A unit of Lasting Forests Sustained forests; sustained profits evolving since March 30, 1999

by Robert H. Giles, Jr., PhD

Preface

This is the book I wish I had had as a state wildlife biologist in the late 50's and early 60's ... and ever since. After that experience, I've edited the Techniques Manual (1971) of the Wildlife Society, authored Wildlife Management (Freeman Co., 1978), wrote a "principles " textbook (in Chinese, 1993), and estimated populations in my PH.D. work (Brotzman and Giles 1996). I've also been involved with the topic in consulting and in work with graduate students in Idaho and Virginia. The topic is extremely interesting and offers such a challenge from so many different directions that thousands of people have taken the challenge.

I have prepared this book and computer programs because I have seen other people develop concepts and programs that exceed the abilities of the users. The procedures and models, while splendid in appearance, deep in statistical sophistication, and having vast computer options, just have unrealistic requirements for data -- including types, numbers, costs, and risks.

The quality of these few pages in this web space does not come near to the book and journal articles of Seber (1982), Begon (1979), Blower et al. (1981), White et al. (1982), Jolly (1965) and Davis (1983) on the topic. By analogy, I am a first-aider in the presence of medical specialists. I report on the ideas and work of these people and try to present it so that it can take on new meaning for the important work ahead in the field.

Hopefully, the reader will not belabor the meaning of "practical" in the title. That will be impractical. I have presented available techniques and a computer aid for a variety of uses. My opinion is that rough estimates are more frequently needed than very precise ones, that variation and confounding events are common in the field (they cannot be called "unexpected"), that budgets always limit our sampling more than our design demands, and that far too much time is spent sampling in the face of the dangers and costs of doing so and the likely use that will be made of the results. I argue for a rationally robust approach (Giles et al., 1993) to this work.

Included with the text are 66 BASIC programs. (These are included with Capper. Instructions are provided there for getting the programs and loading them onto your computer in a folder. The text describes each program, or they are described within the program itself. Where literature is relevant, it is cited. I have only presented the key references and rarely even the key equations. They are all within the programs and can be read there, copied, modified, or unified. The formulations are often intricate and, while interesting to a few people, get in the way of practical use, which is my intent. No one, no book, can accommodate the extreme differences in needs, standards, and willingness to take risk among the people working in the wildlife resource and ecology field. Programs (if there are no citations) seem to be my own creation. Now old, I discover "new" ideas in my file of letters of 20 years ago - completely forgotten. I hope I have not forgotten the sources of some of these ideas and programs. If so, I do apologize and remain grateful for the help I have gotten from students and others. The "rough and ready" approach here would never be accepted by "wildlife" journals (my first computer- program write-ups on population estimates were rejected by the Journal of Wildlife Management in the late 60's due to a policy against computer- related articles. The first submissions were "lost" by reviewers). The best use for the book and programs, I believe, is to produce ideas and to "bracket-in" on a population size estimate. The programs should be used in the mode of a simulation, or as in trying to answer "what if this ... was the case?" (where this is an hypothesized number or relationship). I suggest using reasonable, made- up numbers to start with. Make quick entries to learn the program; then enter field data, or simply explore options. The programs are tools for discovery, heuristic devices. In the text, I outline the concept of heuristic convergence.

I urge that the programs be copied and used. I shall appreciate the customary courtesies.

