A SELF-REGULATING AND PATCH SUBDIVIDED POPULATION

作者：Belhadji Lamia^*; Bertacchi Daniela; Zucca Fabio

来源：Advances in Applied Probability, 2010, 42(3): 899-912.

DOI：10.1239/aap/1282924068

• Contact process
• restrained branching random walk
• epidemic model
• phase transition
• critical parameters

We consider an interacting particle system on a graph which, from a macroscopic point of view, looks like Z(d) 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 lambda and an intra-patch birth rate phi. 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 lambda(c) (phi, c, N) and a critical value phi(c)(lambda, c, N). We consider a sequence of processes generated by the families of control functions {c(n)}(n is an element of N) and degrees {N-n}(n is an element of N); we prove, under mild assumptions, the existence of a critical value n(c)(lambda, phi, c). Roughly speaking, we show that, in the limit, these processes behave as the branching random walk on Z(d) with inter-neighbor birth rate lambda and on-site birth rate phi. Some examples of models that can be seen as particular cases are given.

• 出版日期2010-9