Urban Land Area and Population Growth: A New Scaling Relationship for Metropolitan Expansion

In most metropolitan regions throughout the globe, urbanised land area is increasing to accommodate increasing population size. This article provides a simple yet powerful mathematical description of this urban expansion. Specifically, the following scaling relationship is proposed: land area (A) increases proportionally to population size (P) raised to a power (n)— i.e. A ∝ P n. During 1950—2000, this relationship is found to hold well for US Census urban areas (UAs) with a greater than 10 per cent increase in population. Values for the parameter n vary among UAs, with a central tendency value of ∼ 2, suggesting that, on average, newcomers to urban areas occupy about twice the land area per capita of existing residents. If n were exactly equal to 2, then the parameter group P/√A (called `linear population density', or LPD) would be constant over time. LPD (units: people per metre) is the number of people in a metre-wide strip across an urban area. LPD is distinct from, and behaves somewhat differently than, population density. Distributions of LPD values among US UAs during 1950—2000 show surprisingly little variability over multidecade time-scales. For example, from 1950 to 2000, average population, land area and population density changed by more than a factor of 2, but average LPD changed less than 10 per cent. Few, if any, other attributes of urban form have remained so constant during this half-century time-period. International data corroborate the finding that LPD distributions are roughly constant over multidecadal time-scales. These results suggest an underlying pattern to how people arrange themselves within and among urban areas. For US UAs, rank—size rules similar to the generalised version of Zipf's rule hold for population, land area, LPD and population density. LPD is an important predictor of the emissions-to-inhalation relationship for motor vehicle emissions. Results presented here are important for theoretical, practical and empirical investigations of urban form and of how urban areas expand over time.