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Understanding Populations
Estimating the Number of Animals
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Estimating the Number of Animals
| Don't! |
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The variety of species and varied importance of each, variability in abundance among years and areas, frequency of catastrophic events, high costs, and the wide array of available sampling resources with unrealistic assumptions required for most of them make generalizing about animal population abundance treacherous.
The treachery and difficulties of estimation may be translated into "high costs per unit of information of low certainty." In periods in which funds for population estimation are limited, alternative concepts and approaches seem needed. Population estimates of game animals may not be needed or, if they are needed, not in a conventional form. A deductive, recursive method for estimating populations may be appropriate for the forest manager. First, the most difficult step to take is to decide whether an estimate is needed. Population estimates are not as important as they intuitively appear (or the time allocated to them in education and research (or in this book) might suggest). This logical step can only be taken if objectives are well defined. (Enough emphasis on objectives!) There are many good reasons why an estimate should not be attempted.
1. Rates of change are often of primary interest (e.g., the number harvested each year and the trend in harvest). The harvest is generally assumed proportional to the population, or sufficiently so over time, that with internal population compensations, the assumption is sufficiently safe. This will be discussed later.
2. Benefits, not animals, are the subject of managerial systems and the benefits are rarely proportional to or linearly related to the number of animals (Fig. 8.5). There are major benefits associated with seeing and supplying the needs of observers of rare species. Modest abundance is not interesting. Great flocks and large herds provide great benefit, but excessive animals are "vertebrate pests " (Chapter 11).
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| Fig 8.5. As population abundance increases, benefits to most groups of people change. There may be great benefits when species are very rare (e.g., bird watching and rare birds); little interest in common species; great enthusiasm for sky-darkening flocks; and real losses when populations are "too abundant." The same benefits (a) can result from several different levels of abundance. |
One reason for the need for population estimates is the assumed relationship of a positive correlation between numbers and benefits. This has been true, generally, for managing game species. Even there, a little study shows the error of the assumption. Because untrue, the wise measurement is of benefits, success, damage, etc., not the intermediary, the population.
To clarify briefly, the premise is that of introductory economics. At zero in Fig. 8.5 for an extinct species such as the passenger pigeon (Ectopistes) there may be some small benefit in the lessons that can be learned from them and the causes of their decline. There is likely to be some small amount of guilt. The net effect is unknown. Rare or threatened and endangered species ("T and E" among the glib) can provide many benefits. There is a level I call "sparse"; abundance is like that of some of the woodland salamanders. They are there but not seen; never in swarms. You can see one if you try, but it probably is not worth the effort and never a "sure thing." Some woodland warblers are equally sparce. You see a particular species every 2 or 3 years. A ruffed grouse or two in the watershed is interesting, not huntable, not even notable. At some point the abundance becomes interesting, and through management, it can reach levels that represent superior hunting. Past a point, another grouse is just another grouse. People become satiated.
Black bears (Ursus americana) are the clearest example of the curve for the woodland animals. A track in the snow gets a front page newspaper spot when they are rare. Hours of recreational hunting are provided during the increase or at mid level abundance. When allowed to become very abundant, then damage of a wide variety is well known and shooting is permitted year around.
While population estimates may be useful, the better approach is to monitor benefits (e.g., by questionnaires and damage surveys).
3. Under most state and federal law,"prevention of extinction" is the primary constraint on faunal management. Game species, almost by definition, do not approach this limit. The great expanse of managerial "space" between extinction thresholds and maximum densities provides managers with little legal risk. Sustained yield concepts in various laws are so vague and easily debated that they provide no genuine legal threat (thought they do constitute nuisance and court-costs threats).
4. Gross estimates which produce bad effects (i.e., expressions of probable disproduct), while unfortunate and to be avoided, are of only temporary consequence. The reasonable manager can only be so wrong in population recommendations; natural reproduction or mortality of populations will usually rectify decision effects in a few years (less than one human generation).
5. It is apparent that average citizens and citizen groups do not have clear objectives for the faunal resources of the forest. Efforts to get them to weight the relative importance of different animals have produced scant results. Weights (see "Values" in Chapter 4) typically will be assigned 0 to 3, perhaps 0 to 10 but rarely 0 - 100. [It seems likely that subgroups can and will assign weights after education, but not for the near future.] When weights will not be assigned by decision makers, it is possible to present several analyses showing results if certain high, low, and medium weights were given. This technique can be used to simulate the range of decision weights among groups.
6. Let us assume you could estimate an animal population exactly and that it was worth something - anything - say 50 cents an animal. Then the manager, rational financial person, must deal with alternatives and the interest rate. What is the proper interest rate to use? What will it be next month? How shall the difference between now and a year ago when the animal was born be accounted? What discount and depreciation rate is appropriate for the next 5 years? The uncertainty in interest rate argues against achieving great precision in the population estimate.
7. The variance associated with such studies, even the well-done ones, is so wide as to provide little power in decisions based on them.
8. Elaborate population estimate efforts rarely can be justified when it is known a priori that a sufficient sample size cannot be gained to provide a population estimate that is
Nevertheless, some small amount of hope is provided by Powers (1984) who argued that knowledge already available to the researcher (a priori) can be used in a Bayesian approach. When done carefully, he concluded, "... one may be almost 95 per cent confident that a calculated 95 per cent confidence interval contains the time value of the parameter under investigation."
As early as 1949 Hickey (1952) observed (as many others since then) that his efforts to study the population dynamics of 10 species of North American birds were limited because "all too often, the samples finally proved to be too small..."
9. Populations are very dynamic. Animals move and hide. They migrate. An estimate today deserves another one tomorrow. The occurrence of natural catastrophes is not unexpected. A poacher, a predator, an aberrant insecticide spray-plane - all make population estimates quite labile.
10. Suppose the truth about animal density could be known. The areas within which management is done is rarely precisely known. Island size is dynamic (Weatherbee et al. 1972) and forested areas change rapidly. Unresolved for most small species is whether ground-distance or horizontal distance should be used. I think volumes should also be used for most life groups; ground distance should be used for species occupying thin volumes. Area surveys are often very crude. Area estimates, even with electronic planimeters, are of limited accuracy. Map paper shrinks and swells during a day with changing moisture and temperature causing changes in large areas of hundreds of acres. True density (animals per unit area) multiplied by a poorly estimated area, results in a poor estimate of abundance. Small populations just introduced or recovering from endangerment can experience decreasing density as animals expand their range. This is merely a geometric relationship but easily overlooked, somewhat counterintuitive, and fodder for a hostile press. Criticism need not be borne exclusively by the population estimators; the surveyors and cartographers must share the burden.
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| Enormous transportation and logistic work is needed for many faunal sampling efforts. Often traps must be custom made. Movement of equipment to exact sampling points within a study area can be difficult. | |
Faunal system managers do not have to be apologetic about their estimates, lack of precision, or always be amused when a forester observes that, unlike animals, "trees do not move." Within the ranks of those working almost exclusively with trees, it is widely known that percent crown closure is a measure of limited precision but is a practical parameter giving insight into changes in crown density. Avery and Burkhart (1983:251) made a related observation about site index. They said that "In spite of the [5] foregoing limitations, site index is a useful tool because it provides a simple numerical value that is easily measured and understood by the practicing forester." There are abundant grounds for humility.
The Fate of an Estimation
In the spirit of "beginning at the end," the reader needs to be aware of the use to which some hard-won estimate will be put. The idle question, repeated over and over: "How many elk (or other species) do you have this year?" can wear on a manager. It can produce costly action to find out. The results can be reduced managerial stress and perhaps increased public support or a favorable attitude.
"How many deer are there?" is also the hidden question: "are there more during the coming hunting season and more this year than last year?" It is an odds-assessment question from hunters. "What are my chances this year?" is the alternative.
"What was the harvest?" seems clear. It hides: "Since I know the harvest last year, I will be able to tell if the population is decreasing (my only real concern) if you tell me the harvest this year." He estimates:
H* = Ht / Ht-1
and only will take action (not hunt in the area) if it is significantly less than 1.0. See CAP 9083. The rate of change is:
H** = (Ht - Ht-1) / Ht-1
If a manager cannot think of one action that will require allocation of some resources to cause a population to change in some desired way, there is no managerial need for an estimate. It is useless knowledge. Nothing will (or can) be done with it. This is the case with many forest birds, amphibians, reptiles, and insects. Even most conspicuous butterflies are out of range of our knowledge for effective action to cause specific, desired change. We just do not know what to do to increase species x. Species x can be dropped from our allocation algorithms.
A manager stocks animals. How many do we have this year? is a reasonable and expected question. It is unreasonable to spend time on making an estimate. It is reasonable to answer with an estimate or the counter question: If I told you precisely, what would you do with the answer? Because the probability is high that the action taken will be something other than "if low, I'll fire you"; it seems reasonable to proceed with caution and with an eye on efficiency, i.e., gaining useful information per unit of effort of scarce resource invested. The usual answer to the abundance question in a stocking situation follows a quick analysis of exponential growth
Nt = N0 (1.0 + r)t.
