[Paper Review] Mathematical modeling of urban sprawl

Marc Barthelemy, Ulysse Marquis|arXiv (Cornell University)|Mar 9, 2026

Land Use and Ecosystem Services0 citations

TL;DR

The paper surveys PDE-based approaches to model the spatio-temporal dynamics of urban expansion, linking urban growth to density fields, transport networks, and non-equilibrium processes, and outlines a research agenda for empirical, dynamic modeling.

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.

Motivation & Objective

• Motivate the use of partial differential equations to model the growth and shape of urban areas as non-equilibrium systems.
• Identify core challenges in defining the city, describing urban evolution, and choosing fundamental variables.
• Review existing modeling paradigms and their limitations, especially Alonso–Muth–Mills (AMM) and dynamic extensions.
• Propose a research agenda that bridges remote sensing, urban economics, and complexity science for dynamic, empirically grounded models.
• Highlight how PDEs can integrate density, geometry, and infrastructure feedbacks to reproduce stylized facts.

Proposed method

• Introduce a local density field ρ(x,t) to represent urban quantities such as population or built-up density.
• Discuss the general PDE form ∂ρ/∂t = F(ρ, x, t, …) as a framework for urban growth dynamics.
• Review AMM dynamics and its limitations, motivating PDE-based alternatives that allow for non-equilibrium, path-dependent evolution.
• Describe PDE approaches inspired by surface growth physics, including diffusion-like terms, anisotropy, and stochasticity.
• Present early PDE models of isolated cities incorporating diffusion, crowding, and central attraction, and extend to congestion effects via nonlinear terms.

Experimental results

Research questions

• RQ1 How can PDEs capture the spatio-temporal evolution of urban form beyond equilibrium assumptions?
• RQ2 What core variables and terms (density, potential, diffusion, congestion) best describe urban expansion and boundary evolution?
• RQ3 To what extent can PDE-based models reproduce empirical stylized facts and universal patterns of urban growth?
• RQ4 How can urban economics, remote sensing, and complexity science be integrated into dynamic, empirical PDE models?
• RQ5 What are the limitations of monocentric, AMM-based frameworks and how can PDEs overcome them?

Key findings

• PDEs offer a flexible framework to model diffusion, anisotropy, stochasticity, and feedbacks between land use and transport networks.
• Historical and contemporary data (e.g., GHSL) enable empirical grounding and validation of urban growth PDEs.
• AMM and its dynamic extensions reveal path dependence and non-universal trajectories that motivate non-equilibrium PDE approaches.
• Early PDEs for isolated cities link central attraction and crowding to density decay patterns similar to Clark’s results, but with richer dynamics.
• A physics-inspired view (surface growth, KPZ/EW universality) suggests cities may exhibit universal-like scaling in boundary fluctuations.
• A research agenda is proposed to bridge remote sensing, urban economics, and complexity science for dynamic, empirically grounded models.