Extinction and coming down from infinity of CB-processes with competition in a Lévy environment

Abstract: In this note, we are interested on the event of extinction and the property of coming down from infinity of continuous state branching (or CB for short) processes with competition in a L\'evy environment whose branching mechanism satisfies the so-called Grey's condition. In particular, we deduce, under the assumption that the L\'evy environment does not drift towards infinity, that for any starting point the process becomes extinct in finite time a.s. Moreover if we impose an integrability condition on the competition mechanism, then the process comes down from infinity regardless the long term behaviour of the environment.