  We study a general class of birth-and-death processes with state space
$\mathbb{N}$ that describes the size of a population going to extinction with
probability one. This class contains the logistic case. The scale of the
population is measured in terms of a `carrying capacity' $K$. When $K$ is
large, the process is expected to stay close to its deterministic equilibrium
during a long time but ultimately goes extinct. Our aim is to quantify the
behavior of the process and the mean time to extinction in the quasi-stationary
distribution as a function of $K$, for large $K$. We also give a quantitative
description of this quasi-stationary distribution. It turns out to be close to
a Gaussian distribution centered about the deterministic long-time equilibrium,
when $K$ is large. Our analysis relies on precise estimates of the maximal
eigenvalue, of the corresponding eigenvector and of the spectral gap of a
self-adjoint operator associated with the semigroup of the process.
