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

Affiliation(s)

Institute of Systems Science and Mathematics, Naval Aeronautical and Astronautical University Yantai, Yantai, China.

Institute of Systems Science and Mathematics, Naval Aeronautical and Astronautical University Yantai, Yantai, China.

ABSTRACT

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:

x_{n+1}={y_{n-2}+y_{n-3}}/x_{n}, y_{n+1}={x_{n-2}+x_{n-3}}/y_{n} , n=o, 1 ...* Where x_{i, }y_{i }∈（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.

Cite this paper

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

D. Zhang, W. Ji, L. Wang and X. Li, "On the Behavior of Positive Solutions of a Difference Equations System x

References

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[5] A. S. Kurbanli, C. Cinar and I. Yalcinkaya, “On the Behavior of Positive Solutions of the System of Rational Difference Equations, x_{n}=x_{n-1}/y_{n}x_{n-1}+1, y_{n}=y_{n-1}/x_{n}y_{n-1}+1 ,” Mathematical and Computer Modelling, Vol. 53, 2011, pp. 1261-1267.

[6] G. Papaschinopoulos, “On a System of Two Nonlinear Difference Equations,” Journal of Mathematicsal Analysis and Applications, Vol. 219, 1998, pp. 415-426.

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[9] [9] E. Camouzis, E. Drymonis, G. Ladas and W. Tikjha, “Patterns of Boundedness of the Rational System x_{n+1}=α_{1}/A_{1}+B_{1}x_{n}+C_{1}y_{n}, y_{n+1}=α_{2}+β_{2}x_{n}+γ_{2}y_{n}/A_{2}+B_{2}x_{n}+C_{2}y_{n},” Journal of Difference Equations and Applications, Vol. 18, 2012, pp. 89-110.

[1] M. R. S. Kulenovi? and G. Ladas, “Dynamics of Second Order Rational Difference Equation with Open Problems and Conjectures,” CRC Press, Chapman Hall, 2002.

[2] C. Cinar, “On the Positive Solutions of the Difference Equation System x

[3] E. Camouzis and G. Papaschinopoulos, “Global Asymptotic Behavior of Positive Solutions on the System of Rational Difference Equations x

[4] A. Y. Ozban, “On the System of Rational Difference Equations x

[5] A. S. Kurbanli, C. Cinar and I. Yalcinkaya, “On the Behavior of Positive Solutions of the System of Rational Difference Equations, x

[6] G. Papaschinopoulos, “On a System of Two Nonlinear Difference Equations,” Journal of Mathematicsal Analysis and Applications, Vol. 219, 1998, pp. 415-426.

[7] 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.

[8] [8] E. Camouzis, C. M. Kent, G. Ladas and C. D. Lynd, “On the Global Character of Solutions of the System: x

[9] [9] E. Camouzis, E. Drymonis, G. Ladas and W. Tikjha, “Patterns of Boundedness of the Rational System x