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Using Foster-Lyapunov techniques we establish new conditions onnon-extinction, non-explosion, coming down from infinity and staying infinite,respectively, for the general continuous-state nonlinear branching processesintroduced in Li et al. (2019). These results can be applied to identifyboundary behaviors for the critical cases of the above nonlinear branchingprocesses with power rate functions driven by Brownian motion and (or) stablePoisson 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 betweencoming down from infinity and staying infinite.

Boundary behaviors for a class of continuous-state nonlinear branching processes in critical cases