Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 283982, 9 pages http://dx.doi.org/10.1155/2014/283982

Research Article Behavior of a Competitive System of Second-Order Difference Equations Q. Din,1 T. F. Ibrahim,2,3 and K. A. Khan4 1

Department of Mathematics, Faculty of Basic and Applied Sciences, University of Poonch Rawalakot, Rawalakot 12350, Pakistan Department of Mathematics, Faculty of Sciences and Arts (S.A.), King Khalid University, Abha, Sarat Abida 61914, Saudi Arabia 3 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt 4 Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan 2

Correspondence should be addressed to Q. Din; [email protected] Received 9 March 2014; Revised 5 April 2014; Accepted 26 April 2014; Published 15 May 2014 Academic Editor: Maoan Han Copyright © 2014 Q. Din et al. 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. 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: 𝑥𝑛+1 = (𝛼1 + 𝛽1 𝑥𝑛−1 )/(𝑎1 + 𝑏1 𝑦𝑛 ), 𝑦𝑛+1 = (𝛼2 + 𝛽2 𝑦𝑛−1 )/(𝑎2 + 𝑏2 𝑥𝑛 ), where the parameters 𝛼𝑖 , 𝛽𝑖 , 𝑎𝑖 , and 𝑏𝑖 for 𝑖 ∈ {1, 2} and initial conditions 𝑥0 , 𝑥−1 , 𝑦0 , and 𝑦−1 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:

of the following second-order system of rational difference equations: 𝑥𝑛+1 =

𝛼1 + 𝛽1 𝑥𝑛−1 , 𝑎1 + 𝑏1 𝑦𝑛

𝑦𝑛+1 =

𝛼2 + 𝛽2 𝑦𝑛−1 , 𝑎2 + 𝑏2 𝑥𝑛

(2)

where the parameters 𝛼𝑖 , 𝛽𝑖 , 𝑎𝑖 , and 𝑏𝑖 for 𝑖 ∈ {1, 2} and initial conditions 𝑥0 , 𝑥−1 , 𝑦0 , and 𝑦−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.

2. Boundedness and Persistence 𝑥𝑛+1 =

𝛼 + 𝛽𝑥𝑛−1 . 𝛾 + 𝑥𝑛

(1)

Motivated by the above study, our aim in this paper is to investigate the qualitative behavior of positive solutions

The following theorem shows the boundedness and persistence of every positive solution of system (2). Theorem 1. Assume that 𝛽1 < 𝑎1 and 𝛽2 < 𝑎2 ; then every positive solution {(𝑥𝑛 , 𝑦𝑛 )} of system (2) is bounded and persists.

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Proof. For any positive solution {(𝑥𝑛 , 𝑦𝑛 )} of system (2), one has 𝑥𝑛+1 ≤ 𝐴 1 + 𝐵1 𝑥𝑛−1 ,

𝑦𝑛+1 ≤ 𝐴 2 + 𝐵2 𝑦𝑛−1 , 𝑛 = 0, 1, 2, . . . ,

(3)

where 𝐴 𝑖 = 𝛼𝑖 /𝑎𝑖 and 𝐵𝑖 = 𝛽𝑖 /𝑎𝑖 for 𝑖 ∈ {1, 2}. Consider the following linear difference equations: 𝑢𝑛+1 = 𝐴 1 + 𝐵1 𝑢𝑛−1 ,

𝑛 = 0, 1, 2, . . . ,

V𝑛+1 = 𝐴 2 + 𝐵2 V𝑛−1 ,

𝑛 = 0, 1, 2, . . . .

𝑛 = 1, 2, . . . ,

𝑛 𝐴2 + 𝑐3 𝐵2𝑛/2 + 𝑐4 (−√𝐵2 ) , V𝑛 = 1 − 𝐵2

𝑛 = 1, 2, . . . ,

(5)

where 𝑐𝑖 for 𝑖 ∈ {1, 2, 3, 4} depend upon initial conditions 𝑢−1 , 𝑢0 , V−1 , and V0 . Assume that 𝛽1 < 𝑎1 and 𝛽2 < 𝑎2 ; then the sequences {𝑢𝑛 } and {V𝑛 } are bounded. Suppose that 𝑢−1 = 𝑥−1 , 𝑢0 = 𝑥0 , V−1 = 𝑦−1 , and V0 = 𝑦0 ; then by comparison we have 𝑥𝑛 ≤

