We study a generalized branching random walk where particles breed at a rate which depends on the number of neighboring particles. Under general assumptions on the breeding rates we prove the existence of a phase where the population survives without exploding. We construct a nontrivial invariant measure for this case.
"Ecological equilibrium for restrained branching random walks." Ann. Appl. Probab. 17 (4) 1117 - 1137, August 2007. https://doi.org/10.1214/105051607000000203