Continuous-state branching processes with competition: duality and reflection at infinity

The boundary behavior of continuous-state branching processes with quadratic competition is studied in whole generality. We first observe that despite competition, explosion can occur for certain branching mechanisms. We obtain a necessary and sufficient condition for infinity to be accessible in terms of the branching mechanism and the competition parameter c > 0. We show that when infinity is inaccessible, it is always an entrance boundary In the case where infinity is accessible, explosion can occur either by a single jump to infinity (the process at z jumps to infinity at rate lambda z for some lambda > 0) or by accumulation of large jumps over finite intervals. We construct a natural extension of the minimal process and show that when infinity is accessible and 0 <= 2 lambda/c < 1, the extended process is reflected at infinity. In the case 2 lambda/c >= 1, infinity is an exit of the extended process. When the branching mechanism is not the Laplace exponent of a subordinator, we show that the process with reflection at infinity gets extinct almost surely. Moreover absorption at 0 is almost sure if and only if Grey's condition is satisfied. When the branching mechanism is the Laplace exponent of a subordinator, necessary and sufficient conditions are given for a stationary distribution to exist. The Laplace transform of the latter is provided. The study is based on classical time-change arguments and on a new duality method relating logistic CSBPs with certain generalized Feller diffusions.