(A) Mechanisms where a mutation affects the fitness of cells in the vicinity. In the direct mechanism (A.I.) a mutated cell (red) generates a molecular signal (source) that affects a region around it. In the indirect mechanism, the signal produced by the mutated cell generates a signal in response also affecting a region around it (sink). The simplest model (A.III.) simplifies both mechanisms. The region affected by the signal has a radius R. Cells affected by the signal get a benefit b, and cells producing the signal pay a cost c. The replacement of cells in the stochastic process is done within a region L, which can be larger or smaller than R. (B) Steps of the stochastic process. (C) Models of benefit accumulation. In the “Non-cumulative model” (C.I.), the amount of benefit that each of the cells affected by the signal obtains is the same, and it is independent of the number of mutated cells. In the “Cumulative model” (C.II.), the amount of benefit that each of the cells affected by the signal obtains depends on the number of mutated cells. In both examples from C.I and C.II. we have assumed R = 2.