A spatial mutation model with increasing mutation rates

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 kth mutation at rate $\mu_k$ . We assume that the mutation rates $\mu_k$ are increasing, and we find the asymptotic waiting time for the first cell to acquire k mutations as the torus volume tends to infinity. This paper generalizes results on waiting for $k\geq 3$ mutations in Foo et al. (2020), which considered the case in which all of the mutation rates $\mu_k$ are the same. In addition, we find the limiting distribution of the spatial distances between mutations for certain values of the mutation rates.