Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-26T16:10:54.984Z Has data issue: false hasContentIssue false
  • English
  • Français

Growth of a Population of Bacteria in a Dynamical Hostile Environment

Part of: Special processes Equilibrium statistical mechanics

Published online by Cambridge University Press:  22 February 2016

Olivier Garet  and
Régine Marchand
Olivier Garet*
Affiliation:
Université de Lorraine
Régine Marchand*
Affiliation:
Université de Lorraine
*
∗ Postal address: Université de Lorraine, Institut Élie Cartan de Lorraine, UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France.
∗ Postal address: Université de Lorraine, Institut Élie Cartan de Lorraine, UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Contact processdirected percolationrenormalizationrandom environmentstochastic dominationblock constructioninteracting particle system

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 82B43: Percolation
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 46 , Issue 3 , September 2014 , pp. 661 - 686
Copyright
© Applied Probability Trust 

References

Bezuidenhout, C. and Grimmett, G. (1990). The critical contact process dies out. Ann. Prob. 18, 1462–1482.CrossRefGoogle Scholar
Bramson, M. and Durrett, R. (1988). A simple proof of the stability criterion of Gray and Griffeath. Prob. Theory Relat. Fields 80, 293–298.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Durrett, R. (1984). Oriented percolation in two dimensions. Ann. Prob. 12, 999–1040.CrossRefGoogle Scholar
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.Google Scholar
Durrett, R. (1992). Multicolor particle systems with large threshold and range. J. Theoret. Prob. 5, 127–152.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Durrett, R. and Møller, A. M. (1991). Complete convergence theorem for a competition model. Prob. Theory Relat. Fields 88, 121–136.CrossRefGoogle Scholar
Durrett, R. and Schinazi, R. (1993). Asymptotic critical value for a competition model. Ann. Appl. Prob. 3, 1047–1066.CrossRefGoogle Scholar
Durrett, R. and Swindle, G. (1991). Are there bushes in a forest? Stoch. Process. Appl. 37, 19–31.CrossRefGoogle Scholar
Garet, O. and Marchand, R. (2012). Asymptotic shape for the contact process in random environment. Ann. Appl. Prob. 22, 1362–1410.CrossRefGoogle Scholar
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.Google Scholar
Garet, O. and Marchand, R. (2014). Large deviations for the contact process in random environment. Ann. Prob. 42, 1438–1479.CrossRefGoogle Scholar
Grimmett, G. R. and Marstrand, J. M. (1990). The supercritical phase of percolation is well behaved. Proc. R. Soc. London A 430, 439–457.Google Scholar
Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin.CrossRefGoogle Scholar
Liggett, T. M., Schonmann, R. H. and Stacey, A. M. (1997). Domination by product measures. Ann. Prob. 25, 71–95.CrossRefGoogle Scholar
Lindvall, T. (1999). On Strassen's theorem on stochastic domination. Electron. Commun. Prob. 4, 51–59.CrossRefGoogle Scholar
Luo, X. (1992). The Richardson model in a random environment. Stoch. Process. Appl. 42, 283–289.CrossRefGoogle Scholar
Remenik, D. (2008). The contact process in a dynamic random environment. Ann. Appl. Prob. 18, 2392–2420.CrossRefGoogle Scholar
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.Google Scholar
Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Statist. 36, 423–439.CrossRefGoogle Scholar
Stroock, D. W. (1993). Probability Theory, an Analytic View. Cambridge University Press.Google Scholar