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Probabilistic Foundations of Spatial Mean-Field Models in Ecology and Applications
SIAM Journal on Applied Dynamical Systems ( IF 2.1 ) Pub Date : 2020-12-08 , DOI: 10.1137/19m1298329 Denis D. Patterson , Simon A. Levin , Carla Staver , Jonathan D. Touboul
SIAM Journal on Applied Dynamical Systems ( IF 2.1 ) Pub Date : 2020-12-08 , DOI: 10.1137/19m1298329 Denis D. Patterson , Simon A. Levin , Carla Staver , Jonathan D. Touboul
SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2682-2719, January 2020.
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.
中文翻译:
空间均值场模型在生态学和应用中的概率基础
SIAM应用动力系统杂志,第19卷,第4期,第2682-2719页,2020年1月。
植被的确定性模型通常以宏观尺度概括发生在微观尺度上的大量内在随机事件。我们通过证明收敛性来弥合这些标度之间的差距,对于代表大系统规模的限制中的每个单独生态事件的一般随机模型的平均场限制。证明依赖于经典的随机耦合技术,我们对此进行了概括以涵盖空间扩展的交互作用。平均场极限是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,该方程描述了一块土地处于给定状态的概率的演变(该过程的广义Kolmogorov方程(GKE))。因此,我们提供了一个可访问的通用框架,用于从生态学和种群动力学空间扩展许多经典的有限状态模型。我们通过对有限空间模型与特定稀树草原森林模型(即所谓的Staver-Levin模型)的有限大小版本进行详细比较,证明了该方法的实际有效性。尽管在有限大小系统中森林几乎灭绝,但GKE与有限大小系统之间仍具有显着的动态一致性。为了解决这一明显的悖论,我们证明了当非平凡的平衡出现在GKE中时,灭绝速率急剧下降,并且有限尺寸系统的准平稳分布(以非灭绝为条件的平稳分布)与GKE的分叉图紧密匹配。此外,极限过程可以支持概率分布的周期性振荡,因此可以提供不收敛到平稳分布的跳跃过程的基本示例。在空间扩展的环境中,环境异质性可能导致入侵波和前钉现象。
更新日期:2020-12-09
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.
中文翻译:
空间均值场模型在生态学和应用中的概率基础
SIAM应用动力系统杂志,第19卷,第4期,第2682-2719页,2020年1月。
植被的确定性模型通常以宏观尺度概括发生在微观尺度上的大量内在随机事件。我们通过证明收敛性来弥合这些标度之间的差距,对于代表大系统规模的限制中的每个单独生态事件的一般随机模型的平均场限制。证明依赖于经典的随机耦合技术,我们对此进行了概括以涵盖空间扩展的交互作用。平均场极限是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,该方程描述了一块土地处于给定状态的概率的演变(该过程的广义Kolmogorov方程(GKE))。因此,我们提供了一个可访问的通用框架,用于从生态学和种群动力学空间扩展许多经典的有限状态模型。我们通过对有限空间模型与特定稀树草原森林模型(即所谓的Staver-Levin模型)的有限大小版本进行详细比较,证明了该方法的实际有效性。尽管在有限大小系统中森林几乎灭绝,但GKE与有限大小系统之间仍具有显着的动态一致性。为了解决这一明显的悖论,我们证明了当非平凡的平衡出现在GKE中时,灭绝速率急剧下降,并且有限尺寸系统的准平稳分布(以非灭绝为条件的平稳分布)与GKE的分叉图紧密匹配。此外,极限过程可以支持概率分布的周期性振荡,因此可以提供不收敛到平稳分布的跳跃过程的基本示例。在空间扩展的环境中,环境异质性可能导致入侵波和前钉现象。