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

of an Alternative Wildlife Resource Management

The Concept of the Maximum

The concept of the maximum is a topic of efficiency. It is grounded in natural or production efficiency, not specifically in economic efficiency (Kopp, R.J. 1981. The measurement of productive efficiency: a reconsideration, Resources for the Future Reprint 203, Washington, DC). The concept suggested is that of rejecting the engineer's criterion of technical efficiency grounded in the "best results observed in practice" and accepting the best theoretically obtainable. This concept seeks to suggest a means to estimate the amount theoretically obtainable.

Crop scientists have asked the question of world wide and historical concern, "how wide is the gap between what is being done and that which is possible?" That gap was very wide between average production and record production (which is likely to increase with research, etc.).

Select yields (bushels per acre) in the US, 1973, of crops potentially used by wild fauna. (Corn will produce more digestible nutrients per unit area of land than any other food crop. Sorghum is a close second.)
Crop	Average	Top	Record
Corn	94	230	306
Wheat	32	135	216
Sorghum	63	200	320
Barley	41	150	212
Oats	49	150	296

Managers are tempted to work from minimums and not maximums. "The populations have improved" may be the claim, but the reasonable skeptic will ask "as compared to what?" The modern manager needs to concentrate on estimating the maximum numbers and densities of a species that may occur within an area. This gives a target, a basis for comparison, and prevents excessive investments to gain numbers of animals, services, or use-units. As an example, a manager may be able to have a maximum of 100 animals, but will seek rewards for doubling a population, once 10 animals. Perhaps an increase of 10% would be an alternative claim.

It is possible to transform an estimate of the most animals of a species seen on a particular day into the product of natural logarithm of the estimate (N) plus 1 and 10. Thus,

aBar = 10 ln (N + 1).

The expression of the maximum number of species seen on a single day is aBar. It is ten times the naperian logarithm of the observed number. Maximizing aBar may be an objective on wildlife areas; minimizing it may be an objective of damage managers.
Estimated N	aBar
equal to or less than 22,027	100
8,104	90
2,982	80
1,098	70
404	60
149	50
56	40
21	30
8	20
3	10
0	0

The resulting number ranges from 0 to 100. This variable, aBar, (read "a-bar"), may be used to provide insight into abundance and may be used as a system performance measure. In general, people desire to see many animals (e.g., birds) ... either richness, already discussed, or great abundance of one species. "We saw thousands", and "The sky was darkened" are expressions capturing the excitement of the view. When the number is very great, a is equal to or greater than 22,026 (see the Table ). Very rare species have a very small aBar. The number is akin to pH in chemistry but varies from 0 to 100 and is equivalent to an abundance score. It is believed to be useful in some situations to compare areas or select one from several options for certain types of projects (e.g., a nature trail).

It also is valuable in determining the theoretical maximum group size or in some situations, the maximum socially relevant population itself. "Socially relevant" is used because it is hypothesized that the human mind does not discriminate well among big numbers. "Many" are many. The difference between 20,000 and 24,000 is probably indistinguishable to more than 90 percent of the people. Between these two numbers, for example, is a difference of between 17 and 20 percent (because even the proper denominator cannot be readily decided). The biological or ecological significance of such differences is not readily apparent. It turns out that if 22,026 birds are seen and each is allocated sky or marsh space of one square meter, then the flock covers 2.2 hectares or 5.4 acres. An aBar of 100 represents many animals. As a performance measure, it does not suggest in any way numbers should be reduced to 22,026. It merely suggests that funds spent purposely to increase the number beyond that one are probably misallocated. The target or maximum value has already been hit. This is a visual, socially relevant index and in no way is an expression of a desired maximum population within an area.

One example and an expansion of the use of aBar is as follows. In a situation where birds are observed every day (e.g., a wildlife refuge) flocks of species A are observed.

Over long periods, aBar will fluctuate, but by plotting progressively over time the maximum aBar values, values will stabilize and will allow a theoretical maximum to be estimated. Where an unexpected change occurs (x), the graph no longer reflects simple dynamics but a major system change.

On each day that a exceeds the previous value, the a value is plotted and points connected. When the curve levels, the maximum may be projected. Many years may pass before the next larger population is observed or the maximum seen. (See Figure.) The process is not unlike estimating the height of the 50- or 100-year flood stage at a stream, or that of the collector's curve (Pielou 1977:285).

Since low-to-high ranked species abundance tends toward a positive linear logarithm relationship (rather than great evenness or equitability), then the relationship of aBar to S (species richness) is equivalent to the surveyor's "rise-over-run" expression of slope. Thus, aBar becomes expressive (generally) of standing biomass, area productivity and area dynamics or vitality; aBar will be correlated with richness, S, but it is not appropriate to calculate ABar (read "cap-a-bar") as ABar = aBar/s for a system performance index.

Thus, when there are few species, say 20 (S = 20 20) and the maximum ever seen on one day of a species is 86, then aBar = 45 and ABar = 2.25. Where aBar is 86 in another habitat but same area, for example, it does not seem unreasonable to expect the number of species to be about 38. Deviation from this amount may suggest habitat factors or mortality factors that explain the difference. Lesser differences are often due to chance, making predictive equations difficult and costly to develop ... or wrong. (Thus, identified, changing such factors may be the means to move ahead, cost effectively.)

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Reference

Wittwer, S.H. 1974. Maximum production capacity of food plants, BioScience 24(4):216-224.

Last revision July 2, 2001.