This paper studies systems of particles following independent random walks
and subject to annihilation, binary branching, coalescence, and deaths. In the
case without annihilation, such systems have been studied in our 2005 paper
"Branching-coalescing particle systems". The case with annihilation is
considerably more difficult, mainly as a consequence of the non-monotonicity of
such systems and a more complicated duality. Nevertheless, we show that adding
annihilation does not significantly change the long-time behavior of the
process and in fact, systems with annihilation can be obtained by thinning
systems without annihilation
