Document: BOR-3-92-3

The general relationship between species-area curves, diversity indices and species abundance distributions: A multifractal analysis.

BORDA-DE-ĮGUA, L.* ¹, S.P.HUBBELL ² and M.K.MCALLISTER ¹

Imperial College of Science, Technology and Medicine, London SW7 2 AZ United Kingdom ¹
University of Georgia, Athens, GA 30602 USA ²

Abstract:
Although fractals have for long been applied in ecology, multifractals have, in contrast, received little attention. In this work we explore the application of the multifractal formalism to the power law form of the species-area relationship. The generalization of the power law species-area relationship by multifractal methods is potentially far reaching: it ties species-area theory to a number of indices used widely to quantify diversity, generates a number of new hypotheses about patterns of biodiversity for further investigation, and it develops new tools with utility in addressing practical issues, such as, conservation. While fractal sets are described in terms of a single number (the fractal dimension), multifractals require an infinite number of "dimensions", usually called a spectrum. The characterization of multifractals can be obtained by two different approaches: the method of moments and the method of histograms. The former deals with the statistical moments of the species abundance distribution, but can only be applied when the moments exhibit power law scaling with area. The latter method is of more general application and uses histograms of the species abundance distribution obtained at different areas. The method of moments shows that the power law form of the species area relationship, and the Shannon, Simpson, and Berger-Parker diversity indices belong to a family of equations relating the species number, species relative abundance and area through the moments of the species abundance distribution. Explicit formulas for these diversity indices, as a function of area, are derived. In addition, the method of moments implies relationships between a species range and its relative abundance. The method of histograms highlights the dependence of the shape of the species relative abundance distribution on area. The application of these methods is illustrated with data on tree and shrub species collected in a 50 ha plot in the Barro Colorado Island, Panama. Results from the method of moments show that some moments of the species abundance distribution have power law scaling with the area. Results from the method of histograms show that after appropriate transformation the species abundance distributions obtained at different areas converge to a single curve: the multifractal spectrum.

Keywords: Species-area relationship, multifractals, diversity indices, species-abundance distributions

This abstract is being presented at: 11:00 AM in session:
Oral Session #39: Theoretical Ecology.