pi
Modern Wild Faunal Resource System Management
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As more people in the eastern U.S. become more urbanized, they forget about or have fewer and fewer outdoor experiences. It is very easy to lose awareness of and appreciation for how variable conditions are in farming, forestry, and outdoor activities. There are also reported conditions on television that suggest that scientists are gaining control and predictability in amazing ways. These achievements can be generalized to "all scientists," but they need to be carefully related to each field of work. Science in wildland work has made great gains but there is now (and it will always remain) great variability due to natural factors (lightning fires; storms; disease epidemics; insect infestations; vandalism; and the "under-coordinated" activities of 1000 species of large animals; 20,000 species of insects, and over 100 species of plants.) Timing of many events will remain unpredictable. There are unpredictable economic and political factors that also play a role in making the wildlands a very variable and highly unpredictable place. Yet some people seem to assume and act like that is not true; that conditions and responses in the wildlands are very predictable. Clearly, at one scale, things are very predictable. Trees grow, water runs down hill, animals eat vegetation, etc. We can predict (with low confidence but "within reason" what a forest will look like in 20 years after trees are removed. Ecology -- the study of plant and animal relationships -- is one field of research, but there are many needed for the forests and wildlands -- geology, ornithology, ichthyology, dendrology, mammalogy, hydrology.., and other equally major areas of knowledge-based-building. Predictability in all of these field, together, is very difficult because of high variability.
Added to the above situation that creates such difficulties for the land and resource manager is the problem of the decision period. Most wildland-related decisions are (or ought to be) long-term. Uncertainty about any future is well known; it is especially great for the distant future. Natural resource work and planning must match the slow processes of tree growth and community energy flow and mineral cycling. We may gain ability to predict events...but not the time of occurrence.
There is great variability, great uncertainty about the future. Judgment is needed and this means using all available knowledge in the most logical and sophisticated ways possible, but it also implies taking risks. In an uncertain situation, everyone cannot be right all of the time in every situation. Errors are natural; they need to be expected. If no errors or mistakes in judgment are made, then no real or relevant decisions are being made. If so, they are made with such fuzzy bounds that any result will be judged "OK." Committees seem to reduce risks because blame for what failures (natural and expected) do occur cannot be assigned. Judgment in the face of variability by knowledgeable, risk-taking managers, using superior aids (e.g., computer maps and optimization) is needed.
Variance a well known word in statistics. It takes more work than the average student gives the concept to master it. (I find that the word and phrase "departure from the middle" seems to help make sense of the variance idea to some people. It is an important concept for many reasons, one being that when people use the term biodiversity they may mean biological variance.("Biodiversity" is more frequently used than understood.)
| A | B |
| 5 | 7 |
| 6 | 5 |
| 4 | 5 |
| 5 | 5 |
| 5 | 3 |
| Total 25 | 25 |
If I measured the content of 5 bags (say of game bird seed mix) and found exactly 5 pounds in each one, what is the variance in their weights? The answer is
zero. The answer (and the word used) is intuitive. Which has the greater variance A or B?
| B | ||
| 7 | - 5 | = 2 |
| 5 | - 5 | = 0 |
| 5 | - 5 | = 0 |
| 5 | - 5 | = 0 |
| 3 | - 5 | = - 2 |
Then each deviation is squared (getting rid of the minus sign), then all are added to get the sum of the deviations that have been squared. The sum of the squared deviations in A is 2 and that in B is 8 (2 2 + (-2) 2 = 8). Since A and B may not always have the same number of bags (or other items) then an average is taken. In this case, the mean squared deviation for A is 0.4 and that for B is 1.6. These two values are the variance of population (or sample) A and B.
To get these squared numbers back into realistic values, we "un-square" them, i.e., take the square root. This value is called the standard deviation, an averaged deviation from the average, plus or minus, having taken a long route to get it. The standard deviation for A is 0.6 and for B, 1.26. The reason that simple deviations cannot be averaged is clear in B (2 + (-2) = 0); the average seems to be zero - clearly not correct and it misrepresents what we want to express about the distribution of the numbers.
For the most "normally distributed" populations or sets of numbers, those having about equal numbers on either side of the average, 2 standard deviations (actually 1.96) will include 95% of the numbers likely to be encountered. There is only about a 5% chance of a number being more than 2 standard deviations away from the mean in either direction when the numbers have the general appearance of a bell.
As noted, when the standard deviation or variance is very small, the numbers are tightly packed near the center (the average). When the variance is large, the "bell" is fat or broad, widely distributed, highly varied.
