  We consider birth-and-death processes of objects (animals) defined in ${\bf
Z}^d$ having unit death rates and random birth rates. For animals with
uniformly bounded diameter we establish conditions on the rate distribution
under which the following holds for almost all realizations of the birth rates:
(i) the process is ergodic with at worst power-law time mixing; (ii) the unique
invariant measure has exponential decay of (spatial) correlations; (iii) there
exists a perfect-simulation algorithm for the invariant measure. The results
are obtained by first dominating the process by a backwards oriented
percolation model, and then using a multiscale analysis due to Klein to
establish conditions for the absence of percolation.
