Motivated by models of cancer formation in which cells need to acquire $k$ mutations to become cancerous, we consider a spatial population model in which the population is represented by the $d$-dimensional torus of side length $L$. Initially, no sites have mutations, but sites with $i-1$ mutations acquire an $i$th mutation at rate $\mu_i$ per unit area. Mutations spread to neighboring sites at rate $\alpha$, so that $t$ time units after a mutation, the region of individuals that have acquired the mutation will be a ball of radius $\alpha t$. We calculate, for some ranges of the parameter values, the asymptotic distribution of the time required for some individual to acquire $k$ mutations. Our results, which build on previous work of Durrett, Foo, and Leder, are essentially complete when $k = 2$ and when $\mu_i = \mu$ for all $i$.

中文翻译：

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进化空间模型中的突变时间

受癌症形成模型的启发，其中细胞需要获得 $k$ 突变才能癌变，我们考虑了一个空间群体模型，其中群体由边长 $L$ 的 $d$ 维环面表示。最初，没有位点有突变，但是具有 $i-1$ 突变的位点以每单位面积 $\mu_i$ 的速率获得第 $i$ 个突变。突变以 $\alpha$ 的速度传播到邻近的位点，因此在 $t$ 时间单位发生突变后，获得突变的个体区域将是一个半径为 $\alpha t$ 的球。对于某些参数值范围，我们计算某些个体获得 $k$ 突变所需时间的渐近分布。我们的结果建立在 Durrett、Foo 和 Leder 之前的工作基础上，当 $k = 2$ 并且当 $\mu_i = \mu$ 对所有 $i$ 时，我们的结果基本上是完整的。