  We study a family of interacting particle systems with annihilating and
coalescing reactions. Two types of particles are interspersed throughout a
transitive unimodular graph. Both types diffuse as simple random walks with
possibly different jump rates. Upon colliding, like particles coalesce up to
some cap and unlike particles annihilate. We describe a phase transition as the
initial particle density is varied and provide estimates for the expected
occupation time of the root. For the symmetric setting with no cap on
coalescence, we prove that the limiting occupation probability of the root is
asymptotic to 2/3 the occupation probability for classical coalescing random
walk. This addresses an open problem from Stephenson.
