Mathematical modelling of collective dynamics in urban systems

城市化正在重塑人类社会和自然环境的许多方面，既带来机遇，也带来挑战。关于城市增长和形成的一般定量理论仍然难以捉摸，这将使我们能够预测未来的人口情况。观察到的人口增长趋势可以用精确的统计规律来表征，例如城市规模的分布、城市的空间分布和人口增长率的时空相关性。对经验数据的分析表明，这些规律在许多国家都是共同的，这表明所观察到的模式的形成可以用一种普遍的机制来解释。特别是，由于自然增长（出生和死亡）和移徙（人口迁移），一国内部人口的空间分布随时间而变化。这两个过程的精确模型应该能够再现观察到的统计模式，并允许我们调查这些模式对特定事件的稳定性，例如自然增长率的变化或迁移范围。在本文中，将发展人口动态的随机模型来模拟城市的形成和增长。这将不同于目前的人口预测方法，后者根据时间序列的外推来预测，而不是根据人口增长和移徙过程的基本特性来预测进化。这些模型将以三个相关领域的进展为基础：1)估算人类迁移的空间流动；ii)人口增长模型，其中随机模型可以再现城市规模的齐夫定律和观察到的增长率的时空相关性；三是城市城市化模式。除了这些模型之外，还将开发单独的模型来预测未来城市化地区的位置。这将通过估算城市化概率来实现；一个非城市化地区变成城市化地区的可能性。基于集群增长和聚集的城市发展模型依赖于城市化概率取决于非城市化地点与其他城市化地点之间的距离这一假设。要考虑的方法将使用机器学习技术来预测城市化概率，不仅考虑到与其他城市化地点的距离，还考虑到其他相关的经济和地理变量。这一方法将用于调查人口稳定下降对人口空间分布的影响。这是日本等发达国家的情况，未来在其他发达国家也可能成为一种增长趋势。人口增长地区聚落分布的空间动力学模型多种多样，但对人口减少地区聚落分布的空间动力学模型研究较少。日本人口数据将用于确定城市化进程是否可逆，即如果人口增长时最后一个成为城市的地区，当总人口减少时，它们是否也是第一个失去人口的地区？

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?