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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: lanchier@math.asu.edu
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Abstract

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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.

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