Abstract
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 who sees it as a rumor propagation model, and the birth-and-assassination process, introduced by Aldous and Krebs. We exhibit a coupling between these processes and branching random walks killed at the origin. 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 the 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.
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Acknowledgments
I am deeply indebted to Itai Benjamini for suggesting that I study, this problem as well as to the Weizmann Institute of Science, where this work began, for their hospitality. I would also like to thank Bastien Mallein and Elie Aïdékon for stimulating discussions. I am also grateful to Elie Aïdékon for the proof of Proposition 10.
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Kortchemski, I. Predator–Prey Dynamics on Infinite Trees: A Branching Random Walk Approach. J Theor Probab 29, 1027–1046 (2016). https://doi.org/10.1007/s10959-015-0603-2
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DOI: https://doi.org/10.1007/s10959-015-0603-2