Abstract
We explore the boundedness and persistence, existence of an invariant rectangle, local dynamical properties about the unique positive fixed point, global dynamics by the discrete-time Lyapunov function, and the rate of convergence of some -type exponential systems of difference equations. Finally, theoretical results are numerically verified.
1. Introduction
Recently, global dynamical properties of as well as -type exponential difference equations or systems of exponential difference equations have widely been explored. In this regard, Ozturk et al. [1] have explored the global dynamical properties of the following -type exponential difference equation:where and are the positive real numbers. Equation (1) may be viewed as a model in mathematical biology where is the immigration rate, is the population growth rate, and is the carrying capacity. Bozkurt [2] has explored the global dynamical properties of the following -type exponential difference equation:where and are the positive real numbers. More precisely, .Bozkurt [2] has explored the local asymptotic stability of the equilibrium point by linearized stability theorem, gasymptotic stability behavior by Lyapunov function, and semicycle analysis of positive solutions of the exponential difference equation, which is depicted in (2). Finally, theoretical results are verified numerically. Motivated from the aforementioned studies, here our purpose is to explore the global dynamical properties of the following -type exponential systems, that is the extension of the work of Bozkurt [2]:where and are the positive real numbers.
The rest of the paper is organized as follows: in Section 2, we explore that every positive solution of systems (3)–(8) is bounded and persists, whereas construction of an invariant rectangle is explored in Section 3. In Section 4, we explore the existence as well as uniqueness of the positive equilibrium point of systems (3)–(8). In Section 5, we explore the local dynamical properties about the unique positive equilibrium point of systems (3)–(8). In Section 6, we explore global dynamics about the positive equilibrium by the discrete-time Lyapunov function. We study the rate of convergence in Section 7, whereas discussion along with numerical simulations is presented in Section 8.
2. Boundedness and Persistence of Systems (3)–(8)
Theorem 1. Every positive solution of systems (3)–(8) is bounded and persists.
Proof. (i)If is a positive solution of (3), then From (3) and (9), one gets So, from (9) and (10), one has(ii)If is a positive solution of (4), then From (4) and (12), one gets So, from (12) and (13), one gets(iii)If is a positive solution of (5), then From (5) and (15), one gets So, from (15) and (16), one gets(iv)If is a positive solution of (6), then From (6) and (18), one gets So, from (18) and (19), one gets(v)If is a positive solution of (7), then From (7) and (21), one gets So, from (21) and (22), one gets(vi)If is a positive solution of (8), then From (8) and (24), one gets So, from (24) and (25), one gets
3. Existence of Invariant Rectangle of Systems (3)–(8)
Theorem 2. If is a positive solution of systems (3)–(8), then their corresponding invariant rectangles, respectively, are , , , , and .
Proof. If is a positive solution with , and , then from (3), one hasHence (27) then implies that . Finally from (3), it is easy to establish that if .
Remark 1. In a similar way, one can prove the invariant rectangle for systems (4)–(8).
4. Existence as well as Uniqueness of Positive Fixed Point of Systems (3)–(8)
Existence as well as uniqueness of a positive fixed point of systems (3)–(8) is explored in this section, as follows.
Theorem 3. (i)System (3) has a unique positive fixed point: if where(ii)System (4) has a unique positive fixed point: if where(iii)System (5) has a unique positive fixed point: if where(iv) System (6) has a unique positive fixed point: if where(v)System (7) has a unique positive fixed point: if where(vi)System (8) has a unique positive fixed point: if where
Proof. (i)From (3), one hasFrom (40), one getsFrom (41), settingDenotewhereand . Here, our finding is that has a unique solution, where . From (43) and (44), one getswhereUsing (44) and (46) in (45), we obtainwhere is depicted in (29). Now, assuming that if (28) along with (29) holds then from (47), one gets .
Remark 2. The proof of (ii)–(vi) is same as the proof of (i). So, it is omitted.
5. Local Dynamics about Unique Positive Fixed Point of Systems (3)–(8)
The local dynamics about , respectively, of systems (3) to (8) is explored in this section, as follows.
Theorem 4. For of (3), the following holds:(i) is a sink if(ii) is a source if
Proof. (i)If is a fixed point of (3), then Moreover, we have the following map for constructing the corresponding linearized form of (3): where The about subject to the map (51) is where The auxiliary equation of about is where Now, Assuming that (48) holds, and then from (57), one gets . Hence, of (3) is a sink.(ii)Using similar manipulation as in the proof of (i), one hasAssuming that (49) holds, and then from (58), one gets . Hence, of (3) is a source.
In similar manner, one can explore the local dynamics about , respectively, of systems (4)–(8), as follows.