𝛼1 = 𝑈1 , 𝑎1 − 𝛽1

𝑦𝑛 ≤

𝛼2 = 𝑈2 , 𝑎2 − 𝛽2

(6)

𝑛 = 1, 2, . . . . Furthermore, from system (2) and (6) we obtain that 𝑥𝑛+1

𝛼1 (𝑎2 − 𝛽2 ) 𝛼1 ≥ ≥ = 𝐿 1, 𝑎1 + 𝑏1 𝑦𝑛 𝑎1 (𝑎2 − 𝛽2 ) + 𝑏1 𝛼2

𝑦𝑛+1 ≥

𝛼2 (𝑎1 − 𝛽1 ) 𝛼2 ≥ = 𝐿 2. 𝑎2 + 𝑏2 𝑥𝑛 𝑎2 (𝑎1 − 𝛽1 ) + 𝑏2 𝛼1

𝐿 2 ≤ 𝑦𝑛 ≤ 𝑈2 ,

𝑛 = 1, 2, . . . .

(7)

(8)

Lemma 2. Let {(𝑥𝑛 , 𝑦𝑛 )} be a positive solution of system (2). Then, [𝐿 1 , 𝑈1 ] × [𝐿 2 , 𝑈2 ] is invariant set for system (2).

The point (𝑥, 𝑦) is also called a fixed point of the vector map 𝐹. Definition 3. Let (𝑥, 𝑦) be an equilibrium point of the system (9). (i) An equilibrium point (𝑥, 𝑦) is said to be stable if for every 𝜀 > 0 there exists 𝛿 > 0 such that, for every initial condition (𝑥𝑖 , 𝑦𝑖 ), 𝑖 ∈ {−1, 0} if ‖ ∑0𝑖=−1 (𝑥𝑖 , 𝑦𝑖 ) − (𝑥, 𝑦)‖ < 𝛿 implies that ‖(𝑥𝑛 , 𝑦𝑛 ) − (𝑥, 𝑦)‖ < 𝜀 for all 𝑛 > 0, where ‖ ⋅ ‖ is usual Euclidian norm in R2 . (ii) An equilibrium point (𝑥, 𝑦) is said to be unstable if it is not stable. (iii) An equilibrium point (𝑥, 𝑦) is said to be asymptotically stable if there exists 𝜂 > 0 such that 󵄩󵄩 0 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∑ (𝑥𝑖 , 𝑦𝑖 ) − (𝑥, 𝑦)󵄩󵄩󵄩 < 𝜂, 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑖=−1 󵄩󵄩

(11)

as 𝑛 󳨀→ ∞.

(iv) An equilibrium point (𝑥, 𝑦) is called global attractor if (𝑥𝑛 , 𝑦𝑛 ) → (𝑥, 𝑦) as 𝑛 → ∞.

Definition 4. Let (𝑥, 𝑦) be an equilibrium point of a map 𝐹 = (𝑓, 𝑥𝑛 , 𝑔, 𝑦𝑛 ), where 𝑓 and 𝑔 are continuously differentiable functions at (𝑥, 𝑦). The linearized system of (9) about the equilibrium point (𝑥, 𝑦) is 𝑋𝑛+1 = 𝐹 (𝑋𝑛 ) = 𝐹𝐽 𝑋𝑛 ,

(12)

𝑥𝑛 𝑦

𝑛 where 𝑋𝑛 = ( 𝑥𝑛−1 ) and 𝐹𝐽 is Jacobian matrix of system (9)

Proof. The proof follows by induction.

𝑦𝑛−1

about the equilibrium point (𝑥, 𝑦).

3. Stability Analysis Let us consider fourth-dimensional discrete dynamical system of the following form: 𝑥𝑛+1 = 𝑓 (𝑥𝑛 , 𝑥𝑛−1 , 𝑦𝑛 , 𝑦𝑛−1 ) ,

𝑛 = 0, 1, . . . ,

(10)

(v) An equilibrium point (𝑥, 𝑦) is called asymptotic global attractor if it is a global attractor and stable.

Hence, theorem is proved.

𝑦𝑛+1 = 𝑔 (𝑥𝑛 , 𝑥𝑛−1 , 𝑦𝑛 , 𝑦𝑛−1 ) ,

𝑦 = 𝑔 (𝑥, 𝑥, 𝑦, 𝑦) .

