Stochastic dynamics of spatially structured evolving populations

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An important scientific challenge is to develop models to gain insight into the behaviour of complex systems of intermediate size and having a multilevel structure. Systems of this type arise in evolutionary biology, ecology, and genetics as well as in many other fields including communication networks, economics and financial markets. The principal objective of this research program is to develop mathematical tools for the modelling and analysis of these systems. Our methodological approach is based on the study of spatially structured stochastic processes including interacting particle systems with finitely or infinitely many types and their measure-valued generalizations. These can model populations of individuals with internal states coding for genealogical and historical information and their analysis involves mean-field and hierarchical mean-field approximations, and multiple space-time scale analysis. In particular interacting Fleming-Viot probability-measure-valued processes and their function-valued and set-valued duals which we have recently introduced play a central role in this research.