具有选择的空间两型Fisher-Wright系统中稀有突变体的入侵
Invasion by rare mutants in a spatial two-type Fisher-Wright system with selection
摘要：
考虑了在$\{1,2,...,N\}$空间上具有选择和突变的Fisher-Wright扩散方程的平均场系统。类型1具有适应度0，类型2具有适应度1，并且（罕见）突变以速率$m...N^{-1}$从类型1到类型2发生，选择以速率$s>0$发生。系统在集中于类型1的状态下启动，即低适应度的状态。我们调查这个系统的$N \to \infty$的原始和大的时间尺度。我们发现，对于一些$α\in (0,s)$在时间$α^{-1} \log N+t, t \in \R, N \to \infty$的出现2型（积极的全球2型强度）在全球范围内发生，而在时间$α^{-1} \log N+t_N$，与$t_N \to \infty$，我们得到固定在2型，另一方面与$t_N \to -\infty$作为$N \to \infty$渐近只有1型存在。我们描述的过渡从出现到固定的时间尺度$α^{-1} \log N+t, t \in \R$在极限$N \to \infty$的McKean-Vlasov随机进入法律。对于正随机变量$^\ast \CW$，该入口定律对于$t \to -\infty$的行为类似于$^\ast\CW e^{-α|t|}$。2型为主的网站在时间$o(\log N)$，或$γ...\log N, γ\in (0,α^{-1})$的小液滴的形成是描述在限制$N \to \infty$由一个测量值的过程驱动的泊松型噪声，我们明确确定的随机方程。这种有限的$(N \to \infty)$液滴过程的总质量像$\CW^\ast e^{αt}$一样增长为$t \to \infty$。我们证明了小时间尺度的退出行为等于大时间尺度的进入行为，即$\CL[^\ast\CW] =\CL[\CW^\ast]$。
英文摘要：
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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