Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 834020 | https://doi.org/10.1155/2010/834020

Tuo Li, Xiu-Mei Jia, "Global Behavior of a Higher-Order Difference Equation", Discrete Dynamics in Nature and Society, vol. 2010, Article ID 834020, 8 pages, 2010. https://doi.org/10.1155/2010/834020

Global Behavior of a Higher-Order Difference Equation

Academic Editor: Elena Braverman
Received18 Mar 2010
Revised28 Apr 2010
Accepted04 May 2010
Published23 Jun 2010

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.

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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.


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