Simple mathematical models for urban growth

Abstract

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.

The angiogenic growth of cities

“…Urban areas grow around their edges primarily due the demand for new space which is translated through the usual process into new development’. Similarly, Bracken & Tuckwell [ 52 ] proposed a simple deterministic mathematical model for urban growth considering ‘local logistic growth, diffusion and an integral (killing) term representing the inhibitory effect of congested central areas on growth’. Yet this assumption is purely theoretical as, strictly speaking, there is no clear evidence that the ‘diffusion’ of urbanization is the result of random walks and mixing of population (e.g.…”

Mapping Urban Transitions in the Greater Beirut Area Using Different Space Platforms

“…Some studies try to understand this urban growth [7]. Others create simple mathematical models [8,9]. Meanwhile, using and developing logistic regression has grown in recent years [10][11][12].…”

Cities as Evolutionary Systems in Random Media

“…There has been a long tradition in urban and regional science to approach the mathematical modelling of urban processes in the spirit of classical mathematical physics, by developing various deterministic (sometimes, quite sophisticated) models, aiming to capture the multi-component, dynamically variable nature of cities by augmenting the equations with suitable feedback mechanisms, akin to reaction-diffusion models or predator-prey models (see, e.g., Wilson 1981;Bracken & Tuckwell 1992, and further references therein). Although often useful in practical terms, the efficiency of such models proved to be limited to short spatio-temporal scales, hence there was the growing understanding among both theorists and practitioners that the deterministic approach was conceptually flawed, as it was missing some intrinsic, significant features of cities as agglomerates with formidably complex structure and intricate dynamics.…”