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Did Something Change? Thresholds in Population Models

  • Frank Hoppensteadt
  • Paul Waltman
Conference paper

Abstract

The goal of this article is to illustrate several interesting bifurcations that can arise in population biology. These are of interest since it is often through bifurcation phenomena that changes significant enough to be measured occur. For example, a minor change in some environmental parameter can cause a system to change from being at rest to oscillating. We illustrate here the role of several canonical types of bifurcations in population modeling.

Keywords

Periodic Orbit Hopf Bifurcation Sample Path Infected Snail Stable Oscillation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    F. Albrecht, H. Gatzke, A. Hadad, and N. Wax, The dynamics of two interacting populations, J.Math. Anal. Appl. 46 (1974), 658–670MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    V.I. Arnol’d, ed., Dynamical Systems V. Bifurcation Theory and Catastrophe Theory. Springer-Verlag, New York, 1994.Google Scholar
  3. 3.
    G. J. Butler and P. Waltman, Bifurcation from a limit cycle in a two predator-one prey ecosysstem modeled on a chemostat, J. Math. Bio. 12 (1981), 295–310MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Cheng, K. -S., Uniqueness of limit cycles for a predator-prey system, SIAM J. Math. Analysis 12 [ 1981 ], 541–548Google Scholar
  5. 5.
    W. Feller, An Introduction to Probability Theory and its Applications, vols. I, II, Wiley-Interscience, New York.Google Scholar
  6. 6.
    H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel Dekker, New York, 1980MATHGoogle Scholar
  7. 7.
    J. Glieck, Chaos: Making of a New Science, Viking, 1987.Google Scholar
  8. 8.
    J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, Springer-Verlag, New York, 1983.Google Scholar
  9. 9.
    J. Hofbauer and J.W.-H. So, Multiple Limit Cycles for Pedator-Prey Models, Math. Biosciences 99 (1990), 71–75MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    F. C. Hoppensteadt, A nonlinear renewal equation with periodic and chaotic solutions, Proc. AMS-SIAM Conf. Appl. Math., New York, April, 1976.Google Scholar
  11. 11.
    F.C. Hoppensteadt, Analysis and Simulation of Chaotic Systems, 2nd ed., Springer Verlag, New York, 2000.MATHGoogle Scholar
  12. 12.
    F.C. Hoppensteadt, Mathematical Methods of Population Biology, Cambridge University Press, Cambridge, 1986.Google Scholar
  13. 13.
    F.C. Hoppensteadt and J.M. Hyman, Periodic solutions to a discrete logistic equation, SIAM Appl. Math. 32(1977)73–81.Google Scholar
  14. 14.
    F.C. Hoppensteadt, E.M. Izhikevich, Weakly Connected Neural Networks, Springer-Verlag, New York, 1997.CrossRefGoogle Scholar
  15. 15.
    F.C. Hoppensteadt, Z. Jacekewicz, Numerical solution of nonlinear Volterra equations, in preparation.Google Scholar
  16. 16.
    F.C. Hoppensteadt, C.S. Peskin, Modeling and Simulation in Medicine and the Life Sciences, Springer-Verlag, in press.Google Scholar
  17. 17.
    F.C. Hoppensteadt, P. Waltman, A problem in the theory of epidemics (1970, 1971)Math. Biosci. 9,71–91, 12, 133–145.Google Scholar
  18. 18.
    S. B. Hsu, S.P. Hubbell, and P. Waltman, Competing Predators, SIAM J. Appl. Math. 35 (1978), 617–625MathSciNetMATHGoogle Scholar
  19. 19.
    S. B. Hsu, S.P. Hubbell, and P. Waltman, A contribution to the theory of competing predators, Ecol. Monographs 48 (1978), 337–349CrossRefGoogle Scholar
  20. 20.
    A. Kolmorgoroff, Sulla teoria di Volterra della latta per l’esistenza, Gi. Inst. Ital. Attuari 7 (1936), 74–80Google Scholar
  21. 21.
    Yu. Kuznetzov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 1995.CrossRefGoogle Scholar
  22. 22.
    J. E. Marsden and M. McCracken, The Hopf bifurcation and its applications, Springer-Verlag, New York, 1976MATHCrossRefGoogle Scholar
  23. 23.
    R. M. May, Limit cycles in predator-prey communities, Science 177 (1972), 900–902.CrossRefGoogle Scholar
  24. 24.
    J. D. Murray, Mathematical Biology, Springer-Verlag, New York, 1989.MATHCrossRefGoogle Scholar
  25. 25.
    W.E. Ricker, Stock and recruitment, J. Fish. Res. Bd. Canada, 11 (1954) 559–623.CrossRefGoogle Scholar
  26. 26.
    A. Riscigno and I.W. Richardson, On the competitive exclusion principle, Bulletin Math. Biophysics, 27 (1965), 85–89CrossRefGoogle Scholar
  27. 27.
    A. Riscigno and I. W. Richardson,The struggle for life I:two species, Bulletin Math. Biphysics, 29 (1967), 377–388CrossRefGoogle Scholar
  28. 28.
    A.V. Skorokhod, F.C. Hoppensteadt, H.S. Salehi, Random Perturbations of Dynamical Systems, Springer-Verlag, 2002.Google Scholar
  29. 29.
    H. L. Smith, The interaction of steady state and Hopf bifurcation in a two-predator, one prey competition model, SIAM J. Applied Math. (1982)27–43.Google Scholar
  30. 30.
    R. Thom, Structural stability and Morphogenesis: An outline of a general theory of models, W.A. Benjamin, Reading, MA, 1975.Google Scholar
  31. 31.
    P. Waltman, Deterministic Threshold Models in the Theory of Epidemics, Springer-Verlag, New York, 1974.MATHCrossRefGoogle Scholar
  32. 32.
    P. Waltman, Competition Models in Population Biology, Soc. Ind. Appl. Math., Philadelphia, 1983.Google Scholar
  33. 33.
    S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 1990.MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Frank Hoppensteadt
    • 1
  • Paul Waltman
    • 2
  1. 1.606 GWC — SSERCASUTempeUSA
  2. 2.Department of Mathematics and Computer ScienceEmory University Suite 148AtlantaUSA

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