias have been described above, but in the absence of suitable supplementary data, about all that can be done is to lump all of these sources of variation in one "error" category and attempt to devise suitable weights which will favor the indices with smallest fluctuations from true deer population levels. It seems desirable to derive the weights independently of the trans— formation (change of scale) inasmuch as a poor choice of weights will not in general bias the results, but will simply give less precise combined index values. The effects of failure to transform values to exactly the same scale are much less certain, and since it seems evident from the 87 beginning that the relationship of the indices to true populations is not exactly linear, one cannot very well expect'to get a transformation to precisely the same scale. The best I hope to do here is to make arstartw a good deal more study and some rather more complicated mathematics will be necessary for full evaluation of the possibilities. If weights are chosen independently of the transformations, they need to be essentially based on "dimensionless" measures. These might include the correlation coefficient, coefficient of variation (variation relative to the mean) and sample sizes. These three items are at least roughly independent of the units of measurement (and hence "dimension- less"). An alternate possibility would be available if actual population levels could be measured, since it would then be possible to consider weights in terms of regression or least-squares relationships of the indices to true population levels. One such measure is given here with the pellet-group counts used as "true" population levels. Some pgssible criteria for combiningiindices. Several possible bases for comparisons of the indices are described in the following paragraphs. (1) Sample size. I do not believe the indices used here are likely to have a common variance, so their reliability probably cannot be judged accurately from the size of samples obtained. However, the following items should yield rough measures of reliability: Average Square Index Unit Used Number Root ‘Height July count Number of deer seen 5,000 .70.0 .409 Archery kill Number of hunters 2,000 44.7 .258 in samples Camp kill Number of parties 200 14.1 .081 in samples - Highway kill Number of deer tallied 1,900 43.6 .252 1.000 88 The weights shown above are proportional to the square root of the average number tallied, in accordance with the principle that the standard error of an estimate decreases as the square root of the sample size increases. Probably more reasons can be proposed for not using the above "sample sizes" than can be arrayed in defense of the choice. The simplest defense is that the only wholly valid criterion would be comparison with actual deer populations; without this, the only recourse is to use such measures as can be shown to have a bearing on the fluctuations of the index. (2) Coefficient of variation. Complete sets of data for the four indices are available for all nine Districts for six years (l953~l958). Deer populations unquestionably vary from District to District and, in at least some Districts, vary among the years covered. I have therefore used the analysis of variance technique to attempt to remove varia— tion due to these differences (between Districts and years) and con- sider the deviations or "error" component as a measure of the sampling variability not accounted for by these two sources. Actually, in terms of the analysis of variance, there will no doubt be a signifie cant "interaction." However, we do not have independent samples within a given District in a particular year and thus cannot obtain a measure of interaction. Its presence seems to be a simple conse— quence of the fact that population levels vary in different manners in different Districts; in at least one District (District 7) there has been a persistent downward trend in the earlier years covered here, while several of the other Districts show little evidence of change of population level. Again, the best I can do now is to consider such variation as is known to exist and is accessible. Also, the error mean-square here does not necessarily measure sampling (3) (4) 89 variation exactly; one could probably find a number of measures with smaller error terms over the area and years considered, but with little or no relation to deer populations. Results of the analysis are shown in Table 9 along with coefe ficients of variation, computed as the square root of the error mean-square divided by the mean value of the index. Eggn-square deviations from regression. Simple linear regressions of pelletmgroup counts on the indices provide a means of measuring variability of the indices in terms of agreement with the pelletu group count results on certain areas. Results are in terms of the same scale of measurement (deer per square mile) since the devise tions~mean~square from regression used here is the average of veru tical deviations squared. Results of that analysis are shown along with those of the coefficient of variation in Figure 23. 