Evolutionary Model of the City Size Distribution

An evolutionary model of the city size distribution is presented that explains the size of a city from the reproduction process and the migration of humans between cities. The model suggests that the city size distribution is a lognormal distribution with a power law tail in agreement with empirical results and computer simulations. The main idea of the model is that the competition between cities in the migration process is the origin of Gibrat's law. While growth rate fluctuations generate the lognormal branch of the size distribution, the power law tail for large cities is caused by a small mean growth rate. 1. Introduction The paper aims at deriving the size distribution of cities from the idea that the evolution of a city is a self-organized process. The term city is used here very loosely to mean any human settlement not distinguishing between villages, towns, cities, or megacities. The size is usually characterized by the urban area or human population. Ranking cities of a finite region, for example, a country, from largest to smallest, Zipf observed in 1949 that the probability to find the size of a city that is greater than some is given by , with a Zipf exponent . This power law relationship is known as Zipf’s law [1]. Using appropriate definitions of the city size the validity of Zipf law was confirmed by the investigation of large cities [2–10]. For smaller cities empirical investigations established considerable deviations from this law. Eeckhout [11, 12] showed that taking all settlements into account the size distribution for small cities can be rather described by a lognormal distribution. Empirical investigations suggest therefore that the size distribution is lognormal with a power law tail [13–15]. Several theoretical models have been established to understand these empirical facts [7, 16–22]. Since classic economic theories fail to explain the size distribution [23] the majority of models have in common to apply Gibrat’s law of proportionate effects [24, 25] taking advantage of geographic and socioeconomic forces in their reasoning. The presented evolutionary theory goes beyond previous research by deriving Gibrat’s law from the main two processes governing the population size of a city: the reproduction of humans and their migration between cities. The key idea is to regard cites as entities of a self-organized system. The dynamics of self-organized systems is governed by positive feedback processes [26, 27]. Both the reproduction of humans and their migration are self-amplifying processes, while the migration flow is shown to be