  We consider a dynamical process on a graph $G$, in which vertices are
infected (randomly) at a rate which depends on the number of their neighbours
that are already infected. This model includes bootstrap percolation and
first-passage percolation as its extreme points. We give a precise description
of the evolution of this process on the graph $\mathbb{Z}^2$, significantly
sharpening results of Dehghanpour and Schonmann. In particular, we determine
the typical infection time up to a constant factor for almost all natural
values of the parameters, and in a large range we obtain a stronger, sharp
threshold.
