Dynamic equation approach to forecast long-range demographic scenarios

我们生活在一个巨大的社会变革时代：经济发展和医疗创新促成了前所未有的人口激增，影响着数十亿人的生活，并以前所未有的速度影响着环境。在不到三个世纪的时间里，世界人口增长了十倍，导致大多数国家的人口分布发生了重大变化，在这些国家，全球都可以看到城市化的持续增长。尽管城市化正在重塑人类社会和自然环境的许多方面，带来了机遇和挑战，但关于城市增长和形成的一般量化理论仍然缺乏，使我们能够预测未来的人口情景。观察到的人口增长趋势可以通过精确的统计规律来定量描述，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，这些统计规律在许多国家都是常见的，这表明观察到的模式的形成可能是由一般机制解释的。特别是，由于自然增长(出生-死亡)和迁移(人口迁移)，一个国家内人口的空间分布会随着时间的推移而变化。这两个过程的精确数学模型应该能够再现观察到的统计模式，并使我们能够调查这些模式对特定事件的稳定性，例如自然增长率或迁徙范围的全球或局部变化。在这项研究中，我的目标是建立一个基于对人口过程的简单而现实的描述的人口动力学模型，并描述正在出现的人口分布模式。准确的移民流动的数学描述对于确定人口在空间中的重新分布是至关重要的。为了对移民进行建模，我提出了一个单一约束重力和空间流动的干预机会模型的推广版本，将对其进行调查，以估计英国和美国的净移民。我将开发不同形式的动态方程来描述人口密度的时间演变，将空间流动模型与随机过程相结合来模拟人口增长。我将评估模型再现关于城市增长的大小、数量、位置和时空相关性的特征统计模式的能力。拟议的研究将提供一个数学框架，将人口分布的新统计模式与潜在微观过程的特征属性联系起来：出生、死亡、迁移。它将有助于阐明各种现象对我们社会的长期影响，从新运输方式的发展到冲突和极端自然事件的后果，并有可能为实现可持续和平衡增长的战略决策提供信息。

We live in a time of big social change: economic development and medical innovations have contributed to produce an unprecedented demographic boom that is affecting the lives of billions of people and impacting the environment at an unprecedented rate. In less than three centuries the world population increased ten times, causing a major shift in the distribution of population in most countries, where a continuing growth of urbanisation is observed globally. Although urbanisation is reshaping many aspects of human societies and the natural environment, presenting both opportunities and challenges, a general, quantitative theory on the growth and formation of cities that would enable us to forecast future demographic scenarios still remains elusive.The observed trends of population growth can be quantitatively 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 statistical laws are common to many countries, suggesting that the formation of the observed patterns 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-deaths) and migrations (people relocating). An accurate mathematical 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 global or local change of the rate of natural increase or the range of migrations. In this research, I aim to develop a dynamical model of population dynamics based on simple yet realistic descriptions of demographic processes and to characterise the emerging patterns of population distribution. An accurate mathematical description of migration flows is of primary importance to determine how population redistributes in space. To model migrations, I propose a generalised version of singly-constrained gravity and intervening opportunities models of spatial flows, which will be investigated to estimate net migration in the UK and the United States. I will develop different forms of dynamic equations to describe the temporal evolution of the density of population, combining models of spatial flows with stochastic processes to model population growth.I will assess the models' ability to reproduce the characteristic statistical patterns about the size, number, position, and spatiotemporal correlations of growth of cities.The proposed research will offer a mathematical framework to relate the emerging statistical patterns of population distribution with the characteristic properties of the underlying microscopic processes: births, deaths, migrations. It will contribute to shed light on the long term effect on our society of various phenomena, from the development of new forms of transportation to the consequences of conflicts and extreme natural events, with the potential to inform strategic decisions toward a sustainable and balanced growth.