We consider two interacting particle systems on $\mathbf{Z}^d$ to
model predator-prey and host-parasite interactions. In both models we have two
types of $\mathbf{Z}^d$ particles (1 and 2) and each site in can be in one of
four states: empty, occupied by a type 1 particle, occupied by a type 2
particle or occupied by two particles (one of each type). Each type gives birth
to particles of the same type on nearest neighbor sites. The interaction
between the two types of particles occurs only when a site is occupied by one
particle of each type. For both models we show that coexistence and
noncoexistence are possible in any dimension.