Theoretical Ecology

, Volume 2, Issue 2, pp 89–103 | Cite as

Invariance in species-abundance distributions

  • Arnošt L. ŠizlingEmail author
  • David Storch
  • Jiří Reif
  • Kevin J. Gaston
Original paper


Many attempts to explain the species-abundance distribution (SAD) assume that it has a universal functional form which applies to most assemblages. However, if such a form does exist, then it has to be invariant under changes in the area of the study plot (the addition of neighboring areas or subdivision of the original area) and changes in taxonomic composition (the addition of sister taxa or subdivision to subtaxa). We developed a theory for such an area-and-taxon invariant SAD and derived a formula for such a distribution. Both the log-normal and our area-and-taxon invariant distribution fitted data well. However, the log-normal distributions of two adjoined sub-assemblages cannot be composed into a log-normal distribution for the resulting assemblage, and the SAD composed from two log-normal distributions fits the SAD for the assemblage poorly in comparison to the area-and-taxon invariant distribution. Observed abundance patterns therefore reveal area-and-taxon invariant properties absent in log-normal distributions, suggesting that multiplicative models generating log-normal-like SADs (including the power-fraction model) cannot be universally valid, as they necessarily apply only to particular scales and taxa.


Scale invariance Area invariance Taxon invariance Population sizes Frequency distributions Relative abundances 



We thank J. Mourková and J. Cepák for their help with the fieldwork. We are grateful to M. Williamson, J. Nekola, B. J. McGill, F. He and anonymous referees for their comments. A.L.Š. was supported by the Marie Curie Fellowship no. 039576-RTBP-EIF. K.J.G. holds a Royal Society-Wolfson Research Merit Award. This work was supported by grants from the Czech Ministry of Education No. LC06073 and MSM0021620845, and Grant agency of the AS CR (IAA601970801).

Supplementary material

12080_2008_31_MOESM1_ESM.doc (2.2 mb)
ESM 1 (DOC 2.18 MB)


