• Bernoulli
  • Volume 1, Number 1-2 (1995), 171-189.

Limits of spatial branching populations(with Discussion)

Anton Wakolbinger

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In critical branching and migrating populations, mobility of the individuals counteracts, and the clumping effect caused by the branching favours local extinction of the population in the large time limit. For example, d-dimensional critical binary branching Brownian motion (d>1) with a spatially inhomogeneous branching rate V(x), when started off with a homogeneous Poisson population, persists if V(x) ~ ||x||d-2(log||x||)-(1+ε), and suffers local extinction if V(x) ~ ||x||d-2(log||x||)-1 as ||x||→∞; this can be derived from a probabilistic persistence criterion (Theorem 2). Besides presenting this result, the paper reviews conditions on the parameters of various other models which are necessary and sufficient for persistence, and discusses related results for superprocesses. Common to the proofs is the method of analysing the backward tree of an individual encountered in an old population, originally due to Kallenberg (1977) and Liemant (1981) in discrete time settings.

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Bernoulli, Volume 1, Number 1-2 (1995), 171-189.

First available in Project Euclid: 2 August 2007

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Zentralblatt MATH identifier

backward trees branching populations large time limits local extinction persistence spatially inhomogeneous critical branching superprocess limits


Wakolbinger, Anton. Limits of spatial branching populations(with Discussion). Bernoulli 1 (1995), no. 1-2, 171--189.

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