General regularities of the distribution of cities by population

We consider mathematical models that describe the distribution of a set of cities by population. The empirical data are derived from population censuses in several countries. We show that the dependence of the number of cities on population within any region or country can be described by the same distribution function. A relatively narrow range of distribution function parameters fit the observed data. We also demonstrate the relationship between the distribution function and the "rank-size" dependence that uses a rank function to describe the ranking of cities by population. The article also examines a mathematical model of interacting cities within a single hierarchical system, which explains the eventual formation of the "rank-size" distribution. The model is based on an ODE system that describes the changes in city population. © 2012 Springer Science+Business Media, Inc.