Duality and Complete Convergence for Multi-type Additive Growth Models

Abstract: We consider a class of multi-type particle systems having similar structure to the contact process and show that additivity is equivalent to the existence of a dual process, extending a result of Harris. We give two additional characterizations of these systems, in spacetime as percolation models, and biologically as population models in which the interactions are due to crowding. We prove a necessary and sufficient condition for the model to preserve positive correlations. We then show that complete convergence on $\mathbb{Z}^d$ holds for a large subclass of models including the two-stage contact process and a household model, and give examples.