Large scale dynamical systems in mathematical population dynamics

大尺度动力系统（或简称大系统）处于理论与应用之间的界面。理论上，系统的大小没有限制。当将应用程序考虑到一个特定的系统时，大小问题就变得相关了。这种限制的一个标准例子是n体问题，它描述了恒星体在其他恒星体的引力相互作用下的运动。常微分方程理论的标准结果证明了解的存在唯一性，以及许多其他的简单性质。然而，当人们考虑到系统的演变时，这个问题就变得棘手了。在数学种群动力学领域，即种群进化的数学方面，大系统也经常出现。它们产生的一个背景是描述传染病在不同地理单位之间的空间传播。在这种情况下，考虑的是区域（或城市）网络，其中每个区域的流行病学参数的演变由一组微分方程建模。然后将这些区域耦合在一起，以描述个体在它们之间的运动。我的项目有两个部分：应用部分和理论部分。我将研究传染病传播的空间离散模型，以及数学种群动力学中的一些其他大型模型。我将尝试在使用的方法中发现规律和模式。这些模式将使我能够概括和考虑大型系统的更多理论方面。

Large scale dynamical systems (or large systems, for short) lie at the interface between theory and applications. Theoretically, there is no limitation to the size of a system. When an application is considered to a particular system, the problem of size becomes relevant.    A standard example of such limitations is the n-body problem, which describes the movement of stellar bodies submitted to the gravitational interactions of the other stellar bodies present. Standard results of ordinary differential equations theory show the existence and uniqueness of solutions, as well as many other simple properties. However, when one considers the evolution of the system, the problem becomes intractable.    In the field of mathematical population dynamics, that is, the mathematical aspects of the evolution of populations, large systems are also frequent. One context in which they are arise is the description of the spatial spread of infectious diseases between distinct geographical units. In this case, what are considered are networks of regions (or cities), where in each region the evolution of epidemiological parameters is modeled by a set of differential equations. The regions are then coupled together to describe the movement of individuals between them.    My project has two components: an applied one, and a more theoretical one. I will study spatially discrete models of infectious disease spread, as well as some other large models in mathematical population dynamics. I will try to detect regularities, patterns, in the methods used. These patterns will allow me to generalise and consider more theoretical aspects of large systems.