FW (ZO) 353
Wildlife Management

ESTIMATING WILDLIFE POPULATIONS

    To establish and to appraise management practices, wildlife managers must estimate the sizes of wildlife populations. For game species, such inventories are ideally taken 3 times a year: during the breeding season, after the young are born or hatched and before the start of the hunting or trapping season, and after the hunting or trapping season. In practice, population estimates are usually done only once a year, at best, because of manpower and funding shortages.
    Wildlife managers use 4 general approaches to estimate population sizes of wildlife: total counts, incomplete counts, indirect counts, and mark-recapture methods. We shall examine each of these methods and detail some of their advantages and disadvantages.

COMPLETE COUNTS OR TOTAL COUNTS

    A complete count, or total count, counts every member of a population. Where populations of large species occur in open areas, such as waterfowl on lakes, seals on breeding beaches, or pronghorns on shortgrass prairie, aerial counts of most individuals are possible, especially with the aid of photography. Sometimes, wildlife managers can count deer in enclosed populations using a drive approach: a large group of people crosses the enclosure in a line, counting all deer that pass in each direction. Distances between the members of the drive crew are critical for success because all deer must be counted, even those hiding. Nonetheless, wildlife managers seldom use this approach because lack of funds or personnel usually make censussing an entire population impractical or impossible and, in addition, such an undertaking disturbs, and can even destroy, the population or its habitat. Even when used, this approach is usually expensive.

INCOMPLETE COUNTS

    An incomplete count involves counting part of a population and then extrapolating to the entire population. Quadrats may be established in a sample area and an attempt made to count all the individuals in each quadrat. A "deer drive" census, using large sized quadrats, can be an effective way to estimate deer populations on wooded areas. Stationary observers stand along 3 sides of a quadrat and count all deer leaving and entering the area in front of a drive crew walking across the quadrat from the 4th side. The total number of animals is then calculated as the sum of the animals leaving the area ahead of the drive crews plus the animals passing back through the drive line minus the animals entering the quadrat through one of the sides or through the drive line. As with complete counts, distances between observers and between members of the drive crew are critical for success.
    Strip censuses, roadside counts, flushing counts and booming or drumming ground counts are all incomplete count methods. A strip census can be used to estimate grouse population sizes. An observer walks a transect through a representative section of habitat and records the distances at which birds flush to either side. The population size, P, is estimated to be

where A is the area of the habitat censussed, Z is the total number of grouse flushed, X is the total distance walked and Y is twice the average distance from the observer to the bird when flushed. The fundamental assumptions of this method are 1) birds vary randomly in distances at which they flush, 2) birds are scattered randomly across the study area and 3) the average flushing distance is a good estimate of the "true" average. Which of these assumptions are likely to be met? What if some birds will not flush? A Wildlife Monograph has dealt extensively with these types of population size estimates (Burnham et al. 1980).

INDIRECT COUNTS

    As it is often impossible to obtain accurate, visual or auditory counts of the animals in a population, wildlife managers use indirect signs of the animals present as indices of relative abundance. An index of population indicates relative size of a population and shows population trends (up, down, stable) but does not provide an actual estimate of the number of animals. Examples of indirect counts include counting numbers of muskrat houses, counting scats (fecal pellets) of deer and rabbits, and counting numbers of nests or den sites in a given area. Sometimes counting the number of birds heard singing is considered an incomplete count and sometimes it is considered an indirect count. Which makes more sense?
    One can count fecal pellets of deer or rabbits along transects or in delineated study plots. In either case, the first thing to do after establishing the transects or plots is to remove all old pellets. Then, at a predetermined interval, count all new piles of fecal pellets. This is an index of the number of deer or rabbits in the area: the more animals, the more pellets produced. What assumptions does this index make?
    In those areas where muskrats build houses of vegetation in marshes, the number of active, maintained houses in a marsh year to year is an index of the number of muskrats: more muskrats make more houses. If, for a given area, one knows the average number of muskrats living in each house, then the number of houses can be used to estimate the population size. It should be remembered, however, that indirect counts are only indices of population sizes unless other information is known, such as the average number of muskrats living in each house.

