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Necessity and chance: deterministic chaos in ecology and evolution


Author: Robert M. May
Journal: Bull. Amer. Math. Soc. 32 (1995), 291-308
MSC: Primary 92D25; Secondary 92D15, 92D40
DOI: https://doi.org/10.1090/S0273-0979-1995-00598-7
MathSciNet review: 1307905
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Abstract | References | Similar Articles | Additional Information

Abstract: This is an outline of my Gibbs Lecture to the American Mathematical Society in January 1994; it is essentially a sign-posted guide to a stilldeveloping literature.


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  • [1] R. M. May, Simple mathematical models with very complicated dynamics, Nature 261 (1976), 459-467.
  • [2] Colin W. Clark, Mathematical bioeconomics, 2nd ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1990. The optimal management of renewable resources; With a contribution by Gordon Munro; A Wiley-Interscience Publication. MR 1044994
  • [3] R. M. Anderson and R. M. May, Infectious diseases of humans: Dynamics and control, Oxford Univ. Press, Oxford, 1991.
  • [4] R. M. May and M. A. Nowak, Superinfection, metapopulation dynamics, and the evolution of diversity, J. Theoret. Biol. 170 (1994), 95-114.
  • [5] M. A. Nowak, R. M. May, and K. Sigmund, Immune responses against multiple epitopes, J. Theoret. Biol. (1995) (in preparation).
  • [6] D. Tilman, R. M. May, C. L. Lehman, and M. A. Nowak, Habitat destruction and the extinction debt, Nature 371 (1994), 65-66.
  • [7] P. A. B. Moran, Some remarks on animal population dynamics, Biometrics 6 (1950), 250-258.
  • [8] W. E. Ricker, Stock and recruitment, J. Fish. Res. Bd. Canad. 11 (1954), 559-623.
  • [9] O. M. Šarkovs′kiĭ, Co-existence of cycles of a continuous mapping of the line into itself, Ukrain. Mat. Ž. 16 (1964), 61–71 (Russian, with English summary). MR 0159905
  • [10] T. Y. Li and James A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), no. 10, 985–992. MR 0385028, https://doi.org/10.2307/2318254
  • [11] R. M. May and G. F. Oster, Bifurcations and dynamic complexity in simple ecological models, Amer. Natur. 110 (1976), 573-599.
  • [12] E. N. Lorenz, Deterministic nonperiodic flow, J. Atmospheric Sci. 20 (1963), 130-141.
  • [13] T. Mullin (ed.), The nature of chaos, Oxford Univ. Press, Oxford, 1993.
  • [14] L. A. Smith, Local optimal prediction: Exploiting strangeness and the variation of sensitivity to initial conditions, Philos. Trans. Roy. Soc. London Ser. A 348 (1994), 371-381.
  • [15] Rodney C. L. Wolff, Local Lyapunov exponents: looking closely at chaos, J. Roy. Statist. Soc. Ser. B 54 (1992), no. 2, 353–371. MR 1160475
  • [16] I. Hanski, P. Turchin, E. Korpimäki, and H. Henttonen, Population oscillations of boreal rodents: Regulation by mustilid predators leads to chaos, Nature 364 (1993), 232-235.
  • [17] G. Sugihara, B. T. Grenfell, and R. M. May, Distinguishing error from chaos in ecological time series, Philos. Trans. Roy. Soc. London Ser. B 330 (1991), 235-251.
  • [18] L. F. Olsen and W. M. Scharfer, Chaos versus noisy periodicity: Alternative hypotheses for childhood epidemics, Science 249 (1990), 499-504.
  • [19] G. Sugihara, Nonlinear forecasting for the classification of natural time series, Philos. Trans. Roy. Soc. London Ser. A 348 (1994), 477-495.
  • [20] Blake LeBaron, Chaos and nonlinear forecastability in economics and finance, Philos. Trans. Roy. Soc. London Ser. A 348 (1994), no. 1688, 397–404. MR 1300159, https://doi.org/10.1098/rsta.1994.0099
  • [21] D. Ruelle, The Claude Bernard Lecture, 1989. Deterministic chaos: the science and the fiction, Proc. Roy. Soc. London Ser. A 427 (1990), no. 1873, 241–248. MR 1039785
  • [22] A. S. Weigend and N. A. Gershenfeld, Time series prediction: Forecasting the future and understanding the past, Addison-Wesley, Reading, MA, 1993.
  • [23] Howell Tong, A personal overview of non-linear time series analysis from a chaos perspective, Scand. J. Statist. 22 (1995), no. 4, 399–445. With discussion and a reply by the author. MR 1363222
  • [24] A. M. Hastings, C. L. Horn, S. Ellner, P. Turchin, and H. C. J. Godfray, Chaos in ecology: Is Mother Nature a strange attractor?, Ann. Rev. Ecol. Syst. 24 (1993), 1-33.
  • [25] Martin Casdagli, Chaos and deterministic versus stochastic nonlinear modelling, J. Roy. Statist. Soc. Ser. B 54 (1992), no. 2, 303–328. MR 1160473
