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Simple mathematical models with very complicated dynamics

Nature 261, 459–467 (10 June 1976) | Download Citation

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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.

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References

  1. 1.

    , and , Am. Nat., 110 (in the press).

  2. 2.

    , , and , Insect Population Ecology (Blackwell, Oxford, 1973).

  3. 3.

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

  4. 4.

    , , and , Theor. Pop. Biol. (in the press).

  5. 5.

    , , and , Theor. Pop. Biol. (in the press).

  6. 6.

    , and , J. theor. Biol. (in the press).

  7. 7.

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

  8. 8.

    , Foundations of Economic Analysis (Harvard University Press, Cambridge, Massachusetts, 1947).

  9. 9.

    , Econometrica, 19, 1–17 (1951).

  10. 10.

    , Economic Dynamics, 3rd ed. (Macmillan, New York, 1970).

  11. 11.

    See, for example, , and , Mathematical Models in the Social Sciences (MIT Press, Cambridge, Massachusetts, 1972).

  12. 12.

    , and , Q. Jl Math. Oxford, 7, 74–80 (1936).

  13. 13.

    , Ann. Akad. Sc. Fennicae, A, I, No. 336/3 (1963).

  14. 14.

    , Ann. Akad. Sc. Fennicae, A, I, No. 259 (1958).

  15. 15.

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

  16. 16.

    , , and , J. Combinatorial Theory, 15(A), 25–44 (1973).

  17. 17.

    , Mathematical Ideas in Biology (Cambridge University Press, Cambridge, 1968).

  18. 18.

    , Ecology (Harper and Row, New York, 1972).

  19. 19.

    , Am. Nat., 107, 46–57 (1972).

  20. 20.

    , and , Am. Math. Monthly, 82, 985–992 (1975).

  21. 21.

    , and , (Courant Institute, New York University: preprint, 1975).

  22. 22.

    , and ,(Department of Mathematics, Berkeley: preprint, 1976).

  23. 23.

    , Science, 186, 645–647 (1974).

  24. 24.

    , Biometrics, 6, 250–258 (1950).

  25. 25.

    , J. Fish. Res. Bd. Can., 11, 559–623 (1954).

  26. 26.

    , Nature, 207, 316 (1965).

  27. 27.

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

  28. 28.

    , J. theor. Biol., 51, 511–524 (1975).

  29. 29.

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

  30. 30.

    , and , Illinois J. Math., 5, 657–667 (1961).

  31. 31.

    , (King's College, Cambridge: preprint, 1976).

  32. 32.

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

  33. 33.

    , Sci. Am., 233(6), 56–69 (1975).

  34. 34.

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

  35. 35.

    , and , Bull. Am. math. Soc. (abstr.), 53, 1120 (1947).

  36. 36.

    , Ann. Math., 47, 33–49 (1946).

  37. 37.

    , Science, 181, 1074 (1973).

  38. 38.

    , J. Anim. Ecol., 44, 283–296 (1974).

  39. 39.

    , , and , J. Anim. Ecol. (in the press).

  40. 40.

    , and , Comm. math. Phys., 20, 167–192 (1971).

  41. 41.

    , and , Fluid Mechanics (Pergamon, London, 1959).

  42. 42.

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

  43. 43.

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

  44. 44.

    , , and , Numer. Math., 10, 1–19 (1967).

  45. 45.

    , and , Automatica, 5, 303–317 (1969).

  46. 46.

    , and , Rosprawy Mat., 39, 1–66 (1964).

  47. 47.

    , , and , Nature, 255, 58–60 (1975).

  48. 48.

    , and , Differential Equations, Dynamical Systems and Linear Algebra (Academic, New York, 1974).

  49. 49.

    , Science, 189, 984–985 (1975).

  50. 50.

    , (Department of Mathematics, Berkeley: preprint, 1976).

  51. 51.

    , and , SIAM J. Appl. Math., 29, 243–253 (1975).

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Affiliations

  1. King's College Research Centre, Cambridge CB2 1ST; on leave from Biology Department, Princeton University, Princeton 08540.

    • Robert M. May

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

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