Abstract
We study the global dynamics of a list of following systems of exponential difference equations: , , , , , , , , , , , , , , where and the initial conditions are arbitrary nonnegative real numbers. This proposed work is considerably extended and improve some existing results in the literature. Some numerical examples are given to not only verify our theoretical results but also provide different types of qualitative behavior of solution of these systems.
1. Introduction
Discrete dynamical systems described by difference equations are more appropriate for population dynamics as compared to continuous ones. Population models involve exponential difference equations and their stability analysis though complicated, but interesting. The beginning of 2 century has witnessed an increasing interest in the population dynamics. Consequently, many works have been appeared on difference equations or systems of difference equations involving exponential terms (see [1–14] and reference cited therein). For instance, Metwally et al. [1] have investigated the dynamics of the following second-order difference equation:which is the solution of the following logistic equation with piecewise constant arguments:where and initial conditions are arbitrary nonnegative real numbers. Equation (1) may be viewed as a model in Mathematical Biology where is immigration rate and is the population growth rate. Moreover it is also pointed out in [2] that this model is suggested by the people from the Harvard school of public health, studying the population dynamics of single-species . Further Papaschinopoulos et al. [2] and Papaschinopoulos and Schinas [3] produced nice results in this direction by investigating the dynamical properties like boundedness and persistence of positive solutions, existence of the unique positive equilibrium, and local and global asymptotic stability of two-species model described by systems of difference equations, which is natural extension of single-species population model depicted in (1).
In [4], Grove et al. have investigated the global dynamics of the positive solution of the following difference equations:where and initial conditions are arbitrary nonnegative real numbers. This equation can be considered as a biological model, since it arises from models studying the amount of litter in perennial grassland (see [6]). After that Papaschinopoulos et al. [5, 6] have studied the asymptotic behavior of the positive solution of two-species model which is also natural extension of single-species model represented in (3). In 2016, Wang and Feng [7] have investigated the dynamics of positive solution for the following difference equation which is naturally a new form of single-species model depicted in (1):where and initial conditions are arbitrary nonnegative real numbers. In biological point of view is immigration rate and is population growth rate.
Ozturk et al. [8] have investigated the global asymptotic stability, boundedness and periodic nature of the following -order exponential difference equation:where α, β, γ and are arbitrary nonnegative numbers. Equation (5) is also viewed as a model in Mathematical Biology where is immigration rate, is population growth rate and is the carrying capacity. Later Papaschinopoulos et al. [9] have investigated boundedness and persistence and local and global asymptotic behavior of two-species model which is natural extension of single-species model (5), represented by the following exponential systems of difference equations: where and initial conditions are nonnegative real numbers. Motivated from aforementioned studies, our aim in this paper is to investigate the global dynamics of three-species models, which is natural extension of single-species model studied by Ozturk et al. [8], represented by the following list of systems of exponential difference equations:where and the initial conditions are arbitrary nonnegative real numbers. Equations (7) to (11) may be viewed as discrete-time population models in Mathematical Biology and biological parameters can be interpretations as shown in Table 1.
The rest of the paper is organized as follows: In Section 2, we investigate the global dynamics and rate of convergence of system (7). This includes boundeness and persistence, existence and uniqueness of positive equilibrium point, local and global stability about the unique positive equilibrium point and rate of convergence that converge to its unique positive equilibrium point of the system. Sections 3, 4, 5 and 6 include same analysis for system (8), (9), (10) and (11), respectively. Section 7 is about the numerical simulations that verify our theoretical results. A brief conclusion is given in Section 8.
2. On the System: , ,
In this section we will study the main results for system (7).
2.1. Boundedness and Persistence
The following theorem shows that every positive solution of system (7) is bounded and persists.
Theorem 1. Every positive solution of system (7) is bounded and persists.
Proof. If is an arbitrary solution of (7), thenIn addition from (7) and (12), one getsHence from (12) and (13), one gets
2.2. Existence of Invariant Set for Solutions
Theorem 2. If is a positive solution of system (7), then is invariant set for system (7), where , , , , and .
