Discrete Dynamics in Nature and Society

Volume 2010, Article ID 834020, 8 pages

http://dx.doi.org/10.1155/2010/834020

## Global Behavior of a Higher-Order Difference Equation

Department of Mathematics, Hexi University, Zhangye, Gansu 734000, China

Received 18 March 2010; Revised 28 April 2010; Accepted 4 May 2010

Academic Editor: Elena Braverman

Copyright © 2010 Tuo Li and Xiu-Mei Jia. 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.

#### Abstract

This paper is concerned with the global behavior of higher-order difference equation of the form , , Under some certain assumptions, it is proved that the positive equilibrium is globally asymptotical stable.

#### 1. Introduction and Preliminaries

Nonlinear difference equations of order greater than one are of paramount importance in applications where the generation of the system depends on the previous generations. Such equations appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations which model various diverse phenomena in biology, ecology, physiology, physics, engineering, and economics [1–8]. The global character of difference equations is a most important topic and there have been many recent investigations and interest in the topic [1, 5–7, 9–14]. In particular, many researchers have paid attention to the global attractivity and convergence of the th-order recursive sequence [2, 10, 14–19] and several approaches have been developed for finding the global character of difference equations; see [2, 6, 7, 9–15, 17, 19–21]. Moreover, we refer to [3, 4, 6, 7, 16] and the references therein for the oscillation and nonoscillation of difference equations. However, a large number of the literatures concerned with the *rational* difference equations and it is not enough to understand the global dynamics of a general difference equations, particularly irrational difference equation.

In this paper we study the global behavior of higher-order difference equation of some genotype selection model: where , , , and initial conditions are . When , (1.1) reduces to which was introduced by [6] as an example of a map generation by a simple mode for frequency dependent natural selection. The local stability of positive equilibrium of (1.2) was investigated by [6].

We note that the appearance of in the selection coefficient reflects the fact that the environment at the present time depends upon the activity of the population at some time in the past and that this in turn depends upon the gene frequency at that time. The points 0, , and 1 are the only equilibrium solutions of (1.1). One can easily see that for all . If for some , then for all and if for some , then for all , So in the following, we will restrict our attention to the difference equation: By introducing the substitution then (1.3) becomes where , , , and are arbitrary initial conditions.

In the sequel we will consider (1.5). It is clear that (1.5) has unique positive equilibrium point

In the following, we give some results which will be useful in our investigation of the behavior of solutions of (1.5), and the proof of lemmas can be found in [6].

*Definition 1.1. *The equilibrium point of the equation is the point that satisfies the condition .

*Definition 1.2. *(a) A sequence is said to be oscillate about zero or simply oscillate if the terms are neither eventually all positive nor eventually all negative. Otherwise the sequence is called nonoscillatory. A sequence is called strictly oscillatory if for every , there exist such that .

(b) A sequence is said to be oscillate about if the sequence oscillates. The sequence is called strictly oscillatory about if the sequence is strictly oscillatory.

*Definition 1.3. *Let be an equilibrium point of equation ; then the equilibrium point is called (a)locally stable if for every there exists such that for all with , we have , for all ;(b)locally asymptotically stable if it is locally stable and if there exists such that for all with , we have (c)a global attractor if for all , we have (d)globally asymptotically stable if is locally stable and is a global attractor.

Lemma 1.4. *Assume that is a positive real number and is a nonnegative integer; then the following statements are true.*(a)*If , then every solution of (1.2) oscillates about if and only if .*(b)*If , then every solution of (1.2) oscillates about if and only if
*

Lemma 1.5. *The linear difference equation
**
where , is asymptotically stable provided that
*

Lemma 1.6. *The linear difference equation
**
where and are positive integers, is asymptotically stable provided that
*

Lemma 1.7. *Consider the difference equation
**
one assumes that the function satisfies the following hypotheses. *()*, and , where
*()* is nonincreasing in *()*The equation
has a unique positive solution *()*Either the function does not depend on or for every and
with
*()*The function does not depend on or that for every with** Furthermore assume that the function is given by
**
and has no periodic orbits of prime period 2; then is a global attractor of all positive solutions of (1.11).*

Lemma 1.8. *Let be a nonincreasing function, and let denote the (unique) fixed point of . Then the following statements are equivalent:*(a)* is only fixed point of in ;*(b)* for .*

#### 2. Main Results

In this section, we will investigate the asymptotic stability and global behavior of the positive equilibrium point of (1.5).

Theorem 2.1. *Assume that is a positive real number and is a nonnegative integer; then the following statements are true. *(a)*If , then every solution of (1.5) oscillates about if and only if .*(b)*If , then is an asymptotically stable solution of (1.5).*(c)*If , then is an asymptotically stable solution of (1.5).*

*Proof. *When , then , (1.5) becomes (1.2), and case (a) follows from Lemma 1.4(a).

When , the linearized equation of (1.5) about equilibrium point is
If , by applying the Lemma 1.6, (2.1) is asymptotically stable provided that
so for
and is an asymptotically stable solution of (1.5). The proof of (b) is completed.

If , by applying Lemma 1.5, (2.1) is asymptotically stable provided that
so for , and is an asymptotically stable solution of (1.5). The proof of (c) is completed.

