Abstract
We study the boundedness and persistence, existence, and uniqueness of positive equilibrium, local and global behavior of positive equilibrium point, and rate of convergence of positive solutions of the following system of rational difference equations: , , where the parameters , , , and for and initial conditions , , , and are positive real numbers. Some numerical examples are given to verify our theoretical results.
1. Introduction
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 [1–3]. In [4–10], applications of difference equations in mathematical biology are given. Nonlinear difference equations can be used in population models [11–17]. 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.
Gibbons et al. [18] investigated the qualitative behavior of the following second-order rational difference equation: Motivated by the above study, our aim in this paper is to investigate the qualitative behavior of positive solutions of the following second-order system of rational difference equations: where the parameters , , , and for and initial conditions , , , and are positive 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 (2) which converge to its unique positive equilibrium point.
2. Boundedness and Persistence
The following theorem shows the boundedness and persistence of every positive solution of system (2).
Theorem 1. Assume that and ; then every positive solution of system (2) is bounded and persists.
Proof. For any positive solution of system (2), one has where and for . Consider the following linear difference equations: Obviously, solutions of these second-order nonhomogeneous difference equations are given by where for depend upon initial conditions , , , and . Assume that and ; then the sequences and are bounded. Suppose that , , , and ; then by comparison we have Furthermore, from system (2) and (6) we obtain that From (6) and (7), it follows that Hence, theorem is proved.
Lemma 2. Let be a positive solution of system (2). Then, is invariant set for system (2).
Proof. The proof follows by induction.
3. Stability Analysis
Let us consider fourth-dimensional discrete dynamical system of the following 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 that 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 (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 the corresponding linearized form of system (2) we consider the following transformation: where , , , and . The linearized system of (2) about is given by where and the Jacobian matrix about the fixed point under the transformation (13) is given by
Lemma 5. 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.
The following theorem shows the existence and uniqueness of positive equilibrium point of system (2).
Theorem 6. Assume that and ; then there exists unique positive equilibrium point of system (2) in , if the following condition is satisfied:
Proof. Consider the following system of equations:
Assume that ; then it follows from (17) that
Take
where and . Then, we obtain that
Hence, it follows that
Furthermore,
Hence, has at least one positive solution in .
Furthermore, assume that condition (16) is satisfied; then one has
Hence, has a unique positive solution in . The proof is therefore completed.
Theorem 7. The unique positive equilibrium point of system (2) is locally asymptotically stable if .
Proof. The characteristic polynomial of Jacobian matrix about is given by Let and . Assume that 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 (24) satisfy , and it follows from Lemma 5 that the unique positive equilibrium point of the system (2) is locally asymptotically stable.
Arguing as in [2], we have following result for global behavior of (2).
Lemma 8. Assume that and are continuous functions and , , , and are positive real numbers with , . Moreover, suppose that and such that following conditions are satisfied: (i) is increasing in and decreasing in , and is decreasing in and increasing in ;(ii)let , , , and be real numbers such that , , , and ; then and .
Then, the system of difference equations , has a unique positive equilibrium point such that .
Theorem 9. The unique positive equilibrium point of system (2) is global attractor if .
Proof. Let and . Then, it is easy to see that is increasing in and decreasing in . Moreover, is decreasing in and increasing in . Let be a solution of the system Then, one has Furthermore, we have From (27), it follows that On subtracting (29), one has Similarly, from (30), we obtain Furthermore, from (31) and (32), we obtain where . Finally, from (33), it follows that . Similarly, it is easy to see that .
Lemma 10. Under the conditions of Theorems 7 and 9 the unique positive equilibrium of (2) is globally asymptotically stable.
4. 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 (2).
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, [19]). Suppose that condition (35) holds. If is a solution of (34), then either for all large or exists and is equal to the modulus of one of the eigenvalues of matrix .
Proposition 12 (see [19]). Suppose that condition (35) holds. If is a solution of (34), then either for all large or exists and is equal to the modulus of one of the eigenvalues of matrix .
Let be an arbitrary solution of the system (2) such that and , where and . To find the error terms, one has from the system (2) 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 (2) about the equilibrium point . Using Proposition 11, one has following result.
Theorem 13. Assume that is a positive solution of the system (2) such that and , where and . Then, the error vector of every solution of (2) satisfies both of the following asymptotic relations: where are the characteristic roots of Jacobian matrix .
5. Existence of Unbounded Solutions of (2)
In this section, we study the behavior of unbounded solutions of system (2).
Theorem 14. Consider system (2). Then, for every positive solution of (2) the following statements are true:(i)let and ; then as ;(ii)let and ; then as .
Proof. (i) Suppose that ; then it follows from Theorem 1 that , . Furthermore, from system (2) it follows that
where
Consider the following second-order difference equation:
The solution of (47) is given by
where , depend on initial values , . Moreover, assume that ; that is, ; then we obtain that is divergent. Hence, by comparison, we have as .
(ii) Assume that ; then from Theorem 1 we obtain that , . Moreover, from system (2) we have
where
Next, we consider the following second-order difference equation:
Then, it is easy to see that solution of (51) is given by
where , depend on initial values , . Furthermore, suppose that ; that is, ; then one has that is divergent. Hence, by comparison we have as .
6. Periodicity Nature of Solutions of (2)
Theorem 15. Assume that and ; then system (2) has no prime period-two solutions.
Proof. On the contrary, suppose that the system (2) has a distinctive prime period-two solutions where , , and , are positive real numbers for . Then, from system (2), one has After some tedious calculations from (54), we obtain where and . From (55), it follows that Similarly, from (56), we have Obviously, from (57) and (58), one has and , respectively, which is a contradiction. Hence, the proof is completed.
7. Examples
Example 1. Let , , , , , , , and . Then, system (2) can be written as
with initial conditions , , , and .
In this case, the unique positive equilibrium point of the system (59) is given by . Moreover, in Figure 1, the plot of is shown in Figure 1(a), the plot of is shown in Figure 1(b), and an attractor of the system (59) is shown in Figure 1(c).
Example 2. Let , , , , , , , and . Then, system (2) can be written as
with initial conditions , , , and .
In this case, the unique positive equilibrium point of the system (60) 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 (60) is shown in Figure 2(c).
Example 3. Let , , , , , , , and . Then, system (2) can be written as
with initial conditions , , , and .
In this case, the unique positive equilibrium point of the system (61) 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 (61) is shown in Figure 3(c).
8. Concluding Remarks
In literature, several articles are related to qualitative behavior of competitive system of planar rational difference equations [20]. It is very interesting mathematical problem to study the dynamics of competitive systems in higher dimension. This work is related to qualitative behavior of competitive system of second-order rational difference equations. We have investigated the existence and uniqueness of positive steady state of system (2). Under certain parametric conditions the boundedness and persistence of positive solutions is proved. Moreover, we have shown that unique positive equilibrium point of system (2) is locally as well as globally asymptotically stable. Furthermore, rate of convergence of positive solutions of (2) which converge to its unique positive equilibrium point is demonstrated. Finally, existence of unbounded solutions and periodicity nature of positive solutions of this competitive system are given.
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 improvement of this paper. For the first author, this work was partially supported by the Higher Education Commission of Pakistan.