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Predator–Prey Dynamics on Infinite Trees: A Branching Random Walk Approach

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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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References

  1. Aïdékon, E., Jaffuel, B.: Survival of branching random walks with absorption. Stoch. Process. Appl. 121, 1901–1937 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aïdékon, E., Hu, Y., Zindy, O.: The precise tail behavior of the total progeny of a killed branching random walk. Ann. Probab. 14, 3786–3878 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aldous, D., Krebs, W.B.: The “birth-and-assassination” process. Stat. Probab. Lett. 10, 427–430 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  4. Athreya, K.B., Karlin, S.: Embedding of urn schemes into continuous time Markov branching processes and related limit theorems. Ann. Math. Stat. 39, 1801–1817 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  5. Biggins, J.D.: Martingale convergence in the branching random walk. J. Appl. Probab. 14, 25–37 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bordenave, C.: On the birth-and-assassination process, with an application to scotching a rumor in a network. Electron. J. Probab. 13(66), 2014–2030 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bordenave, C.: Extinction probability and total progeny of predator-prey dynamics on infinite trees. Electron. J. Probab. 19(20), 1–33 (2014)

    MathSciNet  MATH  Google Scholar 

  8. Daley, D.J., Kendall, D.G.: Stochastic rumours. J. Inst. Math. Appl. 1, 42–55 (1965)

    Article  MathSciNet  Google Scholar 

  9. Feller, W.: An introduction to probability theory and its applications. Vol. II, 2nd edn. Wiley, New York (1971)

    MATH  Google Scholar 

  10. Häggström, O., Pemantle, R.: First passage percolation and a model for competing spatial growth. J. Appl. Probab. 35, 683–692 (1998)

  11. Hu, Y., Shi, Z.: Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. Ann. Probab. 37, 742–789 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jaffuel, B.: The critical barrier for the survival of branching random walk with absorption. Ann. Inst. Henri Poincaré Probab. Stat. 48, 989–1009 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kordzakhia, G.: The escape model on a homogeneous tree. Electron. Comm. Probab. 10, 113–124 (2005). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kordzakhia, G., Lalley, S.P.: A two-species competition model on \({\mathbb{Z}}^{\text{ d }}\). Stoch. Process. Appl. 115, 781–796 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kortchemski, I.: A predator-prey sir type dynamics on large complete graphs with three phase transitions. Stoch. Proc. Appl. 125(3), 886–917 (2015)

  16. Le Gall, J.-F.: Random trees and applications. Probab. Surv. 2, 245–311 (2005)

  17. Lyons, R.: A simple path to Biggins’ martingale convergence for branching random walk, In: Classical and modern branching processes(Minneapolis, MN, 1994), vol. 84 of IMA Vol. Math. Appl., Springer,New York, pp. 217–221 (1997)

  18. Richardson, D.: Random growth in a tessellation. Proc. Camb. Philos. Soc. 74, 515–528 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  19. Tsitsiklis, J.N., Papadimitriou, C.H., Humblet, P.: The performance of a precedence-based queuing discipline. J. ACM 33, 593–602 (1986)

    Article  MathSciNet  Google Scholar 

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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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Correspondence to Igor Kortchemski.

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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

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