A condition of degeneracy of a bounded branching process with continuous time

We consider a bounded branching process with continuous time, i.e. a supercritical branching process with the condition that at any moment the number of particles is bounded from above (the “superfluous” particles are destroyed and have no offspring). A necessary and sufficient condition is given for the process to die (to get into the state when there is no particle at all) with probability one. We find that the “critical levels” of the upper bound are slightly different for discrete and continuous time.