  In a first part, we prove a Lyapunov-type criterion for the $\xi\_1$-positive
recurrence of absorbed birth and death processes and provide new results on the
domain of attraction of the minimal quasi-stationary distribution. In a second
part, we study the ergodicity and the convergence of a Fleming-Viot type
particle system whose particles evolve independently as a birth and death
process and jump on each others when they hit $0$. Our main result is that the
sequence of empirical stationary distributions of the particle system converges
to the minimal quasi-stationary distribution of the birth and death process.