Some Software for Heuristic Convergence

1. Definition of Wildlife Management ----------------------------------- HC01
2. A Memory Aid for the Definition ------------------------------------- HC02
3. Relative Population Abundance Difference ---------------------------- HC03
4. N-Hat Heuristics ---------------------------------------------------- HC04
5. Richness Comparisons ------------------------------------------------ HC05
6. Maximum Level: Leveling-Off of Numbers in Sequence ------------------ HC06
7. Simple Linear Regression -------------------------------------------- HC07
8. Transformation (Log(x+l)) ------------------------------------------- HC08
9. Species-Area Curve Relations ---------------------------------------- HC09
10. Random Numbers ----------------------------------------------------- HC10
11. Calibrating an Observer -------------------------------------------- HC11
12. Multiple Regression ------------------------------------------------ HC12
13. Expert System Introduction ----------------------------------------- HC13
14. Comparing Proportions ---------------------------------------------- HC14
15. Three Similarity Indices ------------------------------------------- HC15
16. General Statistics ------------------------------------------------- HC16
17. Sliding Mean ------------------------------------------------------- HC17
18. Comparing Rates ---------------------------------------------------- HC18
19. Average Growth Rate and Future Projections ------------------------- HC19
20. Time Between Times --------------------------------------------------HC20
21. Zippin Removal Method -----------------------------------------------HC21
22. Maximum Feasible Area or Volume -------------------------------------HC22
23. Influence of Boundary Inaccuracies ----------------------------------HC23
24. Area and Perimeter of Polygon ---------------------------------------HC24
25. Biomass-Based Estimate ----------------------------------------------HC25 
26. Gross Population Estimates Based on Biomass Expected Per
	Unit Area -------------------------------------------------------HC26
27. Weighing Density Estimates from Various Sources ---------------------HC27
28. Relative Abundance --------------------------------------------------HC28
29. Statistics: One and Two Variables -----------------------------------HC29
30. Estimated Home Range from Body-Weight -------------------------------HC30
31. Home Range of Groups of Animals -------------------------------------HC31
32. Bird and Mammal Home Range Related to Body Weight -------------------HC32
33. Mammalian Population Density: The Damuth Equation -------------------HC33
34. Estimation of Population Abundance by Experts -----------------------HC34
35. Application of the Beta Distribution --------------------------------HC35
36. Normally Distributed Random Numbers ---------------------------------HC36
37. Sex Ratio Analyses --------------------------------------------------HC37
38. Population Estimation from Reproductive Rate ------------------------HC38
39. Sample Size: An Aid -------------------------------------------------HC39
40. Sample Rate and Size ------------------------------------------------HC40
41. Stopping Rule Simulator: Sampling Animals and Plants ----------------HC51
42. Estimates Based on Harvests Adjusted for Illegal Kill ---------------HC52
43. Bounded-Count Estimates ---------------------------------------------HC53
44. Deer Density by Track Count -----------------------------------------HC54
45. Deer Density by Pellet-Group Count-I --------------------------------HC55
46. Deer Density by Pellet-Group Count-II -------------------------------HC56
47. A Transect Line Estimate-I ------------------------------------------HC57
48. A Transect Line Estimate-11 -----------------------------------------HC58
49. A Population Estimate Using a King or Hayne Transect ----------------HC59
50. The Lincoln-Petersen Estimate ---------------------------------------HC60
51. Four Capture-Recapture Estimates ------------------------------------HC61
60. The Bailey Estimator ------------------------------------------------HC53
61. Evenness ------------------------------------------------------------HC61
63. The Simpson Index ---------------------------------------------------HC62
64. Comparisons of Items in a List to a Negative Geometric Distribution -HC63
65. Gross Estimate of Deer Population Density ---------------------------HC64
66. Harvest and Density -------------------------------------------------HC65
67. Effectiveness: Performance Measures ---------------------------------HC66
68. Menu----------------------------------------------------------------HC100

Introduction

Population estimates are needed by many groups of people ... from medical staff relating diseases to rodents, to recreationists engaging in new sports of animal watching, to hunters and game managers, to foresters (whose trees are eaten by animals), and to curious scientists, land owners, and citizens.

HC01 and HC02

[The small insert Boxes at the right show the names and numbers of the major computer programs that are related to the topics being presented. Instructions for gaining and using the programs are available at Capper.] I'll not emphasize the uses to which the estimates may be put. I'll not emphasize the intricate mathematical distributions, the realm of probability, the statistics, or the risks that accompany these estimates. I believe this book presents a summary of practical tools, a few novelties, but an important concept that, if not new, at least is worth emphasis. The concept suggested is a large but singular one of "be skillful, be cost- effective, be realistic, use what you can get, be sure results match demands, be timely, adopt tentativeness." Use a "What if ... ?" mode of thought and use more than one estimate. In total, I call the concept one of convergence.

There is a real population in the wildland. Its numbers will change by tomorrow and it will move so you start off realizing that you are trying to estimate the "volume of a cloud." We rarely know why we need a population estimate; our objectives are usually vague or un-stated. There are no excellent universal estimators (equations or models) so we are always handicapped. We need to discover the population. We need an heuristic strategy. Depending on time, money, talent, and imagination, we can explore, test, and try various techniques, tools, and approaches.

We need to eliminate any estimate of a population size that cannot be true (no negatives, no zeroes, no infinite populations). We use past studies. We collect data; we talk to experts. We converge on the real number. We estimate a number and hope it is not too far from the true number. We try diligently in several ways to get an estimate. We discover. We converge, heuristically, on the answer.

heuristic convergence

I'll provide some tools and several alternative patterns of discovery. One I call "backing in." We may attempt to limit progressively or constrain the population as in "It cannot be less than X or more than Y. It has to be less than X₂. At least it is within this ball park." Then, the practical question: "What will a decision maker do with my answer of `between Y₁ and Y₂'." Perhaps more work will be needed; perhaps no more work. My view is that most situations in the wildlife world need gross, robust estimates. Managers, ecologists, zoologists need to try to discover the number, to use several estimates to arrive at, converge on, some useful number that may influence decisions.

Don't!

Implicit within this book is the assumption that an estimate must be made. My strongest recommendation, however is: Don't make as estimate!

