Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Simple mathematical models with very complicated dynamics

Nature volume 261, pages 459–467 (1976)Cite this article

A Corrigendum to this article was published on 01 July 1976

Abstract

First-order difference equations arise in many contexts in the biological, economic and social sciences. Such equations, even though simple and deterministic, can exhibit a surprising array of dynamical behaviour, from stable points, to a bifurcating hierarchy of stable cycles, to apparently random fluctuations. There are consequently many fascinating problems, some concerned with delicate mathematical aspects of the fine structure of the trajectories, and some concerned with the practical implications and applications. This is an interpretive review of them.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Get just this article for as long as you need it

$39.95

Learn more

Prices may be subject to local taxes which are calculated during checkout

References

  1. May, R. M., and Oster, G. F., Am. Nat., 110 (in the press).

  2. Varley, G. C., Gradwell, G. R., and Hassell, M. P., Insect Population Ecology (Blackwell, Oxford, 1973).

  3. May, R. M., (ed.), Theoretical Ecology: Principles and Applications (Blackwell, Oxford, 1976).

  4. Guckenheimer, J., Oster, G. F., and Ipaktchi, A., Theor. Pop. Biol. (in the press).

  5. Oster, G. F., Ipaktchi, A., and Rocklin, I., Theor. Pop. Biol. (in the press).

  6. Asmussen, M. A., and Feldman, M. W., J. theor. Biol. (in the press).

  7. Hoppensteadt, F. C., Mathematical Theories of Populations: Demographics, Genetics and Epidemics (SIAM, Philadelphia, 1975).

    Book  Google Scholar 

  8. Samuelson, P. A., Foundations of Economic Analysis (Harvard University Press, Cambridge, Massachusetts, 1947).

    MATH  Google Scholar 

  9. Goodwin, R. E., Econometrica, 19, 1–17 (1951).

    Article  Google Scholar 

  10. Baumol, W. J., Economic Dynamics, 3rd ed. (Macmillan, New York, 1970).

    Google Scholar 

  11. See, for example, Kemeny, J., and Snell, J. L., Mathematical Models in the Social Sciences (MIT Press, Cambridge, Massachusetts, 1972).

