Bernoulli

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

Limits of spatial branching populations(with Discussion)

Anton Wakolbinger

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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 2 August 2007

Permanent link to this document
https://projecteuclid.org/euclid.bj/1186078366

Mathematical Reviews number (MathSciNet)
MR1354460

Zentralblatt MATH identifier
0868.60069

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

Citation

Wakolbinger, Anton. Limits of spatial branching populations(with Discussion). Bernoulli 1 (1995), no. 1-2, 171--189. https://projecteuclid.org/euclid.bj/1186078366


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