Theorem 2.2. *Assume that is a positive real number and is a positive integer. Then the equilibrium of (1.5) is globally asymptotically stable, when one of the following three cases holds: *(a)*; *(b)*;*(c)* where is a constant with *

*Proof. *If one of the three conditions (a), (b), (c) is satisfied, by applying Theorem 2.1, is an asymptotically stable solution of (1.5). To complete the proof it remains to show that is a global attractor of all positive solutions of (1.5). We will employ Lemma 1.7; set
clearly equation satisfies the hypotheses of Lemma 1.7. Set
If , then
hence, the function does not depend on or that for every with ,
Furthermore assume that the function is given by
Set
then
When , then Hence,
where is a constant with , and clearly So is an increasing function, hence
and so, for , by Lemma 1.8, we know that is only fixed point of in , and by Lemma 1.7, is a global attractor of all positive solutions of (1.5). The proof is complete.

#### References

- R. P. Agarwal,
*Difference Equations and Inequalities*, vol. 228 of*Monographs and Textbooks in Pure and Applied Mathematics*, Marcel Dekker, New York, NY, USA, 2nd edition, 2000. View at Zentralblatt MATH · View at MathSciNet - M. J. Douraki, M. Dehghan, and A. Razavi, “On the global behavior of higher order recursive sequences,”
*Applied Mathematics and Computation*, vol. 169, no. 2, pp. 819–831, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Erbe, J. Baoguo, and A. Peterson, “Nonoscillation for second order sublinear dynamic equations on time scales,”
*Journal of Computational and Applied Mathematics*, vol. 232, no. 2, pp. 594–599, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Erbe and A. Peterson, “Oscillation criteria for second-order matrix dynamic equations on a time scale,”
*Journal of Computational and Applied Mathematics*, vol. 141, no. 1-2, pp. 169–185, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. G. Kelley and A. C. Peterson,
*Difference Equations*, Academic Press, Boston, Mass, USA, 1991, An introduction with application. View at MathSciNet - V. L. Kocić and G. Ladas,
*Global Behavior of Nonlinear Difference Equations of Higher Order with Applications*, vol. 256 of*Mathematics and Its Applications*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993. View at Zentralblatt MATH · View at MathSciNet - M. R. S. Kulenović and G. Ladas,
*Dynamics of Second Order Rational Difference Equations*, Chapman and Hall/CRC, Boca Raton, Fla, USA, 2002, With open problems and conjecture. View at MathSciNet - V. Tkachenko and S. Trofimchuk, “Global stability in difference equations satisfying the generalized Yorke condition,”
*Journal of Mathematical Analysis and Applications*, vol. 303, no. 1, pp. 173–187, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. P. Agarwal, W. T. Li, and P. Y. H. Pang, “Asymptotic behavior of a class of nonlinear delay difference equations,”
*Journal of Difference Equations and Applications*, vol. 8, no. 8, pp. 719–728, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Camouzis and G. Ladas,
*Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures*, vol. 5 of*Advances in Discrete Mathematics and Applications*, Chapman and Hall/CRC, Boca Raton, Fla, USA, 2008. View at MathSciNet - H. El-Owaidy and H. Y. Mohamed, “On the global attractivity of systems of nonlinear difference equations,”
*Applied Mathematics and Computation*, vol. 135, no. 2-3, pp. 377–382, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. L. Kocic and G. Ladas, “Global attractivity in a nonlinear second-order difference equation,”
*Communications on Pure and Applied Mathematics*, vol. 48, no. 9-10, pp. 1115–1122, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Ladas, “On the recursive sequence ${x}_{n+1}=(\alpha +\beta {x}_{n}+\gamma {x}_{n-1)}/(A+B{x}_{n}+{C}_{n-1})$,”
*Journal of Difference Equations and Applications*, vol. 1, no. 3, pp. 317–321, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Papaschinopoulos and C. J. Schinas, “Global asymptotic stability and oscillation of a family of difference equations,”
*Journal of Mathematical Analysis and Applications*, vol. 294, no. 2, pp. 614–620, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Aloqeili, “Global stability of a rational symmetric difference equation,”
*Applied Mathematics and Computation*, vol. 215, no. 3, pp. 950–953, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Dehghan and M. J. Douraki, “The oscillatory character of the recursive sequence ${x}_{n+1}=\alpha +\beta {x}_{n-k+1}/A+B{x}_{n-2k+1}$,”
*Applied Mathematics and Computation*, vol. 175, no. 1, pp. 38–48, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. E. Hamza and S. G. Barbary, “Attractivity of the recursive sequence ${x}_{n+1}=(\alpha +\beta {x}_{n}F({x}_{n-1},\dots ,{x}_{n-k})$,”
*Mathematical and Computer Modelling*, vol. 48, no. 11-12, pp. 1744–1749, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - L. X. Hu, W. T. Li, and S. Stević, “Global asymptotic stability of a second order rational difference equation,”
*Journal of Difference Equations and Applications*, vol. 14, no. 8, pp. 779–797, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Saleh and S. Abu-Baha, “Dynamics of a higher order rational difference equation,”
*Applied Mathematics and Computation*, vol. 181, no. 1, pp. 84–102, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Dehghan and R. Mazrooei-Sebdani, “The characteristics of a higher-order rational difference equation,”
*Applied Mathematics and Computation*, vol. 182, no. 1, pp. 521–528, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Yang, H. Lai, D. J. Evans, and G. M. Megson, “Global asymptotic stability in a rational recursive sequence,”
*Applied Mathematics and Computation*, vol. 158, no. 3, pp. 703–716, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet