The Annals of Applied Probability

Contact and voter processes on the infinite percolation cluster as models of host-symbiont interactions

D. Bertacchi, N. Lanchier, and F. Zucca

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Abstract

We introduce spatially explicit stochastic processes to model multispecies host-symbiont interactions. The host environment is static, modeled by the infinite percolation cluster of site percolation. Symbionts evolve on the infinite cluster through contact or voter type interactions, where each host may be infected by a colony of symbionts. In the presence of a single symbiont species, the condition for invasion as a function of the density of the habitat of hosts and the maximal size of the colonies is investigated in details. In the presence of multiple symbiont species, it is proved that the community of symbionts clusters in two dimensions whereas symbiont species may coexist in higher dimensions.

Article information

Source
Ann. Appl. Probab., Volume 21, Number 4 (2011), 1215-1252.

Dates
First available in Project Euclid: 8 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1312818835

Digital Object Identifier
doi:10.1214/10-AAP734

Mathematical Reviews number (MathSciNet)
MR2857447

Zentralblatt MATH identifier
1234.60093

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Contact process voter model site percolation logistic growth branching random walks random walks host symbiont infrapopulation metapopulation infracommunity component community

Citation

Bertacchi, D.; Lanchier, N.; Zucca, F. Contact and voter processes on the infinite percolation cluster as models of host-symbiont interactions. Ann. Appl. Probab. 21 (2011), no. 4, 1215--1252. doi:10.1214/10-AAP734. https://projecteuclid.org/euclid.aoap/1312818835


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References

  • [1] Barlow, M. T. (2004). Random walks on supercritical percolation clusters. Ann. Probab. 32 3024–3084.
  • [2] Barlow, M., Peres, Y. and Sousi, P. (2009). Collisions of random walks. Available at arXiv:1003.3255.
  • [3] Belhadji, L., Bertacchi, D. and Zucca, F. (2009). A self-regulating and patch subdivided population. Adv. in Appl. Probab. 42 899–912.
  • [4] Bertacchi, D. and Zucca, F. (2009). Approximating critical parameters of branching random walks. J. Appl. Probab. 46 463–478.
  • [5] Bramson, M. and Durrett, R. (1988). A simple proof of the stability criterion of Gray and Griffeath. Probab. Theory Related Fields 80 293–298.
  • [6] Bramson, M. and Griffeath, D. (1980). On the Williams–Bjerknes tumour growth model. II. Math. Proc. Cambridge Philos. Soc. 88 339–357.
  • [7] Bramson, M. and Griffeath, D. (1981). On the Williams–Bjerknes tumour growth model. I. Ann. Probab. 9 173–185.
  • [8] Clifford, P. and Sudbury, A. (1973). A model for spatial conflict. Biometrika 60 581–588.
  • [9] Coulhon, T., Grigor’yan, A. and Zucca, F. (2005). The discrete integral maximum principle and its applications. Tohoku Math. J. (2) 57 559–587.
  • [10] Doyle, P. G. and Snell, J. L. (1984). Random Walks and Electric Networks. Carus Mathematical Monographs 22. Math. Assoc. America, Washington, DC.
  • [11] Durrett, R. (1984). Oriented percolation in two dimensions. Ann. Probab. 12 999–1040.
  • [12] Durrett, R. (1992). Multicolor particle systems with large threshold and range. J. Theoret. Probab. 5 127–152.
  • [13] Durrett, R. and Lanchier, N. (2008). Coexistence in host-pathogen systems. Stochastic Process. Appl. 118 1004–1021.
  • [14] Grimmett, G. (1989). Percolation. Springer, New York.
  • [15] Grimmett, G. R., Kesten, H. and Zhang, Y. (1993). Random walk on the infinite cluster of the percolation model. Probab. Theory Related Fields 96 33–44.
  • [16] Harris, T. E. (1972). Nearest-neighbor Markov interaction processes on multidimensional lattices. Adv. Math. 9 66–89.
  • [17] Harris, T. E. (1974). Contact interactions on a lattice. Ann. Probab. 2 969–988.
  • [18] Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30.
  • [19] Holley, R. A. and Liggett, T. M. (1975). Ergodic theorems for weakly interacting infinite systems and the voter model. Ann. Probab. 3 643–663.
  • [20] Krishnapur, M. and Peres, Y. (2004). Recurrent graphs where two independent random walks collide finitely often. Electron. Comm. Probab. 9 72–81 (electronic).
  • [21] Lanchier, N. and Neuhauser, C. (2006). Stochastic spatial models of host-pathogen and host-mutualist interactions. I. Ann. Appl. Probab. 16 448–474.
  • [22] Lanchier, N. and Neuhauser, C. (2006). A spatially explicit model for competition among specialists and generalists in a heterogeneous environment. Ann. Appl. Probab. 16 1385–1410.
  • [23] Lanchier, N. and Neuhauser, C. (2010). Stochastic spatial models of host-pathogen and host-mutualist interactions II. Stoch. Models 26 399–430.
  • [24] Mathieu, P. and Remy, E. (2004). Isoperimetry and heat kernel decay on percolation clusters. Ann. Probab. 32 100–128.
  • [25] Mooring, M. S. and Samuel, W. M. (1998). Tick-removal grooming by elk (Cervus elaphus): Testing the principles of the programmed-grooming hypothesis. Canadian Journal of Zoology 76 740–750.
  • [26] Moran, P. A. P. (1958). Random processes in genetics. Proc. Cambridge Philos. Soc. 54 60–71.
  • [27] Price, P. W. (1980). Evolutionary Biology of Parasites. Princeton Univ. Press, Princeton.
  • [28] Trappe, J. M. (1987). Phylogenetic and ecologic aspects of mycotrophy in the angiosperms from an evolutionary standpoint. In Ecophysiology of VA Mycorrhizal Plants. ( G. R. Safir, ed.) 5–25. CRC Press, Boca Raton, FL.
  • [29] Woess, W. (2000). Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics 138. Cambridge Univ. Press, Cambridge.