Mathematical Sciences: Dynamical Systems in Structured Populations

9403884研究人员利用微分方程组模型研究了种群内在结构与种群发展之间的相互作用。感兴趣的结构是由(按时间顺序或类别)年龄、大小、位置等个体特征诱导的。一个特定的目标是识别那些结构，这些结构的引入改变了先前非结构化模型解的动力学行为。具体地说，他考察了班级年龄和年代年龄对地方病模型的影响，身体大小结构对浮游动物种群的影响，以及结构化集合种群模型中的进化方面。在数学上，种群结构的引入通常会将小的常微分方程组转化为大的常微分方程组，或者转化为积分、积分-微分、泛函微分方程组或偏微分方程组。这些方程作为演化方程(动力系统)在适当的Banach空间中被研究，该空间通常是无限维的或具有大的有限维。本研究运用了线性和非线性泛函分析方法以及动力系统理论。虽然建立存在性和唯一性是一个必要的步骤，而且通常不是微不足道的一步，但重点是研究解的定性行为，解决稳定性、阻尼性和无阻尼性、持久性和持久性。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之间的相互作用(如年龄、身体大小、位置等)。和人口发展。研究人员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。具体的兴趣点是人口结构如何影响人口的稳定性、恢复力、持久性和灭绝；此外，如何从个人福祉的角度来解释人口数据。后者是相关的，例如，使用微生物来测试洗涤剂。结构化种群的重要例子是结构化集合种群。许多种群生活在零散的环境中，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迁徙过来的个体重新殖民。形成上述集合种群的种群数量增加，因为许多物种以前连续的栖息地是支离破碎的。集合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.