Fractal methods of the analysis of a community structure

The order maintaining fundamental physical and biological processes is the basis of the diversity of life and the complexity of ecosystems. Power laws describe empirical scaling relationships that are interpreted as quantitative characteristics of biodiversity. For these purposes, self-similarity concepts and the elements of fractal geometry are used. Self-similarity suggests that copying and scaling a certain “reference” image allows nature to easily create a complex multi-scale structure. Real objects have a quite clearly limited range of scales, in which they manifest their fractal nature. Power laws allow for extrapolation and prediction in a wide range of scales. Some of them appear to be universal and are found in almost all taxa of organisms and environmental types. We present a brief biography of an outstanding mathematician, Professor Benoit Mandelbrot (1924-2010), the originator of a new direction in geometry – fractal geometry. The theory of fractals developed by him had a significant impact on various areas of human activity and knowledge: computer graphics, finance, ecology, etc. The theory of multi-fractals characterized by an infinite hierarchy of dimensions provides the most general description of the internal structure of self-similar objects. The differences of the mono-fractal and multi-fractal approaches are considered. In applying multi-fractal formalism to a community structure, it is considered as a set consisting of individual fractal subsets which can be interpreted as the sets of individuals belonging to species with similar representation. For such subsets it is possible to calculate the fractal dimension characterizing the species diversity. The results of the application of the Willis rule to the analysis of taxonomic diversity of the flora of Samarskaya Luka (the Volga river basin) are discussed. It is stated that the patterns identified for the multi-fractal spectrum of the species structure of the community are also maintained for the spectrum of a generic structure to a certain extent.