  This article studies the quasi-stationary behaviour of multidimensional birth
and death processes, modeling the interaction between several species, absorbed
when one of the coordinates hits 0. We study models where the absorption rate
is not uniformly bounded, contrary to most of the previous works. To handle
this natural situation, we develop original Lyapunov function arguments that
might apply in other situations with unbounded killing rates. We obtain the
exponential convergence in total variation of the conditional distributions to
a unique stationary distribution, uniformly with respect to the initial
distribution. Our results cover general birth and death models with stronger
intra-specific than inter-specific competition, and cases with neutral
competition with explicit conditions on the dimension of the process.
