We investigate an initial value problem which is closely related to the Williams-Bjerknes tumour model for a cancer which spreads through an epithelial basal layer modeled onI ⊂ Z2. The solution of this problem is a familyp = (pi(t)), where eachpi(t)could be considered as an approximation to the probability that the cell situated ati is cancerous at timet. We prove that this problem has a unique solution, it is defined on [0, +∞[, and, for some relevant situations, limt→∞Pi(t) = 1 for alli ∈ I. Moreover, we study the expected number of cancerous cells at timet.
Volume 130, 2020
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