It can then be said that after 3 years with a growth rate of 0.1, our population of 200 should be 266 animals. If fortunate, and we are getting a rate of 0.2, the population could be as much as 345. We know they are present because we continue to see signs and that is increasing. We have seen young, so r is not zero. Intensive sampling efforts are unlikely to produce an estimate with confidence bounds better than 200 to 345. What might a manager do if they knew the number was 211 after 3 years? There are few forest situations having in their later ages such ponderous rates of ecosystem change that I can image any specific action being taken. Unless the knowledge of abundance is likely to cause a significant change, why"treat" a population of managers or inquirers with such knowledge?
There are some situations where a slow change might prompt action such as anti-poaching efforts, predator control, making food supplements, or trying to add special needs (e.g., water, salt, nutrients, special timely protection).
A population estimate for 1 year is almost meaningless. These are frequently made (or required) for environmental impact assessments. They are almost meaningless because:
None of these include the not-very- surprising occurrence of a local predator, a poacher, a unique storm, or the asserted "unusual" coincidence in time and space of, for example,
| the unusual is usual |
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If
Benefits = f (animals, value)
then when one such vital part of the natural resource equation as "value" has only one significant figure, it is logically and mathematically inappropriate to press to achieve more significant figures in the other parts of the equation, i.e., for an estimate of the present animal population.
I hate to discuss the perils of population estimation. To avoid estimating populations is very frustrating to students, managers, and the general public, and frequently unacceptable (in experiences to date). An alternative, then, is to seek the greatest precision in population estimates possible within the time and resources available. It seems too basic, so logical that population estimates are needed and should be made. The need is illusory; the reasons are often counterintuitive.
| heuristic convergence |
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An alternative called heuristic convergence is suggested. The approach is one of using multiple estimators to converge on the truthfulness of an estimation of abundance. The approach is isomorphic with the discussion in this book about validation of models and with the bases of epistemology. The approach is to employ a set of population estimation techniques, express maximum and minimum, weight the perceived relative goodness of each estimate, arrange estimates in rank order, and develop graphical and mathematical expressions of the population estimate. Downing (1980) used a somewhat related approach to deduce characteristics of a population that produced an observed population. The methods used are taken as a sample from a very large theoretical set of possible methods that might be employed. The weights assigned allow experienced wildlife managers to express the relevance of each factor in a complex set (e.g., truthfulness of reports, adequacy of sample size, relevance to a set of conditions on a particular area, mathematical appropriateness, statistical control, and others).
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| Fig. 8.6. Positioning various population estimation methods (ranking) and graphing the various estimates progressively with their confidence limits can produce a zone of likely population density |
The strategies of heuristic convergence (CAP11) are as follows and each will be discussed in turn:
| richness = number of species and subspecies |
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1. Richness, Abundance, and Density
Some population richness studies falter when the question of numbers arises because people do not distinguish among abundance, density, or richness. Abundance, the total count of all animals, say of birds, may be species specific such as in "what is the abundance of grouse this year in watershed x?" Others want to know density or number of a species per unit area. Even fish are estimated relative to the surface area of lakes and streams when relevant volume seems most reasonable and will reduce the variance in observations. Managers usually want to make a comparison between areas or before and after certain treatments or uses of land. The number wanted may simply be number of species. This is coded as the word richness. When a bird watcher says "We had a great morning! We got 47 birds," that means he or she saw 47 species. Though totals may be tabulated, the real interests are in daily richness or in why one area has a greater richness than another.
The manager may use richness estimates as follows:
The techniques for determining richness are much different than those for estimating numbers within a species. Expert observers usually are used. They go to different habitat types during the major seasons and observe, day and night, using binoculars, tracking, traps, cameras ... anything. The quest is to see one animal of a species, then to move on to the next. A checklist is an example of accumulated observations.
The wildlife manager must be careful in expressing conclusions. Not having observed an animal does not mean it is not present. Care is needed too because having seen an animal once does not mean it is still present. Even the simple question of presence or absence is difficult to answer.
Hayden et al. (1985) found richness well correlated with forest area (1.2 to 341 ha and > 1000 ha). They observed that in theirs and 100 other species-area studies (Conner and McCoy 1979) animal community composition and diversity are related to habitat area and some species may be limited if woodland"islands" are too small. They also observed that bird species response to forest stand sizes in Missouri was not the same as reported elsewhere.
Richness is used synonymously by some (e.g., the Nature Conservancy) with diversity. By preserving natural diversity is meant: do not let richness decline. Richness is discussed as an objective in Chapter 4. See CAP29.
In many faunal studies, it is impossible to get good estimates of abundance. The sightings are sparse, the animals inconspicuous, the observable behavior influenced by phenology, observations influenced by temperature and weather, and abundance may be a function of events 1, 2, or 50-years previous. Identical abundance levels are often achieved by several very different pathways or changes. Thus, counts to assess causative factors are almost meaningless.
Censuses of breeding birds conducted in forests show that a high percent of species are recorded only once, and only about 10 percent occurred in more than half of the surveys. A few species (less than 10) comprised more than 25 percent the total number of individuals in most forests. When placed in rank order of abundance, they are usually distributed as the negative logarithm. One implication of this observation for the manager is to study only the abundance and changes in a select few (10), very abundant species of birds. (Data on more than this number will provide little information for decision-making.)
One technique for estimating richness is to plot the rate at which new species are added to a list by an observer. The results appear as in Fig. 8.7. This is akin to estimating the maximum flood for an area. By plotting over the years the highest flood of record, eventually the curve levels off. Where it levels is the maximum flood depth. Analogously, a person searches through a forest for bird species. On the first day, 36 species are seen, on the next, 5 new species are seen. During the next 2 days no new species are seen but on the third, 2 new ones are picked up. See Kuzyakin (1961). The values of S are plotted and N can be estimated using the equation shown where t is time in days and b the rate at which the curve approaches the upper level. This estimation technique employs the rate of gaining species (new, unmarked animals). This is the catch-effort curve of fisheries and the Davis graph (discussed later) of mammal population estimation. When the rate drops to zero (approximately) it is not a bad assumption that you have seen or captured them all (or all that can be seen or captured).
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| Fig. 8.7. Recording new species seen per day (or year) (B) or per unit effort can provide a sequential maximum and can allow a projection to be made to the probable maximum number that may eventually be seen. |
An alternative version of this approach is to use a linear regression of the daily catch or observations (y) on cumulative catch (x). (See CAP110.) Where y = a - bx then setting y = 0 and solving for X, the intercept, the relationship is: xmaxR = N = a / b. If in one field situation the relationship was
y = 55 - 0.2x
then
N = 55 / 0.2 = 275 animals
The relationship tends to be curvilinear so using the logarithm of the daily catch or observation will probably give better results than shown above. This relationship, when plotted, has been called a Leslie-graph. The projection may suggest how many are really present, how many would be discovered in an area if observations were made for a very long time, and the upper intercept: the number present at time zero when observations began.
Another technique which has fascinated many wildlife managers is the species-area curve. The relationship is
S = cAZ
where the species, S, say of birds (or other taxon) are related to the area by an exponent (Z). C is a coefficient to adjust for the units of measure used for the area. Plotting data from studies of islands and large areas on logarithmic paper is an easy way to obtain this equation. The relationship deserves more study. It may be that number of species is related not to area but to the volume of the habitat (the product of area and height) or the number and volume of layers (Short 1984). For example, there are usually more species in a tall forest than in a short grassland.
What can a forest system manager conclude from the presence of one bird of species x? One that was present last year but not this? Will it be back next year? Even at the relatively constant environment of a back-yard feeder, bird richness varies. In one study of salamanders (Douglas Gill), the marked population disappeared for 7 years then returned to the breeding pond. It had been assumed that they had died! They had reached an adult stage in the surrounding forest. The richness of the area had not changed. The observed animals had.
Some species are seen upon every observation attempt. (Observations are not limited to sight but include sound (CAP66), scent, electronic and other sensors.) Animals are more or less conspicuous. A conspicuousness index can be created for each species, one that relates to the probability of seeing it or one that relates to abundance, or both. (See Seierstad et al. 1965.) Assuming that a study is being made of breeding birds or other animals that stay in an area, then a bird seen every other trip has a conspicuousness of 0.5. It can be refined to hours of watching, use of a mechanical call or"squeaker," and other controls. The objective is to see as many birds (or animals) as are present. What is the area richness? is the question. Using random plots to determine richness is the height of folly and a perversion of statistical as well as ecological understanding. Seek out representative habitats where each species is known! Stratify! Forest animals are clumped. See whether they are where they ought to be. Is vacant habitat filled? Has treatment reduced richness?
The conspicuousness of game animals can be approximated by placing cardboard silhouettes along a road side where a strip transect is run. The number of silhouettes seen compared to the number present gives a relation that can be applied to modify the number of animals seen. Where N is the estimate of the animals in the strip then:
N > A / (s/S)
where A is the animals seen, S is the silhouettes placed, and s is the silhouette seen. Because of movement differences, silhouettes seen will usually be a minimum of those present, so the estimate of the animals will be greater than the number calculated. For example:
N > 16 / (30/55)
N > 16 / 0.5454
N > 29
where 30 out of 55 silhouettes had been seen and 16 animals were seen on the strip. Each person may need to be "calibrated" in ability to see animals. That such differences exist is well known.
| calibrate the observer |
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Conditions during a day influence conspicuousness. Temperature, perhaps barometric pressure, time since rain, cumulative sunlight (CAP109) or degree days, all seem to influence animals. Obviously chronological and phenological time influence observations. The quality of an observation day can be derived using animals known to be present (based on repeated visits as done in breeding bird surveys or transect work) then using the characteristic of the day to estimate the probability of seeing an animal known to be present. This may be approximated by a multiple regression (CAP71) but may be better handled by an expert system (CAP54) using Bayesian probability techniques. Whatever the method, the observation quality or potential of a day needs to be computed.
