Simple temporal models that ignore the spatial nature of interactions and track only changes in mean quantities, such as global densities, are typically used under the unrealistic assumption that individuals are well mixed. These so-called mean-field models are often considered overly simplified, given the ample evidence for distributed interactions and spatial heterogeneity over broad ranges of scales. Here, we present one reason why such simple population models may work even when mass-action assumptions do not hold: spatial structure is present but it relates to global densities in a special way. With an individual-based predator–prey model that is spatial and stochastic, and whose mean-field counterpart is the classic Lotka–Volterra model, we show that the global densities and densities of pairs (or spatial covariances) establish a bi-power law at the stationary state and also in their transient approach to this state. This relationship implies that the dynamics of global densities can be written simply as a function of those densities alone without invoking pairs (or higher order moments). The exponents of the bi-power law for the predation rate exhibit a remarkable robustness to changes in model parameters. Evidence is presented for a connection of our findings to the existence of a critical phase transition in the dynamics of the spatial system. We discuss the application of similar modified mean-field equations to other ecological systems for which similar transitions have been described, both in models and empirical data.

中文翻译：

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复杂系统的简单模型：利用局部和全局密度之间的关系。

简单的时间模型通常会忽略个体的充分混合的不切实际的假设，这些模型会忽略交互的空间性质，而仅跟踪平均量的变化（例如全局密度）。鉴于有足够的证据表明在广泛的尺度范围内有分布的相互作用和空间异质性，这些所谓的均值场模型通常被认为过于简化。在这里，我们提出了即使在大规模行动假设不成立的情况下，这种简单的人口模型仍然可以工作的原因之一：存在空间结构，但它以特殊的方式与全球密度有关。利用具有空间和随机性的基于个体的捕食者－猎物模型，并且其均值场对应物是经典的Lotka–Volterra模型，我们证明了全局密度和成对的密度（或空间协方差）建立了一个双稳态定律及其在稳态时的瞬态方法。这种关系意味着整体密度的动态可以简单地作为这些密度的函数而单独编写，而无需调用对（或更高阶矩）。捕食率的双幂定律的指数对模型参数的变化表现出显着的鲁棒性。提出的证据表明我们的发现与空间系统动力学中关键相变的存在有关。我们讨论了在模型和经验数据中相似修正均值场方程在描述了相似过渡的其他生态系统中的应用。