Multiscale Analysis: Fisher–Wright Diffusions with Rare Mutations and Selection, Logistic Branching System

  • Donald A. Dawson
  • Andreas Greven
Conference paper
Part of the Springer Proceedings in Mathematics book series (PROM, volume 11)


We study two types of stochastic processes, first a mean-field spatial system of interacting Fisher–Wright diffusions with an inferior and an advantageous type with rare mutation (inferior to advantageous) and selection and second a mean-field spatial system of supercritical branching random walks with an additional death rate, which is quadratic in the local number of particles. The former describes a standard two-type population under selection, mutation and the latter model describes a population under scarce resources causing additional death at high local population intensity. Geographic space is modelled by 1, , N. The first process starts in an initial state with only the inferior type present or an exchangeable configuration and the second one with a single initial particle. This material is a special case of the theory developed in and describes the results of Section 7 therein. We study the behaviour in two time windows, first between time 0 and T and second after a large time when in the Fisher–Wright model the rare mutants succeed, respectively, in the branching random walk the particle population reaches a positive spatial intensity. It is shown that asymptotically as N the second phase for both models sets in after time α− 1logN, if N is the size of geographic space and N − 1 the rare mutation rate and α ∈ (0, ) depends on the other parameters. We identify the limit dynamics as N in both time windows and for both models as a nonlinear Markov dynamic (McKean–Vlasov dynamic), respectively, a corresponding random entrance law from time − of this dynamic. Finally, we explain that the two processes are just two sides of the very same coin, a fact arising from a new form of duality, in particular the particle model generates the genealogy of the Fisher–Wright diffusions with selection and mutation. We discuss the extension of this duality in relation to a multitype model with more than two types.


Duality Relation Geographic Space Vlasov Equation Rare Mutation Occupied Site 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCarleton UniversityOttawaCanada
  2. 2.Department MathematikUniversität Erlangen-NürnbergErlangenGermany

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