Abstract
We study the boundedness character and persistence, existence and uniqueness of positive equilibrium, local and global behavior, and rate of convergence of positive solutions of the following system of exponential difference equations: , , where the parameters , and for and initial conditions , and are positive real numbers. Furthermore, by constructing a discrete Lyapunov function, we obtain the global asymptotic stability of the positive equilibrium. Some numerical examples are given to verify our theoretical results.
1. Introduction
Many population models are governed by exponential difference equations. We refer to [1–6] and the references therein. Systems of nonlinear difference equations of higher-order are of paramount importance in applications. Such equations also appear naturally as discrete analogues and as numerical solutions of systems differential and delay differential equations which model diverse phenomena in biology, ecology, physiology, physics, engineering, and economics. For applications and basic theory of rational difference equations we refer to [7–9]. In [10–17] applications of difference equations in mathematical biology are given. It is very interesting to investigate the behavior of solutions of a system of nonlinear difference equations and to discuss the local asymptotic stability of their equilibrium points.
El-Metwally et al. [1] investigated boundedness character, asymptotic behavior, periodicity nature of the positive solutions, and stability of equilibrium point of the following population model: Papaschinopoulos et al. [2] studied the boundedness, the persistence, and the asymptotic behavior of positive solutions of the following two directional interactive and invasive species models: Papaschinopoulos et al. [3] investigated the asymptotic behavior of the solutions of the following three systems of difference equations of exponential form: Recently, Papaschinopoulos and Schinas [4] studied the asymptotic behavior of the positive solutions of the systems of the two difference equations: Motivated by the above study, our aim in this paper is to investigate the qualitative behavior of positive solutions of the following system of exponential difference equations: where the parameters , , , , , and for and initial conditions , , and are positive real numbers.
More precisely, we investigate the boundedness character and persistence, existence and uniqueness of positive steady state, local asymptotic stability and global behavior of unique positive equilibrium point, and rate of convergence of positive solutions of system (5) which converge to its unique positive equilibrium point. Some special cases of system (5) can be treated as population models of two species [3].
2. Main Results
The following theorem shows that every solution of (5) is bounded and persists.
Theorem 1. Every positive solution of system (5) is bounded and persists.
Proof. For any positive solution of system (5), one has Furthermore, from systems (5) and (6) we obtain that From (6) and (7), it follows that Hence, the theorem is proved.
Lemma 2. Let be a positive solution of system (5). Then, is invariant set for system (5).
Proof. The proof follows by induction.
2.1. Stability Analysis
Let us consider four-dimensional discrete dynamical system of the form: where and are continuously differentiable functions and are some intervals of real numbers. Furthermore, a solution of system (9) is uniquely determined by initial conditions for . Along with system (9) we consider the corresponding vector map . An equilibrium point of (9) is a point that satisfies The point is also called a fixed point of the vector map .
Definition 3. Let be an equilibrium point of the system (9).(i)An equilibrium point is said to be stable if for every there exists such that for every initial condition , , if implies for all , where is usual Euclidian norm in .(ii)An equilibrium point is said to be unstable if it is not stable.(iii)An equilibrium point is said to be asymptotically stable if there exists such that and as .(iv)An equilibrium point is called global attractor if as .(v)An equilibrium point is called asymptotic global attractor if it is a global attractor and stable.
Definition 4. Let be an equilibrium point of a map , where and are continuously differentiable functions at . The linearized system of (9) about the equilibrium point is where and is Jacobian matrix of system (9) about the equilibrium point .
To construct corresponding linearized form of system (5) we consider the following transformation: where , , , and . The linearized system of (5) about is given by where and the Jacobian matrix about the fixed point under transformation (12) is given by where
Lemma 5 (see [9]). Assume that , , is a system of difference equations such that is a fixed point of . If all eigenvalues of the Jacobian matrix about lie inside the open unit disk , then is locally asymptotically stable. If one of them has a modulus greater than one, then is unstable.
Theorem 6. System (5) has a unique positive equilibrium point , if the following condition is satisfied: where
Proof. Consider the following system of equations: From (18), it follows that Set where and . Then, it follows that Furthermore, it is easy to see that From (22) it follows that Let . Then can be expressed as Suppose that ; then, it follows that . Furthermore, we have Then it is easy to see that which gives that . Hence, has at least one positive solution in . Moreover, we obtain that where Then, from (28) it follows that and using (26) we obtain Hence, has a unique positive solution in . The proof is therefore completed.
Theorem 7. The unique positive equilibrium point of system (5) is locally asymptotically stable under the following condition:
Proof. The characteristic polynomial of Jacobian matrix about is given by Clearly, one root of is . To check the behavior of the other three roots of , we let and . Assume that (30) holds and ; then, one has Then, by Rouche’s Theorem, and have the same number of zeroes in an open unit disk . Hence, all the roots of (31) satisfy , and it follows from Lemma 5 that the unique positive equilibrium point of the system (5) is locally asymptotically stable.
