Spatial Yule Model for Urban Growth with Shared Resources

We study the competition of cities at a highly urbanized stage through a spatial version of the Yule model with a fixed carrying capacity. The capacity naturally determines two phases of the model, wherein the natural growth phase, we derive and observe the finite Zipf scaling of urban size distributions and the exponential decay of population density in each city; the resource constraint phase corresponds to the stage that the population of the urban system reaches the fixed capacity; thus migration among cities rather than urbanization contributes to the population dynamics of cities. In this phase, we derive the likelihood of a switch in the ranking of the cities. Our model sheds light on the post-urbanization era’s inter-city competition patterns and population dynamics.