(𝑥𝑛 , 𝑦𝑛 ) 󳨀→ (𝑥, 𝑦)

From (6) and (7), it follows that 𝐿 1 ≤ 𝑥𝑛 ≤ 𝑈1 ,

𝑥 = 𝑓 (𝑥, 𝑥, 𝑦, 𝑦) ,

(4)

Obviously, solutions of these second-order nonhomogeneous difference equations are given by 𝑛 𝐴1 + 𝑐1 𝐵1𝑛/2 + 𝑐2 (−√𝐵1 ) , 𝑢𝑛 = 1 − 𝐵1

where 𝑓 : 𝐼2 × 𝐽2 → 𝐼 and 𝑔 : 𝐼2 × 𝐽2 → 𝐽 are continuously differentiable functions and 𝐼, 𝐽 are some intervals of real numbers. Furthermore, a solution {(𝑥𝑛 , 𝑦𝑛 )}∞ 𝑛=−1 of system (9) is uniquely determined by initial conditions (𝑥𝑖 , 𝑦𝑖 ) ∈ 𝐼 × 𝐽 for 𝑖 ∈ {−1, 0}. Along with system (9), we consider the corresponding vector map 𝐹 = (𝑓, 𝑥𝑛 , 𝑔, 𝑦𝑛 ). An equilibrium point of (9) is a point (𝑥, 𝑦) that satisfies

(9)

To construct the corresponding linearized form of system (2) we consider the following transformation: (𝑥𝑛 , 𝑦𝑛 , 𝑥𝑛−1 , 𝑦𝑛−1 ) 󳨃󳨀→ (𝑓, 𝑔, 𝑓1 , 𝑔1 ) ,

(13)

where 𝑓 = 𝑥𝑛+1 , 𝑔 = 𝑦𝑛+1 , 𝑓1 = 𝑥𝑛 , and 𝑔1 = 𝑦𝑛 . The linearized system of (2) about (𝑥, 𝑦) is given by 𝑍𝑛+1 = 𝐹𝐽 (𝑥, 𝑦) 𝑍𝑛 ,

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𝑥𝑛 𝑦

𝑛 where 𝑍𝑛 = ( 𝑥𝑛−1 ) and the Jacobian matrix about the fixed

Hence, it follows that

𝑦𝑛−1

𝐹 (𝐿 1 ) =

point (𝑥, 𝑦) under the transformation (13) is given by 𝐹𝐽 (𝑥, 𝑦)

> 0

−

𝑏2 𝑦 𝑎2 + 𝑏2 𝑥 1 0 (

= (−

𝛽1 𝑏1 𝑥 𝑎1 + 𝑏1 𝑦 𝑎1 + 𝑏1 𝑦 0

0

0 1

0 0

0 𝛽2 ) . 𝑎2 + 𝑏2 𝑥 0 0 )

𝛼1 − 𝐿1 𝑎1 − 𝛽1 + 𝑏1 𝑓 (𝐿 1 ) 𝛼1 (𝑎2 − 𝛽2 ) − 𝐿1 (𝑎1 − 𝛽1 ) (𝑎2 − 𝛽2 ) + 𝑏1 𝛼2

−

Lemma 5. Assume that 𝑋𝑛+1 = 𝐹(𝑋𝑛 ), 𝑛 = 0, 1, . . ., is a system of difference equations such that 𝑋 is a fixed point of 𝐹. If all eigenvalues of the Jacobian matrix 𝐽𝐹 about 𝑋 lie inside the open unit disk |𝜆| < 1, then 𝑋 is locally asymptotically stable. If one of them has a modulus greater than one, then 𝑋 is unstable.

