Analyzing and interpreting spatial and temporal variability of the United States county population distributions using Taylor's law

Autoři: Meng Xu aff001;  Joel E. Cohen aff002
Působiště autorů: Department of Mathematics, Pace University, New York, New York, United States of America aff001;  Laboratory of Populations, The Rockefeller University and Columbia University, New York, New York, United States of America aff002;  Earth Institute and Department of Statistics, Columbia University, New York, New York, United States of America aff003;  Department of Statistics, University of Chicago, Chicago, Illinois, United States of America aff004
Vyšlo v časopise: PLoS ONE 14(12)
Kategorie: Research Article
doi: 10.1371/journal.pone.0226096


We study the spatial and temporal variation of the human population in the United States (US) counties from 1790 to 2010, using an ecological scaling pattern called Taylor's law (TL). TL states that the variance of population abundance is a power function of the mean population abundance. Despite extensive studies of TL for non-human populations, testing and interpreting TL using data on human populations are rare. Here we examine three types of TL that quantify the spatial and temporal variation of US county population abundance. Our results show that TL and its quadratic extension describe the mean-variance relationship of county population distribution well. The slope and statistics of TL reveal economic and demographic trends of the county populations. We propose TL as a useful statistical tool for analyzing human population variability. We suggest new ways of using TL to select and make population projections.

Klíčová slova:

Census – Norwegian people – Population density – Population growth – Test statistics – United States – Urban areas


1. Gordon P, Richardson HW, Wong HL. The distribution of population and employment in a polycentric city: the case of Los Angeles. Environ Plann A. 1986;18: 161–173.

2. Plane DA, Rogerson PA. The geographical analysis of population: with applications to planning and business. London: Wiley; 1994.

3. Alberti M, Marzluff JM, Shulenberger E, Bradley G, Ryan C, Zumbrunnen C. Integrating humans into ecology: opportunities and challenges for studying urban ecosystems. BioScience. 2003;53: 1169–1179.

4. Grieco EM, Trevelyan E, Larsen L, Acosta YD, Gambino C, de la Cruz P, et al. The size, place of birth, and geographic distribution of the foreign-born population in the United States: 1960 to 2010. US Census Bureau, Population Division Working Paper. 2012;96. Available from:

5. Giles-Corti B, Vernez-Moudon A, Reis R, Turrell G, Dannenberg AL, Badland H, et al. City planning and population health: a global challenge. Lancet. 2016;388: 2912–2924. doi: 10.1016/S0140-6736(16)30066-6 27671668

6. Meyer WB, Turner BL. Human population growth and global land-use/cover change. Annu Rev Ecol Syst. 1992;23: 39–61.

7. Cardillo M, Purvis A, Sechrest W, Gittleman JL, Bielby J, Mace JM. Human population density and extinction risk in the world's carnivores. PLoS Biol. 2004;2: e197. doi: 10.1371/journal.pbio.0020197 15252445

8. Cincotta RP, Wisnewski J, Engelman R. Human population in the biodiversity hotspots. Nature. 2000;404: 990–992. doi: 10.1038/35010105 10801126

9. Bowman DMJS, Balch J, Artaxo P, Bond WJ, Cochrane MA, D'antonio CM, et al. The human dimension of fire regimes on Earth. J Biogeogr. 2011;38: 2223–2236. doi: 10.1111/j.1365-2699.2011.02595.x 22279247

10. Hales S, de Wet N, Maindonald J, Woodward A. Potential effect of population and climate changes on global distribution of dengue fever: an empirical model. Lancet. 2002;360: 830–834. doi: 10.1016/S0140-6736(02)09964-6 12243917

11. McGranahan G, Balk D, Anderson B. The rising tide: assessing the risks of climate change and human settlements in low elevation coastal zones. Environ Urban. 2007;19: 17–37.

12. Patz JA, Campbell-Lendrum D, Holloway T, Foley JA. Impact of regional climate change on human health. Nature. 2005;438: 310–317. doi: 10.1038/nature04188 16292302

13. Yntema DB. Measures of the inequality in the personal distribution of wealth or income. J Am Stat Assoc. 1933;28: 423–433.

