Simple mathematical models for urban growth

A simple deterministic mathematical model for urban growth and spread is developed in which there is local logistic growth, diffusion, and an integral (‘killing’) term representing the inhibitory effect of congested central areas on growth. Migration is represented by boundary conditions. The resulting integrodifferential equation has steady-state solutions which satisfy a third-order nonlinear differential equation and are obtained numerically. Time-dependent solutions are obtained by using a modified Crank–Nicolson method. The results indicate that there is a threshold effect with respect to the initial population distribution and that for some parameter values populations do not persist, regardless of the initial numbers. Steady state densities may take an exponentially decaying form as in Clark’s original analysis or there may be a maximum some distance from the city centre.