Global Behavior of a Discrete Anticompetitive System in the Plane

Abstract

The main goal of this paper is to investigate the global asymptotic behavior of the difference system with and the initial condition . We obtain some global attractivity results of this system for different values of the parameters, which answer the open problem proposed in “Rational systems in the plane, J. Difference Equ. Appl. 15 (2009), 303-323”.

1. Introduction and Preliminaries

Difference equations or systems have been attracting more and more attention by authors since these models display some complicated character comparing with its analogue differential equations; see [1–11]. In [12], the authors gave a discussion on the following rational system in the plane:System (1) contains, as special cases, a large number of equations whose dynamics have not been thoroughly understood yet and there still exists research aspect to be further studied. Several global asymptotic results for some special cases of (1) have been obtained in [13–18]. As a special anticompetitive system of (1), the difference systemis considered in this paper, where the parameters and the initial value , .

In [12], system is labeled as (16,16) and is concerned as an open problem (Open Problem 7) which asked for determining the boundedness of its solutions, the local stability of its equilibrium, the existence of prime period-two solutions, and the global character of system (2). To answer the open problem, the main goal of this paper is to study the dynamic behavior of system (2).

As far the definition of stability and the method of linearized stability, one can see [19–21].

2. Linearized Stability

The equilibrium of system (2) is the intersection of the two following curves:The slopes of the tangent line of the two curves at the origin are

When , , the two curves and have a unique intersection in , that is . When , , the curves and have also another intersection locating in the first quadrant, satisfying . So we have the following lemma.

Lemma 1. If , then system (2) has a unique fixed point . If , then system (2) has another positive fixed point locating in the first quadrant, satisfying .

The linearized system of (2) about isand its characteristic equation is So

The linearized system of (2) about isNoticing the fact that , the characteristic equation associated with can be written aswhere , .

A simple calculation shows that Since , we get , and hence . Hence,

Therefore, we have the following results.

Theorem 2. (i) Assume that . Then is the only equilibrium of system (2), and it is locally stable while and is nonhyperbolic while .
(ii) Assume that . Then and are equilibria of system (2) and they are all unstable. In fact, is a repeller, and is a saddle point.

3. Periodic Character

Let , . Then the following statements are true.

Lemma 3. (i) The initiation value generates the solution(ii) The initiation value generates the solution(iii) The initial value with generates the solution with for .

Theorem 4. System (2) has positive prime period-two solution if and only if . Further, the prime period-two solution possesses the form such that
(i) with ;
(ii) with

Proof. Applying Lemma 3 (i)-(ii) and the condition that , it is easy to see that the solution of system (2) which starting on either axis are all of period-two. This establishes the sufficient condition. Moreover, it is also a necessary and vital condition, as may be seen by the following argument.
Let , be a period-two solution of system (2). Then they should satisfy the following equation:from which it follows thatObviously, yields , or vice versa, this is a contradiction. Thus and , and (13) yields
Equation (12) also yieldsfrom which it follows thatSince and , , we obtainand so .
If , then , or vice versa. In this case, we claim that . Otherwise, there would be the fact that holds, a contradiction. So system (2) possesses a period-two solutionwhere .
Similarly, if , then , or vice versa, and system (2) possesses a period-two solutionwhere .
The proof is complete.

4. Global Attractivity

4.1. The Case

Theorem 5. Assume that . Then the unique equilibrium is globally asymptotically stable.

Proof. Let be a solution of system (2). Clearly, implies that . Notice that for yieldandSo . Consequently, is a global attractor and hence by Theorem 2 (i), the conclusion follows.

Consider the map on associated with system (2); that is,

Set

Lemma 6. Assume that . Then , with .

Proof. Using (22) and the condition that ,  , we haveandThe proof is complete.

Theorem 7. Assume that . Then
(i) every solution of system (2) starting on either axis is of period-two;
(ii) every solution of system (2) with converges to .

Proof. (i) It is a direct consequence of Theorem 4.
(ii) Let be a solution of system (2) with initial value . Then by Lemma 6, the subsequence , and , are all strictly decreasing and bounded below by zero. Notice the form of period-two solution of system (2) mentioned by Theorem 4; then there is only one result; that is, . This completes the proof.

4.2. The Case

Let for be the usual four quadrants at and numbered in a counterclockwise direction, e.g., . Then we have the following results.

Lemma 8. Assume that . Then , . Further, the regions and are invariant under the iteration by .

Proof. Let . Then and , sofrom which it follows that .
Further,from which it follows that .
The proof for is similar and is omitted, finishing the proof.

To obtain the global attractivity of the positive equilibrium , we further divide into four distinct regions, which are described as follows:

(i) ,

(ii) ,

(iii) ,

(iv) .

Lemma 9. Assume that . Then
(i) , ;
(ii) ,

Proof. (i) Let . Then and ; thusfrom which it follows that .
The proof of is similar and is omitted.
(ii) Here we only prove that ; the proof for is the same and is omitted.
Set . Then yield if , and if . Hencefrom which it follows that .
The proof is complete.

Lemma 10. Assume that . Then
(i) ,    with ;
(ii) ,    with .

Proof. Using the equalities that , , and , and can be rewritten as follows: and(i) In the region , yield   and ; hence , .
(ii) In the region , yield   and ; hence , .
The proof is complete.

Lemma 11. Assume that . Then

(i)  , with ;

(ii)  , with .

Proof. Here, we only prove (i), the proof of (ii) is similar and is omitted.
(i) Let . Then we haveand , . So From (31), we can get that , and so The proof is complete.

Theorem 12. Assume that . Then every solution with the initial value has the following character: (i) eventually enters the region and satisfies ;(ii) eventually enters the region and satisfies .

Proof. By Lemmas 9(ii) and 8, it easy to obtain that and for with .
In view of Lemma 10 (i), we find that in the region the sequence is strictly decreasing and bounded below by , and the sequence is strictly increasing. Notice that, in this case, system (2) has no prime period-two solution and the other equilibrium point; hence .
From Lemma 10 (ii), we find that in the region the sequence is strictly increasing, the sequence is strictly decreasing and bounded below by . Hence , and , and the proof is complete.

Similarly, we have the following result.

Theorem 13. Assume that . Then every solution with the initial value has the following character: (i) eventually enters the region and satisfies ;(ii) eventually enters the region and satisfies .

Theorem 14. Assume that . Then every solution with the initial value has stability trichotomy; that is, exactly one of the following three cases holds: (i) eventually enters the region satisfying and eventually enters the region satisfying ;(ii) eventually enters the region satisfying and eventually enters the region satisfying ;(iii)it remains in the region forever and satisfies .

Proof. Let . Then Lemma 8 implies that . Hence , , or .
Case (a). If , then Theorem 12 implies that , for . Further, , .
Case (b). If , then Theorem 13 implies that , for . Further, , .
Case (c). If and there exists an integer such that , then by Case (a), we have , for , and , .
If and there exists an integer such that , then by Case (b), we have , for . Furthermore, , .
If , and for all , then by Lemma 11 (i), we have that the sequences , and , are all strictly decreasing, and bounded below by , and , respectively. Hence they are all convergent; moreover, .
The proof is complete.

Similarly, we can show that the orbits starting in have the following character.

Theorem 15. Assume that . Then every solution with the initial value has stability trichotomy; that is, exactly one of the following three cases holds: (i) eventually enters the region satisfying and eventually enters the region satisfying ;(ii) eventually enters the region satisfying and eventually enters the region satisfying ;(iii)it remains in the region forever and satisfies .

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

It is gratefully acknowledged that this research was supported by Department of Mathematics of Tianshui Normal University, Gansu, China.