标题： 发现泰勒功率定律六十年后：从生态学和统计学的普遍性，通过素数分布和临界点信号，到复杂网络的异质性和稳定性 显示英文标题

标题： Six Decades Post-Discovery of Taylor's Power Law: From Ecological and Statistical Universality, Through Prime Number Distributions and Tipping-Point Signals, to Heterogeneity and Stability of Complex Networks

摘要： 首次由L. R. Taylor（1961，Nature）发现，泰勒定律（TPL）使用幂函数（V=aM^b）将一组昆虫种群的平均（M）种群丰富度和相应的方差（V）相关联。TPL在众多科学、社会科学和人文学科领域中表现出其“普遍性”。这种普遍性激发了两个主要的研究方向：一个来自数学家和统计学家，他们可能会本能地用类似于高斯分布中心极限定理的收敛定理来回应；另一个来自生物学家、生态学家、物理学家等，他们更感兴趣的是潜在的生态或组织机制。在过去六十年里，TPL研究在抽象世界和物理世界的两个方向上呈现出一个间断的景观，包括三个相对不同的时期（1960年代-1980年代；1990年代-2000年代，以及2010年代-2020年代）。在这片景观上已经确定并回顾了八个主题，包括种群空间聚集和生态机制，TPL与偏态统计分布，TPL的数学/统计机制，样本与总体TPL，种群稳定性，同步性和临界点的早期预警信号，复杂网络上的TPL，宏观生物群落中的TPL，以及微生物群落中的TPL。三个未来的研究方向包括促进两个方向之间的相互作用，异质性测量，以及在进化背景下的探索。TPL研究的意义包括实践上，TPL捕捉到的种群波动与农业、林业、渔业、野生动物保护、流行病学、肿瘤异质性、地震、社会不平等、股票流动性、金融稳定性和临界点事件等相关；理论上，TPL是幂律的一种形式，与相变、普遍性、尺度不变性等相关。 显示英文摘要

摘要： First discovered by L. R. Taylor (1961, Nature), Taylor's Power Law (TPL) correlates the mean (M) population abundances and the corresponding variances (V) across a set of insect populations using a power function (V=aM^b). TPL has demonstrated its 'universality' across numerous fields of sciences, social sciences, and humanities. This universality has inspired two main prongs of exploration: one from mathematicians and statisticians, who might instinctively respond with a convergence theorem similar to the central limit theorem of the Gaussian distribution, and another from biologists, ecologists, physicists, etc., who are more interested in potential underlying ecological or organizational mechanisms. Over the past six decades, TPL studies have produced a punctuated landscape with three relatively distinct periods (1960s-1980s; 1990s-2000s, and 2010s-2020s) across the two prongs of abstract and physical worlds. Eight themes have been identified and reviewed on this landscape, including population spatial aggregation and ecological mechanisms, TPL and skewed statistical distributions, mathematical/statistical mechanisms of TPL, sample vs. population TPL, population stability, synchrony, and early warning signals for tipping points, TPL on complex networks, TPL in macrobiomes, and in microbiomes. Three future research directions including fostering reciprocal interactions between the two prongs, heterogeneity measuring, and exploration in the context of evolution. The significance of TPL research includes practically, population fluctuations captured by TPL are relevant for agriculture, forestry, fishery, wildlife-conservation, epidemiology, tumor heterogeneity, earthquakes, social inequality, stock illiquidity, financial stability, tipping point events, etc.; theoretically, TPL is one form of power laws, which are related to phase transitions, universality, scale-invariance, etc.