14. Duncan OD. The measurement of population distribution. Pop Stud-J Demog. 1957;11: 27–45.

15. Eisler Z, Bartos I, Kertész J. Fluctuation scaling in complex systems: Taylor's law and beyond. Adv Phys. 2008;57: 89–142.

16. Taylor RAJ. Taylor's power law: order and pattern in nature. Cambridge: Elsevier Academic Press; 2019.

17. Bliss CI. Statistical problems in estimating populations of Japanese beetle larvae. J Econ Entomol. 1941;34: 221–232.

18. Fracker SB, Brischle HA. Measuring the local distribution of Ribes. Ecology. 1944;25: 283–303.

19. Hayman BI, Lowe AD. The transformation of counts of the cabbage aphid (Brevicoryne brassicae (L.)). New Zeal J Sci. 1961;4: 271–278.

20. Taylor LR. Aggregation, variance and the mean. Nature. 1961;189: 732–735.

21. Taylor LR, Woiwod IP, Perry JN. The density-dependence of spatial behaviour and the rarity of randomness. J Anim Ecol. 1978;47: 383–406.

22. Taylor LR, Woiwod IP. Temporal stability as a density-dependent species characteristic. J Anim Ecol. 1980;49: 209–224.

23. Taylor LR, Woiwod IP, Perry JN. Variance and the large scale spatial stability of aphids, moths and birds. J Anim Ecol. 1980;49: 831–854.

24. Taylor LR, Woiwod IP. Comparative synoptic dynamics. I. Relationships between inter-and intra-specific spatial and temporal variance/mean population parameters. J Anim Ecol. 1982;51: 879–906.

25. Anderson RM, Gordon DM, Crawley MJ, Hassell MP. Variability in the abundance of animal and plant species. Nature. 1982;296: 245–248.

26. Hanley QS, Khatun S, Yosef A, Dyer RM. Fluctuation scaling, Taylor’s law, and crime. PLoS One. 2014;9: e109004. doi: 10.1371/journal.pone.0109004 25271781

27. Cai Q, Xu HC, Zhou WX. Taylor’s law of temporal fluctuation scaling in stock illiquidity. Fluct Noise Lett. 2016;15: 1650029.

28. Cohen JE. Statistics of primes (and probably twin primes) satisfy Taylor's law from ecology. Am Stat. 2016;70: 399404.

29. Tippett MK, Cohen JE. Tornado outbreak variability follows Taylor’s power law of fluctuation scaling and increases dramatically with severity. Nat Commun. 2016;7: 10668. doi: 10.1038/ncomms10668 26923210

30. Taylor LR, Taylor RAJ. Aggregation, migration and population mechanics. Nature. 1977;265: 415–421. doi: 10.1038/265415a0 834291

31. Kilpatrick AM, Ives AR. Species interactions can explain Taylor's power law for ecological time series. Nature. 2003;422: 65–68. doi: 10.1038/nature01471 12621433

32. Cohen JE, Xu M. Random sampling of skewed distributions implies Taylor’s power law of fluctuation scaling. P Natl Acad Sci USA. 2015;112: 7749–7754.

33. Benassi F, Naccarato A. Modelling the spatial variation of human population density using Taylor’s power law, Italy, 1971–2011. Reg Stud. 2018;53: 206–216.

34. Cohen JE, Xu M, Brunborg H. Taylor's law applies to spatial variation in a human population. Genus. 2013;69: 25–60.

35. Naccarato A, Benassi F. On the relationship between mean and variance of world's human population density: A study using Taylor's power law. LSRS. 2018;11: 307–314.

36. Xu M, Brunborg H, Cohen JE. Evaluating multi-regional population projections with Taylor’s law of mean–variance scaling and its generalization. J Popul Res. 2017;34: 79–99.

