We consider a multistage cancer model in which cells are arranged in a $d$-dimensional integer lattice. Starting with all wild-type cells, we prove results about the distribution of the first time when two neutral mutations have accumulated in some c
ell in dimensions $dge 2$, extending work done by Komarova [Genetics 166 (2004) 1571-1579] for $d=1$.
Here we will use results of Cox, Durrett, and Perkins for voter model perturbations to study spatial evolutionary games on $Z^d$, $dge 3$ when the interaction kernel is finite range, symmetric, and has covariance matrix $sigma^2I$. The games we consi
der have payoff matrices of the form ${bf 1} + wG$ where ${bf 1}$ is matrix of all 1s and $w$ is small and positive. Since our population size $N=infty$, we call our selection small rather than weak which usually means $w =O(1/N)$. The key to studying these games is the fact that when the dynamics are suitably rescaled in space and time they convergence to solutions of a reaction diffusion equation (RDE). Inspired by work of Ohtsuki and Nowak and Tarnita et al we show that the reaction term is the replicator equation for a modified game matrix and the modifications of the game matrix depend on the interaction kernel only through the values of two or three simple probabilities for an associated coalescing random walk. Two strategy games lead to an RDE with a cubic nonlinearity, so we can describe the phase diagram completely. Three strategy games lead to a pair of coupled RDE, but using an idea from our earlier work, we are able to show that if there is a repelling function for the replicator equation for the modified game, then there is coexistence in the spatial game when selection is small. This enables us to prove coexistence in the spatial model in a wide variety of examples where the replicator equation of the modified game has an attracting equilibrium with all components positive. Using this result we are able to analyze the behavior of four evolutionary games that have recently been used in cancer modeling.
Consider a one-dimensional stepping stone model with colonies of size $M$ and per-generation migration probability $ u$, or a voter model on $mathbb{Z}$ in which interactions occur over a distance of order $K$. Sample one individual at the origin and
one at $L$. We show that if $M u/L$ and $L/K^2$ converge to positive finite limits, then the genealogy of the sample converges to a pair of Brownian motions that coalesce after the local time of their difference exceeds an independent exponentially distributed random variable. The computation of the distribution of the coalescence time leads to a one-dimensional parabolic differential equation with an interesting boundary condition at 0.
