Abstract
In this paper, we deal with the global behavior of the positive solutions of the system of -difference equations where the initial conditions are nonnegative real numbers and the parameters , and are positive real numbers for , by extending some results in the literature. By the end of the paper, we give three numerical examples to support our theoretical results related to the system with some restrictions on the parameters.
1. Introduction
Recently, many works have been published on rational difference equations, which have an important position in applied sciences. In this process, many rational difference equations have been studied by mathematicians. And so, some equations have frequently been the subject of many articles using generalizations. Many typical examples of these can be found in the literature. For example, in [1], El-Owaidy et al. dealt with global behavior of the difference equationwith nonnegative parameters and initial conditions. Gumus and Soykan [2] dealt with the dynamical behavior of the positive solutions for a system of rational difference equations of the following form:where the parameters and initial conditions are positive real numbers. Tollu and Yalcinkaya [3] dealt with the dynamical behavior of the positive solutions for the following three-dimensional system of rational difference equations:where the parameters and initial conditions are positive real numbers. For more papers on this topic, see, for example, [4–29].
In the present paper, we investigate the global behavior of the positive solutions of the -dimensional system of difference equations:where the initial conditions are nonnegative real numbers and the parameters , and are positive real numbers for , by extending some recent results in the literature.
Remark 1. This paper extends the results of studies in the references [1–3]. That is to say, if we take , then system (4) reduces equation (1). If we take , then system (4) reduces system (2). Finally, if we take , then system (4) reduces system (3). So, system (4) is a natural generalization of equation (1), system (2), and system (3).
Note that system (4) can be written asby the change of variables , , …, with for . So, we will consider system (5) instead of system (4) from now.
2. Preliminaries
Let be some intervals of real numbers and be continuously differentiable functions. Then, for initial conditions the system of difference equations,has the unique solution . Also, an equilibrium point of system (6) is a point that satisfies the following system:
We rewrite system (6) in the vector formwhere , is a vector map such that , and
It is clear that if an equilibrium point of system (6) is , then the corresponding equilibrium point of system (8) is the point .
In this study, we denote by any convenient vector norm and the corresponding matrix norm. Also, we denote by a initial condition of system (8).
Definition 1. Let be an equilibrium point of system (8). Then,(i)The equilibrium point is called stable if for every there exists such that implies , for all . Otherwise, the equilibrium point is called unstable.(ii)The equilibrium point is called locally asymptotically stable if it is stable and there exists such that and as .(iii)The equilibrium point is called a global attractor if as .(iv)The equilibrium point is called globally asymptotically stable if it is both locally asymptotically stable and global attractor.The linearized system of (8) evaluated at the equilibrium iswhere is the Jacobian matrix of at the equilibrium . The characteristic equation of system (10) about the equilibrium iswith real coefficients and .
Theorem 1. (see [30]). Assume that is a equilibrium point of system (8). If all eigenvalues of the Jacobian matrix evaluated at lie in the open unit disk , then is locally asymptotically stable. If one of them has a modulus greater than one, then is unstable.
3. Global Stability
In this section, we investigate the stability of the two equilibrium points of system (5). When for , the point is the unique nonnegative equilibrium point of system (5). When for , the unique positive equilibrium point of system (5) is
Theorem 2. The following statements hold:(i)If for , then the equilibrium point of system (5) is locally asymptotically stable(ii)If for , then the equilibrium point of system (5) is unstable(iii)If for , then the positive equilibrium point of system (5) is unstable
Proof. (i)The characteristic equation of is given by It is easy to see that if for , then all the roots of the characteristic equation (13) lie in the open unit disk . So, the equilibrium point of (5) is locally asymptotically stable.(ii)It is clearly seen that if for , then some roots of characteristic equation (13) have absolute value greater than one. In this case, the equilibrium point of (5) is unstable.(iii)The characteristic polynomial of is given by where is the binomial coefficient. It is clear that if is an odd number, then has a root in interval since Also, if is an even number, then has a root in interval since So, from Theorem 1, we can say that if for , then the positive equilibrium point of system (5) is unstable.
Theorem 3. If for , then the equilibrium point of system (5) is globally asymptotically stable.
Proof. From Theorem 2, we know that if for , then the equilibrium point of system (5) is locally asymptotically stable. Hence, it suffices to show thatFrom system (5), we have thatfor . From (18), we have by inductionwhere for are the initial conditions. Consequently, by taking limits of inequalities in (19) when for , we have the limit in (17) which completes the proof.
4. Oscillation Behavior and Existence of Unbounded Solutions
In the following result, we are concerned with the oscillation of positive solutions of system (5) about the equilibrium point .
Theorem 4. Assume that , and let be a positive solution of system (5) such thatorfor . Then, oscillates about the equilibrium point with semicycles of length one.
Proof. Assume that (20) holds. (The case where (21) holds is similar and will be omitted.) From (5), we haveThen, the proof follows by induction.
In the following theorem, we show the existence of unbounded solutions for system (5).
Theorem 5. Assume that for , then system (5) possesses an unbounded solution.
Proof. From Theorem 4, we can assume that, without loss of generality, the solution of system (5) is such that and for and . Then, we havefrom which it follows thatwhich completes the proof.
5. Periodicity
In this section, we investigate the existence of period-two solution of system (5).
Theorem 6. If for , then system (5) possesses the prime period-two solutionwith . Furthermore, every solution of system (5) converges to a period-two solution.
Proof. Assume that for , and let be a solution of system (5). Then, from system (5), we havefor . From (26), we obtainfor . If for and , then for and for . Therefore,is a period-two solution of system (5) with and . Furthermore, from (26), we haveFrom (29) and (30), we obtain and for . That is, the sequences and for are nonincreasing. On the other hand, from (26), we have the inequalitieswhich show the boundedness of the solutions. Hence, the odd-index terms tend to one periodic point and the even-index terms tend to another periodic point. This completes the proof.
6. Numerical Examples
In this section, we give some numerical examples to support our theoretical results related to system (5) with some restrictions on the parameters and for .
Example 1. If , , , , , , and in system (5), we obtain the following system:We visualize the solutions of system (32) in Figures 1–3 for the initial conditions , , , , , , , , and .
Example 2. If , , , , , , , , and in system (5), we obtain the following system:We visualize the solutions of system (33) in Figures 4–6 for the initial conditions , , , , , , , , , , , and .
Example 3. If , , , , , and in system (5), we obtain the following system:with , , , , and . We visualize the solutions of system (34) in Figures 7–9 for the initial conditions , , , , , , , , , , , , , , and .
7. Conclusion
In this study, we have generalized some of the results in the literature. As shown in Section 1, equation (1) was developed systematically. By this study, we ended this development. More concretely, we investigated the local asymptotic stability, global asymptotic stability, periodicity, and oscillation behavior of system (5) which is the -dimensional generalization of equation (1). According to our findings, our results are consistent with the results of the paper [1] in the case of . Similarly, our results are in line with the results of the papers [2, 3] in the case of and , respectively.
Data Availability
The data used to support the findings of this study are available from the first author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest associated with this publication.
Acknowledgments
This research was supported by the National Natural Science Foundation of China (Grant nos. 11971142, 11871202, 61673169, 11701176, 11626101, and 11601485).