  In this paper, we study quasi-stationarity for a large class of Kolmogorov
diffusions. The main novelty here is that we allow the drift to go to $-
\infty$ at the origin, and the diffusion to have an entrance boundary at
$+\infty$. These diffusions arise as images, by a deterministic map, of
generalized Feller diffusions, which themselves are obtained as limits of
rescaled birth--death processes. Generalized Feller diffusions take nonnegative
values and are absorbed at zero in finite time with probability 1. An important
example is the logistic Feller diffusion. We give sufficient conditions on the
drift near 0 and near $+ \infty$ for the existence of quasi-stationary
distributions, as well as rate of convergence in the Yaglom limit and existence
of the $Q$-process. We also show that under these conditions, there is exactly
one quasi-stationary distribution, and that this distribution attracts all
initial distributions under the conditional evolution, if and only if $+\infty$
is an entrance boundary. In particular this gives a sufficient condition for
the uniqueness of quasi-stationary distributions. In the proofs spectral theory
plays an important role on $L^2$ of the reference measure for the killed
process.
