The Annals of Applied Probability

The branching process with logistic growth

Amaury Lambert

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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.

Article information

Ann. Appl. Probab., Volume 15, Number 2 (2005), 1506-1535.

First available in Project Euclid: 3 May 2005

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Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 60J85: Applications of branching processes [See also 92Dxx] 92D15: Problems related to evolution 92D25: Population dynamics (general) 92D40: Ecology

Size-dependent branching process continuous-state branching process population dynamics logistic process density dependence Ornstein–Uhlenbeck type process Riccati differential equation fragmentation–coalescence process


Lambert, Amaury. The branching process with logistic growth. Ann. Appl. Probab. 15 (2005), no. 2, 1506--1535. doi:10.1214/105051605000000098.

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