Extinction times for a birth-death process with weak competition

We consider a birth-death process with 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 where lambda > mu, this process can be viewed as a demographic model for a population with high carrying capacity around (lambda-mu)/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 -> 0. All three reproduction regimes lambda > mu, lambd