MARK-RECAPTURE METHODS

    These methods are used extensively to estimate populations of fish, game animals, and many non-game animals. The approach was first used by Petersen (1896) to study European plaice in the Baltic Sea and later proposed by Lincoln (1930) to estimate numbers of ducks. Petersen's and Lincoln's method is often referred to as the Lincoln-Petersen Index, even though it is not an index but a method to estimate actual population sizes. (Should it not be the Petersen-Lincoln Estimate?) Their method involves capturing a number of animals, marking them, releasing them back into the population, and then determining the ratio of marked to unmarked animals in the population. The population (P) is estimated by the formula:

where M is the number of animals marked in the first trapping session, C is the number of animals captured in a second trapping session, and R is the number of marked animals recaptured in the second trapping session. This is derived from the equation:

which states that the proportion of marked animals captured in the second trapping session is the same as the proportion of total marked animals in the total population. Some of the assumptions behind this method are: 1) mortality is the same for marked and unmarked animals; 2) marked individuals do not lose their marks; 3) marked individuals are caught at the same rate as unmarked individuals (no trap-happy or trap-shy animals); 4) the population has no significant recruitment, or ingress (births or immigration); 5) the population has no significant egress (deaths or emigration); 6) marked animals mixed randomly with unmarked animals; and 7) each trapping session captures a representative sample of various age and sex categories from within the population. Think about these assumptions with respect to wildlife. Assumptions 4) and 5) taken together mean that a population is closed.
The Wildlife Society publication, Wildlife Management Techniques, provides methods of estimating 95% confidence limits for Lincoln-Petersen population estimates. Remember, the Lincoln-Petersen method provides and estimate of the true population size; it does not state the actual, or true, population size. By calculating the 95% confidence interval, a wildlife manage can learn how confident he or she should be of the accuracy of the population estimate. 95% of the time, the true population size will be within the 95% confidence interval.

Example of the Lincoln-Petersen Index Imagine that you set out live traps in a muskrat marsh. On the first day of trapping you capture 10 muskrats and put eartags in all of them; thus M = 10. On the second day of trapping you capture 8 muskrats (C = 8), 4 of which are eartagged (R = 4). So . . .

    To express your confidence in this estimate, you calculate the 95% confidence limits for your estimate. The upper and lower 95% confidence limits are

            upper: 59
            lower: 5.5

This means that if you trap muskrats in this way many, many times, 95% of the time that you obtained an estimate of 20 muskrats, the true population size would be somewhere between 6 and 59 animals. Since you actually captured 14 muskrats, you know that the population size is at least 14.
    Otis et al. (1978) developed sophisticated modifications of the L-P Estimator that attempt to insure that data are consistent with the assumptions. Several modifications construct stratified indices whereby data are collected separately for specific sub-groups of the population, such as age and sex categories or trap-happy and trap-shy animals. Thus, researchers must uniquely mark each individual captured and record information about that individual, such as sex and age. These modifications also insure an order of magnitude increase in the complexity of the mathematics and are available in computer software, such as Capture.
    When wildlife managers or researchers establish long-term population studies with frequent samplings, they can estimate not just the population size but the numbers of animals entering and leaving the population (Jolly 1963, 1965; Seber 1973). The Jolly-Seber Method relaxes the assumption that a population is closed. That is, the population can be open and have ingress (births and immigration) and egress (deaths and emigration). By keeping track of capture histories for individual over many capture sessions, ingress and egress can be estimated. Jolly-Seber Estimates can be calculated by hand but the exercise is complicated. Several software packages provide Jolly-Sever Estimates. The Wildlife Management Techniques manual shows how to make Jolly-Seber Estimates.
    Pollock and his colleagues (Kendall & Pollock 1992, Nichols, et al. 1984, Pollock1991, Pollock & Otto. 1983) developed the Robust Design for estimating animal populations, which incorporates capture-recapture methods for both closed and open populations. In its simplest form, the Robust Design uses an L-P Estimate for total population size during each of several, regularly scheduled trapping sessions and uses of the Jolly-Seber approach to estimate ingress and egress between trapping sessions.
    Krebs (1966) formally introduced the Minimum Number Alive (MNA) method, though it had been used by many researchers for years. The MNA method avoids the use of estimators, using instead the minimum number of animals known to be alive during a sampling period as a biased estimator of the population size. Hilborn et al. (1976) tested the sensitivity of this method to five important population parameters in mouse populations. They used simulation models and actual data to estimate the expected error on the MNA in actual studies. Their results showed that the MNA method, though clearly a biased estimator of population size, is an unbiased estimator of critically important population characteristics such as age distribution, pregnancy rate and lactation rate. In addition, in most cases an MNA population estimate is as good as or better than a Jolly-Seber Estimate.
    The Frequency of Capture Method (Eberhardt 1969) can be used when capture data are available over several trapping days. Plot the number of times that individuals are captured against the numbers of animals captured each number of times. For example, imagine that you live trap gray squirrels on campus over the course of 2 weeks and trap 15 squirrels only once, trap 10 twice, 5 3-times, 3 4-times, 3 5-times, 2 6-times, and 1 7-times. You plot these captures like so:


