Fractional modeling of urban growth with memory effects

The previous urban growth model by L. M. A. Bettencourt was developed under the framework of a constant β scaling law in an ordinary differential equation based model assuming instantaneous dynamic growth. In this paper, we improve the model by considering the memory effects based on fractional calculus. By testing this new fractional model to different urban attributes related to sustainable growth, such as congestion delay, water supply, and electricity consumption for selected countries (the USA, China, Singapore, Canada, Switzerland, New Zealand), this new model may provide better agreement to the annual population growth by numerically finding the optimal fractional parameter for different attributes. Based on the theoretical time-independent scaling of β = 5 / 6 (sub-linear) and β = 7 / 6 (super-linear), we also analyze the population growth of 42 countries from 1960 to 2018. Furthermore, time-dependent scaling law extracted from empirical data is shown to provide further improvements. With better agreement between this proposed fractional model and the collected empirical population growth data, useful parameters can be estimated. For example, the maintenance cost and additional cost related to the sustainable growth (for a given city's attribute) can be quantitatively determined for the informed decision and urban planning for the sustainable growth of cities.