| Abstract | In many complex systems such as an ecosystem, a metabolic system, an immune system, a human social system, an economic system, a linguistic system etc, common "general community structures" have been widely observed. A mechanism to generate such patterns, e.g. abundance distributions of components, is not only a main topic in each discipline but also have aroused controversy independently. The current study aim to explore a uniform view and methodology for the patterns by theoretically approaching to complex systems essentially containing diverse components, e.g. species in an ecosystem, genes in a cell, currencies, products or companies in a economic system, words in a book, etc. First, some examples of the general community structure will be presented: species abundance distributions (SAD) and the species-area relationships (SAR) in ecology, and the Zipf's law in linguistics and social sciences. Second, I relate SAD and SAR to the Zipf's law which may be best known as the power distribution of word frequency. Its characteristic power exponent is ubiquitously observed in many distributions such as in benthos population, proteins or mRNAs in a cell, economic system, population of cities, names in a population, etc. Various mechanisms of the Zipf's law have been suggested in each discipline, while there is few discussions on a general principle to generate such a ubiquitous exponent. Reconsidering the relations between SAD and SAR, I present a novel scenario for the origin of the Zipf's law in connection with the "generalized island biogeography" which can be applied to populations, molecules, products, words, etc. In particular, it is suggested that the ubiquitous value of the power exponent is due to a self-organized critical state at which some function of species richness is maximized, and therefore, the Zipf's law can be explained by a general mechanism which is independent from the details of the characteristics of each system. |
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