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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 $i$.
We consider a spatial model of cancer in which cells are points on the $d$-dimensional torus $mathcal{T}=[0,L]^d$, and each cell with $k-1$ mutations acquires a $k$th mutation at rate $mu_k$. We will assume that the mutation rates $mu_k$ are increasi
We consider the mutation--selection differential equation with pairwise interaction (or, equivalently, the diploid mutation--selection equation) and establish the corresponding ancestral process, which is a random tree and a variant of the ancestral
Using graphical methods based on a `lookdown and pruned version of the {em ancestral selection graph}, we obtain a representation of the type distribution of the ancestor in a two-type Wright-Fisher population with mutation and selection, conditional
Microbial communities are ubiquitous in nature and come in a multitude of forms, ranging from communities dominated by a handful of species to communities containing a wide variety of metabolically distinct organisms. This huge range in diversity is
We reconsider the deterministic haploid mutation-selection equation with two types. This is an ordinary differential equation that describes the type distribution (forward in time) in a population of infinite size. This paper establishes ancestral (r