Perhaps "needs to be" is too strong. The point is that unless it is done, then sources of variation which are known to exist creep into the estimates. Our techniques are not very good; the estimates we get have bounds of plus or minus 50 percent. Almost anything we can do to control the variables will improve our estimates. The number of things to control seems very large and difficult to process but computer aids (some on the CAPPER Disks) now overcome many such difficulties.
Association matrices are valuable for improving the rate (and reducing costs) of faunal surveys (see Green 1979:74). There are some species almost always associated with each other. You seine a creek and capture 7 minnows. The expert is surprised that you did not pull out species z. Why? Because z is almost always present when you have x and y in a seine haul. Expert systems (CAP54) offer a methodology for developing these association. Otherwise, a triangular probability-of-capture table may be created with all species listed in both columns and rows. Whenever species A is seen (or collected in any way) you enter a 1 in row A, column A. You then proceed to enter 1's in all columns for the species taken (seen) when A was taken. After all the samples are entered, the proportion in each cell roughly represents the probability that if A is taken, so also will be taken c, d, x ... or n. Combined probabilities may be computed. (If you see A and B, what is the probability of seeing F?) Later, with the table, a few field samples will allow potential richness to be quickly prepared and a probability curve constructed, if needed. Those few species that were occasional associates can be added, or not, depending on the use to be made of the results. Conspicuousness or ease-of-capture coefficients can be combined with the association matrix. These can be invaluable, when region specific, in making reconnaissance visits for clients, inspecting acquisition options, or in some self-sport, making improvements and assessing what you saw in relation to what you could have seen.
Comparisons of richness are available in CAP123. Perfect similarity is achieved when all species in Area A are in B and the converse is true. The emphasis is on species in common. Similarity indexes are shown in Chapter 6 under a discussion of a diversity index as an objective.
"Plants are indicators" is a tenet of ecologists. "Fauna are integrators" is also needed. The animals reflect ability to resist all combinations of assaults and insults possibly thrown at them over the eons. They have won at the great game (Chapter 17).
| fauna are integrators |
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| an animal population is the animals of a life group or an agreed-upon taxon in a specified area at a specified time. It usually includes a range or likely deviation in number |
What is so badly needed is very careful attention to what is badly needed. What are the basic objectives? What are the minimum data? Then the studies can begin. Usually, in my experience, when population abundance questions are raised, estimates of richness will suffice. In a related way, I hypothesize that the density (D) of select life groups will be shown to be related well to R, richness as in
| richness may suffice |
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2. Historical Data
Photographs of hunting party successes, diaries of travelers, low-volume local publications, and newspapers may provide insight into population levels expected. While the future may not be like the past, the information provided may explain part of the present and, most importantly, provide gross insight into maximum and minimum abundance and a datum for comparison with a gross estimate (made similarly) of the animal abundance today. Recording and storing for retrieval the conditions and methods used are more difficult than recording the numbers of animals. Vast, costly observations of fauna and their spaces are now unavailable due to mismanagement of information.
3. Rates
Simple 2-year comparisons of animal numbers suffice for some purposes. Usually a 3- to 7-year comparison is more useful and relevant than a 2-year comparison since a population or harvest is a function of events several years previous. Area harvests, in particular, are related to reputations and rumors that start in one year and last for surprisingly long periods. (See Harris 1986.)
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| Fig. 8.8. A sliding average can be used to observe general rates of change in harvests and other indexes to population abundance. The span of years (shown here as 1, 2, and 3 years) selected can influence the shape of the curve (CAP108). |
CAP108 allows a sliding average to be calculated. Fig. 8.8 shows how sliding averages suppress the extremes and express relations over a few years (the span). The span of years (say 3 or 4) may be selected. The procedure averages the data for the selected number of years, drops the most distant year, advances one year, and averages the next group of the same number of years.
CAP 110 allows a simple linear regression to be calculated for harvest data. Watt (1973:7) suggested this should be done with the natural logarithms because the data are correlated between and among years. If the objective is to reduce the population (as where tree reproduction is in jeopardy from deer) then the rate should be negative. If harvests have not been excessive in the past, the population should be stable or increasing.
Harvest is used to assess the population trend. An estimate is not needed. An assumption is made that the harvest is linearly related to the population and the slopes of these lines are not significantly different. This is adequate for 2-5 years but for longer periods, the assumption is highly unlikely.
Numerical rates of population change are, frankly, rarely of public interest. Most people want to know if there are"more","less" or"enough", not the change per unit time. Of course some want rates. The manager may wish for a comparison. How fast is the population changing relative to its potential rate. He or she may want
r = (Nt-1 - Ntt) / Nt-1
to compare to a theoretical exponential r*, the logic of which might be, grossly: if a female produces 2 to 3 young per year and I have a sex ratio of 20:100 then 2.5 x 100 are produced by 100 + 20 animals or about 2.08. If half of these die in the first 6 months, then the rate, r*, is 1.04. How different is r than r*? See CAP9084. The comparison may be in the ratio of r/r*. The manager then may seek to manage the many inputs and processes of the total faunal resource system (Chapter 5) to get this index to the desired level. If working with game, it probably needs to be close to 1.0. He or she has "a problem" in the difference or gap between the index and 1.0. "Problems," in general, and their magnitude, I think, are usually explained, for most forest faunal systems (and others as well) in the gap between the condition of the present system and the desired condition (in this example, expressed as 1.0). The greater difference, the gap, the greater the problem. If monetary damage were directly related to animal abundance, and the manager was trying to decrease abundance, then the index should be large and negative. Rates can be used to estimate populations before or after some period, then adjustments attempted.
Only rarely useful, a growth rate approach can, nevertheless, suggest population limits. Assuming historical data on populations such as harvests or introductions (i.e., numbers of animals released into the wild) and knowing maximum rates of production of young, it is possible to estimate limits. For example, CAP84 provides a means to enter known or estimated initial populations for several years. This establishes a pattern of growth. The simple relationship of
Ntt = No (1 + r)t
may be used where the starting population is No, r is the rate of change, and t is time in years. For example, if 100 birds were stocked and they increased at 3%, then, after 15 years, the maximum population would be 155 birds. For the number to be different, then there is likely some error in the numbers stocked, the rate, or time (See CAP114).
The manager needs to be cautious about using the rates above because typically when abundance is decreased (as in the case of damage complaints), the potential rate, R*, may increase (a feedback) due to new resources for the remaining animals. As they say in the woods,"the faster you go, the behinder you get."
A rate relationship for population removal can be used to estimate abundance. In many cases, the animals are actually removed by snap traps, hunting, etc., but they may be considered "removed" after they have been tagged or marked. The negative regression is cast into the future, and where it intercepts the horizontal axis is where there will be no more captures. All animals actually or hypothetically removed (in those last days) can be added. The reverse intercept, the intercept on the y axis, suggests some theoretical number that was present before any trapping was done (i.e., the conceptual time zero).
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| Fig. 8.9. A phase plane diagram of harvests or removals depicts the rates of change between years. Populations tend toward an equilibrium around some point on the 45-degree line. |
Harris (1986) defined a population trend line as the slope of the least squares regression (b) of the logarithm of the observed number of animals over time. He quantified trend line reliability by the standard error of the slope. After many analysis, it was observed that such lines never overestimate and sometimes underestimate the time variability of trend lines. He suggested increasing the number of replicate counts each year above those computed as required.
4. Plant and Animal Biomass
Reaching for outside limits of the population, it may be assumed that the animals in an area cannot exceed the plant biomass present. (Of course, we are not discussing waterfowl resting areas or roosts of migratory forest bird.) The temperate forest maximum net primary productivity is 2500 grams (dry weight) per M2/year. The tropical forest has 3500; the boreal forest, 2000; the woodland-shrubland, 1200; and savanna, 2000 g/M2/year. Lakes and streams have a maximum of 1500 with a lower average of about 250 (Whittaker 1975). Assuming animals might consume or otherwise use the entire amount in a temperate area, assuming 4 kilocalories per gram, and assuming an animal might weight 1 kilogram and metabolic weight is used (expressed as the 0.75 power of weight in kilograms) and there being 365 days per year, then:
N = (B * 4.0)/(70 * W0.75) * 365
= (2500 * 4)/(70 * 10.75) * 365
= 10,000 / 27,375
= 0.365 animals/M2
This results in very high density estimate based on daily maintenance and can be revised downward by incorporating food digestibility. It still yields an upper density estimate based on energy. This sounds absurd, and is, but it sets an upper limit. A manager will be correct in saying the population of the animal described is less than about 3,650 per hectare or 1,520 per acre. Recall that we are attempting to converge from limits (both small and great) onto a good population estimate.
