Multiscale analysis: Fisher-Wright diffusions with rare mutations and selection, logistic branching system

Abstract

We study two types of stochastic processes, a mean-field spatial system of interacting Fisher-Wright diffusions with an inferior and an advantageous type with rare mutation (inferior to advantageous) and a (mean-field) spatial system of supercritical branching random walks with an additional deathrate which is quadratic in the local number of particles. The former describes a standard two-type population under selection, mutation, the latter models describe a population under scarce resources causing additional death at high local population intensity. Geographic space is modelled by $\{1, \cdots, N\}$. 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 \cite{DGsel}.} We study the behaviour in two time windows, first between time 0 and $T$ and secondly 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 the second phase for both models sets in after time $\alpha^{-1} \log N$, if $N$ is the size of geographic space and $N^{-1}$ the rare mutation rate and $\alpha \in (0, \infty)$ depends on the other parameters. We identify the limit dynamics in both time windows and for both models as a nonlinear Markov dynamic (McKean-Vlasov dynamic) respectively a corresponding random entrance law from time $-\infty$ of this dynamic. Finally we explain that the two processes are just two sides of the very same coin, a fact arising from 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 relation to a multitype model with more than two types.