Branching random walks and multi-type contact-processes on the percolation cluster of $\mathbbZ^d$

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 $\Cinf$ 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 $\Cinf$. This implies that the strong critical parameters of the branching random walk on $\Zd$ and on $\Cinf$ 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.