A secondary, also gross, estimate is to divide the plant biomass by the weight of the animal. Physical laws (or the assumptions) would have to be violated to get a greater population density.
The so-called ecological rule-of-10 might be invoked suggesting herbivore energy on an area is 10 percent that of the plant's production due to entropy in the transition. Thus, an outside N might be 250 grams divided by the 10% rule based on energy; your calculations are based on biomass. Energy/unit biomass in plants and animals are not equal. Here we would get 250/1000 or 0.25 or 1 animal per 4 M2. Convergence using these numbers can begin. CAP626 allows these relations to be studied rapidly.
Reported animal biomass may provide other needed outer limits to population estimates. Dasmann (1964:91) presented a table showing wild Caribou on a Canadian tundra at 0.63 kg/ha (0.56 lbs/acre), (of course 0 is the lowest), and 11 ungulate species in Albert Park in African Congo, totaled 243 kg/ha (217 pounds/acre). Most entries in the table suggest that 15.7 kg/ha (14 pounds/acre) may be a reasonable number to use in forests. Errington reported 40.3 kg/ha (36 pounds/acre) at a peak muskrat population and 7846 kg/ha (7000 pounds/acre) was reported during a brown lemming population peak. CAPW01 allows these relations to be studied.
5. Maximum Feasible Volume
Assuming animals are stacked side-by-side, end-to-end over the entire area, another maximum estimate may be obtained. (See CAP11.) Where an animal is 8 cm wide and 15 cm long, then there can only be 83 animals per square meter. Absurd? Perhaps, but the outer ranges of the estimate are being gotten as we continue heuristic convergence.
| estimating area is as important as estimating animal numbers |
Increasingly, state wildlife agencies are developing wildlife information systems. Within these are often reported maximum densities found in the literature. An analysis of an unknown deer herd might proceed as follows:
One study of a local density showed 1 per 533 acres (1/215 ha) shortly after deer stocking began. A nearby study 20 years later showed 1 deer per 7.1 acres (1/2.9 ha) in a large enclosed military area. In similar habitat in a nearby state a density of 1 per 22 acres (1/8.9 ha) was observed. Less than 100 miles, another study suggested 1 per 21 acres. The density must be lower than these numbers since there is little sign of forest reproduction being influenced, no conspicuous browse line, and no reported die-off or disease incidence. Local US Forest Service estimates are given as 1 per 213 acres (1/86 ha) but this seems low given the foraging that is seen and the number of tracks observed. State wildlife agency staff estimated populations on two nearby similar acres as 1 per 128 and 1 per 80 acres. The high and probable density for the entire region has been estimated at between 1 per 26 acres and 1 per 64 acres. Putting these in order (Table 8.6) and assigning a weight representing confidence in each based on the time and means by which the estimate was probably made, it is possible to get an operational estimate. From Table 8.6, the estimate is 1 deer per 81 acres (1 per 33 ha) (using CAP629).
7. Relative Abundance
A population of more than 1000 animals is very hard to comprehend. In general, whether it is 1009 or 1116 is rarely of any interest. It is easy to judge that there is no significant difference. I am not sure what the mental calculus is with large number phenomena. Starr (1969) suggested non-linear mental concepts in his discussion of risks. I suggest that relative population size is relevant, much like that for hydrogen ion concentrations in chemistry and decibel levels in acoustics. The relationship suggested is
P* = logeN
and is shown in Table 8.7. When N is 1, P* is 0; when P* is 1, N is 2.718. When N is greater than 22,026 then P* = 11. The protection needed by animals is inversely related to P* (see Salwasser et al. 1984). P* may be viewed as an expression of rarity or super abundance. It provides a continuous relation, avoids arbitrary grouping, and the groups that do emerge as whole numbers (Table 8.7) have intuitive appeal. It is mathematically tractable. It probably has special relevance when used with diversity estimates when the natural logarithm is used.
| Table 8.6. Deer population estimates demonstrating use of the literature to obtain most-likely estimates. Density is expressed as one deer per x acres. A weighting factor expressing confidence in each estimate may be assigned. | |||
| Number | Area (acres) per deer |
Estimate of Weight |
Comment |
|---|---|---|---|
| 1 | > 550 | 1 | Possible; track and sign abundance |
| 2 | 553 | 1 | Report 1 |
| 3 | 213 | 2 | Report 2 |
| 4 | 128 | 8 | State Game Agency staff |
| 5 | 80 | 10 | State Game Agency staff |
| 6 | 64 | 8 | Regional: likely |
| 7 | 26 | 8 | Regional: upper |
| 8 | 22 | 5 | Nearby state |
| 9 | 21 | 5 | Report 3 |
| 10 | 7.1 | 1 | Local military area |
| Table 8.7. Representative values and boundaries for relevant abundance groups (P* = logeN) | |
|---|---|
| N | P* |
| < 3 | 1 |
| 3 - 7 | 2 |
| 8 - 20 | 3 |
| 21 - 55 | 4 |
| 56 - 148 | 5 |
| 149 - 403 | 6 |
| 404 - 1,097 | 7 |
| 1,098 - 2,980 | 8 |
| 2,981 - 8,103 | 9 |
| 8,104 - 22,026 | 10 |
| > 22,026 | 11 |
(See CAPW02 and Moen and Severinghaus 1985.) When P* is 7 or greater a viable population probably exists; agencies may be expected to maximize P* of endangered species, minimize it with pests.
A key reason for suggesting "relative abundance " is that precise abundance is rarely available and, as suggested, rarely used.
| Note: A set of magic numbers of increasing complexity. |
| deer (number) deer/acre deer/forested acre deer/hardwood forest acre deer/hardwood forest acre/hunter deer/hardwood forest acre/hunter day deer/hardwood forest acre/successful hunter days by people age 20-34 |
Numbers of animals harvested are often a key index to management success. The number is a system performance index but may not be useful alone. It probably needs adjustment based on area, then habitat area, then hunters, and available time and legal weapon type. An index, some K, might be buck harvest per 1000 rifle-using hunter/1000 hours/1000 hectares. The magic number easily becomes so complex that no one is interested in discussing its meaning or derivation. It is symbolized, say by M*, and then the trends, changes over time, or relative magnitude of M* can be discussed because there is tacit group recognition that such an expression has within it the salient factors. The resulting number may have little intuitive meaning, thus its "magical" nature. The phrase "magic number" is used to express the ways of combining variables into a progressively more complex variables, useful for certain specific purposes. Plotting the index over time may provide a notion of how a system is doing. It may be the best that a manager or group for whom he or she works can produce as an objective or one that is communicable. Since data for the modern wildlife area are so abundant, methods of processing them are needed. Screening, collating, arranging, and relating data and information are necessary. Statistics combined with graphs are ways of attaching extra meaning to data and of converting data to information (CAP501). Examples of variables with high-information content are: GNP, kcal/m2/day, forage/acre/year, and game meat/sq. mile/year. These can be considered system performance measures, which are synthetic expressions of system output. Often these are not adequate. Better than nothing, such numbers are highly integrative and will, over time, cause change in action as people begin to work on the components to make K increase. There are major cautions about magic numbers.
9. Home Range
Closely related to both the volumetric and biomass approaches already suggested, using home range to estimate populations gets close to the actual abundance and is not intended merely to establish extreme limits. Home range is the area over which animals normally travel (Burt 1943). Each animal may have a living area and a defended area. "Home range" and "territory" are, respectively, population concepts. All are confounded ideas because of the way the terms are used, sampling difficulties, the amount of time or intensity of animal use in each area, overlaps in space (that may not occur in time), and "holes" in areas not used at all. There are sex, age, as well as seasonal differences. Community succession changes the food base, thus studies of the same area can hardly expect to be repeated for comparison. "Cruising radius" was once used, but it has been largely displaced by area concepts, even though these are, for the above reason, nebulous. A center of activity can be estimated and a pattern will appear as in Fig. 8.10 or as a mountain with highest points being where activity is greatest.
Population estimates based on home range, are thus of limited usefulness but, I think, one of the most practical for most managerial work. This concept, as fuzzy as it may sound, matches well with the fuzziness of objectives, the limits of the group of potential resource users, the boundaries of the relevant area, and the indefinite risks to the decision maker of being wrong about the estimate. At least the approach to estimation should be used to confirm or question other estimates.
Home range is usually measured by trapping over a gridded area or by radiotelemetry (Kenward 1987). Hayne (1949) argued that relative frequency of capture per trap is not a dependable index to normal activity because of traps in one part of a range may interfere with capture in another part (trap "competition"). He advanced the "probability of capture" concept depicted in Fig. 8.9 and developed in Jenrich and Turner (1969), Van Winkle (1975), and related concepts in Dixon and Chapman (1980) and Getty (1981).
 |
| Fig. 8.10.The "height" of each bar represents captures or observations at each point on a trapping grid. A "normal" or bell-shaped distribution can be fit to these points, with the vertical Y as the activity and X and Z as map coordinate axes. The average individual living area, assuming the observations are merely samples from a very large number of points, may be presented as a surface. |
 |
| Fig. 8.11. A general diagram of the relationship between distance from a center of activity and the times an animal is caught. Distance becomes radius in computing occupied area per animal leading to population estimates based on individual or group areas within a larger management area. |
Jenrich and Turner (1969) defined home range as the circular or elliptic area of the smallest region which accounts for 95% of an animal's utilization of a habitat.