𝛼1 (𝑎2 − 𝛽2 ) > 0. 𝑎1 (𝑎2 − 𝛽2 ) + 𝑏1 𝛼2

Furthermore, 𝐹 (𝑈1 ) = =

The following theorem shows the existence and uniqueness of positive equilibrium point of system (2). Theorem 6. Assume that 𝛽1 < 𝑎1 and 𝛽2 < 𝑎2 ; then there exists unique positive equilibrium point of system (2) in [𝐿 1 , 𝑈1 ] × [𝐿 2 , 𝑈2 ], if the following condition is satisfied:

(21)

𝛼1 (𝑎2 − 𝛽2 ) = (𝑎1 − 𝛽1 ) (𝑎2 − 𝛽2 ) + 𝑏1 𝛼2

(15)

𝛼1 − 𝑈1 𝑎1 − 𝛽1 + 𝑏1 𝑓 (𝑈1 ) (𝑎2 − 𝛽2 ) (𝑎1 − 𝛽1 ) + 𝛼1 𝑏2 𝛼1 ( − 1) 𝑎1 − 𝛽1 (𝑎2 − 𝛽2 ) (𝑎1 − 𝛽1 ) + 𝛼1 𝑏2 + 𝑏1 𝛼2

< 0. (22) Hence, 𝐹(𝑥) = 0 has at least one positive solution in [𝐿 1 , 𝑈1 ]. Furthermore, assume that condition (16) is satisfied; then one has 𝐹󸀠 (𝑥)

𝛼1 𝛼2 𝑏1 𝑏2 2

< (𝑎1 𝑎2 + 𝑏2 (𝑎1 − 𝛽1 )𝐿 1 − 𝑎1 𝛽2 − 𝑎2 𝛽1 + 𝛼2 𝑏1 + 𝛽1 𝛽2 ) . (16)

= (𝛼1 𝛼2 𝑏1 𝑏2 ( (𝑎1 𝑎2 + 𝑎1 𝑏2 𝑥 − 𝑎1 𝛽2 − 𝑎2 𝛽1 2 −1

+ 𝛼2 𝑏1 + 𝛽1 𝛽2 − 𝛽1 𝑏2 𝑥) ) ) − 1 (23)

Proof. Consider the following system of equations: 𝛼 + 𝛽1 𝑥 𝑥= 1 , 𝑎1 + 𝑏1 𝑦

𝛼 + 𝛽2 𝑦 𝑦= 2 . 𝑎2 + 𝑏2 𝑥

≤ (𝛼1 𝛼2 𝑏1 𝑏2 ( (𝑎1 𝑎2 + 𝑏2 (𝑎1 − 𝛽1 ) 𝐿 1 − 𝑎1 𝛽2 (17)

Assume that (𝑥, 𝑦) ∈ [𝐿 1 , 𝑈1 ] × [𝐿 2 , 𝑈2 ]; then it follows from (17) that 𝑥=

𝛼1 , 𝑎1 − 𝛽1 + 𝑏1 𝑦

𝑦=

𝛼2 . 𝑎2 − 𝛽2 + 𝑏2 𝑥

(18)

Take 𝐹 (𝑥) =

𝛼1 − 𝑥, 𝑎1 − 𝛽1 + 𝑏1 𝑓 (𝑥)

(19)

where 𝑓(𝑥) = 𝛼2 /(𝑎2 − 𝛽2 + 𝑏2 𝑥) and 𝑥 ∈ [𝐿 1 , 𝑈1 ]. Then, we obtain that 𝑓 (𝐿 1 ) =

𝑎1 (𝑎2 − 𝛽2 ) + 𝑏1 𝛼2 𝛼2 𝛼2 ( )< . 𝑎2 − 𝛽2 𝑎1 (𝑎2 − 𝛽2 ) + 𝑏1 𝛼2 + 𝑏2 𝛼1 𝑎2 − 𝛽2 (20)

2 −1

−𝑎2 𝛽1 + 𝛼2 𝑏1 + 𝛽1 𝛽2 ) ) ) − 1 < 0. Hence, 𝐹(𝑥) = 0 has a unique positive solution in [𝐿 1 , 𝑈1 ]. The proof is therefore completed. Theorem 7. The unique positive equilibrium point (𝑥, 𝑦) of system (2) is locally asymptotically stable if 𝑏1 𝑏2 𝑈1 𝑈2 + 𝛽1 𝛽2 + 𝛽1 (𝑎2 + 𝑏2 𝐿 1 ) + 𝛽2 (𝑎1 + 𝑏1 𝐿 2 ) < (𝑎1 + 𝑏1 𝐿 2 )(𝑎2 + 𝑏2 𝐿 1 ). Proof. The characteristic polynomial of Jacobian matrix 𝐹𝐽 (𝑥, 𝑦) about (𝑥, 𝑦) is given by 𝑃 (𝜆) = 𝜆4 − ( +

