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