Advances in Difference Equations

Advances in Difference Equations This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon.

Dynamics of a discrete Lotka-Volterra model Advances in Difference Equations 2013, 2013:95

doi:10.1186/1687-1847-2013-95

Qamar Din ([email protected])

ISSN Article type

1687-1847 Research

Submission date

3 January 2013

Acceptance date

18 March 2013

Publication date

8 April 2013

Article URL

http://www.advancesindifferenceequations.com/content/2013/1/95

This peer-reviewed article can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Advances in Difference Equations go to http://www.advancesindifferenceequations.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com

© 2013 Din This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ADE_231_edited

[04/05 13:56]

1/15

Dynamics of a discrete Lotka-Volterra model Q. Din Abstract. In this paper, we study the equilibrium points, local asymptotic stability of equilibrium points, and global behavior of equilibrium points of a discrete Lotka-Volterra model given by δyn + xn yn αxn − βxn yn , yn+1 = , xn+1 = 1 + γxn 1 + ηyn where parameters α, β, γ, δ, , η ∈ R+ , and initial conditions x0 , y0 are positive real numbers. Moreover, the rate of convergence of a solution that converges to the unique positive equilibrium point is discussed. Some numerical examples are given to verify our theoretical results. Mathematics Subject Classification (2010). 39A10; 40A05. Keywords. difference equations; equilibrium points; local stability; global character.

1. Introduction and preliminaries Many authors investigated the ecological competition systems governed by differential equations of Lotka-Volterra type. Many interesting results related with the global character and local asymptotic stability have been obtained. We refer to [5, 6] and the references therein. Already, many authors [7, 8] have argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations are of non-overlapping generations. Particularly, the persistence, boundedness, local asymptotic stability, global character, and the existence of positive periodic solutions. The discrete Lotka-Volterra models have many applications in applied sciences. Such models were first established in mathematical biology, and then their applications were spread to other fields [1, 2, 3, 4]. Several variations of the Lotka-Volterra predator-prey model have been proposed that offer more realistic descriptions of the interactions of the populations. If the population of rabbits is always much larger than the number of foxes, then This work was supported by HEC of Pakitan.

ADE_231_edited

[04/05 13:56]

2

2/15

Q. Din

the considerations that entered into the development of the logistic equation may come into play. If the number of rabbits becomes sufficiently great, then the rabbits may be interfering with each other in their quest for food and space. One way to describe this effect mathematically is to replace the original model by the more complicated system. Most predators feed on more than one type of food. If the foxes can survive on an alternative resource, although the presence of their natural prey (rabbits) favors growth, a possible alternative model is the discrete dynamical system xn+1 =

δyn + xn yn αxn − βxn yn , yn+1 = , 1 + γxn 1 + ηyn

(1.1)

where parameters α, β, γ, δ, , η ∈ R+ , and initial conditions x0 , y0 are positive real numbers. It is a well-known fact that the discrete-time type models described by difference equations are more suitable than the continuous-time models. Nonlinear difference equations of order greater than one are of paramount importance in applications. Such equations also appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations which model various diverse phenomena in biology, ecology, physiology, physics, engineering, and economics. Rational difference equations are a special form of nonlinear difference equations. We refer to [9, 10, 11, 12, 13, 14] for basic theory of difference equations and rational difference equations. Recently, many authors have discussed the dynamics of rational difference equations [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28].

2. Linearized stability Let us consider a two-dimensional discrete dynamical system of the form xn+1 yn+1

= f (xn , yn ) = g(xn , yn ), n = 0, 1, · · · ,

(2.1)

where f : I × J → I and g : I × J → J are continuously differentiable functions and I, J are some intervals of real numbers. Furthermore, a solution {(xn , yn )}∞ n=0 of the system (2.1) is uniquely determined by initial conditions (x0 , y0 ) ∈ I × J. An equilibrium point of (2.1) is a point (¯ x, y¯) that satisfies x ¯ = f (¯ x, y¯), y¯ = g(¯ x, y¯). Definition 2.1. Let (¯ x, y¯) be an equilibrium point of the system (2.1). (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 (x0 , y0 ) if k(x0 , y0 )−(¯ x, y¯)k< δ implies k(xn , yn ) − (¯ x, y¯)k< ε for all n > 0, where k.k is the usual Euclidian norm in R2 . (ii) An equilibrium point (¯ x, y¯) is said to be unstable if it is not stable.

ADE_231_edited

[04/05 13:56]

3/15

Dynamics of a discrete Lotka-Volterra model

3

(iii) An equilibrium point (¯ x, y¯) is said to be asymptotically stable if there exists η > 0 such that k(x0 , y0 ) − (¯ x, y¯)k< η and (xn , yn ) → (¯ x, y¯) as n → ∞. (iv) An equilibrium point (¯ x, y¯) is called a global attractor if (xn , yn ) → (¯ x, y¯) as n → ∞. (v) An equilibrium point (¯ x, y¯) is called an asymptotic global attractor if it is a global attractor and stable. Definition 2.2. Let (¯ x, y¯) be an equilibrium point of a map F (x, y) = (f (x, y), g(x, y)), where f and g are continuously differentiable functions at (¯ x, y¯). The linearized system of (2.1) about the equilibrium point (¯ x, y¯) is given by Xn+1 = F (Xn ) = FJ Xn ,

xn and FJ is a Jacobian matrix of the system (2.1) about yn the equilibrium point (¯ x, y¯).