These data are then fit to a statistical distribution to determine how many squirrels were never trapped at all even though they were present. In this case, if we assume that squirrels were captured at random, the estimate of the number never trapped is between 16 and 17. The number never trapped at all but present is added to the MNA to give the Frequency of Capture estimate. In this example, the population size estimate is 55.
    Lastly the DeLury Method, first worked out for fish populations, uses kill data to estimate game populations. The critical assumption is that the number of animals killed per unit of hunting time is proportional to the population density; if this assumption is true, then each unit of hunting effort takes a constant proportion of the population. By plotting the kill rate (number of animals killed per unit hunting effort) against the total kill, it is possible to estimate the total population by extending the line to the X-axis. The value at the point of intersection is the estimate of the original population, Po. The validity of this method rests heavily on the assumption of each unit of hunting effort taking a constant proportion of the population. This DeLury Method also assumes that: 1) the population is closed; 2) animal vulnerability remains constant; 3) variable hunter skills average out; and 4) hunting is done individually. Of these assumptions, the one most likely to be violated is constant vulnerability. This can be affected by factors both intrinsic and extrinsic to the hunted population.

Example of the DeLury Method: Imagine that you are a wildlife biologists monitoring the game populations on a designated Wildlife Management Area. Assume further that your Area allows deer hunting for 7 successive days each year and that hunters must apply for a permit to hunt on the Area. Hunters must check in before and after hunting and must report their kill. On each of the 7 successive days, hunters hunt for a total of about 400 hours each day. You record the hours hunted each day, record the number of deer killed each day, and calculate the cumulative kill, producing a table like the following.
Day Animals Killed Hours Hunted Kill/Gun-Hr Cumulative Kill
1 100 400 .250 100
2 90 375 .240 190
3 81 410 .200 271
4 73 405 .180 344
5 66 390 .170 410
6 59 385 .153 469
7 53 395 .134 522

    You then graph kill/gun-hr against cumulative kill to estimate of the initial population size before hunting began. This it the graph you get.

You draw a line through your data points and extend the line to the X-axis. Your estimate of Po , the estimated population size before the hunting season started, is about 1350 on the graph. You also calculate a linear regression through the data points and calculate the X-intercept. Here you find that your best estimate of Po is actually 1335. Because 522 deer were killed, the population after the hunting season is estimated to be 813.

COMPARISONS

    Many researchers have used more than method of estimating populations on the same population at the same time. Let us look at 3 of these comparisons.
    Morgan & Bourn (1981) compared an Incomplete Count and an L-P Index of the giant tortoise population on Aldabra atoll in the Indian Ocean. To make the incomplete count, the atoll was divided into quadrats 100 m square. All tortoises were counted and marked in 5% of the quadrats and the total number counted was multiplied by 20. The L-P Index was made by counting marked tortoises on transect lines. Morgan & Bourn believed that almost all assumptions for each technique were satisfied, yet the estimates of the population size differed significantly: 87,300 for the incomplete count and 68,100 for the L-P Index. Evidently the assumptions for one or both methods were not met as well as believed. Morgan & Bourn had more confidence in their incomplete count estimate than in their L-P Estimate and cautioned readers about using elaborations on the L-P Index unless all assumptions are completely met.
    Mares et al. (1981) compared the L-P Estimate, the Schnabel estimate (a variation on the L-P Estimate that tends to underestimate population sizes slightly), and a removal estimate on a population of known size of eastern chipmunks in Pennsylvania. The chipmunks, they found, fell into 2 categories: those that readily entered traps and those that were hesitant to enter traps. Thus, all methods tended to estimate the population as being composed mostly of the former group and, thus, all methods tended to underestimate the total population size. The 95% confidence limits for the L-P Estimates on successive days always included the known population size, whereas this was not the case with the Schnabel method. They concluded that, for populations with unequal catchability, the L-P Estimate was the best.
    Boufard & Hein (1978) used 7 different methods concurrently for 6 months during 1976 to estimate the size of a gray squirrel population in Pennsylvania. Four of their methods have been discussed (at least briefly) in this handout: Schnabel, Frequency of Capture, Jolly, and MNA. Their results are as follows:



Month
Schnabel Estimate
Frequency of Capture Estimate
Jolly-Seber Estimate
MNA Estimate
June 115 ± 200 392 54 27
July 76 ± 19 96 38 52
August 85 ± 24 82 51 48
September 118 ± 84 35 31
October 159 ± 97 115 111 32
November 179 ± 59 195 20 34

