Figure 1

Taylor’s fluctuation scaling for systems analyzed in the paper. a, European corn borers (Pyrausta nubilalis). b, Gene structures in human chromosome 7. c, Daily turnovers of stocks traded on the New York Stock Exchange (NYSE). d, Mallards (Anas platyrhynchos) and Blue Jays (Cyanocitta cristata) in all routes populated by these species in 1966–2007. e, Savannah Sparrows (Passerculus sandwichensis) in all routes populated by this species in 1966–2007. f, Savannah Sparrows counted in routes corresponding to two different physiographic conditions located in the Drift Prairie and the Southern Rockies. g, Traffic intensity as measured by all automatic traffic recorders in Minnesota in 2007. h, Daily fluctuations in traffic measured by a single recorder (ATR No. 222) in 2002–2007. In the figure, full points correspond to raw data (i.e., mean and variance calculated according to the description given in the text and in the Table ), the open symbols express logarithmic binning of the data, and the solid lines represent their linear fits (in log-log scale). In the case of birds, the key observation is that the characteristic parameter $b$ may differ not only among the species but also within the same species if one takes into account the physiographic stratification characterizing living conditions in different areas (for that reason for further analysis we have selected Blue Jays as a species that weakly depends on the physiographic stratification). A similar comment is true for the traffic intensity. In the latter case, one can see that although the whole data obey Taylor’s fluctuation scaling with the parameter $b=1.43$, in reality the data are very heterogeneous. The parameter $b$ characterizing traffic intensity recorded by a single ATR may change dramatically from hour to hour.Reuse & Permissions

Figure 2

Comparison of real-world and theoretical frequency distributions $P\left(N\right)$ characterizing the considered systems for the given values of $⟨N⟩$ (left column), and quotients of two distributions fitted by the exponential function predicted by Eq. (18) (right column). The figure consists of 8 panels arranged in 4 rows. Each row corresponds to a different data set and provides information on (for detailed information on the data used see Section II and Appendix , detailed description of how fitting of experimental frequency distributions has been done is given in Appendix ): a,b, European corn borer frequency patterns; c,d, Intensity of car traffic in Minnesota, as measured by a single recorder (ATR No. 222) during night hours (1 a.m.–7 a.m.) in 2002–2007; e,f, Blue Jay (Cyanocitta cristata) abundance in North America in 1966–2007; h,g, Daily activity at the New York Stock Exchange (NYSE) in the period May 8–June 5, 2008. Some discrepancies visible in the last case may be due to the problem with selection of more homogeneous subset of analyzed stocks.Reuse & Permissions

Figure 3

Experimental verification of Eq. (19). Different symbols placed in the graph represent different real-world systems that show Taylor’s fluctuation scaling. Note that the symbols cover the solid theoretical line to an impressive extent of nine orders of magnitude of $\Delta \mu$.Reuse & Permissions

Figure 4

(Color online) Interpretation of the number of states function $g\left(N\right)$ given by Eq. (12). Decomposition of the complete Bell polynomials into partial Bell polynomials for different values of Taylor’s parameter $b$. We have already shown that different values of $b$ result from different free energies $F\left(\mu \right)$ Eq. (15) characterizing the considered systems, which, in turn, provide different sets of the coefficients $f_1,f_2,\dots ,f_N$. Graphs a and b show that the whole set of partitions for a given $b$ is dominated by partitions that consist of a well-defined number of subsets, $k$ (here, $N=50$). This number decreases with $b$ leading to an increase in the average size of these subsets, $N/⟨k⟩$ (cf. graph c). The decomposition analysis clearly shows that higher values of $b$ correspond to aggregation effects. The elements of the original set of size $N$ aggregate into subsets whose sizes increase with $b$.Reuse & Permissions

Figure 5

(Color online). Schematic description of the methodology used in real data analysis. Detailed description is given in the text.Reuse & Permissions

Figure 6

(Color online). Dependence of the quotients, $P_1\left(N\right)/P_2\left(N\right)$, on the parameter $d$. a, Blue Jay (Cyanocitta cristata) populations in North America, $⟨N_1⟩=1.22$ and $⟨N_2⟩=3.71$. b, Dynamics of the NYSE, $⟨N_1⟩=6.62\times {10}^5$ and $⟨N_2⟩=2.26\times {10}^6$. c, Traffic intensity in Minnesota, $⟨N_1⟩=2.16$ and $⟨N_2⟩=8.59$.Reuse & Permissions