Abstract
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 $\infty $ to be accessible in terms of the branching mechanism and the competition parameter $c>0$. We show that when $\infty $ is inaccessible, it is always an entrance boundary. In the case where $\infty $ is accessible, explosion can occur either by a single jump to $\infty $ (the process at $z$ jumps to $\infty $ 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 $\infty $ is accessible and $0\leq \frac{2\lambda } {c}<1$, the extended process is reflected at $\infty $. In the case $\frac{2\lambda } {c}\geq 1$, $\infty $ 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 $\infty $ 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.
Highlights
Continuous-state branching processes (CSBPs for short) have been defined by Jirina [Jir58] and Lamperti [Lam67a] for modelling the size of a random continuous population whose individuals reproduce and die independently with the same law
When the branching mechanism is not the Laplace exponent of a subordinator, we show that the process with reflection at ∞ gets extinct almost surely
A logistic CSBP cannot grow indefinitely without exploding. This is a striking difference with CSBPs where indefinite growth with no explosion can occur when the Lévy process (Yt, t ≥ 0) drifts “slowly” towards ∞
Summary
Continuous-state branching processes (CSBPs for short) have been defined by Jirina [Jir58] and Lamperti [Lam67a] for modelling the size of a random continuous population whose individuals reproduce and die independently with the same law. The main contribution of this article is to answer the following question Is it possible for a logistic continuous-state branching process to leave and return to ∞ in finite time? We shall see that the reproduction can be strong enough for explosion to occur and the quadratic competition strong enough to instantaneously push the population size back into [0, ∞) after explosion This phenomenon is captured by the notion of regular instantaneous reflecting boundary. In the Appendix, we provide the calculations needed for classifying the boundaries of (Ut, t ≥ 0) according to Ψ and the parameter c
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