  Using Foster-Lyapunov techniques we establish new conditions on
non-extinction, non-explosion, coming down from infinity and staying infinite,
respectively, for the general continuous-state nonlinear branching processes
introduced in Li et al. (2019). These results can be applied to identify
boundary behaviors for the critical cases of the above nonlinear branching
processes with power rate functions driven by Brownian motion and (or) stable
Poisson random measure, which was left open in Li et al. (2019). In particular,
we show that even in the critical cases, a phase transition happens between
coming down from infinity and staying infinite.
