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Entropy Maximization and Species Abundance

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

An understanding of the determinants of biodiversity is an important goal of academic ecology and the protection of biodiversity is an importantgoal in conservation. Biodiversity has two components: the number (or density) of species and the relative abundance of species. Both components vary overdifferent spatial and temporal scales. The allocation of limiting resources to different species in an ecological community is reflected in the abundanceof each species. The notion of a “community” in ecology (as in more general usages of this word) is poorly defined but will here mean thetotal number of organisms found within a fixed area of space (a site). Abundance is measured in different ways but the most exact measure is thetotal mass of living tissues (i. e. biomass) found in a given species at the site. Relative abundance is the proportional abundance of eachspecies relative to the total abundance of all species at the site. The species...

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Abbreviations

Species richness:

the number of species per unit area; a synonym is species density.

Species abundance:

the number of individuals or amount of living biomass per species per unit area.

Per capita instantaneous growth rate:

the net number of new individuals produced by an average individual in a population at time t.

Deterministic chaos:

a  pattern of change over time that is perfectly determined by initial conditions, but for which tiny changes in initial conditions result is divergences in the dynamics such that the pattern appears random.

Attractor:

a set of values (a point, a curve or higher – including a fractal – dimensional object) towards which a dynamic trajectory will move.

Basin of attraction:

the set of values from which any dynamic trajectory will move towards the same attractor.

Maximum entropy formalism:

a  formal method for producing probability estimates that agree with certain constraints specifying available information about a system but that are otherwise maximally uninformative.

Maximally uninformative (or ignorance) prior:

a probability distribution that encodes the basic structure of a logical problem as given by the definition of the variable, but that contains no other information.

Bibliography

  1. Darwin C (1859) On the origin of species. John Murray,London

    Google Scholar 

  2. Wallace AR (1858) On the tendency of species to form varieties; and on theperpetuation of varieties and species by natural means of selection. III. On the tendency of varieties to depart indefinitely from the original type. J Proc Linnean Soc Lond 3:53–62

    Google Scholar 

  3. Lotka AJ (1925) Elements of physical biology. Williams and Williams,Baltimore

    MATH  Google Scholar 

  4. Hutchinson GE (1978) An introduction to population ecology. Yale UniversityPress, New Haven

    MATH  Google Scholar 

  5. Volterra V (1926) Variazioni e fluttuazioni del numero d'individui in specieanimali conviventi. Mem R Acad dei Lincei (Ser. 6) 2:31–113

    Google Scholar 

  6. Roughgarden J (1979) Theory of population genetics and evolutionary ecology: Anintroduction. Macmillan Publishing, New York

    Google Scholar 

  7. Tilman D (1982) Resource competition and community structure. PrincetonUniversity Press, Princeton

    Google Scholar 

  8. Vandermeer JH (1969) The competitive structure of communities: An experimentalapproach with protozoa. Ecology 50:362–371

    Google Scholar 

  9. Gilpin ME, Carpenter MP, Pomerantz MJ (1986) The assembly of a laboratorycommunity: Multispecies competition in drosophila. In: Diamond JM, Case T (eds) Community ecology. Harper and Row, Cambridge

    Google Scholar 

  10. May RA (1974) Biological populations with nonoverlapping generations: Stablepoints, stable cycles, and chaos. Science 186:645–647

    ADS  Google Scholar 

  11. Hilborn RC (2004) Seagulls, butterflies, and grasshoppers: A briefhistory of the butterfly effect in nonlinear dynamics. Am J Phys 72:425–427

    ADS  Google Scholar 

  12. Costantino RF et al (2005) Nonlinear stochastic population dynamics: The flourbeetle tribolium as an effective tool of discovery. Advances in ecological research: Population dynamics and laboratory ecology, vol 37. Academic Press, London, pp 101–141

    Google Scholar 

  13. Boltzmann L (1898) Lectures on gas theory. Dover Publications, NewYork

    Google Scholar 

  14. Maxwell JC (1871) Theory of heat. Dover Publications, NewYork

    Google Scholar 

  15. Jaynes ET (1971) Violations of boltzmann's h theorem in real gases. Phys RevA4:747–751

    ADS  Google Scholar 

  16. Evans DJ, Searles DJ (2002) The fluctuation theorem. Adv Phys51:1529–1585

    ADS  Google Scholar 

  17. Wang GM, Sevick EM, Mittag E, Searles DJ, Evans DJ (2002) Experimentaldemonstration of violations of the second law of thermodynamics for small systems and short time scales. Phys Rev Lett 89:50601