Proof. For any positive solution of system (7) with initial conditions and, one has from (7)and AlsoandFinally, andHence and . Similarly one can show that if and , then and .
2.3. Existence and Uniqueness of Positive Equilibrium Point
In the following we will study the existence and uniqueness of positive equilibrium point of system (7).
Theorem 3. If where then system (7) has a unique positive equilibrium point (.
Proof. Consider the following system of equations:From (23) set and Denote whereand . We claim that has a unique solution in From (26) and (27), one getsIf is a solution of then from (26) and (27), one gets where In view of (29), (30), and (31), (28) becomesAfter some manipulation, equation (32) reduces into the following form:where . Now assuming that (21) along with (22) hold, then from (33) we get Hence has a unique positive solution in .
2.4. Local and Global Asymptotic Stability
Theorem 4. For local stability about of system (7), the following statements hold:
(i) Equilibrium of system (7) is locally asymptotically stable if (ii) Equilibrium of system (7) is an unstable if
Proof. If ( is equilibrium point of system (7), thenIn order to construct the corresponding linearized form of system (7), we consider the following transformation:whereThe Jacobian matrix about ( under the transformation (37) is given bywhereThe characteristic equation of about ( is given bywhereNowAssuming condition (34) holds, then from (43) one gets and hence Theorem 1.5 of [15] implies that equilibrium of system (7) is locally asymptotically stable.
Proof (ii). Using same manipulations as in the proof of (i), we haveAssume that (35) hold, then from (44) we get Hence by Theorem 1.5 of [15], of (7) is unstable.
Hereafter we will study the global dynamics of system (7) about the unique positive equilibrium point . This is main component in the theory of dynamical systems based on the knowledge of its present state. In recent years it is challenging task to determine the global dynamics for higher-order difference equations or systems of difference equations. So, in this paper we study the global dynamics of (7) about the by Lyapunov stability theory because semiconjugacy and weak contraction cannot be used to analyze global dynamics of system (7). Here we construct the discrete-time Lyapunov function: whose nonnegativity follows from inequality: (see [12–14]).
Theorem 5. Equilibrium of (7) is globally asymptotically stable if
Proof. Consider the following discrete-time Lyapunov function:where the nonnegativity of follows from the following inequality:Moreover, we have Now,Using (48) in (49) and assume that (45) hold then one getsfor all n . It follows that Hence, we obtain that . Therefore, of (7) is globally asymptotically stable.
In the following we will investigate the rate of convergence that converges to of system (7) by employing the method [16, 17].
2.5. Rate of Convergence
Theorem 6. If is a positive solution of (7) such thatwherethen the error vector of every solution of (7) satisfies both the following asymptotic relations: where are the characteristic root of .
Proof. If is a positive solution of system (7) such that (51) along with (52) holds. In order to find the error terms, one has from (7)that isSet In view of (56), (55) takes the following form:whereTaking the limits of , and , we obtainthat iswhere as . Now we have Poincar difference system (1.10) of [18], whereand and , . Thus the limiting system of error terms about ( can be written aswhich is similar to linearized system (7) about equilibrium (.
3. On the System: , ,
In this section our focus is to investigate the behavior of system (8).
3.1. Boundedness and Persistence
Theorem 7. Every positive solution of system (8) is bounded and persists.
Proof. If is an arbitrary solution of (8), thenAdditionally from (8) and (64), one getsFinally from (64) and (65), one gets
3.2. Existence of Invariant Set for Solutions
Theorem 8. If is a positive solution of system (8), then is invariant set for system (8), where , , , , and .
Proof. For any positive solution of (8) with initial conditions and . We have from system (8)andAlsoand FinallyandHence and . Similarly one can show that if and , then and
3.3. Existence and Uniqueness of Positive Equilibrium Point
Theorem 9. If where then system (8) has a unique positive equilibrium point (.
