Predator-prey and host-parasite spatial stochastic models

We consider two interacting particle systems on Zd to model predator-prey and host-parasite interactions. In both models we have two types of Zd 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.