Range is the difference between the largest and smallest number in the group being studied. The larger the sample, the more likely a larger (or smaller) number is to be encountered. The range may be fixed by biological or natural limits (e.g., when water freezes). The range is important; especially the limiting values. The range can be used to estimate a standard deviation. For reasonable size samples, say 20 to 30, then a multiplier of 0.25 is used. The standard deviation is about one-forth of the range, i.e., range x 0.25 (For small samples, say 10, the range will be within 85% of the standard deviation). It is often more instructive in natural resource issues to concentrate on estimating the limits rather than the central tendency. The median is often more instructive than the mean. Because biological systems are skewed (not bell-shaped), variance is often abnormally high resulting in abnormal requirements for sample sizes and unrealistically-high assumed risks.
The range may be determined by out-liers, perhaps erroneously observed or recorded, but always possible real occurrences. They may be very improbable occurrences, thus emphasis on the "central tendency" of numbers is reasonable... unless you work with long time periods and thousands of tracts. Then.., the chance of encountering an extreme value is very high. For the forester or wildlife manager, concentrating on the range limits, not the central tendency, is a reasonable ploy. The coefficient of variation, CV, is the average deviation contributed by each observation, thus:
CV = (standard deviation / mean), often expressed as a percentage (i.e., x 100). (It's an average variation that can be associated with or "hooked on to"to each member or unit of the population. )
This measure allows populations to be compared, independent of the size of the mean value (e.g., big fish in large and small ponds).
Variety as in biodiversity can be categorized as that for
It can also be categorized as
There is variety everywhere.
Some self-evident, others need comment:
Temporal
Disturbance
There is one well-known measures of variety, a numerical dominance measure under structural diversity. A so-called "diversity" index, the Simpson Index of diversity is one measure of variety is
H = 1.0 - pi
where pi is the proportion in each taxon or equivalent class.
The Shannon Index is
H' = -cpilog pi
When the log to the base 2 is used, C = 3.321928
When log to the base e is used, C = 2.302585, and
When base 10 is used, C = 1.0.
Note the negative value needed to convert the index to a positive expression.
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Potential Richness Approximate number of vertebrate species in the US
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Richness needs to be kept separate from discussions of diversity or variety. It can be estimated using the negative logarithm of ranked abundance (sketched above), typically a line cast to zero by the slope estimated from the 3 most abundant species. A comparison may be made to a "standard" curve or a past set of samples.
Studies suggest optimal plant species richness occurs in intermediate productivity environemnts, i.e., those with moderate competition and abiotic challenges (e.g., drought and frost). For large area richness, as expected more plant richness will be found in forests with openings (gaps) than in uniform communities or large homogeneous stands. Plant species and related processes change across regions and thus suggest the need for several approaches to forest management for richness or several expressions of diversity.
Richness is often expressed as number of taxa per unit area. The selection of the area and scale (or political units vs GIS pixels) can have major effects on the resulting numbers obtained.
There is often more information in the richness number than meets the eye. Two areas may have the same number of species, but quite different lists. The farther apart they are, generally the more different will be the lists. The similarity statistic can be used to express the amount of overlap in the lists or relative differences. Similarities in taxa may be confounded by differences in abundance within each taxon.
See USNEP/CBD/COP/6/20 final report of Convention on Biological Diversity (CBD) Hague, 2002 including international policy framework, Global Taxonomy Initiative, agricultural biological diversity, and invasive species.
Also see Biodiversity Support Program.
Much of landscape ecology is an effort to describe geomorphology and its resulting spatial variety (and super-imposed cultural practices made possible or limited by geomorphology). The four related categories of descriptors are shown at the right. Many in the following lists are inverse correlates of each other or simple transformations. Absurd results result from regression analyses and linear correlations of identical but transformed variables.
Spatial Heterogeneity or Diversity
The classes of descriptors are:
Fragmentation and Patchiness
Edges and Corridors
Connectivity
Recall that one of the elements of the concept of a resource is variety. There are now many laws and policies calling for biodiversity but few provide insights into what is intended. Maintaining it may be one objective. Some have comments about maximizing it. Others simply want no species loss (thus stable biological lists). Working with the topic will help you guard against being trapped by it. Positive leadership for improved use of the concept is needed.
The premises related to the variety concept:
1. Diversity has many meanings.
2. There are over 1000 technical papers on the topic. It is not a new topic.
3. Biological diversity is shortened as "biodiversity." Whether it means plants, animals, communities, genes or all of these must be determined from use and the context. It usually means animals but "phytodiversity" (as in phytomass) may help in some situations.
4. Diversity may mean:
Even similarity (or lack of it) to some people.
There are no clear standards and the "best" expression has not been selected.
5. Diversity may be compared by
The diversity of topics makes the concept almost impossible for the public or the courts to grasp and shifting groups can usually result in a change in a diversity index in an alternative direction
6. Diversity is a very diverse topic; at least 6 x 13 (the product of the 2 above lists) possible relations.
7. Richness, R, means the number of taxa. For example, one use might be to say: This area has 30; that one has 40. The one with 40 is more rich, more diverse, with greater biodiversity.
