Modeling the spatial growth of cities

The growth of cities has traditionally been studied from a population perspective, while urban expansion-its spatial growth-has often been approached qualitatively. However, characterizing and modeling this spatial expansion is crucial, particularly given its parallels with surface growth extensively studied in physics. Despite these similarities, approaches to urban expansion modeling are fragmented and scattered across various disciplines and contexts. In this review, we provide a comprehensive overview of the mathematical modeling of this complex phenomenon. We discuss the key challenges hindering progress and examine models inspired by statistical physics, economics and geography, and theoretical ecology. Finally, we highlight critical directions for future research in this interdisciplinary field.