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In this paper we prove that, under the assumption of quasi-transitivity, if abranching random walk on ${{\mathbb{Z}}^d}$ survives locally (at arbitrarilylarge times there are individuals alive at the origin), then so does the sameprocess when restricted to the infinite percolation cluster${{\mathcal{C}}_{\infty}}$ of a supercritical Bernoulli percolation. When nomore than $k$ individuals per site are allowed, we obtain the $k$-type contactprocess, which can be derived from the branching random walk by killing allparticles that are born at a site where already $k$ individuals are present. Weprove that local survival of the branching random walk on ${{\mathbb{Z}}^d}$also implies that for $k$ sufficiently large the associated $k$-type contactprocess survives on ${{\mathcal{C}}_{\infty}}$. This implies that the strongcritical parameters of the branching random walk on ${{\mathbb{Z}}^d}$ and on${{\mathcal{C}}_{\infty}}$ coincide and that their common value is the limit ofthe sequence of strong critical parameters of the associated $k$-type contactprocesses. These results are extended to a family of restrained branchingrandom walks, that is, branching random walks where the success of thereproduction trials decreases with the size of the population in the targetsite.

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