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: Xt=x+.t0.0(Xs)ds+.t0..1(Xs.)0W(ds,du)+.t0..0..2(Xs.)0zÑ(ds,dz, du),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 .0,.1and .2 are functions on .+ with both .1 and .2 taking nonnegative values. Intuitively, this process can be identified as a continuous-state 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 .i, i=0,1,2 are power functions.