Renormalization and Universality for Multitype Population Models

Summary We are concerned with spatial models of populations, where individuals have a type and a geographic location which both undergo a stochastic dynamic. Typical classes we consider are branching systems, state dependent branching systems, catalytic branching, mutually catalytic branching and we also touch on properties of Fleming-Viot models. Many features of such systems are universal in large classes of possible branching mechanism. Important features are the longtime behavior and the small-scale structure of spatial continuum limits of such systems and structural properties of the historical process associated with them. To exhibit these universal features the systems are analysed by renormalizing them through rescaling space and time according to whole sequences of separating scales. The arising collection of limiting systems can be described in a simpler fashion namely as a Markov chain governed by parameters given as a function on the one-component state space. Two such objects arise, one for the longtime and one for the small-scale behavior of the continuum limit. Hence the universality classes of the stochastic system are associated with universality classes for certain nonlinear maps in function spaces, which can be analysed via analytical tools. We review the progress made from 1993 to 2003 and formulate the problems currently under investigation. The techniques developed in the renormalization analysis also have many applications in the analysis of evolutionary models in population genetics.