For general computations of home range, a circular area may be envisioned where
A = 
(D/2)2
with D being the diameter. A population estimate may be obtained by dividing the average home range size into the total area. Since circles do not "pack," there being overlaps or interspaces, an average hexagon may be used. It can be envisioned as a honey-comb pattern over an area with
A = 3.4641 (D/2)2,
D being the distance from side to side. A 3-ha home range, if the range is circular, has a diameter of 195 meters. The distance from side to side of such a hexagon is 186 meters.
Imagining that an approximate home range area is computed for several animals in an area, that it is 95% and hexagonal, that 2 animals occupy each home range (this varies by species, season, etc., etc. but we need to start somewhere), then the question is: how many of these average size areas can be packed into the total area? The results is a gross population density estimate.
 |
| Custon made traps suitable for feral cats, opossums, raccoons, weasels and cottontails have been used in home range and other studies. |
Lindstedt et al. (1986) studied mammalian carnivore home range areas (A) (from 15 to 38 studies) and found that for
All carnivores: A = 170 M1.03
r2 = 0.66
Carnivores at or below 45° latitude: A = 115 M0.94
r2 = 0.61
Carnivores at or above 45° latitude: A = 339 M1.08
r2 = 0.79
Harestad and Bunnell (1979) studies, (revised as shown below in Lindstedt et al. 1986) provided the relations of:
Carnivores A = 137 M1.37
r2 = 0.81
Herbivores A = 2.71 M1.02
r2 = 0.75
Omnivores A = 3.4 M0.92
r2 = 0.90
CAP630 allows the body weight (M) in kilograms to be related to home range area A in hectares. By entering a species' estimated body weight, an estimate of home range may be determined. McNab (1983:683) observed mammalian home ranges proportional to M1.9 and added that in animal groups (e.g., wolf packs), the life group is the foraging entity, not an individual. Economies are achieved at certain optimum group size. Thus,
f = n (1/n)0.75
where f is the ratio of the metabolism of a group of n animals in a group to that of an individual. When for example, n = 6, then
f = 6 (1/6)0.75 = 1.56.
The home range occupied by 6 individual animals is 1.56 times that of an individual, not 6 times larger. When average group size (n) is estimated, then density estimates based on home range can be increased by the group efficiency modifier as
N = Area / (Home range x (1/n)0.75)
Holling (1992: 472) showed how birds and mammals of all trophic levels in all landscapes relate to their home range directly as body mass, namely in:
A = t M(0.67 + 0.33 D)
where t is a constant based on energy metabolic efficiency and available energy, M is body mass, and D is a fractal dimension, probably 1.3.
| Decision power: Home range is a function of animal weight. Populations are a function of home range. |
Damuth (1981) found very strong relations between reported population densities and body weights of mammals. The relationship was:
log D = -0.75 log W + 4.23
where W is weight in grams and D is the estimated number of animals per square kilometer. (See CAPW03.) Damuth's method perhaps deserves note as a separate population estimation "method" for comparisons and use in heuristic convergence.
10. Subjective Estimates
It seems unwise to disregard years of concern, careful observation, and the integrative efforts of the fertile minds of field workers in their intuitive and directed efforts to estimate populations. There is general disdain for "subjective" work, but it is baseless since the very roots of statistical work are the subjective "confidence level" and "allowable error." Precedence is widespread in questionnaire work. Deciding importance of research topics, on experimental strategy, on risk in decision making - all are subjective. A hasty observation in the field with a limited or faltering instrument seems, somehow, more objective than to ask a well-phrased question of a thoughtful, elderly woods watcher. Perhaps our perception is flawed. Expert judgements have been successfully used in industry and military operations. It is likely they can be used in the convergence strategy being developed herein.
| In my experience, bad experiences result from ignoring the experience of field workers. |
N = (aNa + bNb + cNc) / (a + b + c)
Future work may be justified in studying the weight of 4 used in time-to-complete-a-project studies and assigned as the value of b. It is also likely that low estimates are more precise than high estimates.
Because staff are often transferred, expertise may be limited, so it is appropriate to weight each participant in the activity. A weighted median estimate is obtained and the individual highest and lowest estimates may be compared with other estimates. See CAP632.
11. Harvests
By assuming that a constant proportion of the population is taken each year, or that hunters have a constant average success rate, then hunter and trapper harvests may be used as indexes to the population, especially its trends. The constant proportion assumption is tough to make. Models are clearly needed because annual harvest is not proportionate to the population. Annual harvest is very much a function of the harvest in the previous year, season length and type, number of licensed users, weapons used, weather conditions during the season, land use change since the previous hunt, catastrophic events, and often food supplies in the area two-years previous. Nevertheless, the annual harvest is used, no matter how erroneously, and adequate analyses (e.g., to produce a "relative harvest index," a magic number) are rarely made. That they seem to "work" says more about the managerial system than the population.
One use of such data for an area is to put harvests in rank order. The usual form is that in Fig. 8.12. The projection to a maximum may provide insight into the population. By
 |
| Fig. 8.12. By putting harvests in rank order over many years, a projection to a maximum may be achieved. The projection to the minimum of zero rate of change suggests a fundamental harvest result, no-matter-what management occurs. The lower projection may be the best basis for comparing management effort. The difference produced per management dollar is the important criterion for most evaluations of performance. |
similarly placing harvests in adjacent areas in rank order, comparisons may be made. Often annual results from a management area can be compared to the maximum and minimum of all contiguous or nearby areas.
One obvious result of studying harvest is that the population had to have been at least as large as the reported harvest. If that harvest is relatively constant, then a minimum population estimate is the reported harvest. The density is similarly determined from an estimate of the area from which the harvest was made. Refined estimates of actual areas used by deer or other animals can improve this estimate.
An alternative is to study the dynamics of the population, perhaps with a computer model, and determine the proportion of the population that may be harvested and yet achieve a stable population. For example, if the harvest has been stable over several years and the computations suggest that 20% can be harvested and achieve a stable population, then if the harvest is 1000 animals, the estimated population at the beginning of the hunting season is 1000/0.20 or 5000.
If only the annual "surplus" or production is taken in harvests, then it seems feasible to look at that estimate and then to estimate the pre-season harvest. At its most simple, after mortality and just prior to the hunt, the young:female ratio is 1:1, the sex ratio is 100:100, and 1000 animals are harvested, then the pre-hunt population would have to have been 3000 animals (See CAP633).
These ideas are easily combined with adjustments to produce magic numbers associated with harvest per unit of habitat and hunter days, and with concepts of maximum possible densities (and maximum reported harvests).
12. Representative Habitat Sampling
In some cases there is no information about a species or its preferred habitat. I caught only one jumping mouse (Zapus) on one night in thousands of trap nights of sampling in an Ohio forest. I still do not know about its habitat, except for that one observation. In most cases, however, almost anyone who has worked in an area for a few years and observed and listened, can select a "good" place for sampling a species. They can better pick a bad place (i.e., one of expected low density).
Any estimating procedure can be used. Based on the best perception available, it is best to select 3 good (biologically rich) and 3 bad sites, and use the upper and lower values within each. The intent is to bracket the population density in a realistic fashion, and to narrow the wide bounds usually resulting from mark and recapture estimation procedures.
Note that the procedure is modified stratified sampling depending on knowledge of the population (without an a priori estimate) for deciding on two strata.
The procedure, while of very low cost, can be dangerous. Animals often do not "like" the same areas that people think they do. The method assumes some knowledge of habitat preference based on prior trapping, tracks in sand or snow, and sightings. Recently, after years of experience, I went to a very "bad" place, live trapped, and caught more deer mice, Peromyscus leucopus, than I had ever caught per trap in one trap night in my life! I did not try to trap an especially good spot that night. I cite this only to suggest the hazards of this or any estimation technique. These creatures vary in area, time, phenology, predator pressure, nesting, migration, etc., etc. and most are highly clumped or aggregated. Except in the most homogeneous environments (cultivated) the areas that are used vary exponentially in the pattern of the species-area curve (CAP2041) with a few small areas having excellent resources and then many larger areas having a low probability (0.001) of containing any animals of the species or life group being studied.
13. Habitat Adjustments
When 1000 animals are harvested from an area such as a county, the usual assumption made for the constancy of the habitat is that areas advancing into desirable plant stages are equal to those going out of the stages or are being taken out by developments. This is often difficult to prove but satellite images now provide the means for testing and correcting this assumption. The better analysis is to adjust the harvest data for area. This may be done by reducing county, region, or wildlife management area data to make it as descriptive of realistic usable animal volume as possible. This includes subtracting area of towns and cities, roads and some rights-of-way, mines, lakes, and in some cases military or secure areas.
Another adjustment is to average (having done the above) adjacent area harvests with the area to begin to adjust for harvests at the borders of the areas. A weight can be assigned as a function of the length of each contiguous border.
Cover maps can provide detailed habitats and these may be used in regressions of harvests over 5 to 10 years with habitat type changes over the same period. Once the relations are well established, then adjusted harvest information may be used as described above.
