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.
