Staver-Levin森林模型中草、树苗和树木的共存
Coexistence of grass, saplings and trees in the Staver-Levin forest model
摘要：
本文考虑了两个有吸引力的随机空间模型，其中每个地点可以处于状态0，1或2：Krone模型，其中0 ${}={}$空置，1 ${}={}$少年和2 ${}={}$一个成熟的个体能够生育，和Staver-Levin森林模型，其中0 ${}={}$草，1 ${}={}$树苗和2 ${}={}$树。我们的第一个结果表明，如果$(0,0)$是一个不稳定的不动点的平均场常微分方程的密度为1的和2的，然后当相互作用的范围是大的，有正的生存概率从一个有限集和一个平稳的分布，其中所有三种类型都存在。对于Krone模型，我们用这种方法得到的结果是渐近尖锐的.然而，在Staver-Levin森林模型中，如果$(0,0)$是吸引的，那么也可能有另一个稳定的不动点的常微分方程，在某些情况下，有一个非平凡的平稳分布。
英文摘要：
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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