Asymptotic behavior of a competitive system of linear fractional difference equations
Advances in Difference Equations volume 2006, Article number: 019756 (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.
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
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
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
Elaydi SN: Discrete Chaos. Chapman & Hall/CRC, Florida; 2000:xiv+355.
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
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
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
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
Kulenović MRS, Ladas G: Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures. Chapman & Hall/CRC, Florida; 2001.
Kulenović MRS, Merino O: Discrete Dynamical Systems and Difference Equations with Mathematica. Chapman & Hall/CRC, Florida; 2002:xvi+344.
Kulenović MRS, Nurkanović M: Asymptotic behavior of a two dimensional linear fractional system of difference equations. Radovi Matematički 2002,11(1):59–78.
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
May RM: Stability in multispecies community models. Mathematical Biosciences 1971,12(1–2):59–79. 10.1016/0025-5564(71)90074-5
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
Robinson C: Dynamical Systems. Stability, Symbolic Dynamics, and Chaos, Studies in Advanced Mathematics. CRC Press, Florida; 1995:xii+468.
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
Smith HL: Planar competitive and cooperative difference equations. Journal of Difference Equations and Applications 1998,3(5–6):335–357. 10.1080/10236199708808108
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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