Most efforts to estimate numbers of animals will be time consuming, expensive, use highly-paid staff inefficiently, produce disappointing or inconclusive results, and probably answer questions not being asked in the first place. Estimating them is a natural tendency for people who love animals and "want to work with them." It must be resisted ... strongly.

One reason for population estimates is the assumed relationship of a positive correlation between numbers and benefits. This has been true, generally, for managing game species. 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.

One reason for resisting estimating abundance is that animals differ in importance or value. The reasons for being interested in abundance change with the knowledge of abundance itself. 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 threatened and endangered species ("T and E" among the glib), can provide many benefits. There is a population abundance level that I call "sparse"; such 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 is probably is not worth the effort to find it and never a "sure thing." Some woodland warblers are equally sparse. 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 can result in superior hunting or sighting. Past a point, another grouse is just another grouse. People become satiated.

Black bears (Ursus americana) are the clearest example of the curve for resource benefit and interest in the abundance of 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. Interest wanes. While population estimates may be useful, an alternative statistic is that of benefits associated with the population (e.g., from questionnaires and damage surveys).

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 make threats of encountering court-costs).

Gross population estimates which produce bad effects (i.e., costly mistakes), while unfortunate and to be avoided, are usually 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).

It is apparent that average citizens and citizen groups do not have clear objectives for the faunal resources of public lands, and rarely their own lands. Efforts to get them to weight the relative importance of different animals have produced scant results. Weights typically will be assigned 0 to 3, perhaps 0 to 10 but rarely discriminating within a range of 0 to 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 had been given. This technique can be used to stimulate groups to develop decision weights that will usually differ.

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? Do animals "depreciate" with age? Some may appreciate. The uncertainty in interest rate and value-rate rate argues against achieving great precision in the population estimate or its use.

The statistical variance associated with population abundance studies, even the well-done ones, is invariably so wide as to provide little power in decisions based on them.

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:

• timely,
• has a narrow confidence band,
• provides a suitable grossly estimated benefit-to-cost ratio for the local situation,
• does not cause significant death or critical energy loss in the animals sampled,
• does not significantly reduce the wild quality of the user- perceived population (e.g., pink streamer tapes in ears of marked animals), and
• reported to the professional faunal management wildlife community.

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..."

Populations are very dynamic. Animals move and hide. They migrate. Even the sampling effort itself may result in some animals dying or emigrating (and others filling the void). An estimate today deserves another one tomorrow. The occurrence of natural catastrophes has to be expected. A poacher, a predator, an aberrant insecticide spray-plane - all make population estimates quite labile.

Suppose the truth about animal density (number per unit area) could be known. The areas within which management is done is rarely precisely known. There are few islands, rare "closed" populations. Island size itself is dynamic (Weatherbee et al. 1972) and forested areas change rapidly. It is unresolved for most small species, whether ground- distance or horizontal distance should be used. I think volumes should also be used for most life groups;

A life group is a part of a population of a species requiring significantly different management than other parts. There is more difference between a turkey poult and an adult than between many genera. Amphibian and insect life stages (instars) require very different micro sites. Modern faunal systems management is directed at 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 mapped areas of hundreds of acres. True density (animals per unit area) multiplied by a poorly estimated area, results in a poor estimate of density. 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.

All of the above reasons suggest: do not make a costly field estimate unless it is absolutely necessary and unless large sums of money, labor, and analytical assistance are guaranteed. However, with fair warning and having decided that an estimate is needed, then there are ways to do that.

Faunal system managers do not have to be apologetic about their estimates, their lack of precision, or always to be amused when a forester observes that, unlike animals, "trees do not move." Within the ranks of those working almost exclusively with plants, it is widely known that percent crown closure or coverage is a measure of limited precision but is a practical parameter giving insight into changes in stem density. "Site index" is a useful tool for foresters (even though it is limited) because it provides a simple numerical value that is easily measured and understood by the practicing forester. Index limitations are widespread in resource and environmental fields but rarely the humility to go along with them.

The Fate of an Estimation

"How many deer are there?" is also the hidden question: "are there going to be more deer during the coming hunting season 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 likely harvest this year." He estimates:

H* = H_t / H_t-1

HC03

and only will take action (not hunt in the area) if it is significantly less than 1.0. The rate of change is:

H** = (H_t - H_t-1) / H_t-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 estimates that may be used for 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 may stock 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 or scarce resource invested. The usual answer to a question about abundance in a stocking situation follows a quick analysis

of exponential growth as follows:

N_t = N₀ (1.0 + r)^t.

