Simple mathematical models with very complicated dynamics

  • An Erratum to this article was published on 15 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.

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References

  1. 1

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

  2. 2

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

  3. 3

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

  4. 4

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

  5. 5

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

  6. 6

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

  7. 7

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

  8. 8

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

  9. 9

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

  10. 10

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

  11. 11

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

  12. 12

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

  13. 13

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

  14. 14

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

  15. 15

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

  16. 16

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

  17. 17

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

  18. 18

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

  19. 19

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

  20. 20

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

  21. 21

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

  22. 22

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

  23. 23

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

  24. 24

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

  25. 25

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

  26. 26

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

  27. 27

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

  28. 28

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

  29. 29

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

  30. 30

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

  31. 31

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

  32. 32

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

  33. 33

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

  34. 34

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

  35. 35

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

  36. 36

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

  37. 37

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

  38. 38

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

  39. 39

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

  40. 40

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

  41. 41

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

  42. 42

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

  43. 43

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

  44. 44

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

  45. 45

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

  46. 46

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

  47. 47

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

  48. 48

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

  49. 49

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

  50. 50

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

  51. 51

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

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