Fig. 1 in Cohen and Xu 2015: Taylor's law demonstrated as log(variance) being a linear function of log(mean). Each dot is a pair of log(mean)-log(variance) from a random sample of a probability distribution (A: Poisson, B: negative binomial, C: exponential, D: gamma, E: lognormal, F: normal). All but the normal distribution (with zero skewness) display Taylor's law.
Mechanisms of Taylor's Law of Fluctuation Scaling
Variance of a variable scales with the mean of the variable in a power-function form. In physics, such pattern is called the fluctuation scaling. In ecology, it is called Taylor's power law or Taylor's law (Xu 2015). My research goal is to seek the biological or statistical mechanisms that produce such pattern.
Me and my colleagues found that a stochastic multiplicative model (Cohen et al. 2013) and the random sampling of skewed distributions (Cohen and Xu 2015) can generate Taylor's law. These results suggest that the form of Taylor's law does not rely on system-specific mechanism, but maybe a consequence of some universal rule.
Yet, the parameters of Taylor's law could imbed biological and statistical information that deserves attention to the details in the studied system. A unified mechanism that can explain both the form and the parameters of Taylor's law in all counts remains an open question.
Fig. 3 in Xu et al. 2019. Log(cumulative number of fish) as a function of log(number of samples) for each of 14 sampling years in the fished area of Lake Kariba. Stopping lines were sequentially updated, and the first stopping line that intersects with the empirical cumulative abundance plot is shown (solid line). Desired sampling precision of 0.1 is reached at the intersection point.
Applications of Taylor's Law in Fish Sampling and Population Forecast
Not only Taylor's law is a ubiquitous pattern in natural populations, it can be used to design efficient sampling strategy and to select accurate population projections.
The mean-variance relationship depicted by Taylor's law can be used to design and improve fixed-precision sampling strategy. In Xu et al. 2017, we confirmed Taylor's law for the gillnet catch data of fish in an African lake, and used it to find the number of samples required for the fixed-precision sampling. In Xu et al. 2019, We estimated parameters of Taylor's law from sequentially accumulated samples to update a stopping line of fixed precision after each new sample. Overall, our updated stopping-line method reduces 59%~79% of the number of sampling days and 60%~81% of the number of samples that are planned a priori and performed under systematic sampling.
In Xu et al. 2017, we examined Taylor's law and its quadratic generalization for Norwegian county population size. Based on the similarity in the parameters of Taylor's law between historical data and projection data, we selected successfully the projections that resembled the recent census data.
Fig. 1 in Xu 2020. Number of trees (regardless of species) greater than or equal to d (rescaled diameter at breast height) against d in 12 tree communities of the Black Rock Forest, New York. The open circles are observed data. The curves represent the fitted community-level parameterized METE model (in black) and other models (in other colors). In most communities, the parameterized METE model yields the best fit.
Estimation of Metabolic Scaling using Parameterized Maximum Entropy Theory of Ecology
Individual metabolic rate holds the key for the understanding of physiological and ecological relationships. The famous metabolic scaling theory predicts that the metabolic scaling exponent between metabolic rate and body mass is a universal constant 3/4. Empirical data have contradicted such prediction. However an effective way of estimating the exponent is still lacking.
Maximum Entropy Theory of Ecology (METE) is a unified ecological theory based on ecological constraints and information entropy. METE has been used to predict many ecological patterns and distributions. The energetic prediction of METE inherits the universal metabolic scaling exponent from metabolic scaling theory. I extended METE by treating the exponent as a free parameter in the model, and used model fitting to the individual size distribution to estimate the exponent parameter (Xu 2020).
I am working on a joint project that tests and extends the model developed in Xu 2020. In this collaboration, we used two different parameterized METE models to detect the total metabolic rate energy change and partitioning among trees in a monospecific forest after a major earthquake. We found that the extended parameterized model captures the drastic energy decline after the earthquake and the shift in the energy partitioning.