Where N is the number; r, the rate; and t, time. 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 the signs are 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 field 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 in managerial action, 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: (1) the habitat is changing, thus the population is likely to be unique in time; (2) the population sex and age classes are changing even though the total faunal biomass may be relatively stable; (3) the samples are likely to be small because animal densities are sparse and sampling intensity has to be limited because of limited staff, travel, and processing time; (4) when samples are small, bias, miscounts, births, or escapes can have a large influence on an estimate; and (5) populations are strongly related to past events and environments, often more so than the population in 1 year shows to the manager. None of these include the not-very-surprising occurrence of a local predator, a poacher, a unique storm, or the asserted " unusual"

the usual is unusual

coincidence in time and space of, for example, a dust cloud following a freeze and a small storm that puts unusual, fine, non-site sediments into a stream and influences egg masses. The unusual is usual for animal populations. The presence of the sampler assures everyone that the animal population will not be natural; an estimate will not be of a wild population but of "a wild population in an area having a sampler and all of his or her devices and disruptions of behavior."

If benefits from wild animal populations are a function of the number of animals and their separate and group 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 my 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 not to do so are often counterintuitive.

Heuristic Convergence

An alternative to some past approaches to population estimation is called heuristic convergence. The approach is one of using multiple estimators to converge on the truthfulness of an estimation of abundance. The approach includes employing a set of population estimation techniques, expressing maximum and minimum, weighting the perceived relative goodness of each estimate, arranging estimates in rank order, and developing graphical and mathematical expressions of the population estimate. 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). The approach may be viewed as "bracketing-in" an answer (Fig. 1).

Fig. 1. Positioning various population estimation methods (ranking) and graphing the various estimates progressively with their confidence limits can produce a zone of likely population density.

HC04

The topics of heuristic convergence (HC04) are as follows and discussed in turn:

1. Richness, abundance, and density
2. Historical information
3. Rates
4. Maximum feasible volume
5. Plant and animal biomass
6. Reported densities
7. Relative abundance and expert judgment
8. Magic numbers
9. Home range
10. Subjective estimates
11. Harvest
12. Representative habitat sampling
13. Habitat adjustments
14. Illegal harvest adjustments
15. Total counts (see #6 and 9)
16. Track counts
17. Drives with track counts
18. Pellet group counts
19. Roadside counts
20. Counts along transects without width
21. Counts along transects with width
22. Trap nights
23. Lincoln-Petersen estimates and other capture-recapture estimates
24. Removals and change in ratios
25. Bounded count

Richness, Abundance, and Density

The manager may use richness estimates as follows:

1. To compare areas
2. To compare before and after conditions (e.g.,wars, industrialization, and habitat preservation)
3. To indicate ecosystem vigor or productivity
4. To suggest related conditions such as biomass that is present
5. To express quality of an area for wildlife recreation and tourists
6. To suggest long-term changes in populations and their habitats.

The wildlife manager must be careful in expressing conclusions. Not having observed an animal does not mean it is not present. Care is needed 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. 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.

HC05

Richness is used synonymously by some (e.g., the Nature Conservancy) with diversity. By preserving natural diversity is meant: do not let richness decline.

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 seasonal change in hiding cover, 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 of 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. 2.

Fig. 2. Recording new species seen per day (or year) 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.

Making such plots is akin to estimating the maximum flood depth for an area. By plotting over the years the highest flood of record, eventually the curve levels off. The depth at which it levels is the maximum flood depth. Analogously, a person searches through an area 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) used in estimating mammal population.

HC06

When the rate drops to zero (approximately, say no more than 3 percent) it is not a bad assumption that you have seen or captured them all (or all that can be seen or captured). See HC06.

An alternative version of this approach is to use a linear regression of the daily catch or observations (y) on cumulative catch (x). (See HC07.) Where

y = a - bx

then setting y = 0 and solving for X, the intercept, the relationship is:

x_maxR = N = a / b.

If in one field situation the relationship was

y = 55 - 0.2x

HC07

then

N = 55 / 0.2 = 275

animals

HC08

These relationships are rarely linear or straight-line. They tend to be curvilinear, so using the logarithm of the daily catch or observation will probably give a better equation 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. See HC08.

Another technique which has fascinated many wildlife managers is the species-area curve (See Connor and McCoy 1979). The relationship is

S = cA^z

HC09

where the species, S, say of birds (or other taxon) are related to the area by an exponent (z). C is a coefficient partially to adjust for the units of measure used for the area. (The exponent will probably be found some day to be some energetic ratio to life volume of the creatures in a space). Plotting data from studies of islands and large areas on logarithmic paper is an easy way to obtain this equation. See HC09. The apparent mathematical magic of the relationship deserves more study. It may be that number of species is related not to area (and elevation) but to the volume of the habitat (the product of area and dominant vegetation height) or the number and volume of layers (Short 1984), as well as island age and distance from a mainland. For example, there are usually more species in a tall forest than in a short grassland. Nevertheless, The equation has been a surprisingly good estimator of richness and has led to other insights.

What can a population estimator 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 potential richness of the area had not changed. The observed animals had.

