Mutation timing in a spatial model of evolution

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 ith mutation at rate μ_i per unit area. Mutations spread to neighboring sites at rate α, so that t time units after a mutation, the region of individuals that have acquired the mutation will be a ball of radius αt. 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 μ_i=μ for all i.