Abstract
This paper deals with the boundedness, persistence, and global asymptotic stability of positive solution for a system of third-order rational difference equations , , , where , ; , . Some examples are given to demonstrate the effectiveness of the results obtained.
1. Introduction
It is known that difference equation appears naturally as discrete analogous and as numerical solutions of differential equation and delay differential equation having many applications in economics, biology, computer science, control engineering, and so forth. The study of discrete dynamical systems described by difference equations has now been paid great attention by many mathematical researchers. Particularly, the persistence, boundedness, local asymptotic stability, global character, and the existence of positive periodic solutions can be discussed in many papers.
In 1998, DeVault et al. [1] proved every positive solution of the difference equation:where oscillates about the positive equilibrium of (1). Moreover every positive solution of (1) is bounded away from zero and infinity. Also the positive equilibrium of (1) is globally asymptotically stable.
In 2003, Abu-Saris and DeVault [2] studied the following recursive difference equation:where , are positive real numbers. For similar results the reader can refer to [3–9].
Difference equations or discrete dynamical systems are a diverse field which impact almost every branch of pure and applied mathematics. We refer to [10, 11] for basic theory of difference equations and rational difference equations. It is very interesting to investigate the qualitative behavior of the discrete dynamical systems of nonlinear difference equations. Recently there has been a lot of work concerning the global asymptotic stability, the periodicity, and the boundedness of nonlinear difference equations. Moreover similar results in [12–17] have been derived for systems of two nonlinear difference equations.
Papaschinopoulos and Schinas [12] investigated the global behavior for a system of the following two nonlinear difference equations:where is a positive real number, are positive integers, and are positive real numbers.
In 2005, Yang [13] studied the global behavior of the following system:where , , , , is a positive constant, and initial values are positive real numbers.
In 2012, Zhang et al. [14] investigated the global behavior for a system of the following third-order nonlinear difference equations:where , and the initial values , .
Motivated by the above mentioned discussion, in this paper we study the behavior of solutions of the following system:where is a positive constant. And initial conditions , .
Clearly, if , system (6) has always positive equilibrium. Consider If , system (6) has two additional positive equilibria: Our results in this paper are listed below: (i)If , every positive solution of system (6) is bounded.(ii)If , is locally asymptotically stable.(iii)If , every positive solution of (6) approaches .(iv)If , and are locally asymptotically stable.
2. Boundedness
Theorem 1. Let be a positive solution of (6). Then the following statements hold:(i) and for all .(ii)If , then, for , we have the following:
Proof. Assertion (i) is obviously true. We now prove assertion (ii). From (6) and in view of (i), we have, for all , thatLet be the solution of following system, respectively,such that
We prove by induction thatSuppose that (13) is true for . Then from (10) we get the following:Therefore (13) is true. From (11) we have the following:Then from (10), (13), and (14) the proof of the relation (9) follows immediately.
3. Stability
Theorem 2. Assume that . Then the equilibrium is locally asymptotically stable.
Proof. We can obtain easily that the linearized system of (6) about the positive equilibrium iswhereLet denote the eigenvalues of matrix , let be a diagonal matrix, where , , and Clearly, is invertible. Computing matrix , we obtain thatFrom and imply thatFurthermore,It is well known that has the same eigenvalues as : we have that This implies that the equilibrium of (6) is locally asymptotically stable.
To examine the global attractivity of , we need the following result.
Lemma 3. Assume . Then .
Proof. Consider the function . For , we have So is strictly increasing for . We derive that .
Theorem 4. Assume that . Then every positive solution of (6) converges to .
Proof. Let be an arbitrary positive solution of (6). Let From Theorem 1, we have , . The previous and (6) imply that which can derive that It follows that ; namely,If and , it follows from Lemma 3 and the condition that . Then a contradiction occurs. Thus we have either or .
We assume (the discussion for the case is similar). Then exists. Using (27) and in view of , it is obvious that , so . Then exists. From the uniqueness of the positive equilibrium of (6), we conclude that , .
Combining Theorems 2 and 4, we obtain the following theorem.
Theorem 5. Assume that . Then the positive equilibrium of (6) is globally asymptotically stable for all positive solutions.
Theorem 6. If , then the equilibria and are locally asymptotically stable.
Proof. The proof is similar to the proof of Theorem 2. We can obtain easily that the linearized system of (6) about the positive equilibrium iswhereLet denote the eigenvalues of matrix , let be a diagonal matrix, where , , and Clearly, is invertible. Computing matrix , we obtain thatFrom and we imply thatFurthermore, It is well known that has the same eigenvalues as ; we have that This implies that the equilibrium of (6) is locally asymptotically stable.
Similarly, we can prove that the equilibrium is locally asymptotically stable.
4. Numerical Example
In order to illustrate the results of the previous sections and to support our theoretical discussions, we consider several interesting numerical examples in this section. These examples represent different types of qualitative behavior of solutions to nonlinear difference equations and system of nonlinear difference equations.
Example 1. Consider the following system of difference equations:with and the initial values . Then the solution of system (35) is bounded and persists. And the system has a unique equilibrium , which is globally asymptotically stable (see Figure 1).
Let and the initial values , . Then the solution of system (35) is bounded and persists (see Figure 2).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was financially supported by the National Natural Science Foundation of China (Grant no. 11361012), the China Postdoctoral Science Foundation (no. 2013T60934), and the Scientific Research Foundation of Guizhou Provincial Science and Technology Department ([2013]J2083).