Dynamical Systems of Population Dynamics

After a short review of the the role of population dynamics in the formation of the theory of dynamical systems some recent developments are described, centered around work within the DANSE project: Reaction transport equations which are refinements of reaction diffusion equations, showing travelling front solutions and pattern formation, with applications to chemotaxis and the formation of polarized groups of animals or cells. Populations structured by age are modeled by conservation laws and by delay equations, the thorough investigation of the close connection between these two classes of systems leads to a solid justification of the latter as population models. Particular attention is given to models for the spread of infectious diseases: Models describing outbreaks in closed populations, optimal vaccination and control policies, the conditions for backward bifurcations and (unwanted) hysteresis phenomena in public health. Epidemic spread in space (travelling front problems), simplification of complex models to modulation equations. Modeling the behavior of single individuals connects dynamical systems to stochastic processes: Synchronous and asynchronous cellular automata, interacting particle systems, concrete applications are ring vaccination (in animal husbandry) and contact tracing (in human populations).