Advances in Applied Probability

Growth of a population of bacteria in a dynamical hostile environment

Olivier Garet and Régine Marchand

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Abstract

We study the growth of a population of bacteria in a dynamical hostile environment corresponding to the immune system of the colonized organism. The immune cells evolve as subcritical open clusters of oriented percolation and are perpetually reinforced by an immigration process, while the bacteria try to grow as a supercritical oriented percolation in the remaining empty space. We prove that the population of bacteria grows linearly when it survives. From this perspective, we build general tools to study dependent oriented percolation models issued from renormalization processes.

Article information

Source
Adv. in Appl. Probab., Volume 46, Number 3 (2014), 661-686.

Dates
First available in Project Euclid: 29 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.aap/1409319554

Digital Object Identifier
doi:10.1239/aap/1409319554

Mathematical Reviews number (MathSciNet)
MR3254336

Zentralblatt MATH identifier
1317.60129

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

Keywords
Contact process directed percolation renormalization random environment stochastic domination block construction interacting particle system

Citation

Garet, Olivier; Marchand, Régine. Growth of a population of bacteria in a dynamical hostile environment. Adv. in Appl. Probab. 46 (2014), no. 3, 661--686. doi:10.1239/aap/1409319554. https://projecteuclid.org/euclid.aap/1409319554


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References

  • Bezuidenhout, C. and Grimmett, G. (1990). The critical contact process dies out. Ann. Prob. 18, 1462–1482.
  • Bramson, M. and Durrett, R. (1988). A simple proof of the stability criterion of Gray and Griffeath. Prob. Theory Relat. Fields 80, 293–298.
  • Broman, E. I. (2007). Stochastic domination for a hidden Markov chain with applications to the contact process in a randomly evolving environment. Ann. Prob. 35, 2263–2293.
  • Durrett, R. (1984). Oriented percolation in two dimensions. Ann. Prob. 12, 999–1040.
  • Durrett, R. (1991). The contact process, 1974–1989. In Mathematics of Random Media (Blacksburg, VA, 1989; Lectures Appl. Math. 27), American Mathematical Society, Providence, RI, pp. 1–18.
  • Durrett, R. (1992). Multicolor particle systems with large threshold and range. J. Theoret. Prob. 5, 127–152.
  • Durrett, R. (1995). Ten lectures on particle systems. In Lectures on Probability Theory (Saint-Flour, 1993; Lecture Notes Math. 1608), Springer, Berlin, pp. 97–201.
  • Durrett, R. and Møller, A. M. (1991). Complete convergence theorem for a competition model. Prob. Theory Relat. Fields 88, 121–136.
  • Durrett, R. and Schinazi, R. (1993). Asymptotic critical value for a competition model. Ann. Appl. Prob. 3, 1047–1066.
  • Durrett, R. and Swindle, G. (1991). Are there bushes in a forest? Stoch. Process. Appl. 37, 19–31.
  • Garet, O. and Marchand, R. (2012). Asymptotic shape for the contact process in random environment. Ann. Appl. Prob. 22, 1362–1410.
  • Garet, O. and Marchand, R. (2013). Growth of a population of bacteria in a dynamical hostile environment. Preprint. Available at http://arxiv.org/abs/1010.4618v3.
  • Garet, O. and Marchand, R. (2014). Large deviations for the contact process in random environment.Ann. Prob. 42, 1438–1479.
  • Grimmett, G. R. and Marstrand, J. M. (1990). The supercritical phase of percolation is well behaved. Proc. R. Soc. London A 430, 439–457.
  • Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin.
  • Liggett, T. M., Schonmann, R. H. and Stacey, A. M. (1997). Domination by product measures. Ann. Prob. 25, 71–95.
  • Lindvall, T. (1999). On Strassen's theorem on stochastic domination. Electron. Commun. Prob. 4, 51–59.
  • Luo, X. (1992). The Richardson model in a random environment. Stoch. Process. Appl. 42, 283–289.
  • Remenik, D. (2008). The contact process in a dynamic random environment. Ann. Appl. Prob. 18, 2392–2420.
  • Steif, J. E. and Warfheimer, M. (2008). The critical contact process in a randomly evolving environment dies out. ALEA Lat. Amer. J. Prob. Math. Statist. 4, 337–357.
  • Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Statist. 36, 423–439.
  • Stroock, D. W. (1993). Probability Theory, an Analytic View. Cambridge University Press.