How Fast Does Darwin’s Elephant Population Grow?

Article

Abstract

In “The Origin of Species,” Darwin describes a hypothetical example illustrating that large, slowly reproducing mammals such as the elephant can reach very large numbers if population growth is not affected by regulating factors. The elephant example has since been cited in various forms in a wide variety of books, ranging from educational material to encyclopedias. However, Darwin’s text was changed over the six editions of the book, although some errors in the mathematics persisted throughout. In addition, full details of the problem remained hidden in his correspondence with readers of the Origin. As a result, Darwin’s example is very often misinterpreted, misunderstood or presented as if it were a fact. We show that the population growth of Darwin’s elephant population can be modeled by the Leslie matrix method, which we generalize here to males as well. Darwin’s most often cited figure, about 19 million elephants after 750 years is not a typical outcome, actually a very unlikely result under more realistic, although still hypothetical situations. We provide a recursion formula suggesting that Darwin’s original model corresponds to a tribonacci series, a proof showing that sex ratio is constant over all age classes, and a derivation of a generating function of the sequence.

Keywords

Darwin’s correspondence Leslie matrix Population growth The Origin of Species Tribonacci series 

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References

  1. Agronomof, M. 1914. “Sur une suite récurrente.” Mathesis 4: 125–126.Google Scholar
  2. Al-Suwaiyel, Μ. I., Alani, D. and Al-Swailem, A. 2006. “An Investigation of Fibonacci-Like Sequences in Biology and Mathematics.” International Journal of Nonlinear Sciences and Numerical Simulation 7: 133–136.CrossRefGoogle Scholar
  3. Barlow, N. 1958. The Autobiography of Charles Darwin 18091882. With the Original Omissions Restored. Edited and with Appendix and Notes by His Grand-daughter Nora Barlow. London: Collins.Google Scholar
  4. Barrett, P. H. 1977. The Collected Papers of Charles Darwin. 2 vols. Chicago and London: University of Chicago Press. With a foreword by Theodosius Dobzhansky.Google Scholar
  5. Bashford, A. 2014. Global Population. History, Geopolitics, and Life on Earth. New York: Columbia University Press.CrossRefGoogle Scholar
  6. Beeby, A. and Brennan, A-M. 2008. First Ecology: Ecological Principles and Environmental Issues. 3rd edition. Oxford: Oxford University Press.Google Scholar
  7. Benz, R. 2000. Ecology and Evolution: Islands of Change. Arlington, VA: NSTA Press.Google Scholar
  8. Berg, L. R., Hager, M. C. and Hassenzahl, D. M. 2010. Visualizing Environmental Science. 3rd edition. Hoboken, NJ: Wiley.Google Scholar
  9. Blainey, G. 1988. The Great Seesaw. A New View of the Western World, 17502000. Houndmills, Hampshire: MacMillan.Google Scholar
  10. Brousseau, A. 1969. “Fibonacci Statistics in Conifers.” Fibonacci Quarterly 7: 525–532.Google Scholar
  11. Bulhof, I. N. 1992. “The” Language of Science: A Study of the Relationship Between Literature and Science in the Perspective of a Hermeneutical Ontology. With a Case Study of Darwin’s The Origin of Species. Leiden: E.J. Brill.Google Scholar
  12. Burton, R. F. 1998. Biology by Numbers: An Encouragement to Quantitative Thinking. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  13. Chapman, J. L. and Reiss, M. J. 1999. Ecology: Principles and Applications, 2nd ed. Cambridge: Cambridge University Press.Google Scholar
  14. Costa, J. T. 2011. The Annotated Origin. A Facsimile of the First Edition of On the Origin of Species. Boston: Belknap Press of Harward University.Google Scholar
  15. Cronin, H. 1991. The Ant and the Peacock. Altruism and Sexual Selection from Darwin to Today. Cambridge: Cambridge University Press.Google Scholar