'Both measures are plotted against the square root of 'sample size," since both should be expected to decrease in proportion to true sample size. I have already mentioned several reasons for not expecting particularly good agreement, and the results are perhaps a little better than one might expect. The principal difficulty seems to be the raw versal of the position of the Highway and Archery kill indices between the two criteria. Since an important purpose of the present analysis is to construct an independent measure of population size, I believe the variance about regression should not be used for weighting purposes. Correlation coefficients. Recent trends in statistical work (Tukey, 90 was. ewe. new. ewe. soapsaeee mo nunOHOHmmooo snot.“ 0N3. m .30. o , has. wN acme: SUN. :06 Nanotn mm. 0: 0mm. N NNJZH Nomi} mm.mm~..a . o: mnoapmaeoo No.N mo.oa sNNH 86w omé 8.0: H900 8.30m m name» ome.m mm.Nm mm.mm oo.HeN mo.bNm ma.woe.m on.emm.a me.owo.oa w hsoeteuan not? mm. mam mm. mom. N 3.39”. NH mm Haven. 0.8st no.8 sow phenom monsoon onmsqm no.3 new season mohmsqm Eopoohm coupon nmoz mo 55m now: He saw can: he saw emu: mo.Esm we newsman flow g 3 poses a. mmmaummma 4.53 NWQZH ho mofigkw .mo mmmfidlé m Ham. 91 (DCamp kill 5 60 r— 6 Highway kill .a U) U) (D ‘4 +- t" a . h kill S 40 ___ CDArc ery (3 July counts 8 t... c o -H 4.) m -H 5 o a m 3 “I’ c g o I I I I 20 ,In" 40 60 80 @Camp kill 140 ‘r— .3 CDArchery kill 13 .H A m > a. L—- 3 20 ® Highway kill 6 July counts 5 .a 0 .H s. (H (D 8 10 — o I L I J 20 40 60 80 " Sample size" , Ji- Figure 23. A comparison of two measures of variation in deer population indices. 92 1954) have been towards use of regression, rather than.correla- tion analyses. Regression methods seem preferable in the present case inasmuch as a major interest is in predicting population level from the indices. However, it is difficult to use regression meth- ods here by virtue of the several different scales involved, and the "dimensionless" feature of the correlation coefficient thus becomes valuable. The correlation between indices and the true deer popula- tion is the ideal criterion, but in its absence some notion of the relative value of the various indices may be gained by considering the six possible correlations between the four indices. Presumably the better indices will show the higher correlations. The correla- tion coefficients are shown in the diagram below. July count .73/ l .7. .148 Archery kill/// ' .69-——-::::>Highway kill / .27 .27 / -The several coefficients seem to show much the same ranking as \\\Camp kill before, with the Archery and Highway indices showing little differ- ence; possibly the Archery index has a slight advantage over the Highway kill. 'Attenuation' of coefficients. An important prOblem concerning the regres~ sion and correlation coefficients is that of I'attenuation" of the coeffi- cients (Yates, 1953). Meaning of the term here depends on the situation which is presumed to exist. One can assume that the indices are completely determined by level of deer pepulations, and that any fluctuations are due solely to sample sizes, or that various other factors affect the index so 93 that no matter how large the sample, the correlation will never approach unity. In the latter case, the model usually adopted is that of the bi- variate normal distribution, in which the correlation coefficient appears as a parameter (fixed constant) in the distribution. In the problem dealt with here the following model seems more realistic: P Y = AU + e X BU + d where: Y = value of one index. X = value of a second index. U = true deer population per unit area. A and B = constants transforming the true population level to the scale of the index (fraction of the population killed, etc.). a and d = random error components of the same or different magnitude, but both assumed to have zero means. Mathematical expectations of the usual computations needed for re- gression and correlation coefficients are: E{sy23 " ‘20, 2 + r82 E {3x2} ’ 320712 + 0"d2 E {sxyg - A3052 so that: - 2 8y 536' r = u r/sz 2=20“/(A2 +0-2)(Bza-22+o‘d) s ‘Bdrz A b= xy - u - 2 — 2 2 ‘ 2 — Sx B¢rh +153 B + (rd 914. under the above model, the correlation coefficient (r) approaches unity only when the sampling errors are small relative to