Global Asymptotic Stability of a Rational System

Abstract

The main goal of this paper is to investigate the global asymptotic behavior of the difference equation , , with and the initial value such that . The major conclusion shows that, in the case where , if the unique positive equilibrium exists, then it is globally asymptotically stable.

1. Introduction and Preliminaries

Our aim in this paper is to investigate the dynamics of the following difference equation: with and the initial value such that .

System (1) is a special case of the rational system where all parameters and the initial value are nonnegative such that denominators are always positive. There is some interest in systems of rational and related difference equations, for example, see [1–9]. In this paper, we will determine the global convergence properties of the system (1) under certain conditions.

When , the first component of the solution of the system (1) satisfies the first-order linear difference equation and the second component is constant and equal to for .

If the initial value is given by , then by simple iteration, it is easy to find that is the solution of (3). If , then . If , then for all , and for , we have .

Therefore, in the remaining part, we will assume that .

Clearly, is always an equilibrium, and when

(1) also has a unique positive equilibrium

Equation (1) was investigated in [10] and the main result they obtained is the following.

Theorem 1. (i) Assume that . Then every solution of the system (1) converges to if and only if , and when , the system (1) has unbounded solutions.
(ii) Assume that . Then every positive solution of the system (1) is bounded if and only if . In particular, when , the equilibrium is a global attractor of all solutions of the system (1).

In [10], the author proposed the following conjecture.

Conjecture 2. Assume that Show that the unique positive equilibrium of the system (1) is globally asymptotically stable.

Inspired by Conjecture 2, we investigate the global behavior of the system (1). To start our discussion, some basic results should be presented which will be useful in the sequel.

Consider the system where is continuous and .

A vital tool for dealing with the linearized stability of (8) is the following well-known result which we incorporate in the following lemma (see, e.g., [11, 12]).

Lemma 3. Let be a continuously differentiable function defined on an open set . (a)If the eigenvalues of the Jacobian matrix have modulus less than one, then the equilibrium of (8) is locally asymptotically stable.(b)If at least one of the eigenvalues of the Jacobian matrix has modulus greater than one, then the equilibrium of (8) is unstable.(c)The equilibrium of (8) is locally asymptotically stable if every solution of the characteristic equation of the Jacobian matrix lies inside the unit circle, that is, if In this case, is also called a sink.(d)The equilibrium of (8) is a repeller if every solution of characteristic equation (9) lies outside the unit circle, which is equivalent to the following condition: (e)The equilibrium of (8) is a saddle point if the Jacobian matrix has one eigenvalue that lies inside the unit circle and if the other one lies outside the unit circle, that is, if and only if

The following well-known comparison result will be used in estimating the value of a solution of the system (1).

Lemma 4 (a comparison result). Assume that and . Let and be sequences of real numbers such that and Then for .

Consider the following difference equation:

The following result of Hautus and Bolis [13] (see also [11, 12]) deals with the global attractivity of (14).

Lemma 5. Let be some interval and assume that satisfies the following conditions: (i) is nondecreasing in ;(ii)equation (14) has a unique positive equilibrium and the function satisfies the negative feedback condition: Then every positive solution of (14) with initial conditions in converges to .

To prepare for our major investigation, we consider the following equation: and the following lemma should be mentioned which is from [12].

Lemma 6. Let be an interval of real numbers and assume that is a continuous function satisfying the following properties: (i) is nondecreasing in each of its arguments;(ii)the function has a unique positive solution.Then (16) has a unique equilibrium and every solution of (16) converges to .

2. Linearized Stability

In this section, we will make some conclusions about linearized stability. Consider the map on associated with the system (1), that is, Calculating the partial derivatives of the functions and shows that

The Jacobian matrix of evaluated at is and its eigenvalues are and .

Another equilibrium , namely, (6), exists if and only if (5) holds. Using the equality , the Jacobian matrix of evaluated at is and its characteristic equation associated with is given by where

When , we find that and . Thus and

When , holds and by simple computation, we have

Furthermore, and

Employing Lemma 3, we formulate the results in the following.

Theorem 7. (i) The equilibrium of the system (1) is locally asymptotically stable when , and it is unstable (a saddle point) when , and it is nonhyperbolic when .
(ii) Assume that and (7) holds. Then the unique positive equilibrium of the system (1) is locally asymptotically stable.
(iii) Assume that and . Then the unique positive equilibrium of the system (1) is unstable; further, it is a saddle point.

3. Global Attractivity

In this section, we will commence global asymptotic stability analysis. Let be a solution of the system (1), then it is easy to obtain the following result from the second equation of the system (1).

Theorem 8. (i) Assume that . Then every solution of the system (1) satisfies for .
(ii) Assume that . Then every solution of the system (1) satisfies for .

Theorem 9. Every solution of the system (1) with converges to .

Proof. Since implies that for , thus , finishing the proof.

Theorem 10. Assume that . Then every solution of the system (1) with satisfies , .

Proof. Using Theorem 8, we get that when , and when , since the only equilibrium of the system (1) is when .
Further, using the boundedness of , we have
The proof is complete.

For the case where , the authors had obtained that the unique positive equilibrium is a global attractor of all solutions of the system (1) in [10], see Theorem 1 (ii). Moreover, in view of Theorem 7 (i), we may formulate the result in the following theorem.

Theorem 11. Assume that . Then the unique equilibrium of the system (1) is globally asymptotically stable.

Now, we pay attention to dealing with the global attractivity of the unique positive equilibrium , namely, (6), under the condition that . In this case, exists if and only if (7) holds. To obtain the global attractivity of , the following useful lemma should first be established.

Consider the following difference equation: where and the initial value . Equation (30) possesses two equilibria, namely, zero and .

Lemma 12. Every positive solution of (30) converges to the unique positive equilibrium .

Proof. Clearly, implies that for . Let , then is increasing in for and Thus for by applying Lemma 5.
The proof is complete.

Theorem 13. Assume that (7) holds. Then the unique positive equilibrium of the system (1) is globally asymptotically stable.

Proof. In view of Theorem 7, it is sufficient to show that is a global attractor of all positive solutions of the system (1).
In this case, holds for and thus the system (1) yields for . Let , , then the system (1) becomes
Further, the system (33) may reduce to the following second-order difference equation:
Clearly, zero is always the equilibrium of (34) and when (7) holds, (34) also possesses a unique positive equilibrium Notice that for , and we get and thus
Let , , then and . Hence by Lemma 12, we get that every positive solution of the following difference equation converges to its unique positive equilibrium .
Let , , then , which means that . Similarly, by Lemma 12, we know that every positive solution of the following difference equation converges to its unique positive equilibrium .
Applying Lemma 4 and (37), we find that every solution of (34) with initial value satisfies Hence for , there exists an integer such that for , Moreover,
Let , , then and every solution of (34) eventually enters the invariant interval .
Denote the function and simple computation shows that
Applying Lemma 6, to establish the global attractivity of the equilibrium of (34), it is sufficient to confirm that the following equation has a unique positive solution.
Solving (46), we get from which it follows that
Therefore, , and hence, Furthermore, and thus the result follows.
The proof is complete.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.