Global survival of branching random walks and tree-like branching random walks

Abstract

The reproduction speed of a continuous-time branching random walk is proportional to a positive parameter λ. There is a threshold for λ, which is called λw, that separates almost sure global extinction from global survival. Analogously, there exists another threshold λs below which any site is visited almost surely a finite number of times (i.e. local extinction) while above it there is a positive probability of visiting every site infinitely many times. The local critical parameter λs is completely understood and can be computed as a function of the reproduction rates. On the other hand, only for some classes of branching random walks it is known that the global critical parameter λw is the inverse of a certain function of the reproduction rates, which we denote by Kw. We provide here new sufficient conditions which guarantee that the global critical parameter equals 1/Kw. This result extends previously known results for branching random walks on multigraphs and general branching random walks. We show that these sufficient conditions are satisfied by periodic tree-like branching random walks. We also discuss the critical parameter and the critical behaviour of continuous-time branching processes in varying environment. So far, only examples where λw = 1/Kw were known; here we provide an example where λw > 1/Kw.

Branching random walks with uncountably many extinction probability vectors

“…In particular, if ρ x does not depend on x ∈ X, we say that the BRW can be projected on a branching process (see [1,4] for details). More generally, some BRWs can be projected onto BRWs defined on finite sets as explained in [6, Section 2.3] (see also Section 3.5 for the case of continuoustime BRWs).…”

“…If the process is irreducible then the critical parameters do not depend on x. See [1,2,3,4] for a more detailed discussion on the values of λ w (x) and λ s (x), including their characterizations.…”

On the role of reduced habitat in the phase transition of a stochastic model for seed dispersal

“…This can be formulated as a discrete‐time stochastic process whose space state is described by the collection of possible positions of particles at any time. For an overview of the formulation and recent results of these type of stochastic processes, we refer the reader to [11]. In our model, the space is continuous, and since it is a one‐dimensional model, it is exactly $$$ \mathbb{R} $$$.…”