Theorem 8. The unique positive equilibrium point of system (5) is globally asymptotically stable, if the following condition is satisfied:
Proof. Arguing as in [18], we consider the following discrete time analogue of Lyapunov function: Then nonnegativity of follows from the following inequality: Furthermore, we have Assume that (33) holds true; then, it follows that for all so that is monotonically decreasing sequence. It follows that . Hence, we obtain that Then it follows that and . Furthermore, for all , which gives that is uniformly stable. Hence, unique positive equilibrium point of system (5) is globally asymptotically stable.
2.2. Rate of Convergence
In this section we will determine the rate of convergence of a solution that converges to the unique positive equilibrium point of the system (5).
The following result gives the rate of convergence of solutions of a system of difference equations: where is an -dimensional vector, is a constant matrix, and is a matrix function satisfying as , where denotes any matrix norm which is associated with the vector norm
Proposition 9 ((Perron’s Theorem) [19]). Suppose that condition (40) holds. If is a solution of (39), then either for all large or exists and is equal to the modulus of one of the eigenvalues of matrix .
Proposition 10 (see [19]). Suppose that condition (40) holds. If is a solution of (39), then either for all large or exists and is equal to the modulus of one of the eigenvalues of matrix .
Let be any solution of the system (5) such that and , where and . To find the error terms, one has from the system (5) Let and ; then, one has where Moreover, Now the limiting system of error terms can be written as which is similar to linearized system of (5) about the equilibrium point .
Using Proposition 9, one has the following result.
Theorem 11. Assume that is a positive solution of the system (5) such that , and , where and . Then, the error vector of every solution of (5) satisfies both of the following asymptotic relations: where are the characteristic roots of Jacobian matrix .
2.3. Examples
In order to verify our theoretical results and to support our theoretical discussions, we consider several interesting numerical examples in this section. These examples represent different types of qualitative behavior of solutions to the system of nonlinear difference equations (5). The first and last examples show that positive equilibrium of system (5) is unstable with suitable parametric choices. Moreover, from the remaining examples it is clear that unique positive equilibrium point of system (5) is globally asymptotically stable with different parametric values.
Example 1. Let , , , , , , , , , , , and . Then, system (5) can be written as
with initial conditions and .
In this case the positive equilibrium point of the system (50) is unstable. Moreover, in Figure 1 the plot of is shown in Figure 1(a), the plot of is shown in Figure 1(b), and a phase portrait of the system (50) is shown in Figure 1(c).
Example 2. Let , , , , , , , , , , , and . Then, system (5) can be written as
with initial conditions , and .
In this case the unique positive equilibrium point of the system (51) is given by . Moreover, in Figure 2 the plot of is shown in Figure 2(a), the plot of is shown in Figure 2(b), and an attractor of the system (51) is shown in Figure 2(c).
Example 3. Let , , , , , , , , , , , and. Then, system (5) can be written as
with initial conditions , , , and .
In this case the unique positive equilibrium point of the system (52) is given by . Moreover, in Figure 3 the plot of is shown in Figure 3(a), the plot of is shown in Figure 3(b), and an attractor of the system (52) is shown in Figure 3(c).
Example 4. Let , , , , , , , , , , , and . Then, system (5) can be written as
with initial conditions , and .
In this case the unique positive equilibrium point of the system (53) is given by . Moreover, in Figure 4 the plot of is shown in Figure 4(a), the plot of is shown in Figure 4(b), and an attractor of system (53) is shown in Figure 4(c).
Example 5. Let , , , , , , , , , , , and . Then, system (5) can be written as
with initial conditions , , , and .
In this case the positive equilibrium point of the system (54) is unstable. Moreover, in Figure 5 the plot of is shown in Figure 5(a), the plot of is shown in Figure 5(b), and a phase portrait of system (54) is shown in Figure 5(c).
3. Concluding Remarks
In literature several articles are related to qualitative behavior of exponential systems of rational difference equations. It is a very interesting mathematical problem to study the dynamics of such systems because these are closely related to models in population dynamics and biological sciences. This work is related to qualitative behavior of an exponential system of second-order rational difference equations. We have investigated the existence and uniqueness of positive steady state of system (5). Under certain parametric conditions the boundedness and persistence of positive solutions are proved. Moreover, we have shown that unique positive equilibrium point of system (5) is locally as well as globally asymptotically stable. The main objective of dynamical systems theory is to predict the global behavior of a system based on the knowledge of its present state. An approach to this problem consists of determining the possible global behaviors of the system and determining which parametric conditions lead to these long-term behaviors. By constructing a discrete Lyapunov function, we have obtained the global asymptotic stability of the positive equilibrium of (5). Furthermore, rate of convergence of positive solutions of (5) which converge to its unique positive equilibrium point is demonstrated. Finally, some illustrative examples are provided to support our theoretical discussion.
Conflict of Interests
The authors declare that they have no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors thank the main editor and the anonymous referees for their valuable comments and suggestions leading to improvement of this paper. This work was supported by the Higher Education Commission of Pakistan.