Branching random walks and multi-type contact-processes on the percolation cluster of ${\mathbb{Z}}^{d}$

In this paper we prove that, under the assumption of quasi-transitivity, if a branching random walk on ${{\mathbb{Z}}^d}$ 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 ${{\mathcal{C}}_{\infty}}$ 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 ${{\mathbb{Z}}^d}$ also implies that for $k$ sufficiently large the associated $k$-type contact process survives on ${{\mathcal{C}}_{\infty}}$. This implies that the strong critical parameters of the branching random walk on ${{\mathbb{Z}}^d}$ and on ${{\mathcal{C}}_{\infty}}$ 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.