  We are interested in predator-prey dynamics on infinite trees, which can
informally be seen as particular two-type branching processes where individuals
may die (or be infected) only after their parent dies (or is infected). We
study two types of such dynamics: the chase-escape process, introduced by
Kordzakhia with a variant by Bordenave, and the birth-and-assassination
process, introduced by Aldous & Krebs. We exhibit a coupling between these
processes and branching random walks starting from a random point and killed at
the barrier 0. This sheds new light on the chase-escape and
birth-and-assassination processes, which allows us to recover by probabilistic
means previously known results and also to obtain new results. For instance, we
find the asymptotic behavior of tail of the number of infected individuals in
both the subcritical and critical regimes for the chase-escape process, and
show that the birth-and-assassination process ends almost surely at
criticality.
