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.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Agronomof, M. 1914. “Sur une suite récurrente.” Mathesis 4: 125–126.
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.
Barlow, N. 1958. The Autobiography of Charles Darwin 1809–1882. With the Original Omissions Restored. Edited and with Appendix and Notes by His Grand-daughter Nora Barlow. London: Collins.
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.
Bashford, A. 2014. Global Population. History, Geopolitics, and Life on Earth. New York: Columbia University Press.
Beeby, A. and Brennan, A-M. 2008. First Ecology: Ecological Principles and Environmental Issues. 3rd edition. Oxford: Oxford University Press.
Benz, R. 2000. Ecology and Evolution: Islands of Change. Arlington, VA: NSTA Press.
Berg, L. R., Hager, M. C. and Hassenzahl, D. M. 2010. Visualizing Environmental Science. 3rd edition. Hoboken, NJ: Wiley.
Blainey, G. 1988. The Great Seesaw. A New View of the Western World, 1750–2000. Houndmills, Hampshire: MacMillan.
Brousseau, A. 1969. “Fibonacci Statistics in Conifers.” Fibonacci Quarterly 7: 525–532.
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.
Burton, R. F. 1998. Biology by Numbers: An Encouragement to Quantitative Thinking. Cambridge: Cambridge University Press.
Chapman, J. L. and Reiss, M. J. 1999. Ecology: Principles and Applications, 2nd ed. Cambridge: Cambridge University Press.
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.
Cronin, H. 1991. The Ant and the Peacock. Altruism and Sexual Selection from Darwin to Today. Cambridge: Cambridge University Press.
Crow, W. B. 2013. A Synopsis of Biology. 2nd edition. Bristol: John Wright and Sons.
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.
Dash, M. C. and Dash, S. D. 2009. Fundamentals of Ecology. 3rd edition. New Delhi: Tata-McGraw Hill.
Delson, E., Tattersall, I., Van Couvering, J. and Brooks, A. S. (eds.). 2000. Encyclopedia of Human Evolution and Prehistory. 2nd edition. New York: Routledge.
Dennett, D. C. 1995. Darwin’s Dangerous Idea: Evolution and the Meanings of Life. New York: Simon & Schuster.
Douady, S. and Couder, Y. 1996. “Phyllotaxis as a Dynamical Self Organizing Process.” Journal of Theoretical Biology 178: 255–274.
Duzdevich, D. 2014. Darwin’s On the Origin of Species. A Modern Rendition. Bloomington & Indianapolis: Indiana University Press.
Dyer, A. 2014. Chasing the Red Queen: The Evolutionary Race Between Agricultural Pests and Poisons. Washington: Island Press.
Egerton, F. N. 2012. Roots of Ecology. Antiquity to Haeckel. Berkeley: University of California Press.
Eldredge, N. 1985. Frames: The Evolution of Punctuated Equilibria. Princeton: Princeton University Press.
Eldredge, N. 1997. Dominion. Berkeley: University of California Press.
Eldredge, N. 1998. Life in the Balance. Humanity and the Biodiversity Crisis. Princeton: Princeton University Press.
Eysenck, M. W. 2004. Psychology. An International Perspective. Hove, East Sussex: Psychology Press.
Feinberg, M. 1963. “Fibonacci-Tribonacci.” Fibonacci Quarterly 1: 71–74.
Francis, K. A. 2007. Charles Darwin and the Origin of Species. Westport, CT: Greenwood Press.
Garvey, B. 2014. Philosophy of Biology. New York: Routledge.
Gore, A. and Paranjpe, S. A. 2001. A Course in Mathematical and Statistical Ecology. Dordrecht: Kluwer.
Hale, P. J. 2014. Political Descent. Malthus, Mutualism and the Politics of Evolution in Victorian England. Chicago: University of Chicago Press.
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.
Hardin, G. 1993. Living within Limits: Ecology, Economics, and Population Taboos. Oxford: Oxford University Press.
Johnson, P. 2012. Darwin. Portrait of a Genius. New York: Viking.
Khanna, D. R. 2004. Animal Distribution. New Delhi: Discovery.
Kingsland, S. E. 1995. Modeling Nature. Episodes in the History of Population Ecology, 2nd edition. Chicago: University of Chicago Press.
