\IT .i‘nha—JA... I -’wn“.‘§ LIEAARY Michigan State 2.; Univmty / Ll FHE8I8 This is to certify that the thesis entitled AN EVALUATION OF MARK-RECAPTURE ESTIMATORS UTILIZING FIELD AND COMPUTER TECHNIQUES ON KNOWN POPULATIONS presented by John Frederick Sefcik has been accepted towards fulfillment of the requirements for M. S. degreeinFisheries 8 Wildlife thQfl/W ””9188”! M 2 , 8 Date ay 1 19 O 0-7639 5 u - I‘v‘hne .. I a c l‘l-‘Jfi Will! WWW ”Will!” 293 10255 7745 w: 25¢ per do per in: {fl.l\\\\ ~ RETURNING LIBRARY MATERIAL§z Ptace in book return to mo 3 3'3",” charge from circulation react AN EVALUATION OF MARK-RECAPTURE ESTIMATORS UTILIZING FIELD AND COMPUTER TECHNIQUES ON KNOWN POPULATIONS By John Frederick Sefcik A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Fisheries and Wildlife 1980 ABSTRACT AN EVALUATION OF MARK-RECAPTURE ESTIMATORS UTILIZING FIELD AND COMPUTER TECHNIQUES 0N KNOWN POPULATIONS By John Frederick Sefcik Since many different mark-recapture estimators are avail- able to the biologist, it is desirable to evaluate their performance. Field evaluation of the estimators was possible because four mark-recapture experiments were performed on Microtus pennsylvanicus populations of known size. 0f the nine estimators evaluated, the nonparametric frequency of capture method is recommended for estimating the size of trap- happy mice populations. Besides demonstrating a low sample bias and variance, it has the advantage of not being affected by population stratification. In addition, a computer simulation model for mark-recapture experiments in a closed population was validated for future use. The model incorporates home range movements, spatial patterns, and a learning process (trap-happy or trap-shy), and yields typical mark-recapture data. It should prove to be a useful tool for providing insights into decision-making alternatives in mark-recapture experiments. ACKNOWLEDGEMENTS I would like to extend my sincere appreciation to my major professor, Dr. Stanley Zarnoch, for guidance and in- spiration throughout my masters program” Sincere appreciation is also given to Glenn Dudderar and Dr. John King for their needed assistance. I am grateful to Rich Hoppe and Laurie Lucas for help with the field work. Financial support came from the Michigan State University Agricultural Experiment Station. TABLE OF CONTENTS Page LIST OF TABLES .......................................... .iv LIST OF FIGURES ................................... ‘ ...... v INTRODUCTION .................................. ' .......... 1 STUDY AREA .............................................. 5 METHODS ................................................. 9 Mark-Recapture Estimators ........................ 9 The Mark-Recapture Simulation Model .............. 13 Collection of Data ............................... 21 Data Analysis .................................... 26 RESULTS ................................................. 36 Mark-Recapture Estimators ........................ 38 The Mark-Recapture Simulation Mbdel .............. 52 DISCUSSION .............................................. 71 Mark-Recapture Estimators ........................ 72 The Mark-Recapture Simulation Model .............. 77 LITERATURE CITED ........................................ 82 APPENDIX ................................................ 87 iii Number 10 11 LIST OF TABLES Page The average percent bias in experiments I, III, and IV ....................................... 50 The differences between the combined population estimates (males and females estimated sepa-, rately and added together) and the original field population estimates ....................... Sl Significance of the test to pool the correlation coefficients of several animals, and an estimate of the common rho (r) with a 95% confidence interval ......................................... 54 Experiment I. Estimates of the intial proba- bility of capture (p) and the learning process Experiment II. Estimates of the initial proba- bility of capture (p) and the learning process 57 (a) ............................................... Experiment III. Estimates of the initial probability of capture (p) and the learning process (8) ...................................... 58 Experiment IV. Estimates of the initial proba- bility of capture (p) and the learning process 58 (a) .............................................. The model simulation values of 5, 8, N, the variance of x, and the variance of y ............. 60 Experiment I. Bias in the simulation mark- recapture estimates .............................. 62 Experiment III. Bias in the simulation mark- recapture estimates .............................. 63 Experiment IV. Bias in the simulation mark- recapture estimates .............................; 64 iv Number 10 LIST OF FIGURES Location of the trapping grid, water stations and release points ... ......................... Experiment I. Mark-recapture estimates of population size ............................... Experiment II. Mark-recapture estimates of population size ............................... Experiment III. Mark-recapture estimates of population size ............................... Experiment IV. Mark-recapture estimates of population size ............................... 95% confidence intervals for the day 10 popu- lation estimates .............................. Experiment I (random spatial pattern). Field and simulation nonparametric estimates ........ Experiment III (random.spatial pattern). Field and simulation nonparametric estimates ... Experiment IV (random spatial pattern). Field and simulation nonparametric estimates ........ Experiment III (fixed spatial pattern). Field and simulation nonparametric estimates ........ 'Page 66 67 68 70 INTRODUCTION Mark-recapture techniques have a long history and a ‘wide array of designs and analyses. Since the 1950's, the literature on estimation methods has increased tremendously. Fortunately, the book by Seber (1973) and the review by Cormack (1968) provide comprehensive summaries. Dr. Seber also plans to publish a review of recent developments in the very near future (Eberhardt et al. 1979). Generally, a mark-recapture experiment consists of placing live-traps in a regular grid pattern over the habitat to be studied. The experiment consists of two or more trapping periods, each a random sample of the indi- viduals in the population. In each sample, the previously marked individuals are recorded, unmarked individuals are marked, and all are returned to the population. The next sample is taken after an adequate time has been allowed for the marked and unmarked individuals to randomly mix (Zarnoch and Burkhart 1980). The aim.of the experiment would be to investigate one or more of these properties (Caughley 1977): movement Eggfggeggfic fecundity rates age-specific mortality rates size of the population rate of birth and immigration combined OHflFqfih0h‘ l -7. rate of death.and emigration combined 8. rate of harvesting' 9. rate of increase' An estimate of population size has traditionally been the main objective. In addition to research problems, a wide variety of environmental assessment studies and bio- logical inventory programs require the estimation of animal abundance. These needs have been further emphasized by the requirement for the preparation of Environmental Impact Statements imposed by the National Environmental Protection Act in 1970 (Otis et a1. 