    ADS  Google Scholar 

  18. Gibbs JW (1902) Elementary principles in statistical mechanics. YaleUniversity Press, New Haven

    MATH  Google Scholar 

  19. Jaynes ET (1957) Information theory and statistical mechanics i. Phys Rev106:620–630

    MathSciNet  ADS  MATH  Google Scholar 

  20. Jaynes ET (1957) Information theory and statistical mechanics ii. Phys Rev108:171–190

    MathSciNet  ADS  Google Scholar 

  21. Jaynes ET (2003) Probability theory. The logic of science. CambridgeUniversity Press, Cambridge

    MATH  Google Scholar 

  22. Kullback S, Leibler RA (1951) On information and sufficiency. Ann Math Statist22:79–86

    MathSciNet  MATH  Google Scholar 

  23. Agmon N, Alhassid Y, Levine RD (1979) An algorithm for finding thedistribution of maximal entropy. J Comput Phys 30:250–258

    ADS  MATH  Google Scholar 

  24. Della Pietra S, Della Pietra V, Lafferty J (1997) Inducing features of randomfields. IEEE Trans Pattern Anal Mach Intell 19:1–13

    Google Scholar 

  25. Schuster P, Sigmund K (1983) Replicator dynamics. J Theor Biol100:533–538

    MathSciNet  Google Scholar 

  26. Roff DA (1997) Evolutionary quantitative genetics. Chapman and Hall,NY

    Google Scholar 

  27. Hubbell SP (2001) The unified neutral theory of biodiversity andbiogeography. Princeton University Press, Princeton

    Google Scholar 

  28. Grime JP (2007) Comparative plant ecology. Castlepoint Press,Colvend

    Google Scholar 

  29. Shipley B, Lechoweicz MJ, Wright I, Reich PB (2006) Fundamental trade-offsgenerating the worldwide leaf economics spectrum. Ecology 87:535–541

    Google Scholar 

  30. Wright IJ et al (2004) The worldwide leaf economics spectrum. Nature428:821–827

    ADS  Google Scholar 

  31. Vile D, Shipley B, Garnier E (2006) A structural equation model tointegrate changes in functional strategies during old-field succession. Ecology 87:504–517

    Google Scholar 

  32. Shipley B, Vile D, Garnier E (2006) From plant traits to plant communities:A statistical mechanistic approach to biodiversity. Science 314:812–814

    MathSciNet  ADS  MATH  Google Scholar 

  33. Motomura I (1932) A statistical treatment of associations [injapanese]. Jpn J Zool 44:379–383

    Google Scholar 

  34. Corbet AS (1941) The distribution of butterflies in the malay peninsula(lepid.). Proc Roy Entomol Soc Ser A 16:101–116

    Google Scholar 

  35. Fisher RA, Corbet AS, Williams CB (1943) The relation between the number ofspecies and the number of individuals in a random sample from an animal population. J Animal Ecol 12:42–58

    Google Scholar 

  36. Pueyo S (2006) Diversity: Between neutrality and structure. Oikos112:392–405

    Google Scholar 

  37. Preston FW (1948) The commonness, and rarity, of species. Ecology29:254–283

    Google Scholar 

  38. Preston FW (1962) The canonical distribution of commonness and rarity. Ecology43:185–215

    Google Scholar 

  39. Bell G (2000) The distribution of abundance in neutral communities. Am Nat155:606–617

    Google Scholar 

  40. McGill BJ, Maurer BA, Weiser MD (2006) Empirical evaluation of neutraltheory. Ecology 87:1411–1423

    Google Scholar 

  41. Preston FW (1950) Gas laws and wealth laws. Sci Mon71:309–311

    ADS  Google Scholar 

  42. Limpert E, Stahel WA, Abbt M (2001) Log‐normal distributions across thesciences: Keys and clues. Bioscience 51:341–352

    Google Scholar 

  43. Nekola JC, Brown JH (2007) The wealth of species: Ecological communities,complex systems and the legacy of frank preston. Ecol Lett 10:188–196

