ON THE STABILITY OF A POPULATION-GROWTH MODEL WITH SEXUAL REPRODUCTION ON Z2

• 成果类型: Article
• 署名作者: CHEN, HN
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176989926
• 发表日期: 1992
• 页码: 232-285
• 关键词: percolation; set

摘要

In this paper we study a growth model known as the contact process with sexual reproduction on Z2. We focus on the symmetric model in which a child particle can be produced at a vacant site whenever a pair of its neighboring sites is occupied by parent particles. Two kinds of stability of the absorbing state-phi (i,e., the state in which all the sites are vacant) are investigated in this paper. The first kind of stability concerns the behavior of the system when it starts close to the state-phi. More explicitly, we consider the system starting with a random configuration in which the sites are occupied independently with occupation probability p, where p is a small positive parameter. The system is said to be stable if, for sufficiently small p, the probability that a site is occupied approaches 0 as time approaches infinity. The second kind of stability concerns the behavior of the system under the perturbation of adding a small quantity-beta > 0 to all the birth rates (spontaneous birth at rate-beta). In this case, stability means that there is an equilibrium state which is close to phi when beta is small. It is proven in this paper that in the symmetric model the state-phi is stable under the first kind of perturbation, but it is unstable under the second kind of perturbation.