1978). ZMark-recapture 'methods have been used extensively to estimate fish, insect, and small mammal populations. They have recently found wider application in big game censusing (Rice and Harder 1977). Since many different estimators are available to the biologist, it is desirable to evaluate their performance. The validity of an estimate and its 95% confidence limits depends at least upon the condition that all the assumptions underlying the mathematical model from which the estimator is obtained are upheld (Roff 1973a). The only measure of accuracy generally available is the standard error estimated from the data. Field evaluation of the estimators is possible if mark-recapture experiments are performed in areas where the true sizes of the population are known. However, this information is rarely available. Five mark-recapture studies on known populations were found in the literature. (1) Edwards and Eberhardt (1967) reported the results of a live-trapping study in Ohio on a penned population of 135 wild cottontails (Sylvilagus floridanus). The estimators they examined were the Schnabel, Schumacher- Eschmeyer, two frequency of capture methods (based on the poisson and geometric distributions), and linear regression. (2) Cook et a1. (1967) released 1093 individuals of the North American moth HyalOphora promethea in Trinidad where they do not naturally occur. The method of estimation adopted was that of Fisher and Ford, which is well suited to a census where there are rather few returns and where the estimate sought is the total population of images during the season. (3) Smith (1968) used two sampling methods to determine the number of mice of two species,'Mus“mu8culus and‘Pero- myscus‘polionOtus, in an abandoned field in Florida. He first live-trapped using -the mark-release technique and later captured the mice by digging out their burrows (evidence suggests that the later technique revealed the exact structure of the population). Animal abundance was estimated using the Lincoln and Hayne (Schumacher-Eschmeyer) methods. (4) Carothers (1973) conducted a mark-recapture experiment on the taxicab population of Edinburgh, Scotland. The population is a real one (though not involving animals) and had a known size of 420. Estimators used include the Lincoln (Chapman's modification), Schnabel, Schumacher- Eschmeyer, Marten, Tanaka, geometric frequency of capture, and Jolly-Seber. (5) Rice and Harder (1977) conducted a helicopter- assisted mark-recapture study on a white-tailed deer (Edg- ‘coileus‘virginianus) population in northern Ohio. Part of their study area was a 122 ha Test Area with a known deer population of 155. They used the Lincoln Index to estimate deer density. The need for additional controlled studies where the population size is known has been stressed by Otis et a1. (1978), Begon (1979), and Zarnoch (1979). The objectives of this study are threefold: (1) To conduct mark-recapture experiments on closed populations (no recruitment or losses) of known size. (2) To evaluate nine mark-recapture estimators (these include two modifications of the Lincoln and one of the Schnabel). (3) To calculate additional needed parameters and validate for future use the simulation model for mark-recapture experiments in a closed population developed by Zarnoch (1976). STUDY AREA The study was conducted in a mouse-proof, predator- free enclosure at the MSU Wildlife Research Area in Okemos, ‘Michigan. The enclosure was built in a k acre (0.2 ha) overgrown clearing. The clearing sloped about -5 degrees to the north towards the Red Cedar River. The vegetation was predominantly brome grass (Bromus spp.), Queen Anne's lace (Daucus Carota), and white sweet clover (MeliIOtus alba) (Gleason and Cronquist 1963). During the study, the sweet clover extended over most of the enclosure and was 4-5 ft (1.2-1.5 m) tall. The square enclosure covered 0.27 acres (0.11 ha); each side was 109 ft (33.2 m) long. The fence was con- structed by using 36-inch-wide sheets of 24-gauge steel supported by 2x4's. The walls of the fence extended 22-24 inches above the ground. The bottom six inches of the steel sheets were bent along their length at a 90-degree angle to form a perpendicular shelf. This shelf was buried in the soil 6-8 inches deep so that the shelf projected towards the inside, thus preventing the voles from digging out (Price 1967). A 20-inch border was mowed inside the fence and a 6.5-ft border outside, to aid in detecting digging or disturbances and in keeping voles and mice away from.the fence (Schwartz and Schwartz 1959). 'Both borders were mowed before each of the four experiments began. To prevent mammalian predators from.entering the study area, two strands of electric wire surrounded the enclosure. One strand ran along the top of the fence; the other six inches below. To exclude avian predators, the top of the enclosure was completely covered with Conwed plastic netting. The netting was black, weighed 2.75 pounds] 1000 square feet, and had a strand count of 1.5 x 1.2 strands per inch. At the edges of the enclosure, the netting was attached directly to the top of the fence. In the center, it was supported by 2x2's and l7-gauge wire at an average height of approximately 6.5 ft. Construction of the enclosure began in May 1979. The steel fence was completed on June 27, 1979; movement of small mammals into or out of the study area was restricted. The top was covered with plastic netting on July 6, 1979, completing the enclosure. Throughout the study, an attempt was made to keep the study area as natural and undisturbed as possible. There were a few exceptions. In mid-May, straw was spread over a few areas in the southwest corner where the vegetation was slightly sparser than in the rest of the enclosure. The straw added_ground structure and cover. To facilitate trapping and to provide a measure of location, a 10.x 10 permanent grid of red flags was established inside the enclosure. Each flag was designated by a letter and a number corresponding to its row and column (Figure 1). Flags were spaced 10 ft (3.0 m) apart; the outermost rows and columns were approximately 9.5 ft (2.9 m) from the fence. Another addition was water stations. Getz (1963) and other authors have noted that the meadow vole appears to have a high moisture requirement. To ensure adequate moisture, 16 Little Giant water containers were placed throughout the study area (Figure 1). Each container consisted of an open water-ring and a jar with a one-gallon water reserve. The average home range of Microtus penngylvanigus in southern Michigan has been reported as .10-.50 acres for males and .04-.28 acres for females (Blair 1940, Hayne 1950, and Getz 1961). There- fore, by having 16 water stations throughout the study area, the size and shape of the home ranges should not be affected. ‘3 N (1) A . . . . . . . . . . U A. As O (2) B . . . . . . . . . . u (3) c . ‘. . . . . . . . . a D (4) D . . . . . . . . . . u (5) E . . . . . . . . . . D A A U (5) F . . . . . . . . . . c1 (7) c; . . . . . . . . . D U (8) H . . . . . . . . . . m (9) I . . . . . . . . . . n A A D (10)J . . . . . . . . . . U 1 2 3 4 5 6 7 8 9 10 Figure 1. Location of the trapping grid, water stations, and release points. LEGEND 1 . Trapping grid N 0 Water stations 1 A Release points METHODS Mark-Recapture Estimators The nine mark-recapture estimators of population size that are evaluated in this study are presented below. Lincoln Index'eStimators The Lincoln Index is a two-sample mark-recapture esti- mator. The second sample is usually taken by trapping their? dividuals, but this is not always necessary. Any method for this second sample may be used, provided it is a random sample of the population. Let n1 - number of individuals in sample 1 that were initally taken from the population, marked, and released, n2 = number of individuals in sample 2, m2 - number of marked individuals in sample 2, and N - number of individuals in the total population. Assuming that the ratio of marked to the total number of individuals is constant over the two samples, then the Lin- coln Index estimator is (Lincoln 1930) A modification of the estimator N is given by Chapman (1951) as ( +’1)( ‘+ 1) fi’. 