Theorem 5. (i)For of (4), the following holds: (i.1) is a sink if (i.2) is a source if with where(ii)For of (5), the following holds: (ii.1) is a sink if (ii.2) is a source if with where(iii)For of (6), the following holds: (iii.1) is a sink if (iii.2) is a source if with where(iv)For of (7), the following holds: (iv.1) is a sink if (iv.2) is a source if with where(v)For of (8), the following holds: (v.1) is a sink if (v.2) is a source ifwithwhere
Proof. It is similar to the proof of Theorem 4. So its proof is omitted. □
Hereafter, by constructing a Lyapunov function with discrete time motivated from the work of [2], global dynamics about , respectively, of systems (3)–(8) is explored.
6. Global Dynamics of Systems (3)–(8)
Theorem 6. For global dynamics about , respectively, of systems (3)–(8), following statements hold:(i) of (3) is global asymptotically stable if(ii) of (4) is global asymptotically stable if(iii) of (5) is global asymptotically stable if(iv) of (6) is global asymptotically stable if(v) of (7) is global asymptotically stable if(vi) of (8) is global asymptotically stable if
Proof. (i)Consider the following discrete-time Lyapunov function:NowFrom (80) and (87), one gets . Hence, we obtain that , and thus of (3) is global asymptotically stable. □
Remark 3. The proof of (ii)–(vi) is same as the proof of (i).
7. Rate of Convergence
We will explore the convergence result about the equilibrium point of systems (3)–(8) motivated from the existing literature [3–5], in this section.
Theorem 7. If the positive solution of (3) is , .wherethen the error vector, .,satisfies the following relation:where are the roots of .
Proof. If the positive solution of (3) is , . (88) along with (89) holds. To find the error terms, one hasSo,Similarly,From (92) and (93), one getsDenoteIn view of (95), from (94), one getswhereFrom (97), one getsthat is,where as . Now, we have system 1.10 of [6], where andsuch that . So, about of (3) the limiting system becomeswhich is as about .
In the following theorem, we will summarize the convergence results for systems (4) to (8).
Theorem 8. (i)If the positive solution of (4) is , . (87) along with the following relation holds: then the error vector, which is depicted in (89), satisfies the following relations: where are the roots of .(ii)If the positive solution of (5) is , . (87) along with the following relation holds: then the error vector, which is depicted in (89), satisfies the following relations: where are the roots of .(iii)If the positive solution of (6) is , . (87) along with the following relation holds: then the error vector, which is depicted in (89), satisfies the following relations: where are the roots of .(iv)If the positive solution of (7) is , . (87) along with the following relation holds: then the error vector, which is depicted in (89), satisfies the following relations: where are the roots of .(v)If the positive solution of (8) is , . (87) along with the following relation holds:then the error vector, which is depicted in (89), satisfies the following relations:where are the roots of .
Proof. It is similar to Theorem 7, and hence its proof is omitted.
8. Discussion and Simulations
In the reported work, we have explored the global dynamics of -type exponential systems of difference equations. We have investigated that of systems (3) to (8) is bounded and persists, and the corresponding invariant rectangles, respectively, are , , , , , and . Further, we have explored the existence and uniqueness of the positive equilibrium and the global and local dynamics of systems (3)–(8). We have also investigated the rate of convergence of the positive solution of systems (3)–(8). Finally, some numerical examples are provided to support the theoretical results. For instance, if , respectively, are 13, 24, 319, 12, 0.1, 0.2, , and 0.002, then from Figures 1(a)–1(c), the positive fixed point of (3) is stable and its corresponding attractor is shown in Figure 1(s). Now, if , respectively, are 19, 14, 9, 112, 0.1, 0.2, , and 2, then from Figures 1(d)–1(f), the positive equilibrium point of (4) is stable and its corresponding attractor is shown in Figure 1(t). For (5), if , respectively, are 9, 0.4, 9, 12, 0.1, 0.2, , and 15, then from Figures 1(g)–1(i), its unique positive equilibrium point is stable and its corresponding attractor is shown in Figure 1(u). For (6), if , respectively, are 29, 14, 9, 12, 0.1, 0.2, , and 2, then from Figures 1(j)–1(l), its unique positive equilibrium point is stable and its corresponding attractor is shown in Figure 1(v). For (7), if , respectively, are 29, 14, 1.9, 12, 0.1, 1.2, , and 2, then from Figures 1(m)–1(o), its unique positive equilibrium point is stable and its corresponding attractor is shown in Figure 1(v). Finally, if , respectively, are 9, 4, 129, 12, 0.1, 25, , and 2, then from Figures 1(p)–1(r), the unique positive equilibrium point of system (8) is stable and its corresponding attractor is shown in Figure 1(w). For more results on dynamical properties of difference equations, we refer the reader to [7, 8] and the references cited therein.
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Data Availability
All the data utilized in this article have been included and the sources from where they were adopted are cited accordingly.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.
Acknowledgments
A. Q. Khan and H. M. Arshad research was partially supported by the Higher Education Commission of Pakistan, while the research of B. A. Younis was funded by a Deanship of Scientific Research in King Khalid University, under grant number GRP-326-40.