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

Published

Journal Article

© 2015 Institute of Mathematical Statistics. 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.

Full Text

Duke Authors

Cited Authors

  • Durrett, R; Zhang, Y

Published Date

  • January 1, 2015

Published In

Volume / Issue

  • 25 / 6

Start / End Page

  • 3434 - 3464

International Standard Serial Number (ISSN)

  • 1050-5164

Digital Object Identifier (DOI)

  • 10.1214/14-AAP1079

Citation Source

  • Scopus