111 . n2 . -1 (1112 + l) 9 10 An alternative estimator was developed by Bailey (1951, 1952) by using a binomial approximation.to the conditional hypergeometric distribution of m2. The modification de- veloped under this model is N* .n1(n2.+ 1) (m2 + l) Because the Lincoln estimator is traditionally for a two- sample.experiment, a modification was formulated for a multiple sample mark-recapture experiment. This consisted of considering sample 1 as all trapping periods before the last and sample 2 as the last trapping period. Schnabel'estimators The Schnabel is an S-sample mark-recapture estimator. For each of the S samples, the total sample size and number of previously marked individuals are recorded, all unmarked individuals are marked, and the entire sample is returned to the population. Let nj - number of individuals in the jth sample, mj - number of marked individuals in the Jth sample, j-l M. - 2 (nj’- mj’) - number of marked individuals n the J j’-1 population just before the jt sample ‘ is taken, S = number of samples taken, and N - number of individuals in the total population. 11 M Then assuming N1 is small and mJ follows the poisson dis- tribution, the Schnabel estimator for N is (Schnabel 1938) The Schnabel estimator defined above was modified by Chapman (1952). Chapman's modification is 8 s + 1 z m 1-1 j Schumacher-Eschmgyer estimator The Schumacher-Eschmeyer estimator employs the same estimation techniques as the Schnabel estimators; that is, the same sampling scheme, variables, and data are needed (Schumacher and Eschmeyer 1943). Schumacher and Eschmeyer, however, utilized a regression method weighted by the sample sizes, nj (Seber 1973). The Schumacher-Eschmeyer estimator is N a 19:1 J J S Tanaka estimator For the situation when the probability of capture for marked animals appears to differ from the probability for 12 unmarked, Tanaka (1951, 1952) has prOposed a linear regression mmdel which yields the estimator fi = antilog é Yj = -1og yj, and e g _ 22(Y:I -‘Y)(x%.- x) Variables not defined are the same as in the Schnabel. Geometric estimator The frequency of capture approach has been utilized in estimating population size from.a multiple-sample mark- recapture experiment. It uses the frequency of capture of individuals over all samples. The geometric estimator (Eberhardt et a1. 1963, Edwards and Eberhardt 1967, Nixon et a1. 1967, and Eberhardt 1969) is based on the assumption that the frequency of capture follows a geometric distribution truncated at the zero class. Let I = number of times an individual is caught £=1,2,3,...., f, - number of individuals caught flitimes, S = number of samples taken, and N = total population size. 13 The geometric estimator is then defined as . ( s,f, )( x if, - l) N = '2-1 231 (.2 2f - 2 f ) 2:1 1 i=1 2 where it is obvious that all terms subscripted with £>S are equal to zero and can be ignored. Nonparametric estimator Overton (1969) presented a nonparametric estimator based on the frequency of capture. Each animal has probability Pi’ 033 g). However, he noted that few juveniles are caught in live-traps and that many animals reach stable asympotes of weight at small adult body size. In experiments I-IV of this study, only seven, three, five, and seven voles, respectively, weighed less than 22 g. The smallest overall weighed 16.0 g; all but four under 22 g were females. Because young females begin breeding at about 25 days of age and males at 45 days (Burt 1957), almost all the meadow voles used in this study would probably be considered sexually mature. Although an effort was made to only use animals 24 weighing more than 22 g (to lower variability), the inclusion of some juveniles in the population is a natural phenomenon. Four mark-recapture experiments were performed, each lasting 19 days. The inclusive dates were: (I) July 21 - August 8, 1979 (II) August 8 - August 26, 1979 (III) August 26 - September 13, 1979 (IV) September 13 - October 6, 1979 (snap-trapped five extra days at the end). One day l of each experiment, three gallons of oats were scattered throughout the enclosure to ensure an adequate food supply. Then an approximately equal number of Microtus were released at each of the six points shown in Figure 1. A 4-day interval was allowed for the introduced population to adjust to its new surroundings. This was based on an arbitrary decision which recognized that some time for ad- justment was desirable, but that the longer trapping was delayed the higher the probability of mortality. On the evening of day 5, 100 Longworth traps were set inside the enclosure, one by each flag. The Longworth is an aluminum trap made of two sections, a tunnel and a nest box. The latter contains both nesting material and food (cotton and oats) and therefore not only attracts the small mammal, but also ensures that it is warm, dry, and fed after capture. The attracted animal enters the tunnel, but, by stepping on a sensitive treadle at the nest-box end, closes 25 and locks the tunnel entrance behind it (Begon 1979). Long- worths are one of the most extensively used live-traps in small mammal mark-recapture studies (Morris 1968, Grant 1970). Each trap was covered by a 1x10x12-inch board to shield it from sunshine and rain. Trap spacing was 10 ft (3 m) apart. Tanaka (1966) has recommended a trap spacing of 5 m (15 ft) for ordinary or outbreaking densities of voles; the formula of Otis et a1. (1978) based on home range implies a spacing of 5'm (15 ft) or less. The live-traps were checked each morning at7-8 a.mu for 10 days (days 6-15). Captured animals were identified, recorded by trap coordinates, and released. All the traps were closed during the morning check to prevent captures and, hence, mortality during the day due to the heat; they were reset at approximately 7 p.m. each evening. On day 15, the tenth day of live-trapping, all animals captured were removed from the enclosure. Each live-trap was replaced by a snap-trap (100 total) baited with a mixture of peanut butter and rolled oats. The snap-traps were checked at least once daily through day 19 (4 days). A summarized schedule for each experiment is: Day 1 Microtus released Day 5 Live-traps set Day 6 Live-traps checked for the first time Day 15 Live-trapping ends; snap traps set Day 19 Snap-trapping ends. 26 During each trapping experiment, any "unmarked" animals that were caught were removed from the enclosure. This included animals of other species as well as marked animals from earlier experiments. Snap-trapping at the end of each experiment was necessary to verify that members of the introduced population were still alive and inside the enclosure. All removed members of the experimental population were weighed and positively identified by toe-clip and leg band. Data Analysis Mark-recapture estimatOrs In the four mark-recapture experiments, each animal was considered "marked" after its first capture. Using a program developed on the CDC 750, Model 175, computer at Michigan State University, estimates of population size were computed after trapping periods two through ten of each experiment using the available field data and the nine mark- recapture estimators discussed previously. Since the size of the known population varied between experiments, the percent bias in the population estimates was examined instead of the actual bias. The percent bias is defined as 7 i B = 100(E - K)/K where - number of animals introduced at the start of the m1 experiment, 27 ‘m2 = number of animals known to be alive, E = the mark-recapture population estimate, and ml if E>m1 L_E elsewhere. The average percent bias was computed for each estimator for each of the trapping periods 5-10. Yang et a1. (1970) found