    The Schnabel estimates were the most consistent, despite their variability, and the Frequency of Capture estimates were similar to the Schnabel. Twice (July and November), the Jolly-Seber Estimates were less than the MNA.
    It is obvious from these comparisons that estimating populations is an exercise fraught with inperfections. The best we can do is to choose the method or methods whose assumptions are best met by the population we wish to study. When possible, one should always collect data in such a way that more than one population estimate can be made. Often, the estimate made using the method whose assumptions are best met turns out not to work as well as anticipated. Having other estimates to augment that "best" one can save the day.
    Also note from these comparisons that Lincoln-Petersen Estimates are often reasonably accurate despite violation of their assumptions. I have noted this pattern and, therefore, tend to use L-P Estimates (or variants available through Capture software) over other methods when no other method is an obvious choice (for example, using the DeLury Method to estimate the size of a harvested population). Pollock's Robust Design is consistent with my informal observation.

POPULATION DENSITY

    Up to now, this handout may appear very straightforward. There is a problem, however. When you determine a wildlife population size, you automatically determine density. What I mean is, you determine the population size in a particular area - and that means you have determined

population
area

which is defined as density. Lloyd (1967) noted that the number of animals per unit area is a poor measure of density. Read that last sentence again. On first reading that sounds either downright stupid or at least confusing. But what makes the difference is from whose point of view the population is observed: yours or that of the wildlife. If you want to know how many deer are in such-and-such county so that the number harvested can be compared to the number in the population before harvest, then number of animals per unit area is what you need to know. But if you are interested in what the deer herd will do if you institute a management practice designed to increase the population, you need to know more than number of deer per unit area. Deer do not arrange themselves at random across the countryside. And what you measure as animals per unit area may not be what the deer perceive as the population density. Wolff (1980) showed that habitat patchiness and patchy distributions of snowshoe hares have a significant effect on hare population biology. The same is true of other wildlife. For most wildlife, high "densities" lead to reduced birth rates but the densities that are important are the densities that the wildlife experience, not necessarily what we measure. If deer perceive their population density to be higher than we estimate it to be, they will decrease birth rates more than we anticipate. Similarly, if they perceive population density to be lower than we estimate it to be, they will increase birth rates more than we anticipate. Let's go through an example.
    Let's take 18 points placed at random in 36 quadrats. These are random points, so we can take them from a random number generator which ranges from 0 to 99. Let the points be:
X Y
48 10
58 14
93 8
8 43
35 84
5 76
43 17
97 90
16 86
96 95
23 35
68 96
92 28
4 87
39 78
21 80
62 6
98 96

    If we plot them they look like this:

The points are located completely at random. For example, the point in the upper right corner was determined by the last pair of numbers.
    Notice how, in this example, the points seem to avoid the center of the graph. Something strange like this nearly always happens in a random distribution. The 'luck of the draw' usually leaves some regions relatively unpopulated. The region is (usually) different each time. A priori, each quadrat had an exactly equal chance of receiving each successive point.
    Now, we know that the mean "density", or number of points per unit area of our big square, is 18 points per 36 quadrats or ½ point per quadrat. We can get this in two ways. We can take the total number of points (18) and total number of quadrats (36), divide and get 0.5. Or we can take each quadrat, note its number of points, and then take the average of these numbers. It will come out the same: 0.5


    Suppose that these points represent the locations of animals and suppose that 8 or them decide to move closer to a neighbor (movement shown by arrows). We can show that the animals are no long arranged on the graph in a statistically random manner. The distribution has become patchy. Obviously, what a patchy distribution means from the point of view of the animals is that, on the average, an animal has more neighbors nearby than is would if the distribution were random. Lloyd (1967) developed a simple measure called "mean crowding", which is the average number of other individuals in the same quadrat averaged over all individuals. This is a measure of what each animal perceives as the density of animals around it. Open space with no other animals does little good to the seeker of open space if that open space is far away and hard to find and if all the space close by is filled.






    The animals in the random distribution had the following pattern: 1 quadrat had 3 animals, 2 quadrats each had 2 animals, 11 quadrats had isolated individuals, and 2 quadrats were empty. Thus, 3 animals each had 2 others with them in their quadrat; 4 animals each had 1 other with them; and 11 had no others in their quadrat. Mean crowding, , is therefore:
Number of animals in quadrat
Number of quadrats
0 22
1 11
2 2
3 1
=

    Thus mean crowding for this random distribution of animals is 0.55, which is pretty close to 0.5, which is the mean density. In a random distribution, mean crowding is almost always very close to mean density.
    In the patchy distribution, however, the 18 individuals were grouped differently: 12 animals each had 3 others in its quadrat and 6 animals each had 1 other. For this example, then, mean crowding is:
Number of animals in quadrat Number of quadrats
0 30
1 0
2 3
3 0
4 3
=

which is considerably greater than mean density, which is still 0.5. In a much larger population distributed at random with a mean density of 2.33, the average animal would be no more crowded by others than is an animal in out patchy population whose mean density is only 0.5.
The algebraic expression for mean crowding is
=

where x is the number of individuals in quadrat i and the summation is over all quadrats.