Koch, R. 2014. The 80/20 Principle and Other Powerful Laws of Nature. London: Brealey.
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.
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.
Lénárd, G. 2007. Biológia 12. Budapest: Nemzedékek Tudástára.
Leslie, P. H. 1945. “The Use of Matrices in Certain Population Mathematics.” Biometrika 33: 183–212.
Lyons, S. 2011. Evolution. The Basics. London: Routledge.
McKee, J. K. 2000. The Riddled Chain: Chance, Coincidence, and Chaos in Human Evolution. New Brunswick: Rutgers University Press.
McPherson, G. R. 2005. Killing the Natives: Has the American Dream Become a Nightmare?. Pittsburgh: Whitmore.
Megarry, T. 1995. Society in Prehistory: The Origins of Human Culture. New York: New York University Press.
Moore, J. and Moore, R. 2006. Evolution 101. Westport CT: Greenwood Press.
Moss, C. J. 2001. “The Demography of an African Elephant (Loxodonta africana) Population in Amboseli, Kenya.” Journal of Zoology 255: 145–156.
Nadeau, R. L. 2013. Rebirth of the Sacred. Oxford: Oxford University Press.
Niklas, K. J. 1997. The Evolutionary Biology of Plants. Chicago: University of Chicago Press.
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.
Overy, C. 1997. Charles Darwin: His Life, Journeys and Discoveries. A Teacher's Guide. London: English Heritage Education Service.
Pandey, G. 2010. Bio-cultural Evolution. New Delhi: Concept Publishing.
Pásztor, L., Botta-Dukát, Z., Magyar, G., Czárán, T. and Meszéna, G. 2016. Theory-based Ecology: A Darwinian Approach. Oxford: Oxford University Press.
Peckham, M. (ed.). 1959. The Origin of Species: A Variorum Text by C. Darwin. Philadelphia: University of Pennsylvania Press.
Quammen, D. 2006. The Reluctant Mr. Darwin: An Intimate Portrait of Charles Darwin and the Making of his Theory of Evolution. New York: Norton.
Raven, O. H., Berg, L. R. and Hassenzahl, D. M. 2012. Environment. New York: Wiley.
Richards, J. R. 2000. Human Nature after Darwin. A Philosophical Introduction. London: Routledge.
Russell, P. J., Hertz, P. E. and McMillan, B. 2011. Biology. The Dynamic Science. Vol. 2. Belmont, CA: Thomson.
Sapp, J. 2003. Genesis. The Evolution of Biology. Oxford: Oxford University Press.
Spellman, F. R. 2007. The Science of Water: Concepts and Applications. 2nd edition. Boca Raton: CRC Press.
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.
Stebbing, T. 2011. A Cybernetic View of Biological Growth: The Maia Hypothesis. Cambridge: Cambridge University Press.
Sutherland, J. 2014. Jumbo: The Unauthorised Biography of a Victorian Sensation. London: Aurum Press.
Tattersall, I. 2000. “Time and evolution.” P. J. N. Baert (ed.), Time in Contemporary Intellectual Thought. Amsterdam: Elsevier, pp. 151–170.
Tobin, A. J. and Dusheck, J. 2005. Asking About Life. Belmont, CA: Thomson.
van Wyhe, J. (ed.). 2002. The Complete Work of Charles Darwin Online. (http:// darwin-online.org.uk/). Accessed 15 December 2016.
Walsh, D. M. 2015. Organisms, Agency, and Evolution. Cambridge: Cambridge University Press.
Weisstein, E. W. undated. Tribonacci Constant. From MathWorld – A Wolfram Web Resource. http://mathworld.wolfram.com/TribonacciConstant.html. Accessed 15 December 2016.
Electronic Supplementary Material
The Below is the Electronic Supplementary Material.
About this article
Cite this article
Podani, J., Kun, Á. & Szilágyi, A. How Fast Does Darwin’s Elephant Population Grow?. J Hist Biol 51, 259–281 (2018). https://doi.org/10.1007/s10739-017-9488-5
- Darwin’s correspondence
- Leslie matrix
- Population growth
- The Origin of Species
- Tribonacci series