that both sexes of Microtus ochrogaster were equally susceptible to live-traps. In the present study, the frequency of capture for males and females was compared for each mark-recapture experiment with a goodness of fit test adjusted for small sample size (Gill 1978). . If males and females do not show the same response to live-trapping, part of the bias in the mark-recapture esti- mates may be due to sex. In an attempt to improve the popu- lation estimates, the mark-recapture field data from each experiment was analyzed separately for males and for females. An estimate of population size for trapping periods two through ten was computed for each sex using the nine mark- recapture estimators given previously. Total population size was approximated by adding the male and female population estimates for each trapping period together. The four mark-recapture experiments were "replicates," except for time and the specific experimental animals used. Because the source and trapping background of the voles in 28 the four different experiments varied slightly, the Kolmo- gorov-Smirnov 2-sample test (MSU Computer Laboratory 1978) was used to determine whether the frequencies of capture follow the same distribution. If all four populations follow the same distribution, then the results of all four mark-recapture experiments can be compared and are expected to be very similar. A small probability level indicates that there is some difference in the distribution of the two mark-recapture experiments being tested (MSU Computer Laboratory 1978). The mark-recapture simulation model In order to validate the simulation model, various parameters had to be estimated from the field data. Before utilizing the simulation model, the learning theory option (trap-happy, trap-shy, or no learning), the spatial pattern of the animals, and estimates for the variance of x, the variance of y, the correlation coefficient (p), the parameter in the learning theory model (a), and the initial proba- bility of capture (pi 1) need to be determined. Spatial Pattern Because there was no prior reason for believing that the animals were not distributed in a random spatial pattern, the test due to Hopkins and Skellam (Pielou 1969) was used to check for randomness. Each animal was placed at its center of activity, the average x and y coordinates of its 29 points of capture. For each test, 30 points corresponding to trap locations were choosen from.a random number table (Snedecor and Cochran 1967). The spatial pattern of the animals that was chosen for the simulation model was based on the results of this test. Home Range Parameters All the animals within each population were assumed to have the same bivariate normal home range parameters. An estimate of the pooled variance of x and variance of y were calculated by using the x and y coordinates of all the capture points of all animals. The parameter p, the corre- lation coefficient between x and y, determines the narrowness of the elipse containing the major portion of the observations. In the model, the general form of the bivariate normal proba- bility density function is restricted to the case where p is equal to zero. The validity of this restriction can be examined by testing the estimated correlation coefficient (r) for each animal. However, several sample correlations (for individual animals) may possibly be drawn from.a common p. If this is true, the r's may be combined into an estimate of p which is more reliable than that afforded by any of the separate r's. To test the hypothesis that several r's are from the same p, and to combine them into a pooled estimate of p, the procedures of Snedecor and Cochran (1967) are followed, using the bias correction for averaging large numbers of correlations. The pooled rho with a 95% confidence 30 interval was estimated for each experiment regardless of the results of the pooling test. Only animals live-trapped four or more times could be included in the calculations. Learning Behavior If the catching and handling affect the catchability of marked individuals after their first capture, then the sampling will not be random within the marked population. Testing the assumption of equal catchability in the marked population verifies whether the animals are either trap- happy or trap-shy. Leslie's technique based on thefrequency of recapture of individuals (Seber 1973) was applied with the first two trapping samples consituting the marked population (animals not removed at the end of the experiment were not included). The simulation model option for either no learning, trap-happiness, or trap-shyness is based on the test results. Learning Parameters Estimates of two parameters, a(the learning process) and Pi,l (the initial probability of capture) are still needed (if the no learning option is selected then a ' 1.0). To obtain the estimates, the methods developed by Bush and Mosteller (1955) for free-recall verbal learning experiments were adopted. Their experiments were conducted as follows. A list of N monsyllabic words is read aloud to a subject. The subject is then instructed to write down all the words that he can recall. The experimenter gives him no indication 31 of how well he has performed. Then the order of the words is randomized, and the procedure is repeated. The experiment is continued in this way for many trials until the proportion of words recalled nearly reaches an asymptote. When redefining the model of Bush and Mosteller (1955) for my study, a word becomes a meadow vole, recall of a word is the capture of an animal, and each listing by the subject of the words he can recall is a trapping period. Three basic assumptions made in analyzing the data are: (1) All animals have the same initial probability of capture, Pi,l’ and that all animals have the same learning parameters (i.e., we have a group of N "identical" subjects). (2) Non-capture of an animal does not changes its probability of being captured during the next trapping period. (3) All the animals can be caught (trap-happy) or none of the animals can be caught (trap-shy) during a trapping period. That is, the proportion of animals caught (or not caught) approaches an asymptote of unity. The initial probability of capture, pi,1’ is estimated from that portion of the data which is independent of a, the learning parameters. The data for each animal from trapping period one up through the trapping period in which each animal is first captured is used to estimate pi 1. A unique unbiased estimate, 5, of Pi,l when N is fixed and Z is the only observed statistic, is given by -=‘N-‘1 P z—z—T 32 where N = the number of different animals captured during the trapping experiment, and Z = the sum over all animals of the number of trapping periods preceded by a zero frequency of capture. (If an animal is not caught until trapping period 4, then it has 4 trapping periods preceded by a zero frequency of capture. Unless the investigator has definite knowledge of uncaptured animals present in the population, than N z-zf i=1i where f is the trapping period in which the animal was first captured.) Thus, p will tend to decrease as the number of animals captured increases, until every animal in the population has been captured at least once. The mean total number, T, of non-captures may be used to estimate a as If N1 = the number of known animals in the population on day i, and Ehe number of times an animal j has been captured (if the population is trap-happy) C - or J the number of times animal j has not been captured (if the population is trap shy, then for a given day i 33 For days 6-10 of each.mark-recapture experiment, two estimates of a and pi,1 were computed from two sets of data. First, only the field data available from trapping periods one through the day when