  1. Akaike H (1974) A new look at the statistical model identification. IEEE Trans Automat Contr 19:716–723. doi: 10.1109/TAC.1974.1100705 CrossRefGoogle Scholar
  2. Baker TC, Preston FW (1946) Fatigue of glass under statistic loads. J Appl Phys 17:170–178. doi: 10.1063/1.1707702 CrossRefGoogle Scholar
  3. Bibby CJ, Burgess ND, Hill DA (1992) Bird census techniques. Academic, LondonGoogle Scholar
  4. Borda-de-Água L, Hubbell SP, He F (2007) Scaling biodiversity under neutrality. In: Storch D, Marquet PA, Brown JH (eds) Scaling biodiversity. Cambridge University Press, Cambridge, pp 347–375Google Scholar
  5. Chave J (2004) Neutral theory and community ecology. Ecol Lett 7:241–253. doi: 10.1111/j.1461-0248.2003.00566.x CrossRefGoogle Scholar
  6. Connolly SR, Hughes TP, Belwood DR et al (2005) Community structure of corals and reef fishes at multiple scales. Science 309:1363–1365. doi: 10.1126/science.1113281 PubMedCrossRefGoogle Scholar
  7. Death RG (1996) The effect of habitat stability on benthic invertebrate communities: the utility of species abundance distributions. Hydrobiologia 317:97–107. doi: 10.1007/BF00018733 CrossRefGoogle Scholar
  8. Dewdney AK (2000) A dynamical model of communities and a new species-abundance distribution. Biol Bull 198:152–165. doi: 10.2307/1542811 PubMedCrossRefGoogle Scholar
  9. Engen S (2001) A dynamic and spatial model with migration generating the log-Gaussian field of population densities. Math Biosci 173:85–102. doi: 10.1016/S0025-5564(01)00077-3 PubMedCrossRefGoogle Scholar
  10. Engen S, Lande R (1996) Population dynamic models generating the lognormal species abundance distribution. Math Biosci 132:169–183. doi: 10.1016/0025-5564(95)00054-2 PubMedCrossRefGoogle Scholar
  11. Etienne RS, Olff H (2004) A novel genealogical approach to neutral biodiversity theory. Ecol Lett 7:170–175. doi: 10.1111/j.1461-0248.2004.00572.x CrossRefGoogle Scholar
  12. Fisher RA, Corbet AS, Williams CB (1943) The relation between the number of species and the number of individuals in a random sample of an animal population. J Anim Ecol 12:42–58. doi: 10.2307/1411 CrossRefGoogle Scholar
  13. Golicher DJ, O’Hara RB, Ruiz-Montoya L et al (2006) Lifting a veil on diversity: a bayesian approach to fitting relative abundance models. Ecol Appl 16:202–212. doi: 10.1890/04-1599 PubMedCrossRefGoogle Scholar
  14. Gray JS, Bjorgesaeter A, Ugland KI (2005) The impact of rare species on natural assemblages. J Anim Ecol 74:1131–1139. doi: 10.1111/j.1365-2656.2005.01011.x CrossRefGoogle Scholar
  15. Green JL, Plotkin JB (2007) A statistical theory for sampling species abundances. Ecol Lett 10:1037–1045. doi: 10.1111/j.1461-0248.2007.01101.x PubMedCrossRefGoogle Scholar
  16. Green J, Harte J, Ostling A (2003) Species richness, endemism and abundance patterns: test of two fractal models in a serpentine grassland. Ecol Lett 6:919–928. doi: 10.1046/j.1461-0248.2003.00519.x CrossRefGoogle Scholar
  17. Gregory RD (2000) Abundance patterns of European breeding birds. Ecography 23:201–208. doi: 10.1111/j.1600-0587.2000.tb00276.x CrossRefGoogle Scholar
  18. Harte J, Conlisk E, Ostling A et al (2005) A theory of spatial structure in ecological communities at multiple spatial scales. Ecol Monogr 75:179–197. doi: 10.1890/04-1388 CrossRefGoogle Scholar
  19. He F (2005) Deriving a neutral model of species abundance from fundamental mechanisms of population dynamics. Funct Ecol 19:187–193. doi: 10.1111/j.0269-8463.2005.00944.x CrossRefGoogle Scholar
  20. Hubbell SP (2001) The unified theory of biodiversity and biogeography. Princeton University Press, PrincetonGoogle Scholar
  21. Kammesheidt L (1998) The role of tree sprouts in the restorations of stand structure and species diversity in tropical moist forest after slash-and-burn agriculture in Eastern Paraguay. Plant Ecol 139:155–165. doi: 10.1023/A:1009763402998 CrossRefGoogle Scholar
  22. Kempton RA, Taylor LR (1974) Log-series and log-normal parameters as diversity discriminants for the lepidoptera. J Anim Ecol 43:381–399. doi: 10.2307/3371 CrossRefGoogle Scholar
  23. Loehle C, Hansen A (2005) Community structure and scaling relations for the avifauna of the US pacific and inland northwest. Ecol Complex 2:59–70. doi: 10.1016/j.ecocom.2004.09.001 CrossRefGoogle Scholar
  24. Magnussen S, Boyle TJB (1995) Estimating sample-size for inference about the Shannon-Weaver and the Simpson index of species-diversity. For Ecol Manage 78:71–84. doi: 10.1016/0378-1127(95)03596-1 CrossRefGoogle Scholar
  25. Magurran AE (1988) Ecological diversity and its measurement. Croom Helm Australia, New South WalesGoogle Scholar
  26. Magurran AE, Henderson PA (2003) Explaining the excess of rare species in natural species abundance distributions. Nature 422:714–716. doi: 10.1038/nature01547 PubMedCrossRefGoogle Scholar
  27. Mandelbrot BB (1963) New methods in statistical economics. J Polit Econ 71:421–440. doi: 10.1086/258792 CrossRefGoogle Scholar
  28. Marquet PA, Keymer JE, Cofré H (2003) Breaking the stick in space: of niche models, metacommunities and patterns in the relative abundance of species. In: Blackburn TM, Gaston KJ (eds) Macroecology: Concepts and consequences. British Ecological Society and Blackwell Science, Oxford, pp 64–81Google Scholar
