Journal of Dynamics and Differential Equations

, Volume 2, Issue 4, pp 367–421 | Cite as

The Poincare-Bendixson theorem for monotone cyclic feedback systems

  • John Mallet-Paret
  • Hal L. Smith
Article

Abstract

We prove the Poincare-Bendixson theorem for monotone cyclic feedback systems; that is, systems inRn of the form
$$x_i = f_i (x_i , x_{i - 1} ), i = 1, 2, ..., n (\bmod n).$$
We apply our results to a variety of models of biological systems.

Key words

Cellular control system cyclic system monotonicity negative feedback Poincare-Bendixson theorem 

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References

  1. 1.
    D. J. Allwright, “A Global Stability Criterion for Simple Control Loops,”J. Math. Biol. 4 (1977), 363–373.Google Scholar
  2. 2.
    U. an der Heiden, “Delays in Physiological Systems,”J. Math. Biol. 8 (1979), 345–364.Google Scholar
  3. 3.
    U. an der Heiden, “Existence of Periodic Solutions of a Nerve Equation,”Biol. Cybern. 21 (1976), 37–39.Google Scholar
  4. 4.
    U. an der Heiden and H. O. Walther, “Existence of Chaos in Control Systems with Delayed Feedback,”J. Diff. Eq. 47 (1983), 273–295.Google Scholar
  5. 5.
    H. T. Banks and J. M. Mahaffy, “Mathematical Models for Protein Biosynthesis,” Technical Report, Lefschetz Center for Dynamical Systems, Brown University, Providence, R. I., 1979.Google Scholar
  6. 6.
    H. T. Banks and J. Mahaffy, “Stability of Cyclic Gene Models for Systems Involving Repression,”J. Theor. Biol. 74 (1978), 323–334.Google Scholar
  7. 7.
    H. T. Banks and J. M. Mahaffy, “Global Asymptotic Stability of Certain Models for Protein Synthesis and Repression,”Q. App. Math. 36 (1978), 209–220.Google Scholar
  8. 8.
    C. Berding and T. Harbich, “On the Dynamics of a Simple Biochemical Control Circuit,”Biol. Cybern. 49 (1984), 209–219.Google Scholar
  9. 9.
    P. Brunovsky and B. Fiedler, “Zero Numbers on Invariant Manifolds in Scalar Reaction Diffusion Equations,”Nonlin. Anal. TMA 10 (1986), 129–194.Google Scholar
  10. 10.
    S. N. Chow, O. Djekmann, and J. Mallet-Paret, “Stability, Multiplicity, and Global Continuation of Symmetric Periodic Solutions of a Nonlinear Volterra Integral Equation,”Jap. J. Appl. Math. 2 (1985), 433–469.Google Scholar
  11. 11.
    B. Fiedler and J. Mallet-Paret, “A Poincaré-Bendixson Theorem for Scalar Reaction Diffusion Equations,”Arch. Rat. Mech. Anal. 107 (1989), 325–345.Google Scholar
  12. 12.
    A. Fraser and J. Tiwari, “Genetical Feedback-Repression. II. Cyclic Genetic Systems,”J. Theor. Biol. 47 (1974), 397–12.Google Scholar
  13. 13.
    G. Fusco and W. Oliva, “Jacobi Matrices and Transversality,”Proc. Roy. Soc. Edinburg Sect. A 109 (1988), 231–243.Google Scholar
  14. 14.
    B. C. Goodwin,Temporal Organization in Cells, Academic Press, New York, 1963.Google Scholar
  15. 15.
    B. C. Goodwin, “Oscillatory Behavior in Enzymatic Control Processes,”Adv. Enzyme Reg. 3 (1965), 425–439.Google Scholar
  16. 16.
    J. S. Griffith, “Mathematics of Cellular Control Processes I, II,”J. Theor. Biol. 20 (1968), 202–208, 209–216.Google Scholar
  17. 17.
    J. K. Hale,Ordinary Differential Equations, R. E. Krieger, 1980.Google Scholar
  18. 18.
    S. P. Hastings, “On the Uniqueness and Global Asymptotic Stability of Periodic Solutions for a Third Order System,”Rocky Mt. J. Math. 7 (1977), 513–538.Google Scholar
  19. 19.
    S. P. Hastings, J. Tyson, and D. Webster, “Existence of Periodic Solutions for Negative Feedback Cellular Control Systems,”J. Diff. Eq. 25 (1977), 39–64.Google Scholar
  20. 20.
    M. W. Hirsch, “The Dynamical Systems Approach to Differential Equations,”Bull. A.M.S. 11 (1984), 1–64.Google Scholar
  21. 21.
    M. W. Hirsch, “Stability and Convergence in Strongly Monotone Dynamical Systems,”J. Reine Angewendte Math. 383 (1988), 1–53.Google Scholar
