Spatial fleming-viot models with selection and mutation

The monograph constructs a rigorous framework to analyse some phenomena in evolutionary theory of populations arising due to the combined effects of migration, selection and mutation in a spatial stochastic population model, namely the evolution towards fitter and fitter types through punctuated equilibria. The discussion is based on some new methods, in particular multiple scale analysis, nonlinear Markov processes and their entrance laws, atomic measure-valued evolutions and new forms of duality (for state-dependent mutation and multitype selection) which are used to prove ergodic theorems in this setting and are applicable for many other questions and renormalization analysis to analyse some phenomena (stasis, punctuated equilibrium, failure of naive branching approximations, biodiversity) which occur due to the combination of rare mutation, mutation, resampling, migration and selection and require mathematically to bridge between (in the limit) separating time and space scales. Consider the following spatial multitype population model. The state of a single colony is described by a probability measure on some countable type space, the geographic space is modelled by a set of colonies and the colonies (or demes) are labelled with the countable hierarchical groupωN D ZN, with Z the cyclic group of order N. This set mimics Z 2 as N → ∞. The stochastic dynamics is given by a system of interacting measure-valued diffusions and the driving mechanisms include resampling (pure genetic drift) as diffusion term, migration, selection and mutation as drift terms. Resampling is modelled in each colony by the usual Fleming-Viot diffusion. The haploid selection is based on a fitness function on the space of types, and the mutation is type-dependent and has the feature that fitter types are more stable against deleterious mutation an