  29. May R (1975) Patterns of species abudance and diversity. In: Cody ML, Diamond JM (eds) Ecology and evolution of communities. The Belknap Press of Harvard University Press, Cambridge, pp 81–120Google Scholar
  30. May RM, Crawley JM, Sugihara G (2007) Communities: Patterns. In: May RM, McLean A (eds) Theoretical ecology. Oxford University Press, Oxford, pp 111–131Google Scholar
  31. McGill BJ (2003) Strong and weak tests of macroecological theory. Oikos 102:679–685. doi: 10.1034/j.1600-0706.2003.12617.x CrossRefGoogle Scholar
  32. McGill BJ, Etienne RS, Gray JS et al (2007) Species abundance distributions: moving beyond single prediction theories to integration within an ecological framework. Ecol Lett 10:995–1015. doi: 10.1111/j.1461-0248.2007.01094.x PubMedCrossRefGoogle Scholar
  33. Motomura I (1932) A statistical treatment of associations. Zool Mag Tokyo 44:379–383 in JapaneseGoogle Scholar
  34. Nee S, Harvey PH, May RM (1991) Lifting the veil on abundance patterns. Proc R Soc Lond B Biol Sci 243:161–163. doi: 10.1098/rspb.1991.0026 CrossRefGoogle Scholar
  35. Preston FW (1948) The commonness, and rarity, of species. Ecology 29:254–283. doi: 10.2307/1930989 CrossRefGoogle Scholar
  36. Preston FW (1981) Pseudo-lognormal distributions. Ecology 62:355–364. doi: 10.2307/1936710 CrossRefGoogle Scholar
  37. Pueyo S (2006) Diversity: between neutrality and structure. Oikos 112:392–405. doi: 10.1111/j.0030-1299.2006.14188.x CrossRefGoogle Scholar
  38. Pueyo S, He F, Zillio T (2007) The maximum entropy formalism and the idiosyncratic theory of biodiversity. Ecol Lett 10:1017–1028. doi: 10.1111/j.1461-0248.2007.01096.x PubMedCrossRefGoogle Scholar
  39. Salvador-Van-Eysendore D, Bogaert J, Zak-Mnacek V et al (2003) Sapling diversity in canopy gaps in an Equadorian rain forest. For Sci 49:909–917Google Scholar
  40. Sichel HS (1997) Modelling species-abundance frequencies and species-individual functions with the generalized inverse Gaussian-Poisson distribution. South African Statist J 31:13–37Google Scholar
  41. Šizling AL, Storch D (2007) Geometry of species distributions: Random clustering and scale invariance. In: Storch D, Marquet PA, Brown JH (eds) Scaling biodiversity. Cambridge University Press, Cambridge, pp 77–100Google Scholar
  42. Storch D, Šizling AL (2008) The concept of taxon invariance in ecology: Do diversity patterns vary with changes in taxonomic resolution? Folia Geobot. doi: 10.1007/s12224-008-9015-8
  43. Storch D, Gaston KJ, Cepák J (2002) Pink landscapes: 1/f spectra of spatial environmental variability and bird community composition. Proc R Soc Lond B Biol Sci 269:1791–1796. doi: 10.1098/rspb.2002.2076 CrossRefGoogle Scholar
  44. Sugihara G (1980) Minimal community structure: an explanation of species abundance patterns. Am Nat 116:770–787. doi: 10.1086/283669 CrossRefGoogle Scholar
  45. Syrek D, Weiner WM, Wojtylak M et al (2006) Species abundance distribution of collembolan communities in forest soils polluted with heavy metals. Appl Soil Ecol 31:239–250. doi: 10.1016/j.apsoil.2005.05.002 CrossRefGoogle Scholar
  46. Tokeshi M (1999) Species coexistence: Ecological and Evolutionary perspectives. Blackwell, OxfordGoogle Scholar
  47. Ulrich W, Ollik M (2005) Limits to the estimation of species richness: the use of relative abundance distributions. Divers Distrib 11:265–273. doi: 10.1111/j.1366-9516.2005.00127.x CrossRefGoogle Scholar
  48. Visser S (1995) Ectomycorrhizal fungal succession in jack pine stands following wildfire. New Phytol 129:389–401. doi: 10.1111/j.1469-8137.1995.tb04309.x CrossRefGoogle Scholar
  49. Volkov I, Banavar JR, Hubbell SP et al (2003) Neutral theory and relative species abundance in ecology. Nature 424:1035–1037. doi: 10.1038/nature01883 PubMedCrossRefGoogle Scholar
  50. Walla TR, Engen S, DeVries PJ et al (2004) Modeling vertial beta-diversity in tropical butterfly communities. Oikos 107:610–618. doi: 10.1111/j.0030-1299.2004.13371.x CrossRefGoogle Scholar
  51. Wasserman LA (2004) All of statistics: A concise course in statistical inference. Springer, BerlinGoogle Scholar
  52. Whittaker RH (1970) Communities and Ecosystems. Macmillan, New YorkGoogle Scholar
  53. Wilson JB (1993) Would we recognise a broken-stick community if we found one? Oikos 67:181–183. doi: 10.2307/3545108 CrossRefGoogle Scholar
  54. Williamson M, Gaston KJ (2005) The lognormal distribution is not an appropriate null hypothesis for the species-abundance distribution. J Anim Ecol 74:409–422. doi: 10.1111/j.1365-2656.2005.00936.x CrossRefGoogle Scholar
  55. Yin ZY, Peng SL, Ren H et al (2005) LogCauchy, log-sech and lognormal distributions of species abundances in forest communities. Ecol Modell 184:329–340. doi: 10.1016/j.ecolmodel.2004.10.011 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  • Arnošt L. Šizling
    • 1
    • 2
    Email author
  • David Storch
    • 2
    • 3
  • Jiří Reif
    • 3
  • Kevin J. Gaston
    • 1
  1. 1.Biodiversity and Macroecology Group, Department of Animal and Plant SciencesUniversity of SheffieldSheffieldUK
  2. 2.Center for Theoretical StudyCharles University in Prague and the Academy of Sciences of the Czech RepublicPraha 1Czech Republic
  3. 3.Department of Ecology, Faculty of ScienceCharles UniversityPraha 2Czech Republic

Personalised recommendations