14. Illegal Harvest Adjustments
It is truly amazing how little research has been done on illegal kill, either in- or out-of-season. (See Chapter 14 on Wildlife Law Enforcement.) People have opinions about it, but they are almost groundless.
One assumption usually made is that illegal kill is proportional to the harvest. To know harvest is to know the illegal kill. Another hypothesis is that it is proportional to the hunting population, another to the total human population. Another is that it is proportional to the population abundance. It is probably a function of all of these and, itself a system, as complex as the system by which harvest estimates are made.
Having adjusted for habitat and other factors, it is feasible for a faunal system manager to inquire: if the harvest is stable and I estimate that a number of animals equivalent to the annual production is being legally harvested and I know poaching does occur, then where is my model in error?
Harvest data need to be expanded to include illegal kills as well as crippling loss. McCaffery (1985) told of the need to partition causes of hunting-inflicted losses, especially crippling mortality, in a systematic way. Reducing the various losses requires different solutions. Using losses to adjust population estimates also requires precise definitions. McCaffery provided these as:
| Crippling mortality (or irretrievable legal kills) includes losses of legal ... [animals] as a result of delayed mortality or the hunter's inability to find a dead ... [animal]. (A legal ...[animal] is one the shooter is eligible to bag.) When stated as a proportion of the retrieved harvest, these losses are an expression of relative hunter skill or equipment efficiency. Crippling losses may account for the majority of dead ... [animals] remaining in the field after an either-sex hunt, but may be only a small proportion of the unrecovered ... [animals] following a restrictive ... hunt. Abandoned kills include otherwise legal ... [animals] that were found by a hunter but deliberately not retrieved. This situation usually occurs for the purpose of hunting a larger ... [animal] or discarding a 'shot-up' ... [animal]. Reliably separating abandoned kills from crippling mortality in the field may be difficult, but a brain-shot ... [animal] is clearly not counted as a cripple. Abandoning kills is unethical, if not illegal, in most states. Illegal kills include all dead ... [animals] of a protected sex, age, or ... condition that are accidentally or deliberately shot during the hunting season. Illegal and abandoned kills, when stated as a proportion of the legal harvest, are an expression of ethical conduct and effectiveness of hunting season regulations. Quantifying illegal kills (except in enclosures) can be difficult because many may be removed from the field depending on local hunting situations. [Out-of-season removals require other clarification.] Reported hits or wounding on hunter questionnaires include actual and suspected hits. Actual hits may range in severity from superficial (e.g., hair or antlers) to mortal. These types of data are not very useful for projecting losses from the population. |
| The possible expressions of illegal take and how population estimates might be adjusted | |
| Estimated Total Population1 = | (estimated population x (1.0 + estimated proportion in legal harvest)) + (legal harvest x (1.0 + estimated proportion of the legal harvest also taken and not reported)) |
| Estimated Total Population2 = | (population x (1.0 + proportion in legal harvest)) + (population x proportion of population taken illegally) |
| Estimated Total Population3 = | (population + (hunters x proportion successful)) + ((hunters x proportion taking one illegal animal) x average number of animals taken per hunter-poacher)) + ((non-hunters x proportion taking illegal animals) x average number of animals taken per non-hunter poacher)). |
CAP634 allows adjustments of estimates based on poaching to be compared and studied. Research on this important illegal harvest factor is badly needed.
15. Total Counts
Although people have stated the need for known populations against which population estimation techniques may be tested, few such observations have been found or created. Where efforts have been made (e.g., deer enclosures) the results have been expensive and the management of the herd and fence (preventing ingress or egress) have been difficult (e.g., trees fall on the fence and a two-tree-wide right-of-way inside and outside the fence is expensive to create and to maintain).
Deer and other forest animals cannot be sensed, cost-effectively, from aerial monitors. Perhaps infrared may be perfected, but even seals on snow and ice have been difficult to count. Aerial surveys are widely used and a vast literature on flight widths, lengths, altitude, and sampling strategies has been developed. Aerial survey has been extensively used for sea mammals, elephants, and plains animals, but forested animals are covered and some reports suggest that they "hide." Repeated photographs may be usefully interpreted in some areas. (See bounded-count method later and CAP122). Animals in migration may be counted by devices or from aerial photographs. The situations for total counts are very few. They are so few and the chances for gaining resources for a thorough test of a variety of techniques such as discussed herein are so slim that wishing seems naive, akin to hoping for an antigravity drug.
Beaver food caches are counted from aircraft, then each is assumed to support a fixed number of animals. Cotton (1990) studied reports and found the number varied quite widely.
16. Track Counts
Overton (1971) showed the derivation of the track count method that evolves to the simple relation of
D = t/d,
where density (D) is a function of tracks per mile and d is the diameter of a circle describing an average area of activity of each animal. The problem shifts from population estimation to estimating d but this is possible using radio telemetry, grid trapping, and other techniques. Roads in an area are dragged smooth and all tracks counted to get t. Density is estimated in animals per square mile. See CAP631.
In another approach, tracking stations are placed in the field, usually 30 to 50 every mile along secondary forest roads. They are smooth areas, even with placed sand or agricultural ground limestone, with a scent-bearing stake driven in the center. Johnson et al. (1987) placed plots along 4 transects cross-country, about 160 m apart. Trends in numbers of tracks are observed. The relationship between track frequency (F) (plots with tracks/total plots) and track density (d). Thus,
F = 1.0 - e-d
and e is the base of the natural logarithm. Scenting is a problem because some animals may be attracted, others repelled by scents selected. Johnson et al. (1987) found no significant difference between scented and unscented stations. Density can probably be related in a simple regression to F. There are strong seasonal differences in tracks suggesting factors other than mortality of animals is involved. Species-specific maximums may be most appropriately used to assess and compare area populations. About 200 stations placed cross-country observed for 5 days were recommended.
17. Drives With Track Counts
While "drives" may be made with other animals, they are primarily used with deer. People form a line through the forest (minimum of 340 acres, 137 ha). A supervisor or team leader(s) walks behind them and supervises the alignment. They walk a straight route, counting all animals that they see that run past them to one side (to avoid double counts). They drive animals to a row of stationary observers at the end and sides of an area. These people also count animals that move past them. Where roads are in blocks, roads may be cleared of tracks and then tracks counted there, reducing the number of observers needed. Keeping a line, preventing gaps through which deer may escape uncounted, and "horse play" among people driving are major problems. Where roads are not present, a light guide-string run through the forest before the drive is made is helpful, almost essential to maintain control.
18. Pellet Group Method
Abundance of deer and other animals may be grossly estimated for short periods by counting defecations observed in the field and then dividing by the average defecations made per animal per day. [Note the wording; such counts have been improperly used.] Deer (Odocoileus) are often the subject of such studies. There are problems with the technique but no more than other population estimation techniques (Eberhardt and Van Etten 1956).
The procedure uses:
N = area x ((pellet groups per plot x plot size x plots per unit area) / (days since leaf-fall x defecations per day))
Leading to the results are assumptions and conditions that are very important for proper application of the method. These are:
1. The area must be accurately determined. Fuzzy management unit borders may exist. (My deer can't tell the color of the boundary paint!) If the area is not precisely known, then massive estimates to achieve a precise estimate or to achieve high precision in parts of the equation seem poorly spent.
2. Where deer congregate (winter range, key or critical areas for energy intake or insect avoidance), counts are misleading. It is generally best to"stratify" samples, i.e., take samples representative of the differences among areas. When used where animals congregate, an index to activity or area use, (not area-wide density), is obtained. "Density on winter range" or "winter density in hardwoods below 400 meters" may be legitimate expressions.
3. Pellet groups in some areas and seasons persist for 3 to 4 years; in others they are removed (buried or consumed) by insects (Ferguson 1955). Local conditions need to be studied and the procedure adjusted to make sense in each area. In most areas, pellets on top of leaves should be counted. Circular plots are in common use (a 1/50-acre plot has a diameter of 33.3 feet or one-half chain). These plots can be carefully studied by people on hands and knees looking under leaves. The method and plot size selected needs to fit the local conditions and objectives.
Plot size and configuration (for any type of study) are no longer the problem they once were. Plot size was once determined on the basis of ease of later computation. With computers readily available, the manager should select a plot easily handled, as large as reasonable, and one that has easily determined units (like the length of a boot or the length of a comfortable walking stick). This number can be entered into a computer (See CAP410). Repeatability is one criterion for good scientific work not to be overlooked, but potential differences in results from a 0.206-acre plot and a 0.191-acre plot hardly seem worth a quibble.
I suggest a long narrow plot for ease of observation and for standing outside (not destroying things inside if later counts are to be made) and looking into an area for a comfortable distance.
In detailed studies I recommend collecting 30 to 40 fresh dropping groups, placing them out in representative habitat, marking them, and observing 1) the proportion, p, that can be seen by observers (the observer is being calibrated) and 2) observing the rate (d) they disappear (for whatever reason). Then
p = pellet groups observed/pellet groups placed
d = 1 - ((pellet greoups placed - pellet groups remaining)/pellet groups placed)) / days
4. The leaf fall that covers pellet groups needs to be accounted properly. In general, groups found in spring and summer have been deposited since leaf fall, thus, deposited during a specific number of days. Those days need to be known for an area. Start of leaf fall may vary as much as 2 weeks in an area. A "boiling," unstable leaf mass and warm autumn days with active insects makes the count impossible or infeasible in some areas with low deer densities.
| under-estimating daily pellet groups over-estimates populations or their behavior |
The correction of 2.5 times is impressive. Faunal managers have used 13 groups per day, drastically overestimating deer (at least deer activity). One conclusion may be that deer population estimates are robust; others are that managers' knowledge makes little difference to these populations (for deer have generally increased impressively since 1940); another could be that population management is not sensitive to knowledge of animal abundance as estimated using pellet groups. There are others.