Proof. ConsiderFrom (75) set andDenote whereand . We claim that has a unique solution in From (78) and (79) one getsAndNow if is a solution of , then from (78) and (79) one gets where In view of equation (81), (82), and (83), (80) becomes After some manipulation, (84) reduces into the following form:where . Now assuming that (73) along with (74) hold then from (85) one gets Hence has a unique positive solution in
3.4. Local and Global Asymptotic Stability
Theorem 10. For local stability about of system (8), the following statements hold:
(i) Equilibrium of system (8) is locally asymptotically stable if(ii) Equilibrium of (8) is unstable if
Proof. If ( is equilibrium point (8), thenTo construct the corresponding linearized form (8), we consider the following transformation:whereThe Jacobian matrix about ( under the transformation (89) is given by where The characteristic equation of about ( is given bywhereNow compute as follows:Assuming that condition (86) holds, then from (95) one gets and hence by Theorem 1.5 of [15] equilibrium of system (8) is locally asymptotically stable.
Proof (ii). Using similar calculation one can prove that of system (8) is unstable.
Hereafter we will prove that of system (8) is globally asymptotically stable.
Theorem 11. Equilibrium of system (8) is globally asymptotically stable if
Proof. Using similar arrangements as for the proof of Theorem 5 and from (48) we obtainfor all n . In view of (96) it follows that Thus we obtain that and hence of system (8) is globally asymptotically stable.
3.5. Rate of Convergence
Theorem 12. If is a positive solution of system (8) such that (51) along with the following holds:then the error vector of every solution of (8) satisfies asymptotic relations (53), where are the characteristic root of Jacobian matrix .
Proof. If is a positive solution of (8) such that (51) along with (98) holds, then in order to find the error terms, one has from (8) that is,SetIn view of (101), (100) take the following form:whereTaking the limits one getsThat is,where as . Now we have system 1.10 of [18], where and . Thus the limiting system of error terms about ( can be written aswhich is similar to linearized system (8) about equilibrium (
4. On the System: , ,
In this we will investigate the dynamics of system (9). If ( is equilibrium point of system (9), thenTo construct the corresponding linearized form of system (9), we consider the following transformation:where
The Jacobian matrix about ( under the transformation is given bywhere
Theorem 13. Every positive solution of system (9) is bounded and persists.
Proof. If is an arbitrary solution of (9), thenNow from (9) and (113) we have Hence from (113) and (114) one gets
Theorem 14. If is a positive solution of system (9) then is invariant set for system (9) where , , , , and .
Proof. Follows by induction.
Note. In remaining section, we will only state other results for system (9) and their proofs are similar as we have done in Sections 2 and 3.
Theorem 15. If wherethen system (9) has a unique positive equilibrium point .
Theorem 16. For local stability about of system (9), the following statements hold:
(i) Equilibrium of system (9) is locally asymptotically stable if(ii) Equilibrium of system (9) is unstable if
Theorem 17. Equilibrium of system (9) is globally asymptotically stable if
Theorem 18. If is a positive solution of system (9) such that (51) along with the following holdsthen the error vector of every solution of (9) satisfies asymptotic relations (53), where are the characteristic root of .
Proof. If is a positive solution of system (9) such that (51) along with (121) holds then in order to find the error terms, one has from (9).that is,Set Using (124), (123) become whereTaking the limits one getsthat is,where as . Now we have system 1.10 of [18], whereand . Thus the limiting system of error terms about ( iswhich is similar to linearized system (9) about equilibrium (.
5. On the System: , ,
This section deals with the study of main results about system (10). If ( is equilibrium point of system (10), thenTo construct the corresponding linearized form of system (10), we consider the following transformation: whereThe Jacobian matrix about ( under the transformation is given bywhere
Theorem 19. Every positive solution of system (10) is bounded and persists.
Proof. If is an arbitrary solution of (10) thenFrom (10) and (136) one getsLastly from (136) and (137) one gets
Theorem 20. If is a positive solution of system (10), then is invariant set for system (10), where , , , and .
Remark 21. In the following we will state the results for the existence and uniqueness of positive equilibrium, condition under which the unique positive equilibrium is locally asymptotically stable, unstable, and globally asymptotically stable for the system (10).
Theorem 22. If wherethen system (10) has a unique positive equilibrium point .
