Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-23T13:29:45.805Z Has data issue: false hasContentIssue false
  • English
  • Français

Contact Process with Destruction of Cubes and Hyperplanes: Forest Fires Versus Tornadoes

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

N. Lanchier
N. Lanchier*
Affiliation:
Arizona State University
*
∗Postal address: School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA. Email address: [email protected]
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.

Nonspatial stochastic models of populations subject to catastrophic events result in the common conclusion that the survival probability of the population is nondecreasing with respect to the random number of individuals removed at each catastrophe. The purpose of this paper is to prove that such a monotonic relationship is not true for simple spatial models based on Harris' contact processes, whose dynamics are described by hypergraph structures rather than traditional graph structures. More precisely, it is proved that, for a wide range of parameters, the destruction of (infinite) hyperplanes does not affect the existence of a nontrivial invariant measure, whereas the destruction of large (finite) cubes drives the population to extinction, a result that we depict by using the biological picture: forest fires are more devastating than tornadoes. This indicates that the geometry of the subsets struck by catastrophes is somewhat more important than their area, thus the need to consider spatial rather than nonspatial models in this context.

Keywords

Contact processcatastrophehypergraphforest firetornado

MSC classification

Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 2 , June 2011 , pp. 352 - 365
Copyright
Copyright © Applied Probability Trust 2011 

Footnotes

Research supported in part by NSF grant DMS-10-05282.

References

[1] Belhadji, L. and Lanchier, N. (2006). Individual versus cluster recoveries within a spatially structured population. Ann. Appl. Prob. 16, 403–422.CrossRefGoogle Scholar
[2] Belhadji, L. and Lanchier, N. (2008). Two-scale contact process and effects of habitat fragmentation on metapopulations. Markov Process. Relat. Fields 14, 487–514.Google Scholar
[3] Bezuidenhout, C. and Grimmett, G. (1990). The critical contact process dies out. Ann. Prob. 18, 1462–1482.CrossRefGoogle Scholar
[4] Durrett, R. (1995). Ten lectures on particle systems. In Lectures on Probability Theory (Saint-Flour, 1993; Lecture Notes in Math. 1608), Springer, Berlin, pp. 97–201.CrossRefGoogle Scholar
[5] Durrett, R. and Remenik, D. (2009). Chaos in a spatial epidemic model. Ann. Appl. Prob. 19, 1656–1685.CrossRefGoogle Scholar
[6] Harris, T. E. (1974). Contact interactions on a lattice. Ann. Prob. 2, 969–988.CrossRefGoogle Scholar
[7] Kang, H.-C., Krone, S. M. and Neuhauser, C. (1995). Stepping-stone models with extinction and recolonization. Ann. Appl. Prob. 5, 1025–1060.CrossRefGoogle Scholar
[8] Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes Wissenschaften (Fundamental Principles Math. Sci. 324), Springer, Berlin.CrossRefGoogle Scholar
[9] Liggett, T. M., Schinazi, R. B. and Schweinsberg, J. (2008). A contact process with mutations on a tree. Stoch. Process. Appl. 118, 319–332.CrossRefGoogle Scholar
[10] Richardson, D. (1973). Random growth in a tessellation. Proc. Camb. Phil. Soc. 74, 515–528.CrossRefGoogle Scholar
[11] Schinazi, R. B. (2002). On the role of social clusters in the transmission of infectious diseases. Theoret. Pop. Biol. 61, 163–169.CrossRefGoogle ScholarPubMed
[12] Schinazi, R. B. (2005). Mass extinctions: an alternative to the Allee effect. Ann. Appl. Prob. 15, 984–991.CrossRefGoogle Scholar
[13] Schinazi, R. B. and Schweinsberg, J. (2008). Spatial and non-spatial stochastic models for immune response. Markov Process. Relat. Fields 14, 255–276.Google Scholar
[14] Van den Berg, J., Grimmett, G. R. and Schinazi, R. B. (1998). Dependent random graphs and spatial epidemics. Ann. Appl. Prob. 8, 317–336.Google Scholar