the parameters were estimated was utilized; these will be referred to as the incomplete esti- mates. These estimates are calculated from the amount of information normally available to a biologist conducting a mark-recapture study. The second set of data also included animals known to be present but not captured during the live- trapping. The only animals from the introduced population not included were those never captured during the live- trapping and not removed during the snap-trapping phase. These estimates will be referred to as the complete estimates. The complete estimates may be an improvement over the in- complete estimates because they are based on the entire population, not just the. marked animals. Model Validation After the model's parameters were estimated from the field data, the validity of the model was checked by simulating ‘mark-recapture experiments. If the model is valid, the population estimates calculated from.the simulated mark- recapture data should closely approximate the population estimates calculated from the original field data. Two simulations were run for each mark-recapture experi- ment. The first used the incomplete estimates of a and Pi,1 (based only on the field trapping data); the second, the 34 complete estimates (based on all known data). Because each simulation costs 10-13 dollars, replications were not made initially. In all the simulations, 100 traps were arranged in a 10 x 10 regular spatial pattern. The number of animals placed on the computer grid for each simulation was approxi- mately the average of the introduced population size and the minimum.number of animals known to be present in the corresponding field experiment. The bias in the model's estimate was found by subtracting the original estimate from the simulation value. After the initial exploratory simulations, each experi- ment was replicated ten times with the incomplete estimates of'c and pi,1 from day 10. The incomplete estimates were chosen because they typify most mark-recapture studies; the estimates from day 10 were thought to be the best because they were based on the most information. The population size in the simulations was the minimum.number of animals known to be present on day 10, a verified true population size. The animals were placed in a random spatial pattern. The bias in the model's estimate was calculated by subtracting the original field estimate from the average of the ten simulation values . When attempting to validate the model, the best test should result from the utilization of all possible information available, even information not normally available to the investigator during a mark-recapture study. In addition to 35 using the.complete estimates of a and pi,l' the animals were placed in a fixed pattern on the computer grid. All animals known to be present on trapping day 10 were placed on the 10 x 10 computer grid at their center of activity. Because of the high cost of simulations, ten replications were ,generated for experiment III only. Experiment III was chosen because the true population size was closely known and because all the animals had a very similar capture and lab history prior to being released into the enclosure. A- gain, the bias in the simulation estimates was calculated by subtracting the original field estimate from the average simulation value. ‘RESULTS In the initial trap-out period prior to experiment I, one MicrOtus and one mole (Scalopus'aquaticus) were removed from the enclosure. Moles continued to be a problem during the study, especially when they dug near or under the fence, because mice, shrews, and other animals often use the tunnels more than the moles do (Schwartz and Schwartz 1959). Three more moles were later removed, one during experiment I and two during III. During the study, the snap traps in the mowed border outside the fence captured ten deermice (Perg- 'mzscu3'maniculatus), six meadow voles, one eastern chipmunk (Tamias striatus), and a variety of birds. None of the Microtus were marked animals from the experimental populations. Inside the enclosure, a deermouse and two shorttail shrews (Blarina brevicauda) were captured during the removal period of experiment III; five shorttail shrews were captured during experiment IV. Throughout the entire study, no unmarked 'Microtus (animals not marked and introduced during the mark- recapture experiments) were trapped inside the enclosure. I Part of the field data (tag number, sex, and capture dates) for experiments I-IV is given in the Appendix. Animals not captured on trap day 10 or during the snap-trapping period were not removed from the enclosure at the end of the 36 37 experiment; they are not known to be alive or inside the enclosure past their last dateof capture. This includes one animal (1023) from experiment I, one (1049) from II, two (1110, 1123) from 111, and four (1137, .1141, 1146, 1173) from IV. One animal frmm III (1095) was found dead in the enclosure on day 19 of that experiment. One animal intro- duced in experiment IV was found dead prior to the start of live-trapping; therefore the introduced population of IV is considered to be 39. No other mortality was known to have occurred during the initial adjustment or the live-trapping phase of any experiment. Based on their points of capture during the study, the voles seem to have established their home ranges by the start of the live-trapping. There are no examples of a dramatic shift in an animal's capture locations. In all four ex- periments, the animals showed a positive average weight gain from.the time of introduction to the time of removal. This indicates there was sufficient food available to sustain the Microtus population. The average weight gain was4.12, 4.04, 4.08, and 1.97 grams in experiments I-IV respectively. Throughout the study, very few closed Longworth traps were empty when checked (average of 1.3 traps/day).' During experiments I and II, the weather was mostly cloudy and humid with rain falling several days in both I and II. During experiments III and IV, the weather was sunny and warm. The temperature throughout the study was fairly uniform, 38 reaching between 65 and 85 degrees most afternoons. The coolest daytime temperatures occurred during experiment 11. 'Mark-Recapturg,Estimators Themark-recapture estimates are plotted for trapping periods two through ten (Figures 2-5). ’Because all three Lincoln estimators gave similar values, only Chapman's modi- fication is shown and is simply called the Lincoln. For the same reason, only Chapman's modification of the Schnabel is shown and.is just referred to as the Schnabel. The Tanaka estimate is never shown for day 2 because it is undefined. The shaded area in Figures 2-5 is the "known" number of animals present in the enclosure (true population size). The.number of animals released at the start of the experiment forms the upper boundary. The.verified number of animals present (by capture or a later trapping) forms the lower boundary. In experiment II, 26 voles out of the population of 31 were captured the first day; all had been captured at least once by day 4. Although these animals were not truly tame (Hediger 1954), they seem to have lost most of their inhi- bitions towards humans and traps during their history of captivity. On several occassions, some of these voles were caught only a few minutes after the traps were opened. They were also much less aggressive than the other vole populations when handled. For these reasons, experiment II has been 39 Legend for figures 2-5 M - Number of animals marked L 8 Lincoln ..... S - Schnabel ___._.___ SE = Schumacher-Eschmeyer _. ...... J T - Tanaka ............ G =- Geometric ...—...... Ni- Nonparametric .......... W""il’l’.'.”f!