    MATH  Google Scholar 

  12. Chaundy, T. W., and Phillips, E., Q. Jl Math. Oxford, 7, 74–80 (1936).

    Article  Google Scholar 

  13. Myrberg, P. J., Ann. Akad. Sc. Fennicae, A, I, No. 336/3 (1963).

  14. Myrberg, P. J., Ann. Akad. Sc. Fennicae, A, I, No. 259 (1958).

  15. Lorenz, E. N., J. Atmos. Sci., 20, 130–141 (1963); Tellus, 16, 1–11 (1964).

    Article  ADS  Google Scholar 

  16. Metropolis, N., Stein, M. L., and Stein, P. R., J. Combinatorial Theory, 15(A), 25–44 (1973).

    Article  Google Scholar 

  17. Maynard Smith, J., Mathematical Ideas in Biology (Cambridge University Press, Cambridge, 1968).

    Book  Google Scholar 

  18. Krebs, C. J., Ecology (Harper and Row, New York, 1972).

    Google Scholar 

  19. May, R. M., Am. Nat., 107, 46–57 (1972).

    Article  Google Scholar 

  20. Li, T.-Y., and Yorke, J. A., Am. Math. Monthly, 82, 985–992 (1975).

    Article  Google Scholar 

  21. Hoppensteadt, F. C., and Hyman, J. M., (Courant Institute, New York University: preprint, 1975).

  22. Smale, S., and Williams, R.,(Department of Mathematics, Berkeley: preprint, 1976).

  23. May, R. M., Science, 186, 645–647 (1974).

    Article  ADS  CAS  Google Scholar 

  24. Moran, P. A. P., Biometrics, 6, 250–258 (1950).

    Article  CAS  Google Scholar 

  25. Ricker, W. E., J. Fish. Res. Bd. Can., 11, 559–623 (1954).

    Article  Google Scholar 

  26. Cook, L. M., Nature, 207, 316 (1965).

    Article  ADS  Google Scholar 

  27. Macfadyen, A., Animal Ecology: Aims and Methods (Pitman, London, 1963).

    Google Scholar 

  28. May, R. M., J. theor. Biol., 51, 511–524 (1975).

    Article  CAS  Google Scholar 

  29. Guckenheimer, J., Proc. AMS Symposia in Pure Math., XIV, 95–124 (1970).

    Google Scholar 

  30. Gilbert, E. N., and Riordan, J., Illinois J. Math., 5, 657–667 (1961).

    MathSciNet  Google Scholar 

  31. Preston, C. J., (King's College, Cambridge: preprint, 1976).

  32. Gumowski, I., and Mira, C., C. r. hebd. Séanc. Acad. Sci., Paris, 281a, 45–48 (1975); 282a, 219–222 (1976).

    Google Scholar 

  33. Layzer, D., Sci. Am., 233(6), 56–69 (1975).

    Article  Google Scholar 

  34. Ulam, S. M., Proc. Int. Congr. Math. 1950, Cambridge, Mass. ; Vol. II, pp. 264–273 (AMS, Providence R. I., 1950).

  35. Ulam, S. M., and von Neumann, J., Bull. Am. math. Soc. (abstr.), 53, 1120 (1947).

    Google Scholar 

  36. Kac, M., Ann. Math., 47, 33–49 (1946).

    Article  MathSciNet  Google Scholar 

  37. May, R. M., Science, 181, 1074 (1973).

    Google Scholar 

  38. Hassell, M. P., J. Anim. Ecol., 44, 283–296 (1974).

    Article  Google Scholar 

  39. Hassell, M. P., Lawton, J. H., and May, R. M., J. Anim. Ecol. (in the press).

  40. Ruelle, D., and Takens, F., Comm. math. Phys., 20, 167–192 (1971).

    Article  ADS  MathSciNet  Google Scholar 

  41. Landau, L. D., and Lifshitz, E. M., Fluid Mechanics (Pergamon, London, 1959).

    Google Scholar 

  42. Martin, P. C., Proc. Int. Conf. on Statistical Physics, 1975, Budapest (Hungarian Acad. Sci., Budapest, in the press).

  43. Southwood, T. R. E., in Insects, Science and Society (edit. by Pimentel, D.), 151–199 (Academic, New York, 1975).

    Book  Google Scholar 

  44. Metropolis, N., Stein, M. L., and Stein, P. R., Numer. Math., 10, 1–19 (1967).

    Article  MathSciNet  Google Scholar 

  45. Gumowski, I., and Mira, C., Automatica, 5, 303–317 (1969).

    Article  Google Scholar 

  46. Stein, P. R., and Ulam, S. M., Rosprawy Mat., 39, 1–66 (1964).

    Google Scholar 

  47. Beddington, J. R., Free, C. A., and Lawton, J. H., Nature, 255, 58–60 (1975).

    Article  ADS  Google Scholar 

  48. Hirsch, M. W., and Smale, S., Differential Equations, Dynamical Systems and Linear Algebra (Academic, New York, 1974).

    MATH  Google Scholar 

  49. Kolata, G. B., Science, 189, 984–985 (1975).

    Article  ADS  CAS  Google Scholar 

  50. Smale, S., (Department of Mathematics, Berkeley: preprint, 1976).

  51. May, R. M., and Leonard, W. J., SIAM J. Appl. Math., 29, 243–253 (1975).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. King's College Research Centre, Cambridge, CB2 1ST

    Robert M. May

  2. on leave from Biology Department, Princeton University, Princeton, 08540

    Robert M. May

Authors
  1. Robert M. May
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

May, R. Simple mathematical models with very complicated dynamics. Nature 261, 459–467 (1976). https://doi.org/10.1038/261459a0

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1038/261459a0

This article is cited by

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing