  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 \infty. 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.