£"l ill “MW! .311 Known number of animals WJJJ (see text) JW 40 Figure 2. Experiment I. Mark-recapture estimates of of population size. 60 55 50 45 4O 35 Estimated P0pu1ation Size 30 25 20 Figure 41 _ (74) , (503) (54X : \ I l 1 \ ' \ \ I Z - '\ \ \ I ./\ . / \ ‘ l/ \ _ -— \ ./.I \ / ._._ ’ l l l I /. . / \- “‘"IIIIIIIIIIIIIIIIIIIII IIIIIII" IIIIIIIIIII IIIIIIII IIIII IIIII IIIIIIII IIIIIIIIIIII IIIIIII l I T I f I l I l 2 3 4 5 6 7 8 9 10 Number of Trapping Periods 42 Figure 3. Experiment II. Mark-recapture estimates of population size. 43 65 - (58) \ l l \ 60 "‘ \ \ \ \ 55 - l \ \ g . ”.4 50 "" \ U) \ 8 '\ a: \ ,3 . \ g. 45 - \ o . 9' \ '° \ 3 .\ g \, fa 4O - \,\ a: ‘\ \ \ \ 35 ‘. , i O ...o“./'.\. ... ....... No. "//“’~..-.._:..:-'..—. 1n":...u 0 mm u now ° mm " ..., " Hm I , um “mumml: 3o — "H, .. - "/'"' "I III" II“ IIIIII‘IIIIII W“ WW I I L M/ ’ S 25 r r T l I I r l l 2 3 4 5 6 7 8 9 10 Number of Trapping Periods Figure 3. 44 Figure 4. .Experiment III. Mark-recapture estimates of population size. 45 (136) (62) (61) 55 ‘ \ 50 ‘ 45 ‘ 40 EIIEI‘EII‘II III'IIIII‘ "'I III. III I" 35 ‘ Estimated Population Size 30 ‘ 25 ‘ 20 ‘ 15 I I I I I I T "l.. I 2 3 4 5 6 7 8 9 10 Number of Trapping Periods Figure 4. 46 Figure 5. Experiment IV. Mark-recapture estimates of population size. so 55 so 0) N -.-I m 45 {3 C -.-I U (U H 8. 3 40 'U Q) U CU E -I-I E 35 30 25 20 Figure 47 (81) (52) \ \ \ \ " \ \ \ \ \’\~ ’ - ' \ \ \ \‘ IIII'W II III III I III IIIiII I III :1» “III III. W III III [III WIN III “I J} WW IN MI IIIIIIIIIII I II ,I o ’ .-.—0 L . .. N M I I I I I I I I I 2 3 4 5 6 7 8 9 10 Number of Trapping Periods 5. 48 excluded from the rest of the analysis except where it is used for illustrative purposes. For experiments L III, and IV, 95% confidence limits on the day 10 population estimates are shown for the Lincoln (Adams 1951), Schnabel, Schumacher-Eschmeyer, and Tanaka (Seber 1973) (Figure 6). Accurate methods for calculating the confidence intervals for the geometric and nonparametric frequency of capture estimators were not available. The average percent bias of the population mark-recapture esti- mators for experiments I, III, and IV is shown in Table 1. To test for a difference in frequency of capture between males and females in each mark-recapture experiment, 2x4 contingency tables were constructed. In all three cases (I, III, and IV) there was no significant difference between sexes in the frequency of capture (P>,30). Experiment II also showed no significant difference (P=.20). Although males and females showed the same response to live-trapping, the field data from each experiment was analyzed separately for males and females in an attempt to improve the papulation estimates by stratifying. The combined population sizes, approximated by adding the male and female population estimates together, are compared to the original field estimates in Table 2. Note that for the nonparametric frequency of capture estflmator, the combined population estimates and the original field estimates are always exactly the same. In experiment II, none of the differences were Population Size Population Size 49 Experiment I SS-W Experiment III 45 o “5' ‘ '3'? ' - mm 'W a WWII WWWWW WWW WWW. 1.; “WWW .3 3 5 .. t; L 3 | 8‘ 9* s SE S 25‘ T SE 25- Experiment IV 4 iWWWWW WWWWWWW WWW WWW! WWW WWW. WW LEGEND 351 WWW“ “WWWWWiMWIIHWW' L 3 Lincoln L T S = Schnabel SE = Schumacher- Eschmeyer 254 T = Tanaka Known population size * The confidence interval for the Tanaka is not shown because of computational difficulties. The Tanaka estimate is 55. Figure 6. 95% confidence intervals for the day 10 popu- "1ation estimates. 50 Table l. The average percent bias in experiments I, III, and IV. Trap Schumacher- Geo- Nonparap Day Lincoln Schnabel Eschmeyer Tanaka. .metric metric 5 -23.99 -26.14 -23.57 -25.22 24.96 -13.82 6 -l4.37 -23.23 920.76 -18.95 22.78 -8.48 7 -8.11 -20.72 918.25 -2.82* 21.47 -6.04 8 -4.27 -17.29 -14.60 441.91 24.01 -0.36 9 -8.10 -15.17 '.-12.75 12.66* 18.94 0.53 10 -2.06 -11.48 -9.04 6.52* 20.28 3.90 *The average of large positive and negative biases. Table.2. III IV I III IV I III IV I III IV I III IV I 51 The differences between the.combined population estimates (males and females estimated separately and added together) and the original field popu- 1ation estimates. Trapping Periods '2‘-,-3'* .4.. .5.. .6.. .7.. .8....9, 10 I Lincoln** 3* 8+ Schnabel 4- 5.. 4- Schumacher- Eschmeyer ~-- 13+ 3+ 3+ ‘_ Tanaka --- --- --- 23* 530+ 28- 31* --- --- 5+ --- 5+- 589- 38- 29- 11* 8- 5* 4* Geometric --- 40- 9-. _ 5- 3- 4- Nonpara- metric*** III IV ** All differences of less than 3 are omitted for all estimators *** All values are exactly the same LEGEND Combined estimate is closer to the true value Combined estimate is farther from the true value The two estimates are approximately equidistant from the true value (on opposite sides) Estimator is undefined. 52 greater than 2.5; none of the differences after.day 4 were greater than 1.0. The Kolmogorov-Smirnov 2-sample test was employed to determine whether the frequencies of capture for experiments I, III, and IV follow the same distribution (are the popu- lations "the same"). This test is sensitive to any type of difference in the two distributions - median, dispersion, skewness, etc. (MSU Computer Laboratory 1978). The following results were obtained by using an SPSS file (Nie et al. 1975) on the CDC 750 computer: ‘Populations cOmpared 2-tai1ed‘probability I and III .2194 I and IV .1893 III and IV .9969 Only frequencies of capture for animals removed from the enclosure were entered into the tests. Since 95% confidence intervals are common in the literature and research (rejecting only if the 2-tailed probabilityngS), it seems reasonable to conclude that all three papulations have the same distribution. Thus, the results of these three mark- recapture experiments can be compared and are expected to be quite similar. 'The Mark-Recapture_§imulationgypdel ”Parameter'estimation In order to use the simulation model, the spatial pattern 53 of the population must be determined. The.Hopkins and Skellam.method tested the randomness of the distribution of the animals throughout the enclosure, based on their centers of activity. The results from experiments I, III, and IV all showed the animals to be distributed in a random spatial pattern (23.65). All of the animals in each population were assumed to have the same bivariate normal home range parameters. Estimates of the pooled variances of x and y are: 'Experiment "variance'of‘x "Variance of y I 1.224 2.034 III 1.482 2.565 IV 1.660 2.018 This seems to indicate more of a circular than an ellip- tical home range. In the model, the general form of the bivariate normal probability density function is restricted to the case where p, the correlation coefficient between x and y, is equal to zero. The hypothesis that several sample correlations are possibly drawn from a common 9 was tested for each experiment (Table 3). Regardless of the results of the pooling test, a pooled rho was estimated and its 95% confidence interval calculated (Table 3). Emphasis falls on two facts (Snedecor and Cochran 1967): (1) In small samples the estimate, 5, is not very reliable. (2) The_1imits are not equally spaced on either side of r, a consequence of its skewed distribution. Table 3. 