The Fractal Model:: a new model to describe the species accumulation process and relative abundance distribution (RAD)

Quantifying a sampled collection of taxa is a widespread problem in ecology. A number of diversity indices have been proposed and used in numerous works in spite of a lack of statistical characteristics and tests of comparison. These Relative Abundance Distributions (RAD) can also be described using graphic representations exhibiting both the number of taxa and the quantitative distribution of the individuals which constitute these taxa. Fitting these RADs to deterministic or stochastic models remains problematic for different reasons like scale dependence or complexity of the numerical procedures. In our study, we introduce a new model to describe RAD and ecological succession. This model is a Fractal Model based on the assumption that during an ecological succession/accumulation process, at each step of the succession, K new species appear which are k times more abundant with K = k(d) (where d is a fractal dimension). We use a Monte-Carlo procedure with a Kolmogorov-Smirnov distance to fit this model and to estimate parameters. Through the study of entomological data from a Mediterranean Man And Biosphere reserve, we demonstrate the qualities of the new Fractal Model. Thus, the Fractal Model was rejected for none of the 13 RADs tested. In addition, the parameters (K, k and d) are independent of the Hill family diversity indices and of three evenness indices and are able to provide additional information concerning both the diversity of the sampled ecosystem and development of this biodiversily during the species accumulation process in this ecosystem.