HC10

Some species are seen upon every observation attempt. (Observations are not limited to sight but include sound, 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 (see HC10) 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! Wild 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 is greater than A / (s/S)

HC11

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. (See HC11) For example:

N is greater than 16 /(30/55)

N is greater than 16 /0.5454

N is greater than 29

calibrate the observer

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. One person seeing 32 silhouettes, another 29, can make a difference in N, the animals in the estimate).

HC12 HC13

Conditions during a day can influence the conspicuousness of animals. Temperature, perhaps barometric pressure, time since rain, fog, cumulative sunlight 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 (HC12) but may be better handled by an expert system (HC13) 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 such weights or scoring mechanisms are used, the sources of variation which are known to exist creep into the estimates. Population estimation techniques are not very good; the estimates we get often 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 with this book, now overcome some of the difficulties.

HC13

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 (HC13) 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. See also HC14. 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.

HC15

fauna are integrators

Comparisons of richness are available in HC05. 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 also of HC15 value and computation aids are provided here. See HC 15. "Plants are indicators" is a tenet of ecologists. "Fauna are integrators" is also needed. The animals present, the richness and similarity of area populations, reflect ability to resist all combinations of assaults and insults possibly thrown at them over the eons. They have won at the great game.

It is possible to determine, for certain human objectives, whether population abundance is excessive or not, based on food and related conditions, by observing the characteristics of animals themselves. The count per se may not be as important as the characteristics of the animals themselves or the environment in which they live. Kirkpatrick (1980: 99-112) and Kie (1990) summarized the useful physiological and anatomical indices relating animals to their environment (and of course to other animals foraging in the same areas.) What is the best index is a leading question. A set used in an expert system will probably yield good results. The parts of such a set, depending on availability of laboratory and other facilities and advice, are:

• Eviscerated carcass weight of females.
• Femur marrow fat (the last fat reserve to be depleted).
• Kidney plus kidney fat weight at a constant time each year of animals in a standard age class.
• Reproductive measures (corporea lutea and placental scars).

The apparent absence (absence cannot be proven)of a species can also communicate some information.

What is so badly needed is very careful attention to what is badly needed. What are the basic objectives? What are the minimum data? After the answers, then studies can begin. Usually, in my experience, when population abundance questions are raised, estimates of richness (R) will suffice. In a related way, I hypothesize that the density (D) of select life groups will be shown to be related well to richness as in

D = a + b(logR+1).

The transformation of R to log(R+1) is available in HC08.

Historical Information

Rates

HC17

A sliding average or moving mean is a useful device for expressing change in a population over time. HC17 allows a sliding average to be calculated. Fig. 3 shows how sliding averages suppress the extremes and express relations over a few years (the span).

Fig. 3 A sliding average can be used to observe general rates of change in harvests and other indexes to population abundance. The span of years selected can influence the shape of the graphed line curve.

The span of years (say 3 or 4) may be selected. The procedure averages the data for the selected number of years, say 2 or 3, then drops the most distant year, advances one year, and averages the next group of the same number of years. Since a population is a function of the numbers in previous years, there may be ecological meaning to the results in addition to its value for "rounding" and generalizing from a set of numbers.

HC07 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 probably is the case where tree reproduction is in jeopardy from foraging 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 often used to assess population trends or rates. 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 and can lead population managers into difficulties.

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 for some year, that is

r_t = (N_t-1 - N_t) / N_t-1

HC18

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 animals are produced by 100 + 20 animals or at a rate of about 2.08. If half of these die in the first 6 months, then the rate, r*, is 1.04. How different is r_t than r*? See HC18.

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 to get this index to the desired level. If working with game, the manager will probably want the rate 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 faunal systems (and others as well) in the gap between the condition of the present system and the desired condition (in this example, implied as 1-0). The greater the 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. If a base is known, rates can be used to estimate populations before or after some period.

HC19

Of limited usefulness, 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, HC19 provides a means to enter known or estimated initial populations for several years. This establishes a pattern of growth. The simple relationship of

N_t = N₀ (1 + r)^t

may be used where the starting population is

N_o,

HC20

r is the rate of change, and t is time in years. For example, if 100 birds were stocked and they increased at 3 percent, 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 HC20 for additional insights into these relations.)

The manager needs to be cautious about using estimated population rates because typically when abundance is decreased (as in the case of damage complaints), the potential rate, R*, may increase (a feedback) due to new or extra resources for the remaining animals. As they say in the woods, "the faster you go, the behinder you get."

HC21

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 in some sampling situations 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 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). The Zippin removal method (HC21) achieves this estimate.

A phase plane can depict the progression of populations over time, typically spiraling toward some equilibrium. By expressing the population,

N_t,

as a function of

N_t-1

(or N_t-1 + N_t-2) (along the X axis of a graph)

insights into the population may be gained. A 45-degree diagonal line shows a zero rate of change. Low harvests in one year may result in high harvests the next, etc.

