Published online by Cambridge University Press: 14 July 2016
We show that a branching random walk that is supercritical on is also supercritical, in a rather strong sense, when restricted to a large enough finite ball of
This implies that the critical value of branching random walks on finite balls converges to the critical value of branching random walks on
as the radius increases to infinity. Our main result also implies coexistence of an arbitrary finite number of species for an ecological model.