Positivity in natural sciences

Abstract

Many problems in natural sciences require a notion of positivity: only non-negative densities, population sizes or probability make sense in real life. This imposes certain constrains in the modelling but also, coupled with metric structure of underlying spaces, offers new powerful tools for analyzing complex systems. The presented lectures describe such an intertwining of topological and order structures in analysis of linear dynamical systems, mainly arising in mathematical biology and kinetic theory. Among covered topics the reader will find a survey of classical topics such as well-posedness and detailed analysis of long-time and asymptotic behaviour of infinite dimensional linear systems as well as recent results on emergence of chaos and phase transitions in such systems. The lectures are concluded by multiple scale asymptotic analysis of singularly perturbed kinetic problems leading to various diffusion equations which preserve the coarse structure of the original models. © 2008 Springer-Verlag Berlin Heidelberg.