    Google Scholar 

  44. Pueyo S, He F, Zillio T (2007) The maximum entropy formalism and theidiosyncratic theory of biodiversity. Ecol Lett 10:1017–1028

    Google Scholar 

  45. MacArthur RH (1972) Geographical ecology: Patterns in the distribution ofspecies. Harper and Row, NY

    Google Scholar 

  46. Volkov I, Banavar JR, He FL, Hubbell SP, Maritan A (2005) Density dependenceexplains tree species abundance and diversity in tropical forests. Nature 438:658–661

    ADS  Google Scholar 

  47. Hubbell SP, Foster RB (1990) Structure, dynamics and equilibrium status ofold‐growth forest on barro colorado island. In: Gentry A (ed) Four neotropical forests. Yale University Press, NewHaven

    Google Scholar 

  48. Plank M (1995) Hamiltonian structures for the n‑dimensional Lotka–Volterra equations. J Math Phys 36:3520–3534

    MathSciNet  ADS  MATH  Google Scholar 

  49. Plank M (1996) Bi‐hamiltonian systems and Lotka–Volterraequations: A three‐dimensional classification. Nonlinearity 9:887–896

    MathSciNet  ADS  MATH  Google Scholar 

  50. Plank M (1999) On the dynamics of Lotka–Volterra equations having aninvariant hyperplane. Siam J Appl Math 59:1540–1551

    MathSciNet  MATH  Google Scholar 

  51. Kerner EH (1964) Gibbs ensemble: Biological ensemble. Gordon and Brench, NewYork

    Google Scholar 

  52. Kerner EH (1979) The gibbs grand ensemble and the eco‐genetic gap. In:Levine RD, Tribus M (eds) The maximum entropy formalism. Massachusetts Institute of Technology, Massachusetts, pp 468–476

    Google Scholar 

  53. Wagensberg J, Valls J (1987) The [extended] maximum entropy formalism and thestatistical structure of ecosystems. Bull Math Biol 48:531–538

    MathSciNet  Google Scholar 

  54. Lurie D, Wasenburg J (1985) An extremal principle for biomass diversity inecology. In: Lamprecht I, Zotin AI (eds) Thermodynamics and regulation of biological processes. De Gruyter, Berlin,pp 577–271

    Google Scholar 

  55. Levich AP (1988) What are the possible theoretical principles in the ecologyof communities? In: Kull K, Tiivel T (eds) Lectures in theoretical biology. Valgus, Tallinn, pp 121–127

    Google Scholar 

  56. Alexeyev VL, Levich AP (1997) A search for maximum species abundances inecological communities under conditional diversity optimization. Bull Math Biol 59:649–677

    MATH  Google Scholar 

  57. Levich AP (2000) Variational modelling theorems and algocoenoses functioningprinciples. Ecol Model 131:207–227

    Google Scholar 

  58. Dewar R (2003) Information theory explanation of the fluctuation theorem,maximum entropy production and self‐organized criticality in non‐equilibrium stationary states. J Phys Math Gen36:631–641

    MathSciNet  ADS  MATH  Google Scholar 

  59. Dewar R (2004) Maximum entropy production and non‐equilibriumstatistical mechanics. In: Kliedon A, Lorenz RD (eds) Non‐equilibrium thermodynamics and the production of entropy: Life, earth andbeyond. Springer, Berlin, pp 41–56

    Google Scholar 

  60. Dewar R (2005) Maximum entropy production and the fluctuation theorem. J PhysMath Gen 38:L371–L381

    MathSciNet  ADS  MATH  Google Scholar 

  61. Peters RH (1983) The ecological implications of body size. CambridgeUniversity Press, Cambridge

    Google Scholar 

  62. Al-Mufti MM, Sydes CL, Furness SB, Grime JP, Band SR (1977)A quantitative analysis of shoot phenology and dominance in herbaceous vegetation. J Ecol 65:759–791

    Google Scholar 

  63. Currie DJ, Paquin V (1987) Large‐scale biogeographic patterns of speciesrichness of trees. Nature 329:326–327

    ADS  Google Scholar 

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Author information

Authors and Affiliations

  1. Département de Biologie, Université de Sherbrooke, Sherbrooke, Canada

    Bill Shipley

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  1. Bill Shipley
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Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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© 2009 Springer-Verlag

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Shipley, B. (2009). Entropy Maximization and Species Abundance. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_174

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