Abstract
In this paper, global dynamical properties of rational higher-order system are explored in the interior of . It is explored that under certain parametric conditions, the discrete-time system has at most eight equilibria. By the method of linearization, local dynamics has been explored. It is explored that positive solution of the system is bounded, and moreover fixed point is globally stable if , . It is also investigated that the positive solution of the system under consideration converges to . Lastly, theoretical results are confirmed by numerical simulation. The presented work is significantly extended and improves current results in the literature.
1. Introduction
It is a well-known fact that difference equations arise naturally as discrete analogues and as numerical solutions of differential as well as delay differential equations having applications in many fields like physics, biology, economy, and ecology. Recently, a lot of studies have been conducted concerning the global dynamics of difference equations and their systems [3–19]. It is really not easy to understand global dynamics of difference equations along their systems; particularly, investigating the global behavior of higher-order equations is a challenging job in recent years. Therefore, investigating the global dynamics of such difference equations along their systems is worth further consideration. For illustration, Gibbons et al. [20] explored global dynamics of the following equation:where and are positive constants. Çinar [21] investigated the dynamics of the following equations:where are positive constants. Shojaei et al. [22] investigated global dynamics of the following difference equations:where and are positive constants. Bajo and Liz [23] investigated the dynamics of the following difference equation:where and are positive constants. Zhang et al. [24] have extended the work explored by numerous authors [21–23] to investigate the dynamics of the following rational system:where and are positive constants. Recently, Qureshi and Khan [25] investigated the global dynamics of the following rational system, which is extension of the work [21–24]:where and are positive constants. Inspired from aforesaid studies, we will extend the work studied by numerous authors [21–25] to investigate global dynamics of the following -dimensional system:where and are positive constants.
The organization of this paper is as follows. In Section 2, existence of equilibria in and corresponding linearized form are investigated. Section 3 deals with the study of local dynamics about equilibrium points. Boundedness of positive solution for the discrete-time system is studied in Section 4. Further, global dynamics about is explored in Section 5. In Section 6, we studied the rate of convergence which converges to of the system. Theoretical results are numerically verified in Section 7, while concluding remarks are given in Section 8.
2. Equilibria and Linearized Form of System (7)
The existence of equilibrium solution in the interior of and linearized form about of system (7) are investigated in this section. So, existence of equilibrium solution can be summarized as the following lemma.
Lemma 1. In the interior of , discrete-time system (7) has at most eight equilibria. More precisely,(i), is the unique boundary point of discrete-time system (7).(ii) is the boundary equilibrium point of system (7) if .(iii) is the boundary equilibrium point of system (7) if .(iv) is the boundary equilibrium point of system (7) if .(v) is the boundary equilibrium point of discrete-time system (7) if .(vi) is the boundary equilibrium point of system (7) if .(vii) is the boundary equilibrium point of discrete-time system (7) if .(viii) is the unique positive equilibrium point of system (7) if .
Hereafter, we establish the corresponding linearized form of (7). For this, one has the following map in order to construct the corresponding linearized form:where
Finally, about under map (8) becomes
3. Local Dynamics about Equilibria
By Theorem 1.5 of [1], the detailed local stability analysis about boundary equilibria , , , , , , and and the positive equilibrium point will be investigated in this section.
3.1. Local Dynamics about Boundary Points
Theorem 1. of system (7) is a sink if
Proof. About the equilibrium point , (10) takes the following form:Now, if denotes characteristic roots of , the diagonal matrix , whereNow,So,From (16), one obtainsFrom (13) and (14), one obtainsIn view of (17) and (18), one obtainsFrom (19), one gets the required statement.
Theorem 2. of (7) is unstable.
Proof. About the equilibrium point , (10) takes the following form:whereNow, if denotes characteristic roots of , the diagonal matrix , where (13) holds. Moreover,where . From (16) and (17), we obtainFrom (23), one gets . But , and hence of (7) is unstable.
Similarly, local dynamics about and of system under consideration can be summarized as follows.
Theorem 3. (i) of system (7) is unstable.(ii) of system (7) is unstable.
Proof. Same as proof of Theorems 1 and 2.
Theorem 4. of system (7) is unstable.
Proof. About the equilibrium point , (10) takes the following form:Now, if denotes characteristic roots of , the diagonal matrix , where (13) holds. Moreover,From (16) and (17), we obtainwhich gives of (7) is unstable.