the variability of true population values, and the poorer indices necessarily show the lower correlations. A similar situation exists with regard to the regres- sion coefficientu-if sampling error is small then the coefficient measures what it is supposed to; otherwise it is "attenuated," and this may prevent the regression line from passing through the origin as it should if the two measures are proportional expressions of exactly the same quantity. There is thus the uncertainty about whether the relations are truly'curviw linear, or whether attenuation of the regression coefficient makes them appear to be so. These difficulties cannot be resolved absolutely without a knowledge of true deer populations. For the present the fair degree of concord among the data as to relative value of the four indices may suffice. Since the relative value of the Highway and Archery indices is not completely clear, I have chosen to weight them equally, and it seems that the "sample size" values provide such an equal weighting and place the indices in ape proximate accord with the other measures of reliability. The step remain— ing is to bring the four indices to a: common scale, as suggested before. Tgapgformgtion to a common scale. Choice of a scale is largely a matter of convenience. I have used 10 as the mid-point here, and prefer units sufficiently small so as to avoid confusion with deer per square mile; larger units (say midepoint of 100) might achieve this purpose more surely, but may also imply a false sense of reliability. In terms of the thermometer example mentioned before, I have here chosen a "zero" point (10). It seems inappropriate to use a transformu ation which may yield negative or zero values. 95 The remaining task is to bring the indices to the same range of values. I have arbitrarily chosen to transform the available data to have a common variance, but other criteria might be used (e.g., requir- ing that a certain proportion of the transformed values fall within a fixed range). Ideally, the transformations ought to be based on the same set of data for all indices (same years and areas), but the Highway kill fig- ures are not available by District for 1952. I have, however, included the 1952 data for the other indices, in order to use the combined index in reference to that year. The actual transformations proceed as follows: To transform a distribution with frequencies f(x), mean i, and variance, 32 to one with mean 2 and variance 52: Let: 2 = Bx + A E = Bi + A Then: i tthOanmna moHHE MMmmmm 00H hem HHHx WHAH: mm nosnommnwua wazooo mama MHDH NH mqm<9 100 :505.5 u 5000.00 550:. u 00 u 0 000.0 u m 5555.5 u m 550:. u 050mm u m u 0 555.5 u . 0555.0 n 00 50 n : 000.:5 u 00 mmwumeuomeMMp :o x0030 QOHmepommnmpp 0mm,m:009m0:mvum :5.0 00.0 05.0 55.0 55.5 55.0 :0.00 :.5 5.0 5.5 0.0 :.: 5.0 0.0 5 05.0 55.5 50.5 50.0 50.0 05.0 50.0 5.5 0.0 0.: :.0 :.0 5.0 5.0 0 05.00 00.00 55.50 00.50 55.:0 :0.50 00.50 0.00 0.00 0.50 5.00 0.00 0.50 0.00 5 00.5 50.0 55.5 00.5 50.0 :0.5 00.0 0.5 0.0 5.: 0.0 5.0 5.5 5.0 0 00.00 50.00 05.00 5:.00 00.0» 0:.00 05.00 5.00 5.0 0.00 0.00. 0.5 5.5 5.0 5 00.00 05.0 00.0 05.0 00.0 55.0 50.0 5.0 0.5 0.5 5.5 0.5 0.0 5.0 : 0:.0 00.5 5:.5 50.5 :5.0 05.0 55.5 0.0 0.5 5.: 0.:. :.5 0.0 0.0 5 :0.00 50.0 05.0 :0.5 50.5 00.5 50.5 0.0 5.0 0.0 5.5 0.5 0.0 5.0 0 50.5 50.0 50.0 05.0 00.0 50.5 55.5 0.0 5.0 0.0 5.0 0.0 0.0 5.0 0 0550 5550 0550 5550 :550 5550 0550 0550 5550 0550 5550 :550 5550 0550 00000000 000H0> GQEAOMmcmne n0 u0mor k), thus appears to depend on the vulner— ability per encounter and a poorly defined measure of a “probability" of encountering a particular deer. At any rate, the notion of area seems implicit in the constant. The derivation above suggests an approach to the problem of changing vulnerability (see kill-effort section) wherein one would evaluate the frequency of encounters with deer (sightings) and the proportion of encounters resulting in a kill as separate items. An opportunity for field study of the above model exists in the con- trolled hunts in a fenced square-mile area (Van Etten, 1957). The unpublished data now available from this area indicate that the notion of a Poisson frequency of encounters, and constant probability of kill per encounter, may be realistic. However, too few deer are killed in any one season to give much information on changes in vul- nerability. LITERATURE CITED Andrewartha, H. 0.. and L. C. Birch. 1954. The distribution and abundance of animals. University of Chicago Press. Pp. xv+782. Bartlett, 1. H. 1938. Whitetails, presenting Nfichigan's deer problem. 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Wildl. Mgt., 23(3): 315-3%. "‘7'1’ifll'filujilii'lfi'ifil'flifilflT