SPECIES-DIVERSITY AS A FRACTAL PROPERTY OF BIOMASS

Species diversity is commonly characterized by a species distribution, and represented by a retrocumulative curve or Rank-Frequency Diagram or RFD. Species are ranked following decreasing order of frequency in the community, and ranks are plotted against the frequency or abundancy of species in the community, often on a log-log scale. A number of models of distributions have been yet proposed. The Zipf-Pareto model, F-r = Cte.(r+beta)(-gamma) (initially derived from socio-economic and linguistic theories), fit to ecological data in certain stages of the evolution of ecosystems and at certain observation scales. A fractal derivation of the Pareto model is proposed here, following Mandelbrot's demonstration that d = 1/gamma is a fractal dimension (d < 1 as a Cantor set). Admit that, running an ecological succession, when more and more specialized and rare species appear, at each step of the succession K more species appear which are k times less abundant on an average, and K = k (d). Then, reparametrizing,a the equation by putting gamma = 1/d and beta = 1/(K-1), the Pareto equation is exactly found. For example with gamma = 2 and beta = 3 (values sometimes observed in natural mature communities), the succession of species is as if at each stage, on an average, 1.333 more species appear, 1.778 less abundant. It appears not unlikely that etch regularities denote an optimization on the management of the set of interspecific interactions in the ecosystem, as it occurs in socio-economic and linguistic systems when a Pareto distribution applies.