  The asymptotic behavior of a stochastic network represented by a birth and
death processes of particles on a compact state space is analyzed. Births:
Particles are created at rate $\lambda_+$ and their location is independent of
the current configuration. Deaths are due to negative particles arriving at
rate $\lambda_-$. The death of a particle occurs when a negative particle
arrives in its neighborhood and kills it. Several killing schemes are
considered. The arriving locations of positive and negative particles are
assumed to have the same distribution. By using a combination of monotonicity
properties and invariance relations it is shown that the configurations of
particles converge in distribution for several models. The problems of
uniqueness of invariant measures and of the existence of accumulation points
for the limiting configurations are also investigated. It is shown for several
natural models that if $\lambda_+<\lambda_-$ then the asymptotic configuration
has a finite number of points with probability 1. Examples with
$\lambda_+<\lambda_-$ and an infinite number of particles in the limit are also
presented.
