Williams and Bjerknes introduced in 1972 a stochastic model for the spread of cancer cells; independently, this model has since surfaced within the field of interacting particle systems as the biased voter model. Cells, normal and abnormal (cancerous), are situated on a planar lattice. With each cellular division, one daughter stays put, while the other usurps the position of a neighbor; abnormal cells reproduce at a faster rate than normal cells. We treat here the long-term behavior of this system. In particular, we show that, provided it lives forever, the tumour will eventually contain a ball of linearly expanding radius. This also demonstrates the ergodicity of the interacting particle system, the coalescing random walk with nearest neighbor births. Our techniques include the use of dual processes, and of different numerical computations involving the use of imbedded processes.
"On the Williams-Bjerknes Tumour Growth Model I." Ann. Probab. 9 (2) 173 - 185, April, 1981. https://doi.org/10.1214/aop/1176994459