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

TL;DR

This paper investigates boundary behaviors of critical continuous-state nonlinear branching processes, establishing new conditions for extinction, explosion, and infinity, revealing phase transitions even in critical cases.

Contribution

It provides novel conditions for boundary behaviors in critical nonlinear branching processes, addressing open questions from prior work and identifying phase transitions.

Findings

Conditions for non-extinction and non-explosion established

Boundary behaviors characterized in critical cases

Phase transition between coming down from infinity and staying infinite

Abstract

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.