Analysis and spatio-temporal modelling of spatial patterns in urban systems: agglomeration, segregation and food deserts

Urban population distributions in cities can show structure such as patchy patterning that may relate to important properties such as journey times, quality of life and sustainability. In this thesis we wish to address the question of what patterns exist between and within cities, and highlight the social unsustainability of the current set up. We look firstly at the length scales between cities and model their emergence. We use Fourier transforms and spatial autocorrelation to show that there is a typical length scale between UK cities of around 45km. We use integro-differential equations to model the spatio-temporal dynamics of urban populations and services, under the assumption that they benefit from spatial proximity to one another, as captured via spatial weight kernels. The system may tend towards a homogeneous state or a spatial pattern. With Gaussian spatial weight 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 length scale. Numerical continuation shows how multiple steady states corresponding to different spatial lengths 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. An example kernel is an off-centred Gaussian kernel for which we show that out of phase patterning occurs. We show that this patterning is at a higher density and of shorter length scale than in phase patterning. We also look at household income inequality and segregation at LSOA level (Table 1) in the UK for the first time, as part of work on the first paper to do so [29]. We show that the poorest areas of Nottingham are the most homogeneous with respect to income and that richer areas are comparatively more unequal. Comparisons between cities show that more affluent areas tend to be more unequal but less segregated. We also use the IDE model approach to explain the emergence of `pockets of affluence' as low income residents can become segregated without any desire themselves for this segregation. Lastly, we look to understand one particular effect of unsustainable spatial patterning in cities which is food deserts, areas in cities where there is poor access to affordable, healthy and fresh food. We show that the poorest residents are eating insufficient food and are often located far from supermarkets. We model food purchase and consumption using an agent based model to further highlight areas where there is poor consumption of healthy food. Sensitivity analysis of the parameters in the model highlights that the most telling interventions target the percentage of fruit and vegetables that people want to eat, as well as the strength of this preference. These interventions should take place in the context of also helping the very poorest in society to survive. In summary, this thesis brings the language of spatio-temporal modelling and linear stability analysis into the urban environment, explores data on the patterns of income inequality in cities and models explain some potential reasons why these phenomena may occurs. It also sheds new light on the food desert situation in Salvador as well as developing the ongoing modelling literature in this area