Dynamical Systems in Structured Populations

研究者通过由偏微分方程、常微分方程大系统和Volterra积分方程组成的数学模型研究人口结构与人口发展的相互作用。兴趣结构是由内部特征引起的，如（时间顺序或类别）年龄、大小、位置或/和空间或时间异质性。对于导致永久免疫的传染病，开发了一个模型，允许任意分布不同疾病阶段的长度。提出了一种时间尺度方法，可以用疾病阶段持续时间的前三个累积量来表示地方性平衡的稳定性。此外，还研究了临时变化的疫苗接种策略（包括仅具有短暂效果的疫苗）对季节性疾病（流感）的影响。研究了在核心人群（艾滋病、结核病）中具有耐药菌株的疾病的最佳治疗策略。作为一个重要的工具，研究者扩展了非自治动力系统的持续理论。在种群生态学中，种群应对其栖息地的空间或时间异质性的策略被考虑。特别地，研究了结构化元种群的迁移策略和短暂栖息地种群的变态和同类相食等适应策略，以最大限度地提高生殖比率。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，有必要了解种群发展与种群和生境结构（如年龄或体型或时空异质性）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和栖息地结构如何影响种群的稳定性、恢复力、持久性和灭绝。在传染病领域，数学模型为研究引起季节性疾病暴发（麻疹、流感）的各种机制和确定有效应对这些疾病的适应性疫苗接种战略提供了一个合适的平台。此外，面对逃避或耐药的微生物菌株（艾滋病、结核病），有可能比较具有核心群体的人群的不同疾病治疗计划。生境破碎化是保护生物学中的一个重要问题，它导致局部种群经常灭绝，并被其他种群迁移的个体重新定居。结构化的元种群模型提供了一个框架，以确定生境逐渐恶化导致种群总数突然崩溃的情况，进而确定为了成功地重新引入已经灭绝的种群而必须满足的临界阈值。

Thieme 9706787 The investigator studies the interplay of population structures and population development via mathematical models consisting of partial differential equations, large systems of ordinary differential equations, and of Volterra integral equations. The structures of interest are induced by internal characteristics like (chronological or class) age, size, location or/and by spatial or temporal heterogeneity. For infectious diseases that lead to permanent immunity, a model is developed that allows arbitrary distributions for the lengths of the various disease stages. A time scale method is presented that makes it possible to express the stability of the endemic equilibrium in terms of the first three cumulants of the durations of the disease stages. Further, effects are investigated that temporarily varying vaccination strategies (including vaccines with only transient effects) have on seasonal diseases (influenza). Optimal treatment strategies are studied for diseases with resistant strains in populations with core groups (AIDS, tuberculosis). As an important tool the investigator extends the persistence theory for non-autonomous dynamical systems. In population ecology, strategies are considered by which populations cope with spatial or temporal heterogeneities of their habitats. In particular, migration strategies in structured metapopulations and adaptive strategies like metamorphosis and cannibalism for populations in ephemeral habitats are investigated that maximize the reproductive ratio. 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 population development and population and habitat structure (as given by age or body size or by spatial and temporal heterogeneity). The investigator develops mathematical concepts and tools to formulate and analyze appropriate models for this interdependence. Particular poi nts of interest are how population and habitat structure affect stability, resilience, persistence and extinction of populations. In the area of infectious diseases, mathematical models provide a suitable arena to study various mechanisms that cause seasonal outbreaks (measles, influenza) and to determine adaptive vaccination strategies that deal with them in an effective way. Further, in face of evasive or resistant microbial strains (AIDS, tuberculosis), it is possible to compare different disease treatment schedules for populations with core groups. As an important issue in conservation biology, habitat fragmentation leads to the formation of a patchy environment where local populations often go extinct and are recolonized by individuals immigrating from other local populations. Structured metapopulation models present a framework to identify the circumstances under which gradual habitat deterioration leads to a sudden collapse of the total population and, in turn, to determine critical thresholds that must be met in order to successfully reintroduce populations that have become extinct.