  In this paper, we study the extinction time of logistic branching processes
which are perturbed by an independent random environment driven by a Brownian
motion. Our arguments use a Lamperti-type representation which is interesting
on its own right and provides a one to one correspondence between the latter
family of processes and the family of Feller diffusions which are perturbed by
an independent spectrally positive L\'evy process. When the independent random
perturbation (of the Feller diffusion) is driven by a subordinator then the
logistic branching processes in a Brownian environment converges to a specified
distribution; otherwise, it becomes extinct a.s. In the latter scenario, and
following a similar approach to Lambert (Lambert, Ann. Appl. Probab., 2005), we
provide the expectation and the Laplace transform of the absorption time, as a
functional of the solution to a Ricatti differential equation. In particular,
the latter characterises the law of the process coming down from infinity.