  22. 22.
    M. W. Hirsch, “Systems of Differential Equations which are Competitive or Cooperative, I: Limit Sets,”SIAM J. Math. Anal. 13 (1982), 167–179; “II: Convergence Almost Everywhere,”SIAM J. Math. Anal. 16 (1985) 432–439.Google Scholar
  23. 23.
    G. A. Lenov, “An Analog of Bendixson's Criterion for Third Order Equations,”Diff. Eq. 13 (1977), 367–368.Google Scholar
  24. 24.
    N. MacDonald, “Bifurcation Theory Applied to a Simple Model of a Biochemical Oscillator,”J. Theor. Biol. 65 (1977), 727–734.Google Scholar
  25. 25.
    N. MacDonald, “Time Lag in a Model of a Biochemical Reaction Sequence with End Product Inhibition,” /.Theor. Biol. 67 (1977), 549–556.Google Scholar
  26. 26.
    M. C. Mackey and L. Glass, “Oscillations and Chaos in Physiological Control Systems,”Science 197 (1977), 287–289.Google Scholar
  27. 27.
    J. M. Mahaffy, “Periodic Solutions for Certain Protein Synthesis Models,”JMAA 74 (1980), 72–105.Google Scholar
  28. 28.
    J. M. Mahaffy, “Stability of Periodic Solutions for a Model of Genetic Repression with Delays,”J. Math. Biol. 22 (1985), 137–144.Google Scholar
  29. 29.
    J. Mallet-Paret and G. Sell. In preparation.Google Scholar
  30. 30.
    H. Matano, “Convergence of Solutions of One-Dimensional Semilinear Parabolic Equations,”J. Math. Kyoto Univ. (1978), 221–227.Google Scholar
  31. 31.
    K. Nickel, “Gestaltaussagen über Lösungen parabolischer Differentialgleichungen,”J. Reine Angew. Math. 211 (1962), 78–94.Google Scholar
  32. 32.
    R. Reissig, G. Sansone, and R. Conti,Nonlinear Differential Equations of Higher Order, Rome, 1969.Google Scholar
  33. 33.
    J. F. Selgrade, “Asymptotic Behavior of Solutions to Single Loop Positive Feedback Systems,”J. Diff. Eq. 38 (1980), 80–103.Google Scholar
  34. 34.
    J. Smillie, “Competitive and Cooperative Tridiagonal Systems of Differential Equations,”SIAM J. Math. Anal. 15 (1984), 530–534.Google Scholar
  35. 35.
    H. L. Smith, “Periodic Orbits of Competitive and Cooperative Systems,”J. Diff. Eq. 65 (1986), 361–373.Google Scholar
  36. 36.
    H. L. Smith, “Oscillations and Multiple Steady States in a Cyclic Gene Model with Repression,”J. Math. Biol. 25 (1987), 169–190.Google Scholar
  37. 37.
    H. L. Smith, “Systems of Ordinary Differential Equations which Generate an Order Preserving Flow. A Survey of Results,”SIAM Rev. 30 (1988), 87–113.Google Scholar
  38. 38.
    R. A. Smith, “The Poincaré-Bendixson Theorem for Certain Differential Equations of Higher Order,”Proc. Roy. Soc. Edinburgh 83A (1979), 63–79.Google Scholar
  39. 39.
    R. A. Smith, “Existence of Periodic Orbits of Autonomous Ordinary Differential Equations,”Proc. Roy. Soc. Edinburgh 85A (1980), 153–172.Google Scholar
  40. 40.
    R. A. Smith, “Orbital Stability for Ordinary Differential Equations,”J. Diff. Eq. 69 (1987), 265–287.Google Scholar
  41. 41.
    R. B. Stein, K. V. Leung, D. Mangeron, and M. Oguztöreli, “Improved Neuronal Models for Studying Neural Networks,”Kybernetic 15 (1974), 1–9.Google Scholar
  42. 42.
    J. J. Tyson, “On the Existence of Oscillatory Solutions in Negative Feedback Cellular Control Processes,”J. Math. Biol. 1 (1975), 311–315.Google Scholar
  43. 43.
    J. J. Tyson and H. G. Othmer, “The Dynamics of Feedback Control Circuits in Biochemical Pathways,” inProg. Theor. Biol. (R. Rosen and F. M. Snell, Eds.), Academic Press, New York, 1978.Google Scholar
  44. 44.
    J. K. Hale, “Topics in Dynamic Bifurcation Theory,”CBMS 47 (1980), A.M.S.Google Scholar

Copyright information

© Plenum Publishing Corporation 1990

Authors and Affiliations

  • John Mallet-Paret
    • 1
  • Hal L. Smith
    • 2
  1. 1.Division of Applied MathematicsBrown UniversityProvidence
  2. 2.Department of MathematicsArizona State UniversityTempe

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