𝑏1 𝑏2 𝑥𝑦 𝛽1 𝛽2 + + ) 𝜆2 𝑎 + 𝑏 𝑦 𝑎 + 𝑏2 𝑥 (𝑎1 + 𝑏1 𝑦) (𝑎2 + 𝑏2 𝑥) 1 1 2

𝛽1 𝛽2 . (𝑎1 + 𝑏1 𝑦) (𝑎2 + 𝑏2 𝑥)

(24)

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Let Φ(𝜆) = 𝜆4 and Ψ(𝜆) = ((𝑏1 𝑏2 𝑥𝑦/(𝑎1 + 𝑏1 𝑦)(𝑎2 + 𝑏2 𝑥)) + (𝛽1 /(𝑎1 +𝑏1 𝑦))+(𝛽2 /(𝑎2 +𝑏2 𝑥)))𝜆2 −(𝛽1 𝛽2 /(𝑎1 +𝑏1 𝑦)(𝑎2 +𝑏2 𝑥)). Assume that 𝑏1 𝑏2 𝑈1 𝑈2 + 𝛽1 𝛽2 + 𝛽1 (𝑎2 + 𝑏2 𝐿 1 ) + 𝛽2 (𝑎1 + 𝑏1 𝐿 2 ) < (𝑎1 + 𝑏1 𝐿 2 )(𝑎2 + 𝑏2 𝐿 1 ) and |𝜆| = 1; then one has

and increasing in 𝑦. Let (𝑚1 , 𝑀1 , 𝑚2 , 𝑀2 ) be a solution of the system

|Ψ (𝜆)| 𝑎2 + 𝑏2 𝑈1 ; then 𝑦𝑛 → ∞ as 𝑛 → ∞;

where 𝑟3 , 𝑟4 depend on initial values 𝑧−1 , 𝑧0 . Furthermore, suppose that 𝛽1 > 𝑎1 + 𝑏1 𝑈2 ; that is, 𝑑1 = 𝛽1 /(𝑎1 + 𝑏1 𝑈2 ) > 1; then one has {𝑧𝑛 } that is divergent. Hence, by comparison we have 𝑥𝑛 → ∞ as 𝑛 → ∞.

(ii) let 𝛽2 < 𝑎2 and 𝛽1 > 𝑎1 + 𝑏1 𝑈2 ; then 𝑥𝑛 → ∞ as 𝑛 → ∞.

6. Periodicity Nature of Solutions of (2)

Proof. (i) Suppose that 𝑎1 < 𝛽1 ; then it follows from Theorem 1 that 𝑥𝑛 ≤ 𝛼1 /(𝑎1 − 𝛽1 ) = 𝑈1 , 𝑛 = 1, 2, . . .. Furthermore, from system (2) it follows that 𝑦𝑛+1

𝛼 + 𝛽2 𝑦𝑛−1 = 2 𝑎2 + 𝑏2 𝑥𝑛

Theorem 15. Assume that 𝑎1 > 𝛽1 and 𝑎2 > 𝛽2 ; 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 . . . , (𝑝1 , 𝑞1 ) , (𝑝2 , 𝑞2 ) , (𝑝1 , 𝑞1 ) , . . .

𝛼 + 𝛽2 𝑦𝑛−1 ≥ 2 𝑎2 + 𝑏2 𝑈1

(45)

= 𝑐2 + 𝑑2 𝑦𝑛−1 ,

where 𝑝1 ≠ 𝑝2 , 𝑞1 ≠ 𝑞2 , and 𝑝𝑖 , 𝑞𝑖 are positive real numbers for 𝑖 ∈ {1, 2}. Then, from system (2), one has 𝑝1 =

where 𝑐2 =

𝛼2 , 𝑎2 + 𝑏2 𝑈1

𝑑2 =

𝛽2 . 𝑎2 + 𝑏2 𝑈1

(46)

Consider the following second-order difference equation: 𝑤𝑛+1 = 𝑐2 + 𝑑2 𝑤𝑛−1 ,

𝑛 = 0, 1, . . . .

(47)

The solution of (47) is given by 𝑛 𝑐2 𝑤𝑛 = + 𝑟1 𝑑2𝑛/2 + 𝑟2 (−√𝑑2 ) , 1 − 𝑑2

𝛼1 + 𝛽1 𝑥𝑛−1 𝑎1 + 𝑏1 𝑦𝑛

≥

𝛼1 + 𝛽1 𝑥𝑛−1 𝑎1 + 𝑏1 𝑈2

(49)

𝑑1 =

𝛽1 . 𝑎1 + 𝑏1 𝑈2

𝑛 = 0, 1, . . . .

