Behavior of a Competitive System of Second-Order Difference Equations

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: x n+1 = (α 1 + β 1 x n−1)/(a 1 + b 1 y n), y n+1 = (α 2 + β 2 y n−1)/(a 2 + b 2 x n), where the parameters α i, β i, a i, and b i for i ∈ {1,2} and initial conditions x 0, x −1, y 0, and y −1 are positive real numbers. Some numerical examples are given to verify our theoretical results.


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][2][3]. In [4][5][6][7][8][9][10], applications of difference equations in mathematical biology are given. Nonlinear difference equations can be used in population models [11][12][13][14][15][16][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 ∈ {1, 2} and initial conditions 0 , −1 , 0 , and −1 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.

Boundedness and Persistence
The following theorem shows the boundedness and persistence of every positive solution of system (2). Theorem 1. Assume that 1 < 1 and 2 < 2 ; then every positive solution {( , )} of system (2) is bounded and persists.
Proof. The proof follows by induction.
The point ( , ) is also called a fixed point of the vector map .
) 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 = +1 , = +1 , 1 = , and 1 = . The linearized system of (2) about ( , ) is given by The Scientific World Journal  (13) is given by Lemma 5. Assume that +1 = ( ), = 0, 1, . . ., 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 | | < 1, 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 1 < 1 and 2 < 2 ; 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 ( , ) ∈ [ 1 , 1 ] × [ 2 , 2 ]; then it follows from (17) that Take where ( ) = 2 /( 2 − 2 + 2 ) and ∈ [ 1 , 1 ]. Then, we obtain that Hence, it follows that Furthermore, Hence, ( ) = 0 has at least one positive solution in Furthermore, assume that condition (16) is satisfied; then one has Hence, ( ) = 0 has a unique positive solution in The proof is therefore completed.

Lemma 10.
Under the conditions of Theorems 7 and 9 the unique positive equilibrium of (2) is globally asymptotically stable.
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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 : Z + → × is a matrix function satisfying as → ∞, where ‖ ⋅ ‖ denotes any matrix norm which is associated with the vector norm Proposition 11 (Perron's Theorem, [19] Let {( , )} be an arbitrary solution of the system (2) such that lim → ∞ = and lim → ∞ = , where ∈ [ 1 , 1 ] and ∈ [ 2 , 2 ]. To find the error terms, one has from the system (2) Let 1 = − and 2 = − ; then one has 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.

Existence of Unbounded Solutions of (2)
In this section, we study the behavior of unbounded solutions of system (2).
In this case, the unique positive equilibrium point of the system (59) is given by ( , ) = (0.484974, 0.154921). 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).
In this case, the unique positive equilibrium point of the system (61) is given by ( , ) = (4.88876, 0.10083). 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).

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  The Scientific World Journal 9 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.