Probabilistic Foundations of Spatial Mean-Field Models in Ecology and Applications
Probabilistic Foundations of Spatial Mean-Field Models in Ecology and Applications
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DOI：
10.1137/19m1298329
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发表时间：
2020-01-01
影响因子：
2.1
通讯作者：
Touboul, Jonathan D.
中科院分区：
文献类型：
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作者：
Patterson, Denis D.;Levin, Simon A.;Touboul, Jonathan D.
植被的确定性模型通常在宏观尺度上概括了微观尺度上发生的大量内在随机事件。我们桥接这些尺度之间的差距证明收敛到一个平均场的限制为一般类的随机模型，代表每个单独的生态事件的限制大系统的大小。证明依赖于经典的随机耦合技术，我们推广到涵盖空间扩展的相互作用。平均场极限是一个空间扩展的非马尔可夫过程，其特征在于非局部积分微分方程描述了一片土地处于给定状态的概率的演变（该过程的广义柯尔莫哥洛夫方程（GKE））。因此，我们提供了一个可访问的一般框架，从生态学和种群动力学的许多经典的有限状态模型在空间上扩展。我们证明了我们的方法的实际有效性，通过我们的限制空间模型和有限大小的版本的一个特定的热带草原森林模型，所谓的Staver-Levin模型的详细比较。尽管在有限尺度系统中几乎必然发生森林绝灭，但GKEs与有限尺度系统具有显著的动态一致性。为了解决这一明显的悖论，我们表明，灭绝率急剧下降时，非平凡的平衡出现在GKEs，和有限大小的系统的准平稳分布（平稳分布条件不灭绝）密切匹配的GKEs的分岔图。此外，极限过程可以支持概率分布的周期振荡，因此提供了不收敛于平稳分布的跳跃过程的基本示例。在空间上扩展的设置，环境异质性可以导致入侵和前钉扎现象的浪潮。
Deterministic models of vegetation often summarize, at a macroscopic scale, a multitude of intrinsically random events occurring at a microscopic scale. We bridge the gap between these scales by demonstrating convergence to a mean-field limit for a general class of stochastic models representing each individual ecological event in the limit of large system size. The proof relies on classical stochastic coupling techniques that we generalize to cover spatially extended interactions. The mean-field limit is a spatially extended non-Markovian process characterized by nonlocal integro-differential equations describing the evolution of the probability for a patch of land to be in a given state (the generalized Kolmogorov equations (GKEs) of the process). We thus provide an accessible general framework for spatially extending many classical finite-state models from ecology and population dynamics. We demonstrate the practical effectiveness of our approach through a detailed comparison of our limiting spatial model and the finite-size version of a specific savanna-forest model, the so-called Staver-Levin model. There is remarkable dynamic consistency between the GKEs and the finite-size system in spite of almost sure forest extinction in the finite-size system. To resolve this apparent paradox, we show that the extinction rate drops sharply when nontrivial equilibria emerge in the GKEs, and that the finite-size system's quasi-stationary distribution (stationary distribution conditional on nonextinction) closely matches the bifurcation diagram of the GKEs. Furthermore, the limit process can support periodic oscillations of the probability distribution and thus provides an elementary example of a jump process that does not converge to a stationary distribution. In spatially extended settings, environmental heterogeneity can lead to waves of invasion and front-pinning phenomena.