Dynamical Systems in Structured Populations

Thieme 9706787 调查研究人口结构和人口发展的相互作用，通过数学模型组成的偏微分方程，大型系统的常微分方程，和沃尔泰拉积分方程。 感兴趣的结构是由内部特征，如（按时间顺序或类别）年龄，大小，位置或/和空间或时间异质性引起的。 对于导致永久免疫的传染病，开发了一个模型，该模型允许各种疾病阶段的长度的任意分布。 提出了一种时间尺度方法，它可以用疾病阶段持续时间的前三个累积量来表示地方病平衡点的稳定性。 此外，还研究了暂时不同的疫苗接种策略（包括仅具有短暂效应的疫苗）对季节性疾病（流感）的影响。 研究了核心群体（艾滋病、结核病）人群中耐药菌株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久性理论。 在种群生态学中，种群科普其栖息地空间或时间异质性的策略被认为是重要的。 特别是，结构化集合种群的迁移策略和适应策略，如变态和同类相食的人口在短暂的栖息地进行了调查，最大限度地提高了繁殖率。 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栖息地结构（由年龄或体型或空间和时间异质性决定）之间的相互作用。 研究人员开发数学概念和工具，以制定和分析这种相互依赖的适当模型。 特别感兴趣的是人口和栖息地结构如何影响稳定性，恢复力，持久性和种群灭绝。 在传染病领域，数学模型提供了一个合适的竞技场，以研究导致季节性爆发（麻疹，流感）的各种机制，并确定适应性疫苗接种策略，以有效的方式处理它们。 此外，面对逃避或耐药的微生物菌株（艾滋病，结核病），有可能比较不同的疾病治疗方案的人口与核心群体。 作为保护生物学中的一个重要问题，栖息地破碎化导致形成一个斑块状的环境，当地种群往往会灭绝，并被来自其他当地种群的个体所取代。 结构化集合种群模型提出了一个框架，以确定在何种情况下，逐渐恶化的栖息地导致突然崩溃的总人口，反过来，以确定关键的阈值，必须满足，以成功地重新引入已灭绝的人口。

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.