Open Access

Asymptotic behavior of a competitive system of linear fractional difference equations

Advances in Difference Equations20062006:019756

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

Received: 18 July 2005

Accepted: 5 April 2006

Published: 30 August 2006

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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Authors’ Affiliations

(1)
Department of Mathematics, University of Rhode Island
(2)
Department of Mathematics, University of Tuzla

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)00326-1MathSciNetView ArticleMATHGoogle 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)00294-8MathSciNetView ArticleMATHGoogle 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. 10.1080/10236190412331334464MathSciNetView ArticleMATHGoogle Scholar
  4. Elaydi SN: Discrete Chaos. Chapman & Hall/CRC, Florida; 2000:xiv+355.MATHGoogle 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/BF00160333MathSciNetView ArticleMATHGoogle 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-CMathSciNetView ArticleMATHGoogle 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)90045-9MathSciNetView ArticleMATHGoogle 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-KMathSciNetView ArticleMATHGoogle Scholar
  9. Kulenović MRS, Ladas G: Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures. Chapman & Hall/CRC, Florida; 2001.View ArticleMATHGoogle Scholar
  10. Kulenović MRS, Merino O: Discrete Dynamical Systems and Difference Equations with Mathematica. Chapman & Hall/CRC, Florida; 2002:xvi+344.View ArticleMATHGoogle 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.MathSciNetMATHGoogle 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.2005.127MathSciNetMATHGoogle Scholar
  13. May RM: Stability in multispecies community models. Mathematical Biosciences 1971,12(1–2):59–79. 10.1016/0025-5564(71)90074-5MathSciNetView ArticleMATHGoogle 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. 10.1080/10236190211954MathSciNetView ArticleMATHGoogle 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/BF00276194MathSciNetView ArticleMATHGoogle Scholar
  17. Smith HL: Planar competitive and cooperative difference equations. Journal of Difference Equations and Applications 1998,3(5–6):335–357. 10.1080/10236199708808108MathSciNetView ArticleMATHGoogle Scholar

Copyright

© M. R. S. Kulenović and Nurkanović 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.