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The Discrete Dynamics of Monotonically Decomposable Maps

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An Erratum to this article was published on 24 January 2008

Abstract

We extend results of Gouzé and Hadeler (in Nonlinear World 1:23–34, 1994) concerning the dynamics generated by a map on an ordered metric space that can be decomposed into increasing and decreasing parts. Our main results provide sufficient conditions for the existence of a globally asymptotically stable fixed point for the map. Applications to discrete-time, stage-structured population models are given.

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References

  1. Angeli D., Sontag E.D. (2003) Monotone Control Systems. IEEE Trans Autom Control 48, 1684–1698

    Article  MathSciNet  Google Scholar 

  2. Berman A., Plemmons R. (1979) Nonnegative Matrices in the Mathematical Sciences. Academic, New York

    MATH  Google Scholar 

  3. Cosner C. (1997) Comparison Principles for systems that embed in cooperative systems, with applications to diffusive Lotka-Volterra models. Dyn Contin Discrete Impuls Syst 3, 283–303

    MathSciNet  MATH  Google Scholar 

  4. Collatz L. (1966) Functional Analysis and Numerical Mathematics. Academic, NY

    Google Scholar 

  5. Costantino R.F., Cushing J.M., Dennis B., Desharnais R.A. (1995) Experimentally induced transitions in the dynamic behavior of insect populations. Nature 375, 227–230

    Article  Google Scholar 

  6. Cushing J.M. (1988) Nonlinear matrix models and population dynamics. Nat Resour Model 2, 539–580

    MathSciNet  MATH  Google Scholar 

  7. Cushing J.M., Costantino R.F., Dennis B., Desharnais R.A., Henson S.M. (1998) Nonlinear population dynamics: models, experiments and data. J Theor Biol 194, 1–9

    Article  PubMed  Google Scholar 

  8. Cushing J.M., Costantino R.F., Dennis B., Desharnais R.A., Henson S.M. (2003) Chaos in Ecology, Experimental Nonlinear Dynamics. Academic, New York

    Google Scholar 

  9. El-Morshedy H.A., Liz E. (2005) Convergence to equilibria in discrete population models. J Differ Eqns Appl 11, 117–131

    Article  MathSciNet  MATH  Google Scholar 

  10. Enciso, G.A., Sontag, E.D. On the global attractivity of abstract dynamical systems satisfying a small gain hypothesis, with applications to biological delay systems. Discrete Contin Dyn Syst (to appear)

  11. Enciso, G.A., Smith, H.L., Sontag, E.D. Non-monotone systems decomposable into monotone systems with negative feedback. J Diff Eqns (to appear)

  12. Gouzé, J.-L. A criterion of global convergence to equilibrium for differential systems with an application to Lotka-Volterra systems. Rapport de Recherche 894, INRIA (1988)

  13. Gouzé J.-L., Hadeler K.P. (1994) Monotone flows and order intervals. Nonlinear World 1, 23–34

    MathSciNet  MATH  Google Scholar 

  14. Hirsch M.W., Smith H.L. (2005). Monotone dynamical systems. In: Canada A., Drabek P., Fonda A. (eds). Handbook of Differential Equations, Ordinary Differential Equations, vol. 2, Elsevier, Amsterdam, pp. 239-357

    Google Scholar 

  15. Hirsch M.W., Smith H.L. (2005) Monotone Maps: a review. J Differ Eqns Appl 11, 379–398

    Article  MathSciNet  MATH  Google Scholar 

  16. Kulenović M., Ladas G. (2002). Dynamics of Second Order Rational Difference Equations. Chapman & Hall/CRC, Boca Raton

    MATH  Google Scholar 

  17. Kulenović M., Ladas G., Sizer W. (1998). On the recursive sequence x n+1 = (αx n + βx n-1)/(γx n + δx n-1). Math Sci Res Hot-Line 2(5): 1–16

    MathSciNet  MATH  Google Scholar 

  18. Krause U., Pituk M. (2004) Boundedness and stability for higher order difference equations. J Differ Eqns Appl 10, 343–356

    Article  MathSciNet  MATH  Google Scholar 

  19. Schröder J. (1959) Fehlerabschätzungen bei linearen Gleichungssystemen mit dem Brouwerschen Fixpunktssatz. Arch Rat Mech Anal 3, 28–44

    Article  MATH  Google Scholar 

  20. Smith H.L. (1998) Planar competitive and cooperative difference equations. J Differ Eqns Appl 3, 335–357

    Article  MATH  Google Scholar 

  21. Thieme H.R. (1979) On a class of hammerstein integral equations. Manuscripta Math 29, 49–84

    Article  MathSciNet  MATH  Google Scholar 

  22. Thieme H.R. (1980) On a class of hammerstein integral equations. Manuscripta Math 31, 379–412

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Arizona State University, Tempe, AZ, 85287, USA

    H. L. Smith

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  1. H. L. Smith
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Correspondence to H. L. Smith.

Additional information

This paper is dedicated to Karl Hadeler on the occasion of his 70th Birthday

An erratum to this article can be found at http://dx.doi.org/10.1007/s00285-008-0155-5

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Smith, H.L. The Discrete Dynamics of Monotonically Decomposable Maps. J. Math. Biol. 53, 747–758 (2006). https://doi.org/10.1007/s00285-006-0004-3

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  • DOI: https://doi.org/10.1007/s00285-006-0004-3

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