Fig. 4. A phase plane diagram of harvests or removals depicts the rates of change between years. When environmental and other factors are stable, 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) (as in using HC07) of the logarithm of the observed number of animals (HC08) 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 counts each year above those computed as required.

Maximum Feasible Area or Volume

HC23

An unusual blind spot in the eye of many population estimates is that if density is to be estimated, then at least as much emphasis needs to be placed on estimating the area as on the population. A forest wildlife clearing may be estimated at "about a quarter of an acre" and much money is spent in estimating the rodent populations there. Days of effort on the animals need to be balanced with effort on area estimation. Boundaries are set by people; the animals rarely see the same lines or limits. Area estimates are extremely complicated or often counterintuitive. A circular half-acre field, plus or minus 10 feet at the edge, will be at least 0.38 acres or as much as 0.62 acres. The larger area (plus 10 feet) is 1.6 times the size of the smaller area (HC23).

HC24

An animal population area can be accurately measured with satellite technology (the GPS or global positioning satellite equipment). See Figure 5. For practical field work, pacing with a compass can provide good estimates of the areas of field, forest, and the sample areas, all improvements over the gross approximations often used. Rarely is accurately classical surveying cost effective and the accuracy will likely exceed that of the population estimate. HC24 is provided to assist making estimates of the area of a polygon.

Fig. 5: Even a relatively simple polygon of 6 vertices can present a problem for analysis. Readers may wish to experiment with estimating the area of this polygon. The coordinates of the vertices shown here may be assumed to be any units-feet, paces, miles, etc.

Figures 6, 7, and 8 (below), show results from using a geographic information system (GIS).

Fig. 6: A GIS image presents three different aspects or the direction that a land slope faces are shown (north are black, southwest are white, gray are all others). [Figures developed by Khaled Hassouna.

Fig. 7: Two elevation categories for the same area can be shown, suggesting a limit to an animal species' range. [Figures developed by Khaled Hassouna.]

Fig. 8: When brought together, "overlayed", 6 potentially different categories result. Here they are placed into project-specific relevant groups: black=north-facing and high; dark gray=north and low; gray=southwest and high; white=all others. [Figures developed by Khaled Hassouna.]

Estimating area is as important as estimating animal numbers

By overlaying two maps, one with two levels of evaluation, the other with three different aspect categories, then a new map can be created. Fig. 8 above shows the combined areas. The modern population manager will use such maps to make category- and area-proportional populations estimates or use the maps to do area- proportional sampling. There is no longer a need to assume all parts of an area are alike or "on-average" equal.

Plant and Animal Biomass Estimates

N = (B x 4.0)/(70 x W^O.75)x 365

= (2500 x 4)/(70 x 1 ^10.75) x 365

= 10,000/27,375

= 0.365 animals /M²

This computation results in a very high density estimate based on daily maintenance and can be revised downward by incorporating food digestibility. The computation still yields an upper density estimate based on energy. This sounds absurd, but it sets a 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. We are attempting to converge from the outside limits 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.

HC25

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 estimate of the population, N, might be 250 grams divided by the 10 percent 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. HC25 allows these relations to be studied rapidly.

HC26

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. HC26 allows these relations to be studied.

Reported Densities

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 away, 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 I 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 1) and assigning a weight representing confidence in each one, based on the time and means by which the estimate was probably made, it is possible to get an operational estimate.

Table 1. Deer population estimates demonstrating use of the literature to obtain maximum and 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	Weight	Comment
1 2 3 4 5 6 7 8 9 10	>550 553 213 128 80 64 26 22 21 7.1	1 1 2 8 10 8 8 5 5 1	Possible tracks and sign abundance Report 1 Report 2 State game agency staff State game agency staff Regional: likely Regional: upper Nearby state Report 3 Local military area

Relative Abundance

P* = log_eN

It is shown in Table 2.

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.

HC28

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 2) have intuitive appeal. It is mathematically tractable. It probably has special relevance when used with diversity estimates when the natural logarithm is used. (See HC28 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.

Table 2. Representative values and boundaries for relevant abundance groups (P* = logeN)
N	P*
less than 3 3-7 8-20 21-55 56-148 149-403 404-1,097 1,098-2,980 2,981-8,103 8,104-22,026 greater than 22,026	1 2 3 4 5 6 7 8 9 10 11

Magic Numbers

Home Range

Fig 9. 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 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.

Population estimates based on home range are thus of limited usefulness but, I think, one of the most practical for 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. 10 and developed in Jenrich and Turner (1969), Van Winkle (1975), and related concepts in Dixon and Chapman (1980) and Getty (1981).

Fig. 10. 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.

Hayne (1949) suggested two types of home range, one "true", the other "trap-revealed." He said "Even when dealing only with the trap- revealed range, it seems necessary to accept the edges of the apparent area as shading off on the basis of probability of capture and not as discrete lines of biological validity."

