We study a model of growing population that competes for resources. At each time step, all existing particles reproduce and the offspring randomly move to neighboring sites. Then at any site with more than one offspring, the particles are annihilated. This is a nonmonotone model, which makes the analysis more difficult.
We consider the extinction window of this model in the finite mean-field case, where there are $n$ sites but movement is allowed to any site (the complete graph). We show that although the system survives for exponential time, the extinction window is logarithmic.
"Extinction window of mean field branching annihilating random walk." Ann. Appl. Probab. 25 (6) 3139 - 3161, December 2015. https://doi.org/10.1214/14-AAP1069