Discrete Dynamics in Nature and Society

Volume 2014 (2014), Article ID 607281, 9 pages

http://dx.doi.org/10.1155/2014/607281

Research Article

Behavior of an Exponential System of Difference Equations

A. Q. Khan and M. N. Qureshi

Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad 13100, Pakistan

Received 30 April 2014; Accepted 21 July 2014; Published 15 September 2014

Academic Editor: Zhan Zhou

Copyright © 2014 A. Q. Khan and M. N. Qureshi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study the qualitative behavior of the following exponential system of rational difference equations:  = /,   = /,   =  where , , , , , and and initial conditions , , , and are positive real numbers. More precisely, we investigate the boundedness character and persistence, existence and uniqueness of positive equilibrium, local and global behavior, and rate of convergence of positive solutions that converges to unique positive equilibrium point of the system. Some numerical examples are given to verify our theoretical results.

1. Introduction

Mathematical models of population dynamics have created great interest in the field of difference equations. As pointed out in [1, 2], to model biological phenomenon, discrete dynamical systems are more appropriate than continuous time models, being computationally efficient to get numerical results. Difference equations also appear naturally as discrete analogs of differential and delay differential equations and have applications in finance, biological, physical, and social sciences. Nonlinear difference equations and their stability analysis and global and local behaviors are of great interest on their own. For some interesting results in this regard we refer to [3–6] and the references therein. Exponential difference equations made their appearance in population dynamics. Though their analysis is hard, it is very interesting at the same time. Biologists believe that the equilibrium points and their stability analysis are important to understand the population dynamics.

El-Metwally et al. [3] have investigated the boundedness character, asymptotic behavior, periodicity nature of the positive solutions, and stability of equilibrium point of the following population model:

where the parameters , are positive numbers and the initial conditions are arbitrary nonnegative real numbers.

Ozturk et al. [7] have investigated the boundedness, asymptotic behavior, periodicity, and stability of the positive solutions of the following difference equation:

where the parameters , , and are positive numbers and the initial conditions are arbitrary nonnegative numbers.

Bozkurt [8] has investigated the local and global behavior of positive solutions of the following difference equation:

where the parameters ,  , and and the initial conditions are arbitrary positive numbers.

Motivated by the above studies, our aim in this paper is to investigate the qualitative behavior of positive solutions of the following exponential system of rational difference equations:

where the parameters ,  ,  ,  ,  , and are positive numbers and the initial conditions are arbitrary nonnegative real numbers.

More precisely, we investigate the boundedness character, 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 (4) which converge to its unique positive equilibrium point.

2. Boundedness and Persistence

The following theorem shows that every solution of (4) is bounded and persists.

Theorem 1. Every positive solution of the system (4) is bounded and persists.

Proof. Let be an arbitrary solution of (4); then

From (4) and (5), we have

So, from (5) and (6), we get

This proves the statement.

Theorem 2. Let be a positive solution of the system (4). Then, is invariant set for system (4).

Proof. It follows by induction.

3. Linearized Stability

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 (8) is uniquely determined by initial conditions for . Along with system (8) we consider the corresponding vector map . An equilibrium point of (8) 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 (8).

(i)An equilibrium point is said to be stable if for every there exists such that for every initial condition , implies for all , where is the 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 the map

where and are continuously differentiable functions at . The linearized system of (8) about the equilibrium point is

where and is the Jacobian matrix of the system (8) about the equilibrium point .

Let be equilibrium point of the system (4); then

To construct corresponding linearized form of system (4) we consider the following transformation:

where

The Jacobian matrix about the fixed point under the transformation (13) is given by

where

Lemma 5 (see [9]). For the 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.

The following theorem shows the existence and uniqueness of positive equilibrium point of system (4).

Theorem 6. If

where + /, then the system (4) has a unique positive equilibrium point in .

Proof. Consider the following system of algebraic equations:

Assume that ; then it follows from (18) that

Define

where , . It is easy to see that

if and only if

if and only if

Hence, has at least one positive solution in the interval . Furthermore, assume that condition (17) is satisfied; then one has

Hence, has a unique positive solution in . This completes the proof.

Theorem 7. If

where

then, the unique positive equilibrium point of the system (4) is locally asymptotically stable.

