An integro-differential equation model for urban population dynamics predicting emergent pattern formation

Timothy Whitely, University of Nottingham. Part of the applied mathematics Leeds applied nonlinear dynamics seminars series.

Urban population distributions in large cities can show structure such as patchy patterning that may relate to important properties such as journey times, quality of life and sustainability. We use integro-differential equations to model the spatio-temporal dynamics of urban populations and services, under the assumption that they benefit from proximity to one another, as captured via spatial weight kernels. The system may tend towards a homogeneous state or a spatial pattern. With Gaussian kernels, linear stability of the spatially homogeneous steady state depends on a key function in the model, the carrying capacity for services given a local population density. In particular, patterning occurs only where the carrying capacity function is convex with respect to population density. Furthermore, this spatial instability can occur only for perturbations with a sufficiently long lengthscale. Numerical continuation shows how multiple steady states corresponding to different spatial wavelengths can coexist and state transitions may occur as carrying capacity grows.

In urban centres, competition for space may cause services and population to be out of phase with one other. To generate such patterning in our model requires kernels with Fourier transforms that are negative for some wavelengths. With box and off-centre kernels, out of phase patterning can occur. We show that this patterning occurs at a higher density and of a shorter lengthscale than in phase patterning.