Theorem 23. For local stability about of system (10), the following statements hold:
(i) Equilibrium of system (10) is locally asymptotically stable if (ii) Equilibrium of system (10) is unstable if
Theorem 24. Equilibrium of system (10) is globally asymptotically stable if
Theorem 25. If is a positive solution of system (10) such that (51) along with the following relation holds:then the error vector of every solution of (10) satisfies both asymptotic relations defined in (53), where are the characteristic root of .
6. On the System , ,
In this section, we will study the behavior of system (11). If ( is equilibrium point of system (11), thenTo construct the corresponding linearized form of system (11), we consider the following transformation:whereThe Jacobian matrix about ( under is given by where
Theorem 26. Every positive solution of system (11) is bounded and persists.
Proof. If is an arbitrary solution of (11), thenAdditionally from (11) and (150), one getsTherefore from (150) and (151), one gets
Hereafter we will state the main results for system (11) and their proof is similar as we have done in Sections 2 and 3.
Theorem 27. If is a positive solution of system (11), then is invariant set for system (11) where , , , , and .
Theorem 28. If where then system (11) has a unique positive equilibrium point (.
Theorem 29. For equilibrium of system (11), the following statements hold:
(i) Equilibrium of system (11) is locally asymptotically stable if (ii) Equilibrium of system (11) is unstable if
Theorem 30. Equilibrium ( of system (11) is globally asymptotically stable if
Theorem 31. If is a positive solution of system (11) such that (51) holds along with the relations then the error vector of every solution of (11) satisfies (53), where are the characteristic root of .
7. Numerical Simulations
In this section, we will provide some numerical simulations to not only verify obtained theoretical results in Sections 2–6 but also exhibit more interesting behavior of these systems. If then system (7) with initial values can be written as From computations it is easy to verify that for above chosen values of parameters the condition under which the unique positive equilibrium point of (159) is satisfied is as follows:Moreover, in Figure 1, plot of is shown in Figure 1(a), plot of is shown in Figure 1(b), plot of is shown in Figure 1(c) and attractor of system (159) is shown in Figure 1(d). Now if then system (8) with initial values can be written asFor chosen values of parameters the condition under which the unique positive equilibrium point of (161) is satisfied, is as follows:Moreover, in Figure 2, plot of is shown in Figure 2(a), plot of is shown in Figure 2(b), plot of is shown in Figure 2(c) and attractor of system (161) is shown in Figure 2(d). For system (9), we choose if with initial values thenFor these values the condition under which the unique positive equilibrium point of (163) is satisfied is as follows:Moreover, in Figure 3, plot of is shown in Figure 3(a), plot of is shown in Figure 3(b), plot of is shown in Figure 3(c) and attractor of system (163) is shown in Figure 3(d). Again if we choose with initial values , then system (10) can be written asFor parametric values the condition under which the unique positive equilibrium point of (165) is satisfied is as follows:Moreover, in Figure 4, plot of is shown in Figure 4(a), plot of is shown in Figure 4(b), plot of is shown in Figure 4(c) and attractor of system (165) is shown in Figure 4(d). Finally, if we choose with initial values then system (11) can be written asFor parametric values the condition under which the unique positive equilibrium point of (167) is satisfied is as follows:Moreover, in Figure 5, plot of is shown in Figure 5(a), plot of is shown in Figure 5(b), plot of is shown in Figure 5(c) and attractor of system (167) is shown in Figure 5(d).
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8. Conclusions
This work is related to the global dynamics of a list of 3×6 systems of exponential difference equations which is natural extension of single-species model studied by Oztruk et al. [8] to three-species model. For each system, we studied the dynamics including boundedness and persistence, existence of invariant interval, existence and uniqueness of positive equilibrium point, and local and global dynamics about the unique positive equilibrium point and conclusion is presented in Tables 2–5. Furthermore rate of convergence that converges to unique positive equilibrium point for each system is also demonstrated. Finally some numerical simulations are presented to verify theoretical results.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by Higher Education Commission (HEC) of Pakistan.