Mathematical modeling of urban sprawl

Abstract

Urban land cover doubled between 1985 and 2015, yet the spatial dynamics of urban form remain under-quantified, despite its importance for sustainability, infrastructure planning, and climate risk. Urban expansion is a non-equilibrium process shaped by interactions between population growth, infrastructure, institutions, and market failures -- rendering static and equilibrium models inadequate. We review key challenges and modeling approaches, focusing on partial differential equation (PDE) frameworks. Borrowed from statistical physics, PDEs capture spatial heterogeneity, anisotropy, stochasticity, and feedbacks between land use and transport networks. Integrating economic and institutional factors remains a major challenge for policy relevance. We propose a research agenda that bridges remote sensing, urban economics, and complexity science to develop dynamic, empirically grounded models of urban expansion.

Figures (8)

• Figure 1: Illustration of urban sprawl in London from 1800 to 2013. Data from angel2012atlasangel2016atlas; see also AtlasUrbanExpansionHistorical.
• Figure 2: Growth patterns of the largest connected urban area as a function of population (1985–2015). (A) Constant-density scaling (Beijing), (B) density increase (Guatemala City), (C) saturation (Las Vegas). Inset: historical density trends. Figure from marquis2025universal.
• Figure 3: Two-dimensional diffusion-limited aggregates with varying sticking probabilities. As adhesion increases, patterns become more branched. From heath2018visualization.
• Figure 4: Growth contours of a cell colony. The evolving tumor boundary exhibits scale-invariant morphology. Adapted from bru2003universal.
• Figure 5: Time evolution of population density, showing convergence to steady state. Adapted from bracken1992simple.