Extinction times for a birth-death process with weak competition

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