For general computations of home range, a circular area may be envisioned where

A = (D/2)²

with D being the diameter.

1Jenrich and Turner (1969) defined home range as the circular or elliptic area of the smallest region which accounts for 95 percent of an animal's utilization of a habitat. It may be elliptic and was defined as: A = 6 | | 1/2 where the absolute value of the convariance matrix of the bivariate normal utilization distribution is | |. When 9 rather than 6 is used, 99% of the utilization area is accounted.

A population estimate may be obtained by dividing the average home range size into a total area, the larger one of managerial interest. 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. An animal with a 3 ha home range, if it is within a circle, 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 percent 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 hexagonal-shaped areas can be packed into the total area? The results is a gross population density estimate.

The beauty of the procedure described above is that average home range sizes have been computed for many animals over the years, that these have been compiled, and home range has been demonstrated to be strongly related to animal weight. The relation is intuitively evident: the larger the animal, the larger the range. Also as Lindstedt et al. (1986:416) observed: animals set the size of home range "... to ensure adequate energy to last for the duration of critical biological time periods, rather than chronological time of days or years."

Lindstedt et al. (1986) studied mammalian carnivore home range areas (A) (from 15 to 38 studies) and found that for

All carnivores: A = 170 M^1.03
r² = 0.66

Carnivores at or below 45'latitude: A = 115 M^O.94
r² = 0.61

Carnivores at or above 45'latitude: A = 339 M^1.08
r² = 0.79

Harestad and Bunnell (1979) studies, (revised as shown below in Lindstedt et al. 1986) provided the relations of:

Carnivores A = 137 M^l.37

r² = 0.81

Herbivores A = 2.71 M^1.02

r² = 0.75

Omnivores A = 3.4 M^O.92

r² = 0.90

Decision Power: Home range is a function of animal weight. Populations are a function of home range.

HC30

HC30 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 M^1.9 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.

HC31

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 (HC31) can be increased by the group efficiency modifier as

N = Area/ (Home range x (1 /n)^0.75)

HC32

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^{(O.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.

Once A, the home range, is estimated and added to or subtracted from (slightly) to improve it and reduce decision risks, then it can be divided into the total area giving a population estimate. Such a low cost, theoretically sound estimate puts the manager in a very strong position. The fine tuning or adjustments to be made differ little from those made by engineers or other professionals, who, after hours of tedious, precise calculations, multiply by a substantial safety factor (e.g., 2).

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

HC33

where W is weight in grams and D is the estimated number of animals per square kilometer. (See HC33.) Damuth's method deserves note as a separate population estimation "method" for comparisons and use in heuristic convergence.

Subjective Estimates

In my experience, bad experiences result from ignoring the experience of field workers.

Experts may be asked to make a low (a), high (c) and most likely (b) estimate of the population or its density. (See Levin and Kirkpatrick 1978.) The most likely is weighted (b) by 4, a and c by 1.0 so the results is

N = (aNa + bNb + cNc) / (a + b + c).

Future work may be justified in studying the weight of 4 (i.e., the value of b) used in time-to-complete-a-project studies and assigned. It is also likely that low estimates are more precise than high estimates.

HC34 HC35

Because staff are often transferred and expertise may be limited, it is appropriate to weight each participant in the activity (HC34).

A weighted median estimate is obtained and the individual highest and lowest estimates may be compared with other estimates. See HC35.

HC36

Most population estimates do not emerge from normal (bell-shaped) curve distribution. Emphasis on the median is reasonable. Normal- distribution work (e.g., HC36 provides normally-distributed random numbers for sampling, placing animal traps, etc. and making comparisons of field data to a widely-known distribution. Uniformly distributed random numbers are provided by HC10).

Harvests

One use of such data for an area is to put harvests in rank order. The usual form is that in Fig. 11.

Fig. 11. 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.

The projection to a maximum may provide insight into the population. By 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 other animals can improve this estimate. Radio telemetry may assist in estimating such areas, but the data from these studies are at least as difficult to analyze well as are population estimates.

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 about 5000.

HC37 HC38

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, the concept is that after mortality and just prior to the hunt, the young:female ratio for the population is 1:1, the sex ratio is 100:100 (HC37), and 1000 animals are harvested, then the pre-hunt population would have to have been 3000 animals. (See HC38.)

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).

Representative Habitat Sampling

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.

HC39 HC40 HC41

Note that the procedure is modified stratified sampling depending on knowledge of the population (without an a priori estimate) for deciding on two strata. See HC39, HC40 and HC41.

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 (HC09) 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.

Habitat Adjustments

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 (as determined using HC24).

Cover maps (e.g., from GIS) 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. Geographic information systems open unlimited opportunities to do area-proportional sampling, type proportional sampling, and to create maps of probable densities based on known habitat and species associations.