  16. Crow, W. B. 2013. A Synopsis of Biology. 2nd edition. Bristol: John Wright and Sons.Google Scholar
  17. Darwin, C. R. 1859–1872. The Origin of Species by Means of Natural Selection, or the Preservation of Favoured Races in the Struggle for Life. 1st–6th editions. London: John Murray.Google Scholar
  18. Dash, M. C. and Dash, S. D. 2009. Fundamentals of Ecology. 3rd edition. New Delhi: Tata-McGraw Hill.Google Scholar
  19. Delson, E., Tattersall, I., Van Couvering, J. and Brooks, A. S. (eds.). 2000. Encyclopedia of Human Evolution and Prehistory. 2nd edition. New York: Routledge.Google Scholar
  20. Dennett, D. C. 1995. Darwin’s Dangerous Idea: Evolution and the Meanings of Life. New York: Simon & Schuster.Google Scholar
  21. Douady, S. and Couder, Y. 1996. “Phyllotaxis as a Dynamical Self Organizing Process.” Journal of Theoretical Biology 178: 255–274.CrossRefGoogle Scholar
  22. Duzdevich, D. 2014. Darwin’s On the Origin of Species. A Modern Rendition. Bloomington & Indianapolis: Indiana University Press.Google Scholar
  23. Dyer, A. 2014. Chasing the Red Queen: The Evolutionary Race Between Agricultural Pests and Poisons. Washington: Island Press.CrossRefGoogle Scholar
  24. Egerton, F. N. 2012. Roots of Ecology. Antiquity to Haeckel. Berkeley: University of California Press.CrossRefGoogle Scholar
  25. Eldredge, N. 1985. Frames: The Evolution of Punctuated Equilibria. Princeton: Princeton University Press.Google Scholar
  26. Eldredge, N. 1997. Dominion. Berkeley: University of California Press.Google Scholar
  27. Eldredge, N. 1998. Life in the Balance. Humanity and the Biodiversity Crisis. Princeton: Princeton University Press.Google Scholar
  28. Eysenck, M. W. 2004. Psychology. An International Perspective. Hove, East Sussex: Psychology Press.Google Scholar
  29. Feinberg, M. 1963. “Fibonacci-Tribonacci.” Fibonacci Quarterly 1: 71–74.Google Scholar
  30. Francis, K. A. 2007. Charles Darwin and the Origin of Species. Westport, CT: Greenwood Press.Google Scholar
  31. Garvey, B. 2014. Philosophy of Biology. New York: Routledge.Google Scholar
  32. Gore, A. and Paranjpe, S. A. 2001. A Course in Mathematical and Statistical Ecology. Dordrecht: Kluwer.CrossRefGoogle Scholar
  33. Hale, P. J. 2014. Political Descent. Malthus, Mutualism and the Politics of Evolution in Victorian England. Chicago: University of Chicago Press.CrossRefGoogle Scholar
  34. Hale, P. J. 2016. “Finding a Place for the Anti-Malthusian Tradition in the Victorian Evolution Debates.” R. Mayhew (ed.), New Perspectives on Malthus. Cambridge: Cambridge University Press, pp. 182–207.Google Scholar
  35. Hardin, G. 1993. Living within Limits: Ecology, Economics, and Population Taboos. Oxford: Oxford University Press.Google Scholar
  36. Johnson, P. 2012. Darwin. Portrait of a Genius. New York: Viking.Google Scholar
  37. Khanna, D. R. 2004. Animal Distribution. New Delhi: Discovery.Google Scholar
  38. Kingsland, S. E. 1995. Modeling Nature. Episodes in the History of Population Ecology, 2nd edition. Chicago: University of Chicago Press.Google Scholar
  39. Koch, R. 2014. The 80/20 Principle and Other Powerful Laws of Nature. London: Brealey.Google Scholar
  40. Lahdenperä, M., Mar, K. U. and Lummaa, V. 2014. “Reproductive Cessation and Post-reproductive Lifespan in Asian Elephants and Pre-industrial Humans.” Frontiers in Zoology 11: 54–54.Google Scholar
  41. Lee, P. C., Fishlock, V., Webber, C. E. and Moss, C. J. 2016. “The Reproductive Advantages of a Long Life: Longevity and Senescence in Wild Female African Elephants.” Behavioral Ecology and Sociobiology 70: 337–345.CrossRefGoogle Scholar
  42. Lénárd, G. 2007. Biológia 12. Budapest: Nemzedékek Tudástára.Google Scholar
  43. Leslie, P. H. 1945. “The Use of Matrices in Certain Population Mathematics.” Biometrika 33: 183–212.CrossRefGoogle Scholar
  44. Lyons, S. 2011. Evolution. The Basics. London: Routledge.Google Scholar
