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Asymptotic behavior of a competitive system of linear fractional difference equations

Advances in Difference Equations volume 2006, Article number: 019756 (2006) Cite this article

Abstract

We investigate the global asymptotic behavior of solutions of the system of difference equations xn+1 = (a+x n )/(b+y n ), yn+1 = (d+y n )/(e+x n ), n = 0,1,..., where the parameters a, b, d, and e are positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers. In certain range of parameters, we prove the existence of the global stable manifold of the unique positive equilibrium of this system which is the graph of an increasing curve. We show that the stable manifold of this system separates the positive quadrant of initial conditions into basins of attraction of two types of asymptotic behavior. In the case where a = d and b = e, we find an explicit equation for the stable manifold to be y = x.

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References

  1. Clark D, Kulenović MRS: A coupled system of rational difference equations. Computers & Mathematics with Applications 2002,43(6–7):849–867. 10.1016/S0898-1221(01)00326-1

    Article  MathSciNet  MATH  Google Scholar 

  2. Clark D, Kulenović MRS, Selgrade JF: Global asymptotic behavior of a two-dimensional difference equation modelling competition. Nonlinear Analysis. Theory, Methods & Applications 2003,52(7):1765–1776. 10.1016/S0362-546X(02)00294-8

    Article  MathSciNet  MATH  Google Scholar 

  3. Clark CA, Kulenović MRS, Selgrade JF: On a system of rational difference equations. Journal of Difference Equations and Applications 2005,11(7):565–580. 10.1080/10236190412331334464

    Article  MathSciNet  MATH  Google Scholar 

  4. Elaydi SN: Discrete Chaos. Chapman & Hall/CRC, Florida; 2000:xiv+355.

    MATH  Google Scholar 

  5. Franke JE, Yakubu A-A: Mutual exclusion versus coexistence for discrete competitive systems. Journal of Mathematical Biology 1991,30(2):161–168. 10.1007/BF00160333

    Article  MathSciNet  MATH  Google Scholar 

  6. Franke JE, Yakubu A-A: Geometry of exclusion principles in discrete systems. Journal of Mathematical Analysis and Applications 1992,168(2):385–400. 10.1016/0022-247X(92)90167-C

    Article  MathSciNet  MATH  Google Scholar 

  7. Hassell MP, Comins HN: Discrete time models for two-species competition. Theoretical Population Biology 1976,9(2):202–221. 10.1016/0040-5809(76)90045-9

    Article  MathSciNet  MATH  Google Scholar 

  8. Hess P, Lazer AC: On an abstract competition model and applications. Nonlinear Analysis. Theory, Methods & Applications 1991,16(11):917–940. 10.1016/0362-546X(91)90097-K

    Article  MathSciNet  MATH  Google Scholar 

  9. Kulenović MRS, Ladas G: Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures. Chapman & Hall/CRC, Florida; 2001.

    Book  MATH  Google Scholar 

  10. Kulenović MRS, Merino O: Discrete Dynamical Systems and Difference Equations with Mathematica. Chapman & Hall/CRC, Florida; 2002:xvi+344.

    Book  MATH  Google Scholar 

  11. Kulenović MRS, Nurkanović M: Asymptotic behavior of a two dimensional linear fractional system of difference equations. Radovi Matematički 2002,11(1):59–78.

    MathSciNet  MATH  Google Scholar 

  12. Kulenović MRS, Nurkanović M: Asymptotic behavior of a system of linear fractional difference equations. Journal of Inequalities and Applications 2005,2005(2):127–143. 10.1155/JIA.2005.127

    MathSciNet  MATH  Google Scholar 

  13. May RM: Stability in multispecies community models. Mathematical Biosciences 1971,12(1–2):59–79. 10.1016/0025-5564(71)90074-5

    Article  MathSciNet  MATH  Google Scholar 

  14. Pituk M: More on Poincaré's and Perron's theorems for difference equations. Journal of Difference Equations and Applications 2002,8(3):201–216. 10.1080/10236190211954

    Article  MathSciNet  MATH  Google Scholar 

  15. Robinson C: Dynamical Systems. Stability, Symbolic Dynamics, and Chaos, Studies in Advanced Mathematics. CRC Press, Florida; 1995:xii+468.

    Google Scholar 

  16. Selgrade JF, Ziehe M: Convergence to equilibrium in a genetic model with differential viability between the sexes. Journal of Mathematical Biology 1987,25(5):477–490. 10.1007/BF00276194

    Article  MathSciNet  MATH  Google Scholar 

  17. Smith HL: Planar competitive and cooperative difference equations. Journal of Difference Equations and Applications 1998,3(5–6):335–357. 10.1080/10236199708808108

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Rhode Island, Kingston, RI, 02881-0816, USA

    MRS Kulenović

  2. Department of Mathematics, University of Tuzla, Tuzla, 75000, Bosnia and Herzegovina

    M Nurkanović

Authors
  1. MRS Kulenović
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  2. M Nurkanović
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Correspondence to MRS Kulenović.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Kulenović, M., Nurkanović, M. Asymptotic behavior of a competitive system of linear fractional difference equations. Adv Differ Equ 2006, 019756 (2006). https://doi.org/10.1155/ADE/2006/19756

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  • DOI: https://doi.org/10.1155/ADE/2006/19756

Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Asymptotic Behavior
  • Functional Equation