8. When a species becomes extinct, the area has a loss in richness. "Preserve species" is a richness slogan, only one aspect of diversity.
9. Taxonomic "lumpers" reduce richness; taxonomic "splitters" increase richness. Major change may occur with no physical changes in numbers of organisms in the field.
10. Genetics allows "splitting" of taxa or organisms to the individual; demes are analyzable; color phases (as in grouse or screech owls or squirrels) may also display diversity; subspecies (as in birds) may be appropriate analytical categories. There is no standard or criterion for judgement about such categories.
11. Abundance and richness are combined in the Simpson index.
Where the proportion in each ith species is pi and
V1 = The proportion of animals or plants in each ith species is used. Also called a dominance index, the value is 1.0 when all are in one species. This condition of all being in one species does not sound very diverse so the modified Simpson is usually used, namely
V2 = 1.0 - V1
12. The Shannon-Weiner index is frequently used. Named after the two people who independently developed the index, it is:
V3= H = - Note the negative sign that is related to the logarithm of a proportion being negative, (it changes the negative to a positive value) thus, V3 being made positive.
Because both abundance and the numbers of species may change, the index has no standard value or base figure against which a score can be obtained. Thus, the only uses are in knowledge that the larger the value, the "more diverse", at least the more evenly distributed.
The proportions are used as above. Logarithms with different bases, Loge, log10, and log2, are all used. There is no best value; the more evenly distributed, the larger will be the value of V3.
Several sets of proportions can give the same value of V3. V3 is a value in search of a meaning. It is called "average rarity."
13. Equitability, V4, is a statistic comparing V3 to the maximum value that would be obtained if all individuals in the sample studied were evenly distributed among the observed species present. It is relative Shannon-Weiner diversity. Using it results in a score or expression relative to a standard...but the meaning of the standard can shift with the season, sampling, etc. Diversity is a tough topic to grasp and, worse, to explain well in court or to use it as an objective or performance measure. The chief difficulty is that very different counts among species (say comparing before and after treatments) can result in identical numbers, the principle of equifinality.
14. V2 and V4 seem managerially useful.
15. Evenness, has an index
V5 = V3 / log R
It is another expression of evenness, as is V3.
16. I think the ranked abundance (high to low) of species in a natural system will usually
approach the negative exponential or "reversed J" distribution. How well an observed distribution fits this hypothesized distribution may be worth studying. The lack of evenness may be of much more interest than its presence. Only the top three numbers need to be analyzed by the field worker intent upon timely surveys and quick analyses. The negative rate will eventually be found to be significant for in the top most abundant life forms will there be integrated the energy and nutrients of the sampling period. The taxa, themselves are almost irrelevant. The forms are food for other organisms. They are merely temporary nitrogen, calcium, and energy pools, soon dissipated.
17. The manager should pick a system performance measure, say "species X
observations per hour along a transect", then develop a multiple regression, each independent variable being other species, values of each being abundance. Keep the abundance information separately so that effect of change in one can be observed on species. Stop using proportions since each species number is a function of the other numbers sampled, probably not the forces within the ecosystem.
18. Equifinality is a real problem if indices are used. Very different inputs from very different systems, when computed, can result in the same number, e.g., the "biodiversity". Avoid indices; work with regression analyses.
19. Temporal diversity, another potential, means change between and among periods and may include time series analyses.
20. Process diversity is descriptive of behavioral diversity (activity patterns and conspicuousness); production diversity (measured as biomass.unit area/unit time - with equifinality common); and trophic diversity (different feeding levels, e.g., carnivores, herbivores).
It is evident that here are many published and imagined ways to produce an index to biodiversity. In preliminary studies that I have done with computer programs, a loss of one species can cause half of some 18 indices to increase, half to decrease. Arguing for biodiversity can be a useful political position because of this either/or as well as both/and condition.
The manager needs to clarify that change in indices between areas are not comparable. A change proportional change ( some pi)in an abundant species is not equivalent to a change in a rare species. A loss of one abundant species is certainly not comparable to loss of a rare species...as would be reflected in Shannon -related indices. Change over time may be compared. Note that the same index can result if a list of species reverses its order of abundance completely! Interpretation is essential for each site and period.
For visitors and all others interested in "diversity" they need to realize that it is unnatural for even numbers of animals in all species to occur (e.g., equal number of bears and beetles or wood peckers and magnolia warblers). Called "equitability" by some, the equal numbers condition is not useful, even if used as a standard or base to compare how "unequal" the populations are (thus the diversity claim). The major index now reported in the literature is the Shannon-Weiner index which has a maximum value when counts within all species are equal.
Study Suggestion
See The Trevey and its section on Variety.
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This Web site is maintained by R. H.
Giles, Jr.
Spatial Variety
means the process of summing all of the i's, then
pi2
( pilog pi)
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Last revision January 19, 2004.