We study a one dimensional two-type contact process with equal rate of propagation (and death) of the two types. We show that the progeny of a finite number of mutants has a positive probability of survival if and only at time 0 there is at most a finite number of residents on at least one side of the mutant's ``colony''.
"Survival of a Single Mutant in One Dimension." Electron. J. Probab. 15 386 - 408, 2010. https://doi.org/10.1214/EJP.v15-769