Persistence, observability, or double counting (not recognizing not-of-the-current-year pellets) can be adjusted by a coefficient C derived by an experimental set of pellets observed and the proportion of all pellets observed correctly as being of the year recorded for each observer. A person who counts 200 groups and consistently has trouble distinguishing fibrous pellets of this year from 1-year old ones may have C = 0.96 and thus the count is multiplied by C.
The defecation rates vary by sex, age, and area and need to be developed in each region. Such rates can be readily used with CAP110. Until other local rates are developed, the value of 34 should be used. The number should be standardized for each area, or better yet, the data retained in a form so that as more is learned about deer, habitat, and sampling, the estimates may be adjusted later to include this knowledge. This procedure is an example of the denial of"garbage-in, garbage-out" so readily thrown at the faunal system manager working with difficult to obtain, costly data. "Garbage," so perceived, may be recycled or converted to very useful products.
The final equation becomes:
N = A GPC P / K(LW/43,560) DT(1-R)
where
and a unit for its computation is available as CAP410. CAP07 provides an alternative approach to a gross population estimate based on counts observed during a day afield.
19. Roadside Counts
Observers such as rural mail carriers have provided useful information on wildlife seen over regularly driven roads (transects) in all seasons. The most difficult task is standardizing routes, observers, and especially time. Animals seen and width of the zone, are important just as is the relevance of the same animal(s) seen day after day in the same area.
In forested areas, calls or sounds of quail, turkey, grouse, and doves have frequently been used as population trend indexes. A person in a car, usually in early morning, drives a fixed route, stops at 1/2 to 1 mile intervals, gets out, listens for 1 to 5 minutes (a fixed period) and records all wildlife gobbling, cooing, or drumming. These are usually males establishing territory or engaged in reproductive ritual in the spring. The minimum population is that one heard. A portable electric recording or call can be used with a colleague to measure the actual width of an average zone sampled by the listener. Gross male densities can thereby be estimated and for some species an equal sex ratio is a reasonable assumption. Density or abundance may thus be estimated.
Many animals' eyes reflect light from a beam at night. McCullough (1982) studied night spotlighting as a deer study technique. He suggested using a 200,000 candlepower beam and binoculars. The best time was 1 hour after sunset for 2 hours when there was no rain or snow. Dealy (1966) had also used spotlighting on western deer. He noted it was useful in openings in dense forest stands. Working from the back of a truck with safety supports, an observer with a seal-beam light attached to a 12-volt battery or to the truck cigarette lighter scans the opening looking for "eyes." Observers may measure their effectiveness against squares of reflecting tape placed to simulate deer eyes. If deer are present, they will be seen. The primary results usually will be in conclusions about monthly habitat use including slope, aspect, and elevation. Secondary observations may be made on population size.
20. Counts Along Transects Without Width
How far did you move (walk or ride) from a random point until you saw the first animal? From many random points, it seems reasonable that the more dense the population, the less will be the distance. The same analyses can be made for time. How many minutes did you move in a straight line until you saw an animal (or group like a covey, brood, or herd)? If the animals are independent (loners) then an average distance or time can be gotten for each observation. These observations, fairly easily made during the course of field reconnaissance, can provide excellent comparisons between areas and years, particularly if the same routes are used by observers with equal abilities (a low probability). A forest road or trail (with few switchbacks; relatively straight), observed at the same phenological time each year, can provide the basis for good comparisons. Half the average distance between observations provides an index to the radius of the maximum living area of an animal and this can be used in the home-range estimating procedure or as a simple index to density. Generally, the inference would be: the smaller the living area, the better the total habitat. Management efforts directed at increasing a forest species would probably show this index to decline over time.
An alternative use of transects is to measure the distance moved until the xth animal (e.g., the 4th) is seen. Preliminary studies of distance moved to numbers seen will suggest where the curve breaks and where no more information is gained by counting more animals. Once this is determined, the observer may move along a randomly determined line until the xth animal is seen. There are some days (all hunters have them) that are not long enough to see the xth animal! A practical stopping rule is needed.
21. Counts along Transects Having Width
A popular and long-used means of sampling animals is a trap line, an observation line or trail, or a "transect." By cutting across an area several times and making observations, a representation of the area is gotten. The intuitive ease is deceptive because transects only work well when (based on Anderson et al. 1976):
A transect can be used with other estimation techniques. When an observer walks through the forest observing the ground, an average distance (W) on either side of the course is observed. This "walk" can be considered a belt transect. All items such as animal defecations or tracks can be counted. The distance walked (L) times (2W) results in the area observed and this may be extrapolated cautiously for larger areas. See CAP640 and CAP07. Such gross but reasonable surveys, given the shortages of time and staff of the manager, already acknowledged, can be refined (1) by plotting results to obtain maximum values over time per unit of effort, (2) separating observations by vegetation type, and (3) doing refined studies within the same areas to inflate the gross observations using proportions of items missed. I have found a simple hiker's pedometer to be useful in estimating distances walked in such field surveys which are often multi-purpose with many distractions and delays.
CAP640 analyzes a conventional transect. See Fig. 8.13. The area is A. The length of the transect is L. The width is constant and it is 2W. Z is the observer on the line when an animal is seen at X. A line is cast from X, perpendicular to the transect line. The point at which that line touches the transect is P. The distance between the observer and the animal is Ri, and the angle is Oi (theta), and the distance from the animal to the transect is yi.
Assume a compartment is 600 acres and grouse, Bonasa umbellus, abundance is desired. Transects are walked, 10 of them, each 300 feet long. A is 600 acres, L is 3000. Six grouse were flushed. The lines were laid out in the forest with tree blazes and a few stakes (all to reduce the attention the observer must give to following the line, maximize animal observation, and assure accurate measures from bird to the line). At each bird seen, the three measures yi, Oi, and ri are made. Only two are needed but three can be used for checking. A flush is seen, a marker is placed at Z. (Fig. 8.12.) The sighting distance ri is measured and a marker placed at X. The sighting angle Oi is measured, then the right-angle distance Xi is measured. Two people usually
 |
| Fig. 8.13. Many measured, straight lines (a to b) are permanently marked in a study area. The observer moves from a to b. An animal is seen at X. Any two observations of Oi, Ri, and Xi are needed. P is a point determined by a line perpendicular from X to the transect. After many such observations of Xi, the likely width of the area is thereby determined and used with length to get the animals per unit area. |
simplify the procedure, and increase accuracy of measurements, to say nothing of increasing the safety of the operation. It is generally desirable to have a total transect length sufficient to have at least 40 animals observed. Transects running along roads, ridges, or streams should be avoided. The perpendicular distance does not have to be measured since it can be obtained by:
Xi = ri sin (Oi)
The perpendicular distance is the measure of interest because if we place all observations on a graph is in Figure 8.13, we see that there is some distance away from the centerline (either side) where the observations stop (at least 90-95% of them). This lets "the animals decide" on the width of the strip transect.
One approach to solving the density problem is to take the mean width, i.e., of xi and then use it plus one standard deviation (for 90% of the observations) and assume a sharply declining distribution as in Fig. 8.13 due to observer ability and animal behavior. This procedure determines a narrow and wide width and the density becomes:
D = 
X / (2W x L)
where density is the sum of the observed animals per unit area. The upper density Du is achieved when the mean W is used; a lower density DL is obtained when W plus 1 standard deviation is used. See CAP640.
CAP641 uses the robust, non-parametric approach of Burnham et al. (1980) on ungrouped data and the processes and program, TRANSECT, described by them is recommended for transect observation analyses.
The proper length of the line (usually the sum of several line segments) is a great problem because cost effective sampling is needed. Not one cent (or second) more should be spent than
 |
| Fig. 8.14. The shape of the distribution varies depending on animals being studied, the habitat, and probably the observer. Observations are studied to obtain the width within which animals are seen so that an area can be calculated (since length and the number of animals seen are known). |
absolutely necessary to collect the data to make a specific decision, one for which the decision maker specifies appropriate confidence levels and allowable error. Burnham et al. (1980) discussed this problem and provided details. Here only a rudimentary computation is made for line length based on their work. The equation is:
L = b/(C)2 (L1/n1)
where L is length of the line to be studied; C is the coefficient of variation (say 10% or 0.10); Liis the length of a line in a pilot study. n1 is the animals seen along the line in that pilot study. The definition of b is a special problem for it is n1/C and both are fuzzy for one is from a pilot study and the other is subjectively determined. In CAP641, several values of b are shown for select value of C. Those are presented to allow the decision maker to see the effects of the decision about the coefficient of variation. If no data are readily available, using b = 3 seems a safe bet. L1 and n1 need to come from experienced staff if a pilot study is not done. (Elements of CAP632 can be used to make such weighted estimates for an area.)