54 Significance of the test to pool the correlation coefficients of several animals, and an estimate of the common rho (r) with a 95% confidence interval. Test for pooling The common pooled rho ‘Degrees Significance ‘__ 95% Experiment of level of Estflmate‘ confidence freedom. the test (r) interval I 12 .197 .022 -.296,.335 III 16 .044 -.310 -.526,-.058 IV 17 .001 -.085 -.323,.162 55 In experiment I, the animal correlation coefficients can definitely be pooled and the common rho is equal to zero (P>.05). In experiment III, the correlation coefficients can be pooled at the .044 level. Zero is not included in the 95% confidence interval, but is in the 99% confidence interval (-.578, .024). Experiment IV should not be pooled, but if it was the common rho is equal to zero (P>.05). Taking into account the small sample sizes, the assumption in the simulation model that pa 0 seems justifiable for simplicity. If trapping and handling affect the catchability of marked individuals after their first capture, then the sampling will not be random within the marked papulation implying a trap-happy or trap-shy situation. Seber's test for equal catchability among the marked animals was applied to the data frmm experiments I, III, and IV. Only animals that were removed at the end of the experiments were included. The results were highly significant in all three cases (Pg.001 in I, P<.01 in III, and P<.001 in IV). Based on field observations and the estimates of y in the Tanaka estimator (it is a measure of the degree of trap-happiness (Seber 1973)), the trap-happy option, rather than the trap- shy option, was selected for the learning theory model in all three experiments. Estimates of the two parameters,H ucoaanomxm .HH manna 65 For simplicity, only one mark-recapture estimator, the nonparametric, was selected to evaluate the model. The non- parametric estimator was chosen because of its low bias and mean square error (MSE, which incorporates the sample bias and variance) in model simulations with trap-happy populations (Zarnoch 1979, Zarnoch and Burkhart 1980). Zarnoch (1976) tested the same five estimators shown in Tables 9-11 with computer simulations and concluded that, in most situations, the sample bias, variance, and mean square error properties of the nonparametric estimator were at least as good as the next competitor and often better. In simulations with a trap-happy population where 5 a .30 and a = .90 (close to the 5 a .25-.30 and a - .75-.82 in this study), the nonparametric estimator clearly had the lowest bias, variance, and mean square error. The nonparametric field estimates in this study also showed the lowest average percent bias (Table.l). In Figures 7-9, the field nonparametric estimates and the average simulation nonparametric estimates with 95% confidence intervals are plotted for trapping periods 2-10. The confidence intervals were calculated by multiplying the standard error of the bias by 2.262 (t9,.025). In another attempt to validate the model, ten replicate simulations of experiment III were generated utilizing all possible information available. In addition to using the complete estimates of a and pi 1 from day 10 (p a .19 and 66 44-— 40.. 36 32 28 _ Estimated Population Size 24 q l I I 1 T r I I 1 2 3 4 5 6 7 8 9 10 Number of Trapping Periods LEGEND -—- Field nonparametric estimate -— Simulation nonparametric estimate -W-95% confidence interval 4- Population size used in the computer simulation Figure 7. Experiment I (random spatial pattern). Field and simulation nonparametric estimates. 67 40 .. -36 _ 0) — N -.-4 m 32 _ s: o «4 U .- cs '3‘ - o ._‘ D“ o J.) m E. 24'? ti [:1 ..W I 20 _ ’ . I 16 T F l l I I l I 7 2 3 4 5 6 7 8 9 10 Number of Trapping Periods LEGEND —— Field nonparametric estimate - — Simulation nonparametric estimate - 95% confidence interval 4- Population size used in the computer simulation Figure 8. Experiment III (random spatial pattern). Field and simulation nonparametric estimates. 68 40 - 36 _ ‘— m d .5} m 32 .. a o -H .- U .53 a 28 _ 0.. o a. v d 3 g 24 - H U (D - Ed 20 _ l6 _ I I I I I‘ I I 1 I 2 3 4 5 6 7 8 9 10 Number of Trapping Periods LEGEND -——- Field nonparametric estimate -—-— Simulation nonparametric estimate ._ 95% confidence interval 4- Population size used in the computer simulation Figure 9. Experiment IV (random spatial pattern). Field and simulation nonparametric estimates. 69 A a = .73), the animals were placed in a fixed pattern cor- responding to their centers of activity. Otherwise, the model specifications were the same as in the previous ten simulations for experiment III. The field nonparametric estimates and the average simulation nonparametric estimates are plotted in Figure 10 for trapping periods 2-10. Again, the confi- dence intervals were calculated by multiplying the standard error of the bias by 2.262. 70 38 - w b u: c: l N m l 5: n: l Estimated Population Size l8 _ l4 - I I I I I I 1 I 1 2 3 4 5 6 7 8 9 10 Number of Trapping Periods LEGEND -——- Field nonparametric estimate -- Simulation nonparametric estimate -W- 95% confidence interval ‘G- Population size used in the computer simulation Figure 10. Experiment III (fixed spatial pattern). Field and simulation nonparametric estimates. DISCUSSION The mark-recapture experiments were conducted in the enclosure under field conditions as natural as possible. A population size of 40 animals is a density of 148 per acre. Christian (1971) estimated a density of Microtus pennsyl- vanicus in his study of 116-200 per acre; Hamilton (1937) estimated peak densities of g, pennsylvanicus at 160-230 per acre. So although the population density in this study is fairly high, it is certainly not unreasonable. The true population size was known exactly or within at most three animals throughout each experiment (except for four animals on day 10 of experiment IV). The 10 ft. trap spacing is closer than the spacing in many mark-recapture studies, but some animals in the population were still never captured during the ten days of live-trapping. The Kolmogorov-Smirnov 2-sample test showed that there was no significance difference between the frequency of capture distributions for experiments I, III, and IV. The Microtus in experiment IV were an unbiased, untrapped sample because they were born and raised in the lab. Therefore, the live-trapping of the animals released in experiments I and III seems to have produced an unbiased sample of the population. Although the animals had all been trapped once, they shared a common experience of one capture. 71 72 To procure Microtus for future studies, live-trapping, per- haps with a different type of trap than is to be used in the study, is an acceptable and easy method. Mark-Recapture Estimators (The discussion of results includes only experiments I, III, and IV of this study.)‘ The Lincoln estimator generally gave a negative bias, although most of the estimates from day 6-7 on had a bias of less than 10%. This corresponds very well to the fact that N' (Chapman's modification) is unbiased when nli-nzzji (Seber 1973). Using other known populations, Carothers (1973) found an average bias of 15-30%, Rice and Harder (1977) calculated an average of five surveys that gave a result very close to the true value, and Smith (1968) reported that the Lincoln values were high for male mice and low for female mice. For reasonable accuracy, several authors have suggested that at least 50% of the population must be marked (Strand- gaard 1967 and Roff 1973a). This corresponds to about trapping day 4 in my experiments. Seber (1973) concluded that of all methods in his book the Lincoln appears to be the most useful, provided that the assumptions underlying the method are satisfied and there are sufficient recaptures in the second sample. However, Cormack (1968) summarized by saying that although the Lincoln provided a simple and intuitively reasonable estimate of population size, there is a universal lack of faith in the assumptions. 73 The Schnabel is a multiple sample maximum likelihood estimator (Caughley 1977). The Schumacher-Eschmeyer is just a modified version of the Schnabel based on regression. It may often be more accurate (robust) when unequal probabilities of capture cause violation of the assumptions (Caughley 1977, Otis et a1. 