The Annals of Applied Probability

Coexistence of grass, saplings and trees in the Staver–Levin forest model

Rick Durrett and Yuan Zhang

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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.

Article information

Source
Ann. Appl. Probab., Volume 25, Number 6 (2015), 3434-3464.

Dates
Received: August 2014
First available in Project Euclid: 1 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1443703779

Digital Object Identifier
doi:10.1214/14-AAP1079

Mathematical Reviews number (MathSciNet)
MR3404641

Zentralblatt MATH identifier
1329.60333

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C22: Interacting particle systems [See also 60K35]
Secondary: 82B43: Percolation [See also 60K35]

Keywords
Coexistence stationary distribution percolation block construction

Citation

Durrett, Rick; Zhang, Yuan. Coexistence of grass, saplings and trees in the Staver–Levin forest model. Ann. Appl. Probab. 25 (2015), no. 6, 3434--3464. doi:10.1214/14-AAP1079. https://projecteuclid.org/euclid.aoap/1443703779


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References

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