On the Behavior of Positive Solutions of a Difference Equations System xn+1={yn-2+yn-3}/xn, yn+1={xn-2+xn-3}/yn

Motivated by an open problem in the literature “Dynamics of Second Order Rational Difference Equations with Open Problems and Conjecture”, we introduce a difference equation system:

xn+1={yn-2+yn-3}/xn, yn+1={xn-2+xn-3}/yn , n=o, 1 ...* Where xi, yi ∈（0, ∞）， i=-3, -2, -1, 0.

We try to find out some conditions such that the solution of system converges to periodic solution. This model can be applied to the two species competition and population biology.

Share and Cite:

D. Zhang, W. Ji, L. Wang and X. Li, "On the Behavior of Positive Solutions of a Difference Equations System xn+1={yn-2+yn-3}/xn, yn+1={xn-2+xn-3}/yn," Applied Mathematics, Vol. 4 No. 9A, 2013, pp. 13-18. doi: 10.4236/am.2013.49A003.

Conflicts of Interest

The authors declare no conflicts of interest.

  M. R. S. Kulenovi? and G. Ladas, “Dynamics of Second Order Rational Difference Equation with Open Problems and Conjectures,” CRC Press, Chapman Hall, 2002.  C. Cinar, “On the Positive Solutions of the Difference Equation System xn+1=1/yn, yn+1=yn/xn-1yn-1 ,” Applied Mathematics and Computation, Vol. 158, 2004, pp. 303-305.  E. Camouzis and G. Papaschinopoulos, “Global Asymptotic Behavior of Positive Solutions on the System of Rational Difference Equations xn+1=1+xn/yn-m, yn+1=1+yn/xn-m,” Applied Mathematics and Letters, Vol. 17, 2004, pp. 733-737.  A. Y. Ozban, “On the System of Rational Difference Equations xn=a/yn-3, yn=byn-3/xn-qyn-q,” Applied Mathematics and Computation, Vol. 188, 2007, pp. 833-837.  A. S. Kurbanli, C. Cinar and I. Yalcinkaya, “On the Behavior of Positive Solutions of the System of Rational Difference Equations, xn=xn-1/ynxn-1+1, yn=yn-1/xnyn-1+1 ,” Mathematical and Computer Modelling, Vol. 53, 2011, pp. 1261-1267.  G. Papaschinopoulos, “On a System of Two Nonlinear Difference Equations,” Journal of Mathematicsal Analysis and Applications, Vol. 219, 1998, pp. 415-426.  G. Papaschinopoulos, M. Radin and C. J. Schinas, “Study of the Asymptotic Behavior of the Solutions of Three Systems of Difference Equations of Exponential Form,” Applied Mathematics and Computation, Vol. 218, 2012, pp. 5310-5318.   E. Camouzis, C. M. Kent, G. Ladas and C. D. Lynd, “On the Global Character of Solutions of the System: xn+1=α1+yn/xn, yn+1=α2+β2xn+γ2yn/A2+B2xn+C2yn,” Journal of Difference Equations and Applications, Vol. 18, 2012, pp. 1205-1252.   E. Camouzis, E. Drymonis, G. Ladas and W. Tikjha, “Patterns of Boundedness of the Rational System xn+1=α1/A1+B1xn+C1yn, yn+1=α2+β2xn+γ2yn/A2+B2xn+C2yn,” Journal of Difference Equations and Applications, Vol. 18, 2012, pp. 89-110. 