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A self-regulating and patch subdivided population

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Lamia Belhadji ,
Daniela Bertacchi  and
Fabio Zucca
Lamia Belhadji*
Affiliation:
Université de Rouen
Daniela Bertacchi*
Affiliation:
Universitá di Milano-Bicocca
Fabio Zucca*
Affiliation:
Politecnico di Milano
*
∗ Postal address: Laboratoire de Mathématiques Raphaël Salem, UMR 6085, CNRS - Université de Rouen, Avenue de l'Université, BP. 12, 76801 Saint Etienne du Rouvray, France. Email address: [email protected]
∗∗ Postal address: Universitá di Milano-Bicocca Dipartimento di Matematica e Applicazioni, Via Cozzi 53, 20125 Milano, Italy. Email address: [email protected]
∗∗∗ Postal address: Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy. Email address: [email protected]
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Abstract

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We consider an interacting particle system on a graph which, from a macroscopic point of view, looks like ℤd and, at a microscopic level, is a complete graph of degree N (called a patch). There are two birth rates: an inter-patch birth rate λ and an intra-patch birth rate ϕ. Once a site is occupied, there is no breeding from outside the patch and the probability c(i) of success of an intra-patch breeding decreases with the size i of the population in the site. We prove the existence of a critical value λc(ϕ, c, N) and a critical value ϕc(λ, c, N). We consider a sequence of processes generated by the families of control functions {cn}n∈ℕ and degrees {Nn}n∈ℕ; we prove, under mild assumptions, the existence of a critical value nc(λ, ϕ, c). Roughly speaking, we show that, in the limit, these processes behave as the branching random walk on ℤd with inter-neighbor birth rate λ and on-site birth rate ϕ. Some examples of models that can be seen as particular cases are given.

Keywords

Contact processrestrained branching random walkepidemic modelphase transitioncritical parameters

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 3 , September 2010 , pp. 899 - 912
Copyright
Copyright © Applied Probability Trust 2010 

References

Belhadji, L. (2007). Ergodicity and hydrodynamic limits for an epidemic models. Preprint. Available at http://arxiv.org/abs/0710.5185.Google Scholar
Belhadji, L. and Lanchier, N. (2006). Individual versus cluster recoveries within a spatially structured population. Ann. Appl. Prob. 16, 403–422.CrossRefGoogle Scholar
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
Bertacchi, D. and Zucca, F. (2009). Approximating critical parameters of branching random walks. J. Appl. Prob. 46, 463–478.CrossRefGoogle Scholar
Bertacchi, D. and Zucca, F. (2009). Characterization of critical values of branching random walks on weighted graphs through infinite-type branching processes. J. Statist. Phys. 134, 53–65.CrossRefGoogle Scholar
Bertacchi, D., Posta, G. and Zucca, F. (2007). Ecological equilibrium for restrained random walks. Ann. Appl. Prob. 17, 1117–1137.CrossRefGoogle Scholar
Bezuidenhout, C. and Grimmett, G. (1991). Exponential decay for subcritical contact and percolation processes. Ann. Prob. 19, 984–1009.CrossRefGoogle Scholar
Bramson, M., Durrett, R. and Swindle, G. (1989). Statistical mechanics of crabgrass. Ann. Prob. 17, 444–481.CrossRefGoogle Scholar
Durrett, R. and Levin, S. (1994). The importance of being discrete (and spatial). Theoret. Pop. Biol. 46, 363–394.CrossRefGoogle Scholar
Harris, T. E. (1974). Contact interactions on a lattice. Ann. Prob. 2, 969–988.CrossRefGoogle Scholar
Schinazi, R. B. (2002). On the role of social clusters in the transmission of infectious diseases. Theoret. Pop. Biol. 61, 163–169.CrossRefGoogle ScholarPubMed
Woess, W. (2000). Random Walks on Infinite Graphs and Groups (Camb. Tracts Math. 138). Cambridge University Press.CrossRefGoogle Scholar
Woess, W. (2009). Denumerable Markov Chains. European Mathematical Society, Zürich.CrossRefGoogle Scholar