1. In diverse ecosystems, organisms cluster together in such a manner that the frequency distribution of cluster sizes is a power law function. Spatially explicit computational models of ecosystems suggest that a loss of such power law clustering may indicate a loss of ecosystem resilience; the empirical evidence in support for this hypothesis has been mixed. On the other hand, a well‐known dynamical feature of systems with reduced resilience is the slower recovery from perturbations, a phenomenon known as critical slowing down (CSD). Here, we examine the relationship between spatial clustering and CSD to better understand the use of cluster size distributions as indicators of ecosystem resilience.
2. Local positive feedback is an important driver of spatial clustering, while also affecting the dynamics of the ecosystem: Studies have demonstrated that positive feedback promotes abrupt regime shifts. Here, we analyse a spatial model of ecosystem transitions that enables us to disentangle the roles of local positive feedback and environmental stress on spatial patterns and ecosystem resilience.
3. We demonstrate that, depending on the strength of positive feedback, power law clustering can occur at any distance from the critical threshold of ecosystem collapse. In fact, we find that for systems with strong positive feedback, which are more likely to exhibit abrupt transitions, there may be no loss of power law clustering prior to critical thresholds.
4. Our analyses show that cluster size distributions are unrelated to the phenomenon of CSD and that loss of power law clustering is not a generic indicator of ecosystem resilience. Further, due to CSD, a power law feature does occur near critical thresholds but in a different quantity; specifically, a power law decay of spatial covariance of ecosystem state. Our work highlights the importance of links between local positive feedback, emergent spatial properties and how they may be used to interpret ecosystem resilience.

中文翻译：

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聚类和相关性：从生态系统的空间格局中推断复原力

1. 在不同的生态系统中，生物体以集群大小的频率分布为幂律函数的方式聚集在一起。生态系统在空间上的显式计算模型表明，这种幂律聚类的丧失可能表明生态系统的弹性丧失。支持该假说的经验证据参差不齐。另一方面，具有降低的弹性的系统的众所周知的动态特征是从扰动中恢复较慢，这种现象称为临界减速（CSD）。在这里，我们研究了空间聚类与CSD之间的关系，以更好地理解使用聚类大小分布作为生态系统复原力的指标。
2. 局部正反馈是空间聚集的重要驱动力，同时也影响了生态系统的动态：研究表明，正反馈会促进系统的突然转变。在这里，我们分析了生态系统变迁的空间模型，这使我们能够弄清局部正反馈和环境压力对空间格局和生态系统弹性的作用。
3. 我们证明，根据正反馈的强度，幂律聚类可以发生在距生态系统崩溃的临界阈值任何距离处。实际上，我们发现，对于具有强正反馈的系统（更可能表现出突然的跃迁），在临界阈值之前可能不会存在幂律聚类的损失。
4. 我们的分析表明，群集大小分布与CSD现象无关，并且幂律群集的丢失不是生态系统弹性的一般指标。此外，由于CSD，幂定律特征的确在临界阈值附近发生，但数量不同。具体来说，幂律会衰减生态系统状态的空间协方差。我们的工作强调了局部正反馈，新兴的空间特性以及如何将其用于解释生态系统弹性之间的联系的重要性。