𝜇 + √4𝑏2 𝛼1 (𝑎1 − 𝛽1 ) (𝑎2 − 𝛽2 ) + 𝜇2 𝑏2 (𝑎1 − 𝛽1 )

,

𝑝1 𝑝2 = (

𝑞1 + 𝑞2 =

𝜇 + √4𝑏2 𝛼1 (𝑎1 − 𝛽1 )(𝑎2 − 𝛽2 ) + 𝜇2 2𝑏2 (𝑎1 − 𝛽1 )

] + √4𝑏2 𝛼1 (𝑎1 − 𝛽1 ) (𝑎2 − 𝛽2 ) + ]2 𝑏1 (𝑎2 − 𝛽2 )

2𝑏1 (𝑎2 − 𝛽2 )

2

(55)

),

,

] + √4𝑏2 𝛼1 (𝑎1 − 𝛽1 )(𝑎2 − 𝛽2 ) + ]2

2

(56)

),

where 𝜇 = (𝑎2 − 𝛽2 )(𝛽1 − 𝑎1 ) + 𝑏2 𝛼1 − 𝑏1 𝛼2 and ] = (𝑎2 − 𝛽2 )(𝛽1 − 𝑎1 ) − 𝑏2 𝛼1 + 𝑏1 𝛼2 . From (55), it follows that 2

(57)

Similarly, from (56), we have (50)

2

(𝑞1 + 𝑞2 ) − 4𝑞1 𝑞2 = 0.

Next, we consider the following second-order difference equation: 𝑧𝑛+1 = 𝑐1 + 𝑑1 𝑧𝑛−1 ,

(54)

𝛼 + 𝛽2 𝑞2 𝑞2 = 2 . 𝑎2 + 𝑏2 𝑝1

(𝑝1 + 𝑝2 ) − 4𝑝1 𝑝2 = 0.

where 𝛼1 , 𝑎1 + 𝑏1 𝑈2

𝛼1 + 𝛽1 𝑝2 , 𝑎1 + 𝑏1 𝑞1

After some tedious calculations from (54), we obtain

𝑞1 𝑞2 = (

= 𝑐1 + 𝑑1 𝑥𝑛−1 ,

𝑐1 =

𝑝2 =

(48)

where 𝑟1 , 𝑟2 depend on initial values 𝑤−1 , 𝑤0 . Moreover, assume that 𝛽2 > 𝑎2 + 𝑏2 𝑈1 ; that is, 𝑑2 = 𝛽2 /(𝑎2 + 𝑏2 𝑈1 ) > 1; then we obtain that {𝑤𝑛 } is divergent. Hence, by comparison, we have 𝑦𝑛 → ∞ as 𝑛 → ∞. (ii) Assume that 𝑎2 < 𝛽2 ; then from Theorem 1 we obtain that 𝑦𝑛 ≤ 𝛼2 /(𝑎2 − 𝛽2 ) = 𝑈2 , 𝑛 = 1, 2, . . .. Moreover, from system (2) we have 𝑥𝑛+1 =

𝛼1 + 𝛽1 𝑝1 , 𝑎1 + 𝑏1 𝑞2

𝛼 + 𝛽2 𝑞1 𝑞1 = 2 , 𝑎2 + 𝑏2 𝑝2

𝑝1 + 𝑝2 = 𝑛 = 1, 2, . . . ,

(53)

(51)

(58)

Obviously, from (57) and (58), one has 𝑝1 = 𝑝2 and 𝑞1 = 𝑞2 , respectively, which is a contradiction. Hence, the proof is completed.

The Scientific World Journal

7

0.50 0.15 0.14

yn

xn

0.49 0.48 0.47

0.13 0.12

50

100

150

200

250

50

300

100

150

200

250

300

n

n (a) Plot of 𝑥𝑛 for the system (59)

(b) Plot of 𝑦𝑛 for the system (59)

0.15

yn

0.14 0.13 0.12

0.47

0.48 xn

0.49

0.50

(c) An attractor of the system (59)

Figure 1: Plots for the system (59).

