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Asymptotic behavior of a competitive system of linear fractional difference equations
© M. R. S. Kulenović and Nurkanović 2006
- Received: 18 July 2005
- Accepted: 5 April 2006
- Published: 30 August 2006
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.
- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
- Asymptotic Behavior
- Functional Equation
- 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
- 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
- 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
- Elaydi SN: Discrete Chaos. Chapman & Hall/CRC, Florida; 2000:xiv+355.MATHGoogle Scholar
- 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
- 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
- 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
- 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
- Kulenović MRS, Ladas G: Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures. Chapman & Hall/CRC, Florida; 2001.View ArticleMATHGoogle Scholar
- Kulenović MRS, Merino O: Discrete Dynamical Systems and Difference Equations with Mathematica. Chapman & Hall/CRC, Florida; 2002:xvi+344.View ArticleMATHGoogle Scholar
- 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
- 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
- May RM: Stability in multispecies community models. Mathematical Biosciences 1971,12(1–2):59–79. 10.1016/0025-5564(71)90074-5MathSciNetView ArticleMATHGoogle Scholar
- 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
- Robinson C: Dynamical Systems. Stability, Symbolic Dynamics, and Chaos, Studies in Advanced Mathematics. CRC Press, Florida; 1995:xii+468.Google Scholar
- 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
- 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
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