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Unexpected dominance of high frequencies in chaotic nonlinear population models

Abstract

BECAUSSE water has a higher heat capacity than air, large bodies of water fluctuate in temperature more slowly than does the atmosphere1. Marine temperature time series are 'redder' than atmospheric temperature time series by analogy to light: in red light, low-frequency variability has greater amplitude than high-frequency variability, whereas in white light all frequencies have the same amplitude2. Differences in the relative importance of high-and low-frequency variability in different habitats affect the population dynamics of individual species and the structure of ecological communities3–9. Population dynamics of individual species are thought to be dominated by low-frequency fluctuations, that is, to display reddened fluctuations10. Here I report, however, that in eight nonlinear, iterative, deterministic, autonomous, discrete-time population models, some of which have been used to model real biological populations, the power spectral densities of chaotic trajectories are neither white nor reddened but are notably blue, with increasing power at higher frequencies.

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References

  1. Monin, A. S., Kamenkovich, V. M. & Kort, V. G. Variability of the Oceans (Wiley, New York, 1977).

    Google Scholar 

  2. Jenkins, G. M. & Watts, D. G. Spectral Analysis and its Applications (Wiley, New York, 1968).

    MATH  Google Scholar 

  3. Steele, J. H. Nature 313, 355–358 (1985).

    Article  ADS  Google Scholar 

  4. Steele, J. H. J. theor. Biol. 153, 425–436 (1991).

    Article  Google Scholar 

  5. Steele, J. H. & Henderson, E. W. Science 224, 985–987 (1984).

    Article  ADS  CAS  Google Scholar 

  6. Steele, J. H. & Henderson, E. W. Phil. Trans. R. Soc. Lond. B 343, 5–9 (1994).

    Article  Google Scholar 

  7. Pimm, S. L. & Redfearn, A. Nature 334, 613–614 (1988).

    Article  ADS  Google Scholar 

  8. Arino, A. & Pimm, S. L. Evol. Ecol. 9, 429–443 (1995).

    Article  Google Scholar 

  9. Caswell, H. & Cohen, J. E. J. theor. Biol. 176, 301–316 (1995).

    Article  Google Scholar 

  10. Halley, J. M. Trends Ecol. Evol. (in the press).

  11. Lewontin, R. C. & Cohen, D. Proc. natn. Acad. Sci. U.S.A. 62, 1056–1060 (1969).

    Article  ADS  CAS  Google Scholar 

  12. May, R. M. & Oster, G. Am. Nat. 110, 573–599 (1976).

    Article  Google Scholar 

  13. Hassell, M. P., Lawton, J. H. & May, R. M. J. Anim. Ecol. 45, 471–486 (1976).

    Article  Google Scholar 

  14. Austin, A. L. & Brewer, J. W. Technol. Forecast, soc. Change 3, 23–49 (1971).

    Article  Google Scholar 

  15. Cohen, J. E. Science 269, 341–348 (1995).

    Article  ADS  CAS  Google Scholar 

  16. Li, T.-Y., Misiurewicz, M., Pianigiani, G. & Yorke, J. A. Phys. Lett. A 87, 271–273 (1982).

    Article  ADS  Google Scholar 

  17. Farmer, D., Crutchfield, J., Froehling, H., Packard, N. & Shaw, R. Ann. N.Y. Acad. Sci. 357, 453–472 (1980).

    Article  ADS  Google Scholar 

  18. Schaffer, W. M. in Chaos in Biological Systems (eds Degn, H., Holden, A. V. & Olsen, L, F.) 233–248 (NATO Advanced Science Institutes Series A, vol. 138) (Plenum, New York, 1987).

    Book  Google Scholar 

  19. Hastings, A., Hom, C. L., Ellner, S., Turchin, P. & Godfray, H. C. J. A. Rev. Ecol. Syst. 24, 1–33 (1993).

    Article  Google Scholar 

  20. Muratori, S. & Rinaldi, S. SIAM J. appl. Math. 52, 1688–1706 (1992).

    Article  MathSciNet  Google Scholar 

  21. Levin, S. A. & Goodyear, C. P. J. math. Biol. 9, 245–274 (1980).

    Article  MathSciNet  Google Scholar 

  22. Gurney, W. S. C., Nisbet, R. M. & Lawton, J. H. J. Anim. Ecol. 52, 479–495 (1983).

    Article  Google Scholar 

  23. Ruxton, G, D., Bascompte, J. & Solé, R. V. J. Anim. Ecol. 63, 1002–1003 (1994).

    Article  Google Scholar 

  24. Markus, M., Hess, B., Rössler, J. & Kiwi, M. in Chaos in Biological Systems (eds Degn, H., Holden, A. V. & Olsen. L. F.) 267–277 (NATO Advanced Science Institutes Series A, vol. 138) (Plenum, New York. 1987).

    Book  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

  27. Pennycuick, C. J., Compton, R. M. & Beckingham, L. J. theor. Biol. 18, 316–329 (1968).

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

  29. Maynard Smith, J. Models in Ecology (Cambridge Univ. Press, 1974).

    MATH  Google Scholar 

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

    Google Scholar 

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Cohen, J. Unexpected dominance of high frequencies in chaotic nonlinear population models. Nature 378, 610–612 (1995). https://doi.org/10.1038/378610a0

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