7. Examples Example 1. Let 𝛼1 = 0.5, 𝛽1 = 12, 𝑎1 = 13, 𝑏1 = 0.2, 𝛼2 = 0.1, 𝛽2 = 17, 𝑎2 = 17.5, and 𝑏2 = 0.3. Then, system (2) can be written as 𝑥𝑛+1 =

0.5 + 12𝑥𝑛−1 , 13 + 0.2𝑦𝑛

𝑦𝑛+1 =

0.1 + 17𝑦𝑛−1 , 17.5 + 0.3𝑥𝑛

(59)

with initial conditions 𝑥0 = 0.46, 𝑥−1 = 0.5, 𝑦−1 = 0.11, and 𝑦0 = 0.14. In this case, the unique positive equilibrium point of the system (59) is given by (𝑥, 𝑦) = (0.484974, 0.154921). Moreover, in Figure 1, the plot of 𝑥𝑛 is shown in Figure 1(a), the plot of 𝑦𝑛 is shown in Figure 1(b), and an attractor of the system (59) is shown in Figure 1(c). Example 2. Let 𝛼1 = 10, 𝛽1 = 1.5, 𝑎1 = 1.6, 𝑏1 = 0.003, 𝛼2 = 12, 𝛽2 = 23, 𝑎2 = 23.1, and 𝑏2 = 0.02. Then, system (2) can be written as 𝑥𝑛+1 =

10 + 1.5𝑥𝑛−1 , 1.6 + 0.003𝑦𝑛

𝑦𝑛+1 =

12 + 23𝑦𝑛−1 , 23.1 + 0.02𝑥𝑛

(60)

with initial conditions 𝑥−1 = 82, 𝑥0 = 89, 𝑦−1 = 5.9, and 𝑦0 = 6. In this case, the unique positive equilibrium point of the system (60) is given by (𝑥, 𝑦) = (83.0225, 6.81644). Moreover,

in Figure 2, the plot of 𝑥𝑛 is shown in Figure 2(a), the plot of 𝑦𝑛 is shown in Figure 2(b), and an attractor of the system (60) is shown in Figure 2(c). Example 3. Let 𝛼1 = 3.2, 𝛽1 = 8, 𝑎1 = 8.1, 𝑏1 = 5.5, 𝛼2 = 4.2, 𝛽2 = 16, 𝑎2 = 16.1, and 𝑏2 = 8.5. Then, system (2) can be written as 4.2 + 16𝑦𝑛−1 3.2 + 8𝑥𝑛−1 , 𝑦𝑛+1 = , 𝑥𝑛+1 = (61) 8.1 + 5.5𝑦𝑛 16.1 + 8.5𝑥𝑛 with initial conditions 𝑥−1 = 3.9, 𝑥0 = 3.5, 𝑦−1 = 0.1, and 𝑦0 = 0.12. In this case, the unique positive equilibrium point of the system (61) is given by (𝑥, 𝑦) = (4.88876, 0.10083). Moreover, in Figure 3, the plot of 𝑥𝑛 is shown in Figure 3(a), the plot of 𝑦𝑛 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

8

The Scientific World Journal 89

6.8

88 6.6

86

yn

xn

87

6.4

85 6.2

84 83

6.0

50

100

150 n

200

250

300

50

(a) Plot of 𝑥𝑛 for the system (60)

100

150 n

200

250

300

(b) Plot of 𝑦𝑛 for the system (60)

6.8

yn

6.6 6.4 6.2 6.0

83

84

85

86

87

88

89

xn (c) An attractor of the system (60)

Figure 2: Plots for the system (60). 4.8 0.135

4.6

0.130 0.125 yn

4.2

0.120

4.0

0.115

3.8

0.110 0.105

3.6 100

200 n

300

400

100

(a) Plot of 𝑥𝑛 for the system (61)

200 n

(b) Plot of 𝑦𝑛 for the system (61)

0.135 0.130 0.125 yn

xn

4.4

0.120 0.115 0.110 0.105 3.6

3.8

4.0

4.2 xn

4.4

(c) An attractor of the system (61)

Figure 3: Plots for the system (61).

4.6

4.8

300

400

The Scientific World Journal 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

9

[12]

[13]

[14]

The authors declare that they have no conflict of interests regarding the publication of this paper.

[15]

Acknowledgments

[16]

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.

[17]

[18]

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