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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 

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