We consider the spreading dynamics of two nested invasion clusters on an infinite tree. This model was defined as the chase-escape model by Kordzakhia and it admits a limit process, the birth-and-assassination process, previously introduced by Aldous and Krebs. On both models, we prove an asymptotic equivalent of the extinction probability near criticality. In the subcritical regime, we give a tail bound on the total progeny of the preys before extinction.
"Extinction probability and total progeny of predator-prey dynamics on infinite trees." Electron. J. Probab. 19 1 - 33, 2014. https://doi.org/10.1214/EJP.v19-2361