1978, and Overton 1969). 'In this study, the Schnabel and Schumacher-Eschmeyer gave very similar estimates after trap day 2, with the Schumacher-Eschmeyer always being slightly higher. On every trap day in all three experiments, both estimates had a large negative bias; after approximately day 7 they gave estimates below the number of marked animals in the population. Carothers (1973) and Edwards and Eber- hardt (1967) both reported a large negative bias in the Schnabel and Schumacher-Eschmeyer when compared to known populations. Smith (1968) found that the Schumacher-Eschmeyer (Hayne) underestimated his true population of mice. Negative bias in the Schnabel has also been shown to occur in populations with heterogeneity of catchability by, among others, Seber (1970) and Zarnoch and Burkhart (1980). A Schnabel-type study on a closed population is affected more by failure of assumptions than a Lincoln-type study with a single_release of animals (Cormack 1968). The Tanaka estimator is based on departures from the underlying assumption that marked and unmarked animals have the same probability of capture (Seber 1973 and Eberhardt 1978). The overall reSults in this study were highly variable. 74 In experiment I, the Tanaka jumped from a large negative bias on days 4-6 to a large positive bias on days 7-10 (to 580 animals on day 8). In experiment III, the Tanaka con- sistently gave estimates below the number of marked animals in the population. In experiment IV, the estimates were close to the true population after day 5. Carothers (1973) concluded that the Tanaka estimates for his known population, though reasonable, generally have a considerable larger standard error and appear no less biased than "equal catch- ability" estimates. Few other examples are available in the literature. The Tanaka has the disadvantage of being un- defined for day 2 or any other day when ni - 0,mi - 0, the variance of x = 0, or the variance of Y - 0. The frequency of capture methods attempt to correct for violation of the assumption of equal probabilities of capture (Eberhardt 1978). The geometric maximum likelihood estimator gave very large positive biases in all experiments; even in experiment II where all animals in the population were marked by day 4. For known populations, Edwards and Eberhardt (1967) achieved useful estimates while Carothers (1973) concluded that the geometric is positively biased in populations with equal catchability and that the bias decreases as the variance of the distribution of probabilities of capture increases. Using computer simulations, Roff (1973b) and Zarnoch (1976) showed the geometric significantly overestimated population size; Romesburg and Marshall (1979) found the geometric 75 unbiased, but that the approximate confidence intervals for N were inaccurate. The nonparametric frequency of capture method has re- cieved little exposure in the literature. With a nonpara- metric approach, one does not need to assume how capture probabilities are distributed over the population. They are appealing because they are robust to specific assumptions regarding the experiment (Otis et al. 1978). In this study, the nonparametric gave good estimates of the true population after day 5-7. In general, the nonparametric went from a large negative bias at the beginning of the experiment to a slight positive bias at the end. Zarnoch (1976) tested five estimators (Lincoln, Schnabel, Schumacher-Eschmeyer, geo- ‘metric, and nonparametric) by computer simulation. This led to acceptance of the nonparametric as the "best" of the five estimators when the population possesses heterogeneity of capture probabilities. In most situations, its sample bias, variance, and mean square error properties were at least as good as the next competitor and often better. The nonpara- metric also fared well when compared to the geometric, Schnabel, and Schumacher-Eschmeyer in other simulations (Zarnoch 1979, and Zarnoch and Burkhart 1980). Confidence intervals (95%) for four of the estimators were presented in Figure 6. However, the positive correlation between the estimates of population size and their estimated standard errors is such that the variance is an insensitive 76 measure of accuracy of the estimate (Manly 1971 and Roff 1973a). Underestimates will appear more accurate than they really are; confidence limits cannot therefore be placed on the estimates (Roff 1973b). Roff (1973a) suggests that an estimate be considered reliable if its coefficient of variation is less than 0.05 (which it is on almost every day of each experiment in this study) and the confidence limits taken to be N :_O.1N. Robson and Regier (1964) suggest that a 10% level of accuracy be the minimum acceptable for management work. Considering a 10% level of accuracy as the minimum acceptable, the average percent bias in experiments I, III, and IV is examined for the six estimators (Table 1). The Schnabel, Schumacher-Eschmeyer, and geometric estimates are clearly unacceptable. The bias in the Tanaka is below 10% on two days, but this was achieved by averaging large positive and negative biases. The Lincoln has a negative bias of less than 10% on days 7-10; the nonparametric has a bias of less than 10% on days 6-10. There is a good deal of evidence that sex and age influence catchability, so independent estimates for such categories should realistically be made whenever possible (Eberhardt 1969). Although males and females in this study did not differ significantly (P>.30) in their frequency of capture, the total population size was estimated by combining the separate estimates of the male and female populations. The 77 combined estimates should be expected to be very close to the original field estimates (Table 2). The Tanaka shows a large discrepancy throughout all ten trapping periods, perhaps due to a large variability with small sample sizes. The combined estimates for the geometric give a slightly larger bias during the early trapping days. The estimates for the Lincoln, Schnabel, and Schumacher-Eschmeyer show a difference of less than 3.0 after day 4. The combined popu- lation estimates and the original field population estimates for the nonparametric are always exactly the same. This is because when the sexes are estimated separately, 6 is the 2. same for the males and the gemales for each value of 2. There- fore, for each value of 2, 5£< X Captured * Live-trapped during Experiment II 87 EXPERIMENT II .d Pe aP nP X X X X Sa r T m X.XXXXX XXXXX XXXXXXXXXXXXX X X 9 X XX XXXXXXXXX X XXX XXXXXXX X 8 X X X XXX XX X X X XXXX d m7 X XX X XXXX X X X XXXXXXX r e P6 X XX XXX XXX XX XXXXXXX n fiS X X XXXXX XX XX X XXXXX X a I. R4 X X XXXXXXXX X X XXX XX 3 XXX XXXXX X XXXX XXXX X 2 XXX X X XXXXXXX XXX XXXXXXXXX 1.. XXXX XXXXXXXXXXX X XXXXXXXXXX m FFMFFMMMFFFFMMMMMMMMMMFFMFFFMFF RXvnYl04174R167/829n31n4174R1611829n3104174§10716 g 4.455555555556666666666777777777 a#r1 0000000000000000000000000000000 ml .1111:11i1fl111:11i1f11i1tlai1fl1i1:11ilfl111fl1ilrl X Captured 88 EXPERIMENT III Tag Sex Tra in: Period Snap- # 1 2 3 4 5 6 7 8 9 10 Trapped 1081 X* 1082 1083 1084 1085 1086 1087 1088 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1116 1117 1118 1120 1121 1122 1123 1124 1125 1126 N NNN NNNN NN NNNN NNNN N N NNNNN N NN NN N N N NNNNN N NNNN N N N NNNNNNNN NNNNNNNNNNN NN NN N N N NN NNNN NNNNN NN wmmzmzmwzwzmzwzmzzzmmwxzmzzzwzmzmzzzwzwm xx xx xx xxx xx xx xx xx xx xx xxxxxx NNN N X Captured * Live-trapped during experiment IV 89 EXPERIMENT IV Tag Sex Tra opingVPeriod N 5 6 7 10 Snap- Trapped H H U! 51 333'31ZZZMWMMWKZKSNN'HMZKSK’UKZKzzzzzflimzflizgx NNNN N N NNNN NNNN N NNN NNNNN NNNNNNNNN N NNN N N N NN N N NNNN NN NN NN NNN N NNNNNNNNNNNN NN N N NNNNNN N NNNN NNNNNN N NN NN N NNNNNN N N NNNN N N NN NN NN X Captured 9O