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December 2015 Coexistence of grass, saplings and trees in the Staver–Levin forest model
Rick Durrett, Yuan Zhang
Ann. Appl. Probab. 25(6): 3434-3464 (December 2015). DOI: 10.1214/14-AAP1079

Abstract

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.

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Rick Durrett. Yuan Zhang. "Coexistence of grass, saplings and trees in the Staver–Levin forest model." Ann. Appl. Probab. 25 (6) 3434 - 3464, December 2015. https://doi.org/10.1214/14-AAP1079

Information

Received: 1 August 2014; Published: December 2015
First available in Project Euclid: 1 October 2015

zbMATH: 1329.60333
MathSciNet: MR3404641
Digital Object Identifier: 10.1214/14-AAP1079

Subjects:
Primary: 60K35, 82C22
Secondary: 82B43

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.25 • No. 6 • December 2015
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