The branching process with logistic growth

Lambert, Amaury

Ann. Appl. Probab., Tome 15 (2005) no. 1A, p. 1506-1535 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 2005-05-14
Classification:  Size-dependent branching process,  continuous-state branching process,  population dynamics,  logistic process,  density dependence,  Ornstein–Uhlenbeck type process,  Riccati differential equation,  fragmentation–coalescence process,  60J80,  60J70,  60J85,  92D15,  92D25,  92D40