A general continuous-state nonlinear branching process

In this paper, we consider the unique nonnegative solution to the following generalized version of the stochastic differential equation for a continuous-state branching process: [Equation presented here], where W(dt, du) and Ñ(ds, dz, du) denote a Gaussian white noise and an independent compensated spectrally positive Poisson random measure, respectively, and γ₀,γ₁ and γ₂ are functions on R₊ with both γ₁ and γ₂ taking nonnegative values. Intuitively, this process can be identified as a continuousstate branching process with population-size-dependent branching rates and with competition. Using martingale techniques we find rather sharp conditions on extinction, explosion and coming down from infinity behaviors of the process. Some Foster-Lyapunov-type criteria are also developed for such a process. More explicit results are obtained when γ_i, i = 0, 1,2 are power functions.