Modeling Morphology of Cities and Towns

Predicting urban growth is important for practical reasons, and also for the challenge it presents to theoretical frameworks for cluster dynamics [1–3]. Recently, the model of diffusion limited aggregation (DLA) [4,5] has been applied to describe urban growth [1], and results in tree-like dendritic structures which have a core or “central business district” (CBD). The DLA model predicts that there exists only one large fractal cluster that is almost perfectly screened from incoming “development units” (people, capital, resources, etc), so that almost all the cluster growth occurs in the extreme peripheral tips. Here we propose and develop an alternative model to DLA that describes the morphology and the area distribution of systems of cities, as well as the scaling of the urban perimeter of individual cities. Our results agree both qualitatively and quantitatively with actual urban data. The resulting growth morphology can be understood in terms of the effects of interactions among the constituent units forming a urban region, and can be modeled using the correlated percolation model in the presence of a gradient. In the model we now develop we take into account the following points: (i) Urban data on the population density ρ(r) of actual urban systems are known to conform to the relation [6], ρ(r) = e, where r is the radial distance from the CBD situated at the core, and λ is the density gradient. Therefore, in our model the development units are positioned with an occupancy probability p(r) ≡ ρ(r) that behaves in the same