We study two-type branching random walks in which the birth or death rate of each type can depend on the number of neighbors of the opposite type. This competing species model contains variants of Durrett's predator-prey model and Durrett and Levin's colicin model as special cases. We verify in some cases convergence of scaling limits of these models to a pair of super-Brownian motions interacting through their collision local times, constructed by Evans and Perkins.
"Competing super-Brownian motions as limits of interacting particle systems." Electron. J. Probab. 10 1147 - 1220, 2005. https://doi.org/10.1214/EJP.v10-229