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May 2005 The branching process with logistic growth
Amaury Lambert
Ann. Appl. Probab. 15(2): 1506-1535 (May 2005). DOI: 10.1214/105051605000000098

Abstract

In order to model random density-dependence in population dynamics, we construct the random analogue of the well-known logistic process in the branching process’ framework. This density-dependence corresponds to intraspecific competition pressure, which is ubiquitous in ecology, and translates mathematically into a quadratic death rate. The logistic branching process, or LB-process, can thus be seen as (the mass of ) a fragmentation process (corresponding to the branching mechanism) combined with constant coagulation rate (the death rate is proportional to the number of possible coalescing pairs). In the continuous state-space setting, the LB-process is a time-changed (in Lamperti’s fashion) Ornstein–Uhlenbeck type process. We obtain similar results for both constructions: when natural deaths do not occur, the LB-process converges to a specified distribution; otherwise, it goes extinct a.s. In the latter case, we provide the expectation and the Laplace transform of the absorption time, as a functional of the solution of a Riccati differential equation. We also show that the quadratic regulatory term allows the LB-process to start at infinity, despite the fact that births occur infinitely often as the initial state goes to ∞. This result can be viewed as an extension of the pure-death process starting from infinity associated to Kingman’s coalescent, when some independent fragmentation is added.

Citation

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Amaury Lambert. "The branching process with logistic growth." Ann. Appl. Probab. 15 (2) 1506 - 1535, May 2005. https://doi.org/10.1214/105051605000000098

Information

Published: May 2005
First available in Project Euclid: 3 May 2005

zbMATH: 1075.60112
MathSciNet: MR2134113
Digital Object Identifier: 10.1214/105051605000000098

Subjects:
Primary: 60J80
Secondary: 60J70, 60J85, 92D15, 92D25, 92D40

Rights: Copyright © 2005 Institute of Mathematical Statistics

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Vol.15 • No. 2 • May 2005
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