Invasion by rare mutants in a spatial two-type Fisher-Wright system with selection

We consider a meanfield system of interacting Fisher-Wright diffusions with selection and rare mutation on the geographic space $\{1,2,...,N\}$. The type 1 has fitness 0, type 2 has fitness 1 and (rare) mutation occurs from type 1 to 2 at rate $m...N^{-1}$, selection is at rate $s>0$. The system starts in the state concentrated on type 1, the state of low fitness. We investigate this system for $N \to \infty$ on the original and large time scales. We show that for some $α\in (0,s)$ at times $α^{-1} \log N+t, t \in \R, N \to \infty$ the emergence of type 2 (positive global type-2 intensity) at a global level occurs, while at times $α^{-1} \log N+t_N$, with $t_N \to \infty$ we get fixation on type 2 and on the other hand with $t_N \to -\infty$ as $N \to \infty$ asymptotically only type 1 is present. We describe the transition from emergence to fixation in the time scale $α^{-1} \log N+t, t \in \R$ in the limit $N \to \infty$ by a McKean-Vlasov random entrance law. This entrance law behaves for $t \to -\infty$ like $^\ast\CW e^{-α|t|}$ for a positive random variable $^\ast \CW$. The formation of small droplets of type-2 dominated sites in times $o(\log N)$, or $γ...\log N, γ\in (0,α^{-1})$ is described in the limit $N \to \infty$ by a measure-valued process following a stochastic equation driven by Poissonian type noise which we identify explicitly. The total mass of this limiting $(N \to \infty)$ droplet process grows like $\CW^\ast e^{αt}$ as $t \to \infty$. We prove that exit behaviour from the small time scale equals the entrance behaviour in the large time scale, namely $\CL[^\ast\CW] =\CL[\CW^\ast]$.

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