  • [26] G. Sugihara and R. M. May, Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series, Nature 344 (1990), 734-741.
  • [27] J. Doyne Farmer and John J. Sidorowich, Exploiting chaos to predict the future and reduce noise, Evolution, learning and cognition, World Sci. Publ., Teaneck, NJ, 1988, pp. 277–330. MR 1036562
  • [28] Floris Takens, Detecting strange attractors in turbulence, Dynamical systems and turbulence, Warwick 1980 (Coventry, 1979/1980), Lecture Notes in Math., vol. 898, Springer, Berlin-New York, 1981, pp. 366–381. MR 654900
  • [29] Peter Grassberger and Itamar Procaccia, Measuring the strangeness of strange attractors, Phys. D 9 (1983), no. 1-2, 189–208. MR 732572, https://doi.org/10.1016/0167-2789(83)90298-1
  • [30] B. T. Grenfell, A. Kleczkowski, S. P. Ellner, and B. M. Bolker, Measles as a case study in nonlinear forecasting and chaos, Philos. Trans. Roy. Soc. London Ser. A 348 (1994), 515-530.
  • [31] J. D. Murray, Mathematical biology, Biomathematics, vol. 19, Springer-Verlag, Berlin, 1989. MR 1007836
  • [32] Michael P. Hassell, The dynamics of arthropod predator-prey systems, Monographs in Population Biology, vol. 13, Princeton University Press, Princeton, N.J., 1978. MR 508052
  • [33] A. J. Nicholson and V. A. Bailey, The balance of animal populations. Part I, Proc. Zoolog. Soc. London 1 (1935), 551-598.
  • [34] M. P. Hassell and R. M. May, Stability in insect host-parasite models, J. Animal Ecol. 42 (1973), 693-726.
  • [35] S. W. Pacala, M. P. Hassell, and R. M. May, Host-parasitoid associations in patchy environments, Nature 344 (1990), 150-153.
  • [36] M. P. Hassell, R. M. May, S. W. Pacala, and P. L. Chesson, The persistence of host-parasitoid associations in patchy environments, Amer. Natur. 138 (1991), 568-583.
  • [37] M. P. Hassell, H. N. Comins, and R. M. May, Spatial structure and chaos in insect population dynamics, Nature 353 (1991), 255-258.
  • [38] H. N. Comins, M. P. Hassell, and R. M. May, The spatial dynamics of host-parasitoid systems, J. Animal Ecol. 61 (1992), 735-748.
  • [39] R. V. Solè and J. Valls, Spiral waves, chaos and multiple attractors in lattice models of interacting populations, Phys. Lett. A 166 (1992), 123-128.
  • [40] M. P. Hassell, H. N. Comins, and R. M. May, Species coexistence and self-organizing spatial dynamics, Nature 370 (1994), 290-292.
  • [41] D. A. Rand, Measuring and characterizing spatial patterns, dynamics and chaos in spatially extended dynamical systems and ecologies, Philos. Trans. Roy. Soc. London Ser. A 348 (1994), 497-514.
  • [42] R. Axelrod, The evolution of cooperation, Basic Books, New York, 1984.
  • [43] Robert Axelrod and William D. Hamilton, The evolution of cooperation, Science 211 (1981), no. 4489, 1390–1396. MR 686747, https://doi.org/10.1126/science.7466396
  • [44] M. A. Nowak and K. Sigmund, Tit for tat in heterogeneous populations, Nature 355 (1992), 250-253.
  • [45] -, Chaos and the evolution of cooperation, Proc. Nat. Acad. Sci. U.S.A. 90 (1993), 5091-5094.
  • [46] R. M. May, More evolution of cooperation, Nature 327 (1987), 15-17.
  • [47] M. A. Nowak and R. M. May, Evolutionary games and spatial chaos, Nature 359 (1992), 826-829.
  • [48] Martin A. Nowak and Robert M. May, The spatial dilemmas of evolution, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 3 (1993), no. 1, 35–78. MR 1218718, https://doi.org/10.1142/S0218127493000040
  • [49] Martin A. Nowak, Sebastian Bonhoeffer, and Robert M. May, More spatial games, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 4 (1994), no. 1, 33–56. MR 1276803, https://doi.org/10.1142/S0218127494000046
  • [50] -, Spatial games and the maintenance of cooperation, Proc. Nat. Acad. Sci. U.S.A. 91 (1994), 4877-4881.
  • [51] A. V. M. Herz, Collective phenomena in spatially extended evolutionary games, J. Theoret. Biol. 169 (1994), 65-87.
  • [52] B. A. Huberman and N. S. Glance, Evolutionary games and computer simulations, Proc. Nat. Acad. Sci. U.S.A. 90 (1993), 7712-7715.
  • [53] J. Maynard Smith, Evolution and the theory of games, Cambridge Univ. Press, Cambridge, 1982.
  • [54] T. Stoppard, Arcadia, Faber and Faber, London, 1993.

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DOI: https://doi.org/10.1090/S0273-0979-1995-00598-7
Article copyright: © Copyright 1995 American Mathematical Society