  45. McKee, J. K. 2000. The Riddled Chain: Chance, Coincidence, and Chaos in Human Evolution. New Brunswick: Rutgers University Press.Google Scholar
  46. McPherson, G. R. 2005. Killing the Natives: Has the American Dream Become a Nightmare?. Pittsburgh: Whitmore.Google Scholar
  47. Megarry, T. 1995. Society in Prehistory: The Origins of Human Culture. New York: New York University Press.CrossRefGoogle Scholar
  48. Moore, J. and Moore, R. 2006. Evolution 101. Westport CT: Greenwood Press.Google Scholar
  49. Moss, C. J. 2001. “The Demography of an African Elephant (Loxodonta africana) Population in Amboseli, Kenya.” Journal of Zoology 255: 145–156.CrossRefGoogle Scholar
  50. Nadeau, R. L. 2013. Rebirth of the Sacred. Oxford: Oxford University Press.Google Scholar
  51. Niklas, K. J. 1997. The Evolutionary Biology of Plants. Chicago: University of Chicago Press.Google Scholar
  52. Noe, T., Piezas, T. III and Weisstein, E. W. Undated. Tribonacci Number. From MathWorld – A Wolfram Web Resource. http://mathworld.wolfram.com/ TribonacciNumber.html. Accessed 15 December 2016.
  53. Overy, C. 1997. Charles Darwin: His Life, Journeys and Discoveries. A Teacher's Guide. London: English Heritage Education Service.Google Scholar
  54. Pandey, G. 2010. Bio-cultural Evolution. New Delhi: Concept Publishing.Google Scholar
  55. Pásztor, L., Botta-Dukát, Z., Magyar, G., Czárán, T. and Meszéna, G. 2016. Theory-based Ecology: A Darwinian Approach. Oxford: Oxford University Press.CrossRefGoogle Scholar
  56. Peckham, M. (ed.). 1959. The Origin of Species: A Variorum Text by C. Darwin. Philadelphia: University of Pennsylvania Press.Google Scholar
  57. Quammen, D. 2006. The Reluctant Mr. Darwin: An Intimate Portrait of Charles Darwin and the Making of his Theory of Evolution. New York: Norton.Google Scholar
  58. Raven, O. H., Berg, L. R. and Hassenzahl, D. M. 2012. Environment. New York: Wiley.Google Scholar
  59. Richards, J. R. 2000. Human Nature after Darwin. A Philosophical Introduction. London: Routledge.Google Scholar
  60. Russell, P. J., Hertz, P. E. and McMillan, B. 2011. Biology. The Dynamic Science. Vol. 2. Belmont, CA: Thomson.Google Scholar
  61. Sapp, J. 2003. Genesis. The Evolution of Biology. Oxford: Oxford University Press.CrossRefGoogle Scholar
  62. Spellman, F. R. 2007. The Science of Water: Concepts and Applications. 2nd edition. Boca Raton: CRC Press.CrossRefGoogle Scholar
  63. Stauffer, R. C. (ed.). 1975. Charles Darwin’s Natural Selection; Being the Second Part of his Big Species Book Written from 1856 to 1858. Cambridge: Cambridge University Press.Google Scholar
  64. Stebbing, T. 2011. A Cybernetic View of Biological Growth: The Maia Hypothesis. Cambridge: Cambridge University Press.Google Scholar
  65. Sutherland, J. 2014. Jumbo: The Unauthorised Biography of a Victorian Sensation. London: Aurum Press.Google Scholar
  66. Tattersall, I. 2000. “Time and evolution.” P. J. N. Baert (ed.), Time in Contemporary Intellectual Thought. Amsterdam: Elsevier, pp. 151–170.CrossRefGoogle Scholar
  67. Tobin, A. J. and Dusheck, J. 2005. Asking About Life. Belmont, CA: Thomson.Google Scholar
  68. van Wyhe, J. (ed.). 2002. The Complete Work of Charles Darwin Online. (http:// darwin-online.org.uk/). Accessed 15 December 2016.
  69. Walsh, D. M. 2015. Organisms, Agency, and Evolution. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  70. Weisstein, E. W. undated. Tribonacci Constant. From MathWorld – A Wolfram Web Resource. http://mathworld.wolfram.com/TribonacciConstant.html. Accessed 15 December 2016.

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of Plant Systematics, Ecology and Theoretical Biology, Institute of BiologyEötvös UniversityBudapestHungary
  2. 2.MTA-ELTE-MTM Ecology Research GroupEötvös UniversityBudapestHungary
  3. 3.MTA-ELTE Theoretical Biology and Evolutionary Ecology Research GroupEötvös UniversityBudapestHungary

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