Proof. The characteristic equation of the Jacobian matrix about equilibrium point is given by

where ,  ,  , and .

Assuming condition (25) one has

Therefore, inequality (28) and Remark of reference [10] imply that the unique positive equilibrium point of the system (4) is locally asymptotically stable.

4. Global Character

Lemma 8 (see [11]). Let and be real intervals, and let and be continuous functions. Consider the system (8) with initial conditions . Suppose that the following statements are true.

(i) is nonincreasing in both arguments .(ii) is nonincreasing in both arguments .(iii)If is a solution of the system

such that and , then there exists exactly one equilibrium point of the system (8) such that .

Theorem 9. The unique positive equilibrium point of the system (4) is a global attractor.

Proof. Define and . It is easy to see that and are nonincreasing in both arguments and . Let be a solution of the system

Then, one has

Furthermore, arguing as in the proof of Theorem 1.16 of [11], it suffices to suppose that

Moreover,

From (31), we have

Using the fact that , one gets

From (36), we conclude that . Similarly, from (32), it is easy to show that . Hence, from Lemma 8, the unique positive equilibrium point of the system (4) is a global attractor.

Corollary 10. If condition (25) of Theorem 7 is satisfied, then the unique positive equilibrium point of the system (4) is globally asymptotically stable.

Proof. The proof is a direct consequence of Theorems 7 and 9.

5. 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 (4).

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 11 (Perron’s theorem [12]). Suppose that condition (38) holds. If is a solution of (37), then either for all large or

exists and is equal to the modulus of one of the eigenvalues of matrix .

Proposition 12 (see [12]). Suppose that condition (38) holds. If is a solution of (37), 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 (4) such that and . To find the error terms, one has from the system (4)

So,

Similarly,

From (43) and (44), we have

Let and . Then system (45) can be represented as

where

Moreover,

So, the limiting system of error terms can be written as

which is similar to linearized system of (4) about the equilibrium point . Using Proposition 11, one has the following result.

Theorem 13. Assume that is a positive solution of the system (4) such that and , with in and in . Then, the error vector of every solution of (4) satisfies both of the following asymptotic relations:

where are the characteristic roots of Jacobian matrix .

6. 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 (4). All plots in this section are drawn with Mathematica.

Example 1. Let ,  ,  ,  ,  , and . Then, the system (4) can be written as

with initial conditions ,  ,  , and .

The plot of for the system (51) is shown in Figure 1, plot of for the system (51) is shown in Figure 2, and its global attractor is shown in Figure 3. The unique positive equilibrium point of the system (51) is given by .

Figure 1: Plot of for the system (51).

Figure 2: Plot of for the system (51).

Figure 3: An attractor of the system (51).

Example 2. Let ,  ,  ,  ,  , and . Then, the system (4) can be written as

with initial conditions ,  ,  , and .

The plot of for the system (52) is shown in Figure 4, plot of for the system (52) is shown in Figure 5, and its global attractor is shown in Figure 6. The unique positive equilibrium point of the system (52) is given by .

Figure 4: Plot of for the system (52).

Figure 5: Plot of for the system (52).

Figure 6: An attractor of the system (52).

Example 3. Let ,  ,  ,  ,  , and . Then, the system (4) can be written as

with initial conditions ,  ,  , and .

The plot of for the system (53) is shown in Figure 7, plot of for the system (53) is shown in Figure 8, and its global attractor is shown in Figure 9. The unique positive equilibrium point of the system (53) is given by .

Figure 7: Plot of for the system (53).

Figure 8: Plot of for the system (53).

Figure 9: An attractor of the system (53).

7. Conclusion

This work is related to the 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 (4). Under certain parametric conditions the boundedness and persistence of positive solutions are proved. Moreover, we have shown that unique positive equilibrium point of system (4) 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 in determining the possible global behaviors of the system and determining which initial conditions lead to these long-term behaviors. Furthermore, rate of convergence of positive solutions of (4) which converge to its unique positive equilibrium point is demonstrated. Finally, some numerical examples are provided to support our theoretical results. These examples are experimental verification of our theoretical discussions.

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 anonymous referees for their valuable comments and suggestions leading to the improvement of this paper. This work was supported by the Higher Education Commission of Pakistan.

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