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

摘要

We study two types of stochastic processes, a mean-field spatial system ofinteracting Fisher-Wright diffusions with an inferior and an advantageous typewith rare mutation (inferior to advantageous) and a (mean-field) spatial systemof supercritical branching random walks with an additional deathrate which isquadratic in the local number of particles. The former describes a standardtwo-type population under selection, mutation, the latter models describe apopulation under scarce resources causing additional death at high localpopulation intensity. Geographic space is modelled by $\{1, \cdots, N\}$. Thefirst process starts in an initial state with only the inferior type present oran exchangeable configuration and the second one with a single initialparticle. {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 $T$ andsecondly after a large time when in the Fisher-Wright model the rare mutantssucceed respectively in the branching random walk the particle populationreaches a positive spatial intensity. It is shown that the second phase forboth models sets in after time $\alpha^{-1} \log N$, if $N$ is the size ofgeographic space and $N^{-1}$ the rare mutation rate and $\alpha \in (0,\infty)$ depends on the other parameters. We identify the limit dynamics inboth time windows and for both models as a nonlinear Markov dynamic(McKean-Vlasov dynamic) respectively a corresponding random entrance law fromtime $-\infty$ of this dynamic. Finally we explain that the two processes are just two sides of the very samecoin, a fact arising from duality, in particular the particle model generatesthe genealogy of the Fisher-Wright diffusions with selection and mutation. Wediscuss the extension of this duality relation to a multitype model with morethan two types.