22. Trap Nights
Traps are placed in the forest to trap animals. Some are designed to capture them alive for photography, marking, and other studies. Some are placed to remove the animal for later study, for measurements or body parts, and as one strategy used in the population estimation.
The usual suggestion given in sampling work is that traps be placed at random. This is very difficult in the forest, collectively to place them as well as to find them, check them, eventually re-collect them, and it is definitely difficult if maps are to be prepared of their location. Grids placement of traps is usually used (assuring a poor match-up with clustered and clumped vegetation and animal group activity). Otherwise transect trapping is used; the trapline.
Data from such efforts can be used in many ways and as Begon (1979:92-94) described, imaginative use is often needed and may result in better or more interesting answers than would have been provided if the original study design had materialized.
One such use of data is to calculate the number of animals captured per trap night (or 100 trap nights). There are several uses of such data. One is to determine the optimum number of traps to operate. By plotting number caught against trapping effort (effective traps operated for one night or for one 24-hour period), an asymptote can often be seen. More traps (effort) will not yield greater returns.
Areas can also be compared using trap night success. Before and after treatments can be compared. Bait preference can also be established.
When traps are placed along a trail or road side (open or closed can be seen from a vehicle but whether a trap is open and inoperable cannot be determined), then it is desirable to express animals per 100 trap nights per unit area. The "width" of the area sampled is half the average distance between trap groups. (A trap group is 2 to 4 traps at one spot, 2-3 meters apart, so an available open trap exists after one is occupied.)
23. Lincoln-Petersen and Other Capture-Recapture Techniques
Fundamental to almost all capture-recapture or mark-resight estimation is the procedure called Lincoln-Petersen index (White et al. 1982; Pollock et al. 1990; Seber 1982). (See CAP120 and CAP5018.) The index has been used with terrestrial and aquatic forms and extended into other areas. I used the concept for estimating the numbers of poachers in Idaho and it was applied by Vilkitis (1968). (See Smith et al. 1989.) The logic is simply that a large number of animals marked (M), released, and randomly scattered into a population (N) is probably related in the same way as the number marked (m) found in a sample of that population (n). Thus,
and N = Mn/m
If there are no marked animals in the sample (m = 0) there is a problem (division by zero). The sampling is usually done repeatedly; the first samples rarely yield convincing results and after sampling, new animals are marked so M becomes larger. The later methods, some described below, help overcome some of these problems and address the assumptions which need to be made, or the estimates adjusted to accommodate them. The assumptions:
(Also see Robson and Regier 1964, 1968.)
After obtaining a population estimate, an estimate of the"standard error" is usually sought. Such error expressions are used as in: My estimate of the population is N and 95 times out of 100 it lies within 2 standard error intervals from this number. Where N is 36, then the upper limit is 45 and the lower one 27.
CAP11 allows use of the Schnabel, the modified Bailey estimates, and Schumacher-Eschmeyer.
Overton (1971) discussed the non-normal distribution of estimators as have others (the lower confidence limit is not equal to the upper one) and used graphs and tables to compute these estimates. The limits are often very large, sometimes so much so as to produce "negative animals" at the lower limits, obviously impossible, but resulting from assuming a normal rather than a strongly skewed distribution.
Primarily to create an unbiased estimator (where division by zero creates an infeasible solution), but also useful with small samples, the Bailey (1951) estimator was created. (See Begon 1979:7.) The equation is:
N = M (n + 1) / (m + 1).
CAP124 provides a means to compute it, and compare its confidence band with that from the Lincoln-Petersen estimates. Confidence limits (standard error) are calculated by using:
SE = (M2 (n + 1) (n - m) / ((m + 1)2 (m + 2)) 0.5
See Begon 1979, Seber 1982, White et al. 1982, and Pollock et al. 1990.
24. Removals and Change in Ratios
The manager does not really have to kill animals in order for the removal rate concept to work. After live-trapping and marking animals, each marked animal can be considered to have been removed from the [unmarked] population.
The "bounded count" method (to be described next) is analogous to the "Davis graph" approach. An observer keeps looking until two very large numbers of animals are seen in a period, then a line is cast to the likely next-highest population.
Smirnov (1961) use a 2-day removal rate by hunters in an analogous way to estimate the initial population. The relationship is
N = A2 / (A - B)
where A is the animals killed in the first day and, B, the animals killed in the second day by one hunter in the same area. The area-wide population was based on questioning many (squirrel) hunters and summarizing their results.
Where there were distinct age differences in animals taken by hunters (as in foxes), he used:
N = Nk N = n ma ( (md - ma) / (mb - ma) )
whereN is the population before the hunting season
n is the number killed in the season
mb is the young:adult ratio before the first hunt
md is the ratio during the hunt
ma is that ratio after the hunt.
The application requires hundreds of observations and is for large areas and for gross estimates since migration, seasons, and timing is highly variable. A major advantage is that observations can be made by hunters over extensive areas.
The removal rate is an ingenious idea and probably has other extensions. It has a general application as a change in ratio (sex, age, antlers, color, or other). The population estimate is
N = n/Q
where the population before the removal process begins is N and n is the number taken in the season and Q is the proportion taken. Reformulated: where there are x-type animals (e.g., antlered) and y-type animals (e.g., antlerless) then the proportion changes due to removal by hunting, trapping or other means, and N may be estimated by
N = Rx - R (P2) / (P1 - P2)
where
Rx = the number of x-type animals removed (known from hunting check station, etc.)
R = Rx + Ry,
the total known removed P1 = X1/N1,
i.e., the proportion of X type animals before removal
P2 = X2/N2,
i.e., the proportion of X type animals after removal.
Where 1300 antlered and 140 antlerless deer were removed from a forest district, and where the male proportion of all deer seen before the season was 0.4, the proportion after the harvest was 0.3, then
N = 130 - (1300 + 140) (0.3) / (0.4 - .3)
N = (1300 - 432)/.1
N = 8,680.
Conner et al. (1986) discussed the method and its many limitations and assumptions. Very large sample sizes are needed. When conditions remain stable, it can provide a good index to population trends. It will probably tend to underestimate a population, thus we converge toward an estimate of a larger population.
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| Fig. 8.15. A population of potentially removable animals before any removals (day zero) may be projected. |
Animals do not have to be killed or physically removed to obtain a useful estimate. To estimate bird species in an area (richness) there is a conceivable list to be produced. At each day (or unit effort) of sampling, new birds are seen and listed "captured" or removed from the yet-unknown list. Progressively birds are removed from the theoretical maximum possible list. The estimate of the maximum can be projected based on the cast of the line as in Leslie or Davis curve procedures.
25. Bounded Count
Overton (1971:426) described a bounded-count method and Routledge (1982:757-761) provided a critique, suggesting that it likely underestimates populations. Under the concept of heuristic convergence being developed here, and assuming at least several approaches will be used to make an estimate, this one provides another useful lower limit. As with other estimators, more work needs to be done but, in parallel, the faunal system manager has to be realistic in naming the zone or area in which the animals are observed, stratifying faunal spaces (not just multiplying all area by one density estimate), and ... as important as ever ... clarifying objectives. I think it (or its successors) has great potential in forest faunal work.
The bounded count procedure is as follows. Animals in an area are observed day after day (a bird transect, a horseback ride) and the numbers collected are 36, 49, 54, 16, 54, 61, 22. There is plenty of variability but that is common in the forest. Only guesses about the causes are possible: weather, disturbance, movement patterns, etc. Only the uneducated act as if they can explain the daily difference.
The estimator is
N = 2 A - B
where A is the maximum seen during an observation period (e.g., 61) and B is the next largest number seen (e.g., 54). The estimate is 68 (i.e., 122-54). The manager knows there are at least 61 since this many was seen once. The concept is (approximately) that if the observer worked the area long enough, all animals (68) would be seen. If the observations were stopped on the fifth day, then A would be 54 as would B; then N would be 54. The observations stabilized. The proper sample size, when to stop observing, remains as difficult as in other methods.
The lower limit is known. The upper confidence limit (alpha = 0.20 or the 80 percent level) is computed as:
Nu = A + ((1 - alpha) / alpha) (A - B)
Nu = 61 + (0.80/0.20) (61 - 54) = 89
The manager will be safe in concluding that the population in the observed area lies between 61 and 89 animals. Only about one time in five, there is a good chance that it may be greater than 89. The rate at which the number grows larger, the relationship between A and B, seems to me to be critical. When it approximates 1.0 after many days of constant conditions, the estimate of N is the number to use (as in the Davis graph discussed above). When A/B is 1.0 after 8 or 10 observations, the manager has probably spent too much time observing animals. An estimate, not a total count (census) of the population, is usually the only requirement given funding, time, etc.
There are many ways to estimate populations. None is best; convergence on an estimate is needed. Let it be remembered that after the population is estimated it will have changed before the analysis is completed. Area must also be estimated to get density, and it will probably be poorly estimated. Density is probably poorly correlated with potential benefits from the animals, and people only dimly perceive their objectives for achieving such benefits. The faunal resource manager must work simultaneously with people and faunal space as well as with populations. Having dealt with population structure, we can now examine the artificial but useful category of dynamics.
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