    So, you can see that for species that react to crowding, measuring mean density (the number of animals per unit area) can give you a very different measure of crowding than the animals actually perceive.
Now there is a major problem with mean crowding. Since Lloyd published his paper on mean crowding years ago, few wildlife biologists have realized its implications. Therefore, little work has been done to develop methods of measuring mean crowding in wildlife populations. Here are some major questions that, at present, have not been answered satisfactorily.

    How can wildlife populations be sampled to measure mean crowding?

    What is that proper quadrat size?

    How can we determine whether a wildlife species reacts to crowding?

LITERATURE CITED

Burnham, K. P., D. R. Anderson and J. L. Laake. 1980. Estimation of density from line transect samplingof biological populations. Wildlife Monographs 72. 202 pp.

Chapman, D. G. 1948. A mathematical study of confidence limits of salmon populations calculated fromsample tag ratios. Internattional Pacific Salmon Fisheries Comm. Bulletin 2: 69-85.

Davis, D. E. and R. L. Winstead, 1980. Estimating the numbers of wildlife populations. Pp. 221-247. In S.D. Schemnitz (editor). Wildlife Management and Techniques Manual. The Wildlife Society,Washington, D.C.

DeLury, D. B. 1954. A the Assumptions underlying estimates of mobile populations. In: O. Kempthorne (ed). Statistics and Mathematics in Biology. Iowa St. College Press. Ames.

Erhardt, L. L. 1969. Population estimates from recapture frequencies. Journal of Wildlife Management 33: 28-39.

Hilborn, R., T. A. Redfield, & C. J. Krebs. 1976. On the reliability of enumerations for mark andrecapture census of voles. Canadian Journal of Zoolology 54: 1019-1024.

Jolly, G. M. 1963. Estimates of population parameters from multiple recapture data with both death and dilutions - deterministic model. Biometrika 50: 113-128.

Jolly, G. M. 1965. Explicit estimates from capture-recapture data with low death and immigration - stochastic model. Biometrika 52: 315-337.

Kendall, W. L. & K. H. Pollock. 1992. The robust design in capture-recapture studies: a review and evaluation by monte carlo simulation. Pp 31-43. In McCullough, D.R. & R.H. Barrett (editors) Wildlife 2001: Populations.

Krebs, C. J. 1966. Demographic changes in fluctuating populations of Microtus californicus. Ecological Monographs 36: 239-273.

Lincoln, F. C. 1930. Calculating waterfowl abundance on the basis of banding returns. USDA Circular 118: 1-4.

Lloyd, M. 1967. Mean crowding. Journal of Animal Ecology 36: 1-30.

Mares, M. A., K. E. Streilein, & M. R. Willig. 1981. Experimental assessment of several population estimation techniques on an introduced population of eastern chipmunks. Journal of Mammalogy 62: 315-328.

Morgan, D. D. V. & D. M. Bourn. 1981. A comparison of two methods of estimating the size of a population of giant tortoises of Aldabra. Journal of Applied Ecology 18: 37-40.

Nichols, J. D., K. H. Pollock & J. E. Hines. 1984. The use of a robust capture-recapture design in small mammal population studies: a field example with microtus pennsylvanicus. Acta Theriologica, 29: 357-365.

Otis, D. L., K. P. Burnham, G. C. White, and D. R. Anderson. 1978. Statistical reference from capture dataon closed animal populations. Wildlife Monographs 62: 1-135 pp.

Petersen, C. G. T. 1896. The yearly immigration of young plaice into the Limfjord from the German Sea. Report of the Danish Biological Station 6: 1-48.

Pollock, K. H. 1991. Modeling capture, recapture, and removal statistics for estimation of demographic parameters for fish and wildlife populations: past, present, and future. American Statistical Association. Journal, 86: 225-238.

Pollock, K. H. & M. C. Otto. 1983. Robust estimation of population size in closed animal populations from capture-recapture experiments. Biometrics Journal, 39: 1035-1049.

Seber, G. A. F. 1973. The estimation of animal abundance and related parameters. Hafner Press, New York. 506 pp.

Wolff, J. O. 1980. The role of habitat patchiness in the population dynamics of snowshoe hare. Ecological Monographs 50: 111-130.