Mathematical models of urban growth

The article models the distribution of cities by population. Two approaches are considered to mathematical modeling of urban growth: a probability model in which the number of cities depends on the population and the rank model of distribution of cities by their population. Five population censuses are analyzed for Russia’s cities. The probability density function n(x, α) for the number of cities as a function of their population x is fitted to all the available censuses with a time-dependent coefficient α . The function α(t ) is approximated and a prediction for the nearest future is computed. In particular, it is shown that in 2010 compared with 2002 the number of large cities should increase, while the number of small town should decrease. A model is also proposed for the interaction of urban areas linked into a single hierarchical system. The model is based on a system of ordinary differential equations describing the change in urban population. Independently of the initial distribution, all the cities and town line up by the rank–size law and deviations from this law, as in real life, are observed only for some large and very small cities. Model parameters are fitted for Russia’s cities.