37. Bliss CI, Owen ARG. Negative binomial distributions with a common k. Biometrika. 1958;45: 37–58.

38. Manson S, Schroeder J, Riper DV, Ruggles S. PUMS National Historical Geographic Information System: Version 13.0; 2018 [cited 2018 July 19]. Minneapolis: University of Minnesota. Available from:

39. Champion T. Urbanization, suburbanization, counterurbanization and reurbanization. In: Paddison R editor. Handbook of urban studies. SAGE Publications Ltd; 2000. pp. 143–161.

40. Taylor LR, Perry JN, Woiwod IP, Taylor RAJ. Specificity of the spatial power-law exponent in ecology and agriculture. Nature. 1988;332: 721–722.

41. R Core Team. R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. 2018. URL

42. Coon RC, Leistritz FL. The state of North Dakota: economic, demographic, public service, and fiscal conditions. Department of Agribusiness and Applied Economics, North Dakota State University, Fargo, ND. 1998.

43. Murphy W. fiftystater: map data to visualize the fifty U.S. states with Alaska and Hawaii insets. R package version 1.0.1. 2016. URL

44. Schwieder D. History of Iowa. Iowa Official Register. 2015;7:1.

45. McLeman R. Migration out of 1930s rural eastern Oklahoma: insights for climate change research. Great Plains Quart. 2006;26: 2740.

46. Bonifazi C, Heins F. Long-term trends of internal migration in Italy. Int J Popul Geogr. 2000;6: 111–131.

47. Renkow M, Hoover D. Commuting, migration, and rural-urban population dynamics. J Reg Sci. 2000;40: 261–287.

48. Ambinakudige S, Parisi D. A spatiotemporal analysis of inter-county migration patterns in the United States. Appl Spat Anal Policy. 2017;10: 121–137.

49. Berry BJ, Dahmann DC. Population redistribution in the United States in the 1970s. Popul Dev Rev. 1977;3: 443–471.

50. Long L, DeAre D. Repopulating the countryside: a 1980 census trend. Science. 1982;217: 1111–1116. doi: 10.1126/science.217.4565.1111 17740957

51. Long L, DeAre D. US population redistribution: a perspective on the nonmetropolitan turnaround. Popul Dev Rev. 1988;14: 433–450.

52. Fuguitt GV. The nonmetropolitan population turnaround. Annu Rev Sociol. 1985;11: 259–280. doi: 10.1146/ 12313950

53. Johnson KM, Beale CL. The recent revival of widespread population growth in nonmetropolitan areas of the United States. Rural Sociol. 1994;59: 655–667.

54. Lee S, Seo JG, Webster C. The decentralising metropolis: economic diversity and commuting in the US suburbs. Urban Stud. 2006;43: 2525–2549.

55. Lang R, Knox PK. The new metropolis: Rethinking megalopolis. Reg Stud. 2009;43: 789–802.

56. Vining DR Jr, Strauss A. A demonstration that the current deconcentration of population in the United States is a clean break with the past. Environ Plan A. 1977;9: 751–758. doi: 10.1068/a090751 12310795

57. Gordon P. Deconcentration without a 'clean break'. Environ Plan A. 1979;11, 281–289. doi: 10.1068/a110281 12262970

58. Johnson KM, Lichter DT. Rural depopulation: growth and decline processes over the past century. Rural Sociol. 2019;84: 3–27.

59. Jiang J, DeAngelis DL, Zhang B, Cohen JE. Population age and initial density in a patchy environment affect the occurrence of abrupt transitions in a birth-and-death model of Taylor's law. Ecol Modell. 2014;289: 59–65.

60. Tweedie MCK. An index which distinguishes between some important exponential families. Proceedings of the Indian Statistical Institute Golden Jubilee International Conference on Statistics: Applications and New Directions. 1984;579: 579–604.

61. Jørgensen B. The theory of dispersion models. London: CRC Press; 1997.

62. Kendal WS, Jørgensen B. Taylor's power law and fluctuation scaling explained by a central-limit-like convergence. Phys Rev E. 2011a;83: 066115.

63. Kendal WS, Jørgensen B. Tweedie convergence: A mathematical basis for Taylor's power law, 1/f noise, and multifractality. Phys Rev E. 2011b;84: 066120.

Článek vyšel v časopise


2019 Číslo 12