Extinction times for a birth-death process with weak competition
We consider a birth-death process with the birth rates $iλ$ and death rates $iμ+i(i-1)θ$, where $i$ is the current state of the process. A positive competition rate $θ$ is assumed to be small. In the supercritical case when $λ>μ$ this process can be viewed as a demographic model for a population with a high carrying capacity around $λ-μ\overθ$. The article reports in a self-contained manner on the asymptotic properties of the time to extinction for this logistic branching process as $θ\to0$. All three reproduction regimes $λ>μ$, $λ<μ$, and $λ=μ$ are studied.
math.PR↗