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

Publication ,  Journal Article
Durrett, R; Zhang, Y
Published in: Annals of Applied Probability
December 1, 2015

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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Published In

Annals of Applied Probability

DOI

ISSN

1050-5164

Publication Date

December 1, 2015

Volume

25

Issue

6

Start / End Page

3434 / 3464

Related Subject Headings

  • Statistics & Probability
  • 4905 Statistics
  • 4901 Applied mathematics
  • 0104 Statistics
  • 0102 Applied Mathematics
 

Citation

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Durrett, R., & Zhang, Y. (2015). Coexistence of grass, saplings and trees in the Staver-Levin forest model. Annals of Applied Probability, 25(6), 3434–3464. https://doi.org/10.1214/14-AAP1079
Durrett, R., and Y. Zhang. “Coexistence of grass, saplings and trees in the Staver-Levin forest model.” Annals of Applied Probability 25, no. 6 (December 1, 2015): 3434–64. https://doi.org/10.1214/14-AAP1079.
Durrett R, Zhang Y. Coexistence of grass, saplings and trees in the Staver-Levin forest model. Annals of Applied Probability. 2015 Dec 1;25(6):3434–64.
Durrett, R., and Y. Zhang. “Coexistence of grass, saplings and trees in the Staver-Levin forest model.” Annals of Applied Probability, vol. 25, no. 6, Dec. 2015, pp. 3434–64. Scopus, doi:10.1214/14-AAP1079.
Durrett R, Zhang Y. Coexistence of grass, saplings and trees in the Staver-Levin forest model. Annals of Applied Probability. 2015 Dec 1;25(6):3434–3464.

Published In

Annals of Applied Probability

DOI

ISSN

1050-5164

Publication Date

December 1, 2015

Volume

25

Issue

6

Start / End Page

3434 / 3464

Related Subject Headings

  • Statistics & Probability
  • 4905 Statistics
  • 4901 Applied mathematics
  • 0104 Statistics
  • 0102 Applied Mathematics