where Xn =

Let (¯ x, y¯) be an equilibrium point of the system (1.1), then δ y¯ + ¯ xy¯ α¯ x − βx ¯y¯ , y¯ = . x ¯= 1 + γx ¯ 1 + η y¯ Hence, O = (0, 0), P = β−βδ+(−1+α)η , γ(−1+δ)+(−1+α) , Q = −1+α β+γη β+γη γ ,0 , and R = 0, −1+δ are equilibrium points of the system (1.1). Then, clearly, η γ(−1+δ)+(−1+α) is the unique positive equilibrium P = β−βδ+(−1+α)η , β+γη β+γη point of the system (1.1), if α > 1, δ ≤ 1, > γ−γδ α−1 or α > 1, δ > 1, −β+βδ η > −1+α . The Jacobian matrix of the linearized system of (1.1) about the fixed point (¯ x, y¯) is given by " # α−¯ yβ x ¯β − 1+¯ (1+¯ xγ)2 xγ FJ (¯ x, y¯) = . y¯ δ+¯ x 1+¯ yη

(1+¯ y η)2

Theorem 2.3. For the system Xn+1 = F (Xn ), n = 0, 1, · · · , of difference ¯ is a fixed point of F . If all eigenvalues of the Jacobian equations such that X ¯ ¯ is locally matrix JF about X lie inside the open unit disk |λ|< 1, then X ¯ asymptotically stable. If one of them has a modulus greater than one, then X is unstable.

3. Main results Theorem 3.1. Assume that α < 1 and δ < 1, then the following statements are true. (i) The equilibrium point O = (0, asymptotically stable. 0) is locally (ii) The equilibrium point Q =

−1+α γ ,0

(iii) The equilibrium point R = 0, −1+δ η

is unstable.

is unstable.

ADE_231_edited

[04/05 13:56]

4

4/15

Q. Din

Proof. (i) The Jacobian matrix of the linearized system of (1.1) about the fixed point (0, 0) is given by α 0 FJ (0, 0) = 0 δ Moreover, the eigenvalues of the Jacobian matrix JF (0, 0) about (0, 0) are λ1 = α < 1 and λ2 = δ < 1. Hence, the equilibrium point (0, 0) is locally asymptotically stable. (ii) matrix of the linearized system of (1.1) about the fixed The Jacobian , 0 is given by point −1+α γ # " 1 β−αβ −1 + α α αγ ,0 = . FJ γ 0 (α−1)+δγ γ −1+α The eigenvalues of the Jacobian matrix JF ( −1+α γ , 0) about ( γ , 0) are

> 1 and λ2 = γδ−+α . γ (iii) The Jacobian matrix of the linearized system of (1.1) about the −1+δ is given by fixed point 0, η # " β−βδ+αη 0 −1 + δ η FJ 0, = . (−1+δ) 1 η δη δ λ1 =

1 α

−1+δ The eigenvalues of the Jacobian matrix JF (0, −1+δ η ) about (0, η ) are λ1 = 1 δ

> 1 and λ2 =

β−βδ+αη . η

Theorem 3.2. The following statements are true. (i) If α > 1, δ < 1, and < γ−γδ α−1 , then the equilibrium point Q = −1+α γ , 0 is locally asymptotically stable. (ii) If δ > 1t and α < 1, then the equilibrium point R = 0, −1+δ is η locally asymptotically stable. Theorem 3.3. Assume that α > 1, δ > 1, and η > −β+βδ , then the unique −1+α β−βδ+(−1+α)η γ(−1+δ)+(−1+α) equilibrium point P = , is locally asymptotβ+γη β+γη ically stable if Ω < (β(γ − γδ + ) + αγη)2 (γδη + (β + (−1 + α)η)), where Ω = β 3 γ 2 δ 3 + αγδ 2 + (5α + δ)2 + + β 2 γ 3 δ 2 + γ(1 + α(5 + 2α + 7δ))2 + 3α2 3 η + + βγ γ 2 2α + δ 2 + αγ(7 + (3 + α)δ) + 1 + α2 + α3 2 η 2 + αγ 2 (γ(1 + α + δ) + α)η 3 . Proof. Assume that α > 1, δ > 1, and η > −β+βδ −1+α . Let L = β − βδ + (−1 + α)η > 0. Then a characteristic polynomial of the Jacobian matrix FJ (P )

ADE_231_edited

[04/05 13:56]


5/15

5

L is given about the unique equilibrium point P = β+γη , γ(−1+δ)+(−1+α) β+γη by Υ(λ) = λ2 − (A − B + C)λ + D − E + F − G + H, where α β(γ(−1 + δ) + (−1 + α))(β + γη) A= , 2 , B = (β + γ(L + η))2 Lγ 1 + β+γη (β + γη)(βδ + γδη + Lλ) , (γδη + (β + (−1 + α)η))2 αδ(β + γη)4 , (β + γ(L + η))2 (γδη + (β + (−1 + α)η))2 βδ(γ