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

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.