Branching random walks and multi-type contact-processes on the percolation cluster of ℤd

In this paper we prove that, under the assumption of quasi-transitivity, if a branching random walk on ℤd survives locally (at arbitrarily large times there are individuals alive at the origin), then so does the same process when restricted to the infinite percolation cluster C∞ of a supercritical Bernoulli percolation. When no more than k individuals per site are allowed, we obtain the k-type contact process, which can be derived from the branching random walk by killing all particles that are born at a site where already k individuals are present. We prove that local survival of the branching random walk on ℤd also implies that for k sufficiently large the associated k-type contact process survives on C∞. This implies that the strong critical parameters of the branching random walk on ℤd and on C∞ coincide and that their common value is the limit of the sequence of strong critical parameters of the associated k-type contact processes. These results are extended to a family of restrained branching random walks, that is, branching random walks where the success of the reproduction trials decreases with the size of the population in the target site.