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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 θ . The article reports in a self-contained manner on the asymptotic properties of the time to extinction for this logistic branching process as θ → 0. All three reproduction regimes λ > µ, λ < µ, and λ = µ are studied.Mathematics Subject Classification: 60J80

It is well known that a supercritical single-type Bienaymé-Galton-Watson process can be viewed as a decomposable branching process formed by two subtypes of particles: those having infinite line of descent and those who have finite number of descendants. In this paper we analyze such a decomposition for the linear-fractional Bienaymé-GaltonWatson processes with countably many types. We find explicit expressions for the main characteristics of the reproduction laws for so-called skeleton and doomed particles.

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