COEXISTENCE OF GRASS, SAPLINGS AND TREES IN THE STAVER-LEVIN FOREST MODEL

• 成果类型: Article
• 署名作者: Durrett, Rick; Zhang, Yuan
• 署名单位: Duke University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/14-AAP1079
• 发表日期: 2015
• 页码: 3434-3464
• 关键词: savanna

摘要

In this paper, we consider two attractive stochastic spatial models in which each site can be in state 0, 1 or 2: Krone's model in which 0 = vacant, 1 = juvenile and 2 = a mature individual capable of giving birth, and the Staver-Levin forest model in which 0 = grass, 1 = sapling and 2 = tree. Our first result shows that if (0, 0) is an unstable fixed point of the mean-field ODE for densities of 1's and 2's then when the range of interaction is large, there is positive probability of survival starting from a finite set and a stationary distribution in which all three types are present. The result we obtain in this way is asymptotically sharp for Krone's model. However, in the Staver-Levin forest model, if (0, 0) is attracting then there may also be another stable fixed point for the ODE, and in some of these cases there is a nontrivial stationary distribution.