Mathematical modelling of collective dynamics in urban systems

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Urbanisation is reshaping many aspects of human societies and the natural environment, presenting both opportunities and challenges. A general quantitive theory on the growth and formation of cities remains elusive, and would enable us to forecast future demographic scenarios. The observed trends of population growth can be characterised by precise statistical laws, such as the distribution of city sizes, the spatial distribution of cities, and the spatiotemporal correlations of population growth rates. The analysis of empirical data reveals that these laws are common to many countries, suggesting that the formation of observed patters might be explained by a general mechanism. In particular, the spatial distribution of population within a country changes over time due to natural increase (births and deaths) and migrations (people relocating). An accurate model of these two processes should be able to reproduce the observed statistical patterns and allow us to investigate the stability of these patterns to specific events, such as the change of the rate of natural increase or the range of migrations.In this thesis stochastic models of human population dynamics will be developed to simulate the formation and growth of cities. This will differ from current population projection methods, which base predictions from extrapolations of time series, instead predicting evolution from the fundamental properties of human demographic growth and migration processes. These models will build upon advancements in three related areas - i) estimation of spatial flows of human migration; ii) models of population growth, where stochastic models can reproduce Zipf's Law of city sizes and the observed spatiotemporal correlations of growth rates; iii) models of city urbanisation. In addition to these models, separate models will be developed to predict the location of future urbanised areas. This will be done by estimating the urbanisation probability; the chance that a non-urbanised location will become urbanised. Models of urban development based on cluster growth and aggregation rely on the assumption that the urbanisation probability depends on the distance between the non-urbanised location and other urbanised locations. The approach to be considered will use machine learning techniques to predict the urbanisation probability taking into account not only the distance to other urbanised locations, but also other relevant economic and geographic variables. This approach will be used to investigate the effect of a steady demographic decline on the spatial distribution of population. This is the case of developed countries such as Japan, and will likely be an increasing trend in other developed countries in the future. There are various dynamical models to describe the spatial dynamics of the distribution of settlements in areas with growing population, however models able to describe the spatial dynamics when the total population is decreasing is not well studied. Japanese population data will be used to determine whether the process of urbanisation is reversible i.e. if the last areas to become urban when the population is growing, are they also the first to lose population when the total population decreases?