Illegal Harvest Adjustments

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 the illegal kill is proportional to the hunting population; another one is that it is related 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, it 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.

Estimated Total = (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))

Total Population₂ = (population x (1.0 + proportion in Legal harvest)) +
(population x proportion of population taken illegally)

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)).

Total Counts

HC43

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 were once difficult to count. Aerial surveys are widely used and a vast literature on sensors, flight widths, lengths, altitude, andsampling 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 the bounded-count method discussed later and HC43).

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, usually 2. Cotton (1990) studied reports and found the number varied quite widely.

Track Counts

In another approach, tracking stations are placed in the field, usually 30 to 50 every mile along secondary forest roads. They are smooth areas, some with sand or agricultural ground limestone place over them, with a scent-bearing stake driven in the center. For example, HC44 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) is determined. 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 (using HC07 and HC08). 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 have been recommended.

Drives With Track Counts

Pellet Group Method

The procedure uses:

N = 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.

HC45 HC46

Plot size and configuration 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 HC 45 and HC 46). 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 counting easily 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 groups 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.

5. Deer defecation rates of about 12.7 groups per day are known from several studies and this number base has been used since at least 1940. Along came L.L. Rogers (1987) showing rates of active deer on leashes fed native vegetation defecate 34 times per day! It is exciting that the method has now been revised and local efforts can be encouraged to discriminate among deer of different ages and sex and among seasonal deposition. Similar large numbers were found in Georgia (Sawyer et al. 1990). A rate of 25 was suggested for autumn, 34 for spring counts.

The correction of 2.5 times (over the long-accepted 12.5 to 13) 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.

under-estimating daily pellet groups overestimates populations or their behavior

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 (see HCH) 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 HC45 and HC46. Until other local rates are developed, the value of 34 may 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 and those who are anti- computer-use. "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(I-R)

where

A is the area in acres

G is the sum of the groups counted in all plots

K is the number of plots counted

C is the correction factor for the observer adjusting for proper age of the groups observed

P is the proportion of the groups present that are actually observed

L is the length of the plot (in feet)

W is the width of the plot (in feet), or a diameter for a circular plot may be used

R is the rate of removal by insects, etc.

P is the days since leaf-fall

D is the defecation rate, typically 34 per day

and a unit for its computation is available as HC45.

HC46 provides an alternative approach to a gross population estimate based on pellet-group counts observed during a day afield.

Roadside Counts

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 drunu-ning. 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 I hour after sunset for 2 hours when there was no rain or snow. Dealy (1966) had also used spotlighting on western deer in open areas. He noted that the procedure 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, it is likely that they will be seen. The primary results usually will be in conclusions about monthly habitat use (separated by slope, aspect, and elevation for analysis and GIS display). Secondary observations may be made on population size.

Counts Along Transects Without Width

An alternative use of transects is to measure the distance moved until the xth animal (e.g., the 4th) is seen. Preliminary studies and plotting 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 (e.g., HC41).

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):

1. All animals (or sign) on the line are always seen.

2. Animals near the line are seen; they do not hide.

3. They do not move sufficiently to be counted twice.

4. There are no measurement errors (distance, direction, etc.).

5. No rounding is done. Exact measures are made.

6. Sightings are independent; seeing one animal does not decrease or increase the chance of seeing others.

7. Lines or line segments must be straight and the observer must stay on it. Walking toward animals off the line tends to cause inflated populations.

The probability of a match with all of these even if the probability for each is 0.95, is only 0.7 (i.e., 0.95⁷=0.698). The odds of having a flawed procedure are about 1 in 3, even with an absurdly-high confidence in achieving each criterion.

HC47 HC48 HC49

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 HC47, HC48 and HC49. 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 and have many distractions and delays that prevent accurate counts.

HC47 analyzes a conventional transect. See Fig. 12.

Fig. 12. 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 0_j, R_i, and X_i are needed. P is a point determined by a line perpendicular from X to the transect. After many such observations of X_i the likely width of the area is thereby determined and used with length to get area, then the animals per unit area.

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 R_i, and the angle is O_i (theta), and the distance from the animal to the transect is y_i.

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 y_i, 0_j, and r_i are made. Only two are needed but three can be used for checking. A flush is seen, a marker is placed at Z. (Fig. 12.) The sighting distance, r_i, is measured and a marker placed at X. The sighting angle O_i is measured, then the right-angle distance X_i is measured. Two people usually 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:

X_i = r_i sine (0_i)

The perpendicular distance is the measure of interest because if we place all observations on a graph as in Fig. 13, we see that there is some

Fig. 8.13. 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).

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 x_i and then use it plus one standard deviation (for 90% of the observations) and assume a sharply declining distribution as in Fig. 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 D_u is achieved when the mean W is used; a lower density D