A stochastic equation for modeling population growth in cities

A team of researchers from Université Paris-Saclay, CNRS, CEA, developed a stochastic equation to model population growth in cities. In their article published in the journal Nature, the group describes the creation of its own equation to account for “intercity migratory shocks” on population changes and the factors involved in producing the results.

Over the past hundreds of years, mathematicians have attempted to create formulas to describe population growth or reduction in major cities in a given country. But so far, the best they have managed to find is Zipf’s law, also known as the Gabaix model, which uses the regularity of city growth to estimate future growth. However, subsequent efforts using the model have uncovered several flaws, particularly when random events occur that can have a dramatic impact on a given city’s population, such as a war.

In this new effort, the researchers took some of the important parts of Zipf’s law and added three important factors to introduce randomness: demographics, departures and arrivals, and long-distance migration. They define long-distance migration as the movement from rural areas to cities or from one city to another. To create and test their equation, they used city population data from France, the United Kingdom, the United States and Canada.

In this effort, they discovered something new about the growth or decline of the cities’ population: migration shocks are important. They define such shocks as the rare movement of people in or out of a city due to social, economic or climatic events. They note that history is full of examples of such shocks that lead to the explosive growth of a city or its demise. The early cities of the American West, they note, are good examples of both. The gold rush in the late 1800s led to rapid population growth in some cities and then sudden crashes when the gold ran out.

The researchers suggest that their equation can be used by urban planners to estimate the population and distribution of cities, and also to predict changes in a city’s hierarchy.

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