Similarly, local dynamics about of system under consideration can be summarized as follows.
Theorem 5. (i) of system (7) is unstable.(ii) of system (7) is unstable.
Proof. Same as proof of Theorems 1, 2, and 4.
3.2. Local Dynamics about Positive Point
Theorem 6. of (7) is unstable.
Proof. About the equilibrium point , (10) takes the following form:whereand is depicted in (21). Moreover,where . From (16) and (17), we obtain From (30), one can conclude that of (7) is unstable.
4. Boundedness
The boundedness of positive solution of (7) is investigated in this section, as follows.
Theorem 7. If is a positive solution of (7), then following holds for :
Proof. For , (31)–(33) are true trivially. Suppose that (31)–(33) are true for , that is,Finally for and using (7), one obtainsFrom (37)–(39), one has required result.
Lemma 2. If (11) holds, then of (7) is bounded.
Proof. It is consequence of Theorem 7.
Remark 1. From above theoretical studies, we conclude that no equilibria are locally stable except , and hence in the following section we will investigate that it is globally asymptotically stable under certain parametric conditions.
5. Global Dynamics about
Theorem 8. of system (7) is globally asymptotically stable if conditions, which are depicted in (11), hold.
Proof. By Theorem 1, of (7) is a sink and also is bounded by Lemma 2. In order to get the required statement, it is adequate to prove is nonincreasing. In view of (11), from (7), one gets , which again gives and . Thus, is decreasing because are nonincreasing. Similarly, it is easy to investigate that and are also decreasing. This implies that .
Lemma 3. Ifthen for of (7), the following holds:(i) if .(ii) if .(iii) if .
Lemma 4. If (38) holds, then for , one gets the following invariant intervals:(i) .(ii) .
Proof. This follows by induction.
6. Convergence Rate
Theorem 9. If (11) is true, then error vectorof positive solution of (7) satisfies the following relations:where is equivalent to modulus of one of the characteristic roots of calculated at .
Proof. If is an arbitrary solution of (7) such that and , thenSimilarly from (7), we obtainNow, setUtilizing (43) in (41) and (42), we getwhereFrom (45) and (46), we getSo, we have the limiting system [26]:where is depicted in (39) and is same as about . In particular, about it becomeswhich is same as linearized system of (7) about .
7. Numerical Simulations
We will present numerous numerical examples, which represent different dynamical behaviors of system (7), in this section.
Example 1. Figure 1 indicates the dynamics of (7) about if , , respectively, are and initial values , , , respectively, are 0.07, 0.9, 0.02, 0.9, 0.9, 1.4, 4.9, 0.9, and 0.8. More precisely, Figures 1(a)–1(c) show that of (7) is a sink while its corresponding global attractor is shown is Figure 1(d). Figure 1(d) clearly indicates that all solutions go towards ; this means that is globally asymptotically stable. Hence, this simulation agrees with the obtained results.
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Example 2. Figure 2 indicates the same dynamics of (7) about if , , respectively, are 14, 15, 33, 6, 5, 2, 11, 13, and 3 and initial values , , , respectively, are 0.07, 0.9, 0.07, 0.9, 0.02, 0.9, 0.9, 0.9, 0.9, 1.4, 4.9, 1.4, 4.9, 0.9, and 0.8.
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Example 3. Figure 3 indicates the dynamics of (7) about if , , respectively, are 14, 15, 33, 6, 5, 2, 11, 13, and 3 and initial values , , , respectively, are 0.7, 1.9, 0.07, 0.9, 0.07, 0.9, 0.02, 1.2, 2.9, 0.9, 1.4, 2.9, 0.29, 4.9, 1.4, 4.9, 0.9, and 0.8.
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8. Concluding Remarks
The presented work is a natural extension of [21–25]. In this work, we have investigated the global dynamics of higher-order rational system (7) which is our main contribution to dynamical system. We explored that system (7) has eight equilibria , and under certain conditions on parameters . We investigated that only is locally stable under the condition, which is depicted in (11), while the rest of the equilibria are locally unstable. Further, we have studied the global dynamics, boundedness, and rate of convergence of positive solution of (7). Finally, we have verified theoretical results numerically [2].
Data Availability
All the data utilized in this article have been included, and the sources from where they were adopted were cited accordingly.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.
Acknowledgments
This study was partially supported by the Higher Education Commission of Pakistan.