Mathematical Sciences: Dynamical Systems in Structured Populations

[403884]研究者通过微分方程模型研究内在人口结构与人口发展的相互作用。兴趣结构是由个体特征引起的，如（时间顺序或阶级）年龄、大小、位置等。一个特定的目标是识别那些将其引入先前非结构化模型并改变模型解的动态行为的结构。具体来说，他研究了类龄和实足年龄对地方病模型的影响，体型结构对浮游动物种群的影响，以及结构化元种群模型中的进化方面。在数学上，人口结构的引入通常将小的常微分方程系统转换为大的ode系统或积分、积分-微分、泛函微分或偏微分方程。这些方程作为演化方程（动力系统）在适当的巴拿赫空间中进行研究，巴拿赫空间通常是无限维或具有较大的有限维。本研究运用线性与非线性泛函分析方法及动力系统理论。虽然建立存在性和唯一性是必要的，而且往往不是微不足道的步骤，但重点是研究解决稳定性，阻尼和无阻尼振荡，持久性和永久性的解决方案的定性行为。为了解释、预测、管理和控制微生物、植物、动物和人类种群的动态，有必要了解固有种群结构（如年龄、体型、位置等）与种群发展之间的相互作用。研究者开发数学概念和工具来制定和分析这种相互依赖的适当模型。特别感兴趣的是种群结构如何影响种群的稳定性、恢复力、持久性和灭绝；进一步，如何从个人福祉的角度解释人口数据。后者是相关的，例如，在测试洗涤剂中使用微生物。结构化种群的重要例子是结构化元种群。许多种群生活在一个不完整的环境中，导致当地种群的形成，这些种群经常灭绝，并被从其他当地种群移民过来的个体重新定居。由于许多物种以前连续的栖息地被破碎，形成上述类元种群的种群数量增加了。超种群动态是保护生物学和自然灭绝物种重建中的重要问题。

9403884 Thieme The investigator studies the interplay of intrinsic population structures and population development via differential equations models. The structures of interest are induced by individual characteristics like (chronological or class) age, size, location etc. A particular goal is to identify those structures the introduction of which into a previously unstructured model changes the dynamical behavior of the model solutions. Specifically, he examines the influence of class age and chronological age on models of endemic diseases, the impact of body size structure on populations of zoo-plankton, and evolutionary aspects in structured metapopulation models. Mathematically, the introduction of population structure typically converts small systems of ordinary differential equations into large systems of ODEs or into integral, integro-differential, functional differential or partial differential equations. These equations are studied as evolution equations (dynamical systems) in an appropriate Banach space, which typically is infinite dimensional or has a large finite dimension. The study uses methods of linear and nonlinear functional analysis and dynamical systems theory. While establishing existence and uniqueness is a necessary and often not trivial step, the focus is on studying the qualitative behavior of solutions addressing stability, damped and undamped oscillations, persistence, and permanence. In order to interpret, forecast, manage and control the dynamics of microbial, plant, animal and human populations it is necessary to understand the interaction between inherent population structures (as given by age, body size, location etc.) and population development. The investigator develops mathematical concepts and tools to formulate and analyze appropriate models for this interdependence. Particular points of interests are how population structure affects stability, resilience, persistence and extinction of populations; fu rther, how to interpret population data in terms of individual well-being. The latter is relevant, e.g., for the use of micro-organisms in testing detergents. Important examples of structured populations are structured metapopulations. Many populations live in a patchy environment, leading to the formation of local populations that often go extinct and are recolonized by individuals immigrating from other local populations. The number of populations forming metapopulations of the above kind increases because the formerly continuous habitats of many species are fragmented. Metapopulation dynamics are important issues in conservation biology and re-establishment of naturally extinct species.