## Diversity index - Fisher's alpha parameter

Fisher's logarithmic series model (Fisher et.al., 1943) represented the first attempt to describe mathematically the relationship between the number of species and the number of individuals in those species. Although originally used as an appropriate fit to empirical data, its wide application, especially in entomological research, has led to a thorough examination of its properties (Taylor, 1978). The small number of abundant species and the large proportion of 'rare' species (the class containing one individual is always the largest) predicted by the log series model suggest that, like the geometric series, it will be most applicable in situations where one or a few factors dominate the ecology of a community. For instance Magurran (1981) showed that species abundances of ground flora in an Irish conifer plantation (in which ligth is greatly limited) followed a log series distribution.

The log series takes the from:

x, x2 /2, x3 /3, .. xn /n

x being the number of species predicted to have one individual, x2 /2 those with two and so on (Fisher et al., 1943; Poole, 1974).
The total number of species, S, is obtained by adding all the terms in the series which reduces to the following equation

S = [ - ln(1-x)]

x is estimated from the iterative solution of

S/N = (1-x)/x[- ln(1-x)]

where N is the total number of individuals.
In practice x is almost always > 0.9 and never > 1.0. If the ratio N/S > 20 then x > 0.99 (Poole, 1974). Two parameters, , the log series index, and N, summarize the distribution completely, and are related by

S = ln(1 + N/)

is the index of diversity. It has been widely used, and remains popular, despite the vegaries of index fashion. The index may be obtained from the equation

= N(1-x)/x

## Javascript program

Following is a Javascript program for estimating Fisher's from a species count and a stem count using Newton's method.

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### Summary of the performance and characteristics of Fisher's diversity index

• Discriminant ability = Good
• Sensitivity to the sample = Low
• Richness or evenness dominance = Richness
• Calculation = Simple
• Widely used ? = Yes

### References

Fisher, R. A., Corbet, A. S., Williams, C. B., 1943 "The relation between the number of species and the number of individuals in a random sample of an animal population." J. Anim. Ecol., 12, 42-58.
Magurran, A. E., 1981 "Biological diversity and woodland management." Unpublished D.Phil.thesis, New University of Ulster.
Poole, R. W., 1974 "An introduction to quantitative ecology". McGraw-Hill Kogakusha, Tokio.
Taylor, L. R., 1978 "A variety of diversities. In Diversity of Insect Faunas" 9th Symposium of the Royal entomology Society (eds. L. A. Mound and N. Warloff), Balckwell, Oxford, pp. 1-18.

Non nobis

Quantitative ecology rationale: g.d.campetella
Javascript source and page web master: m.angeletti