Extinction and coming down from infinity of continuous-state branching processes with competition in a Lévy environment

We are interested in the property of coming down from infinity of continuous-state branching processes with competition in a Levy environment. We first study the event of extinction for such a family of processes under Grey’s condition. Moreover, if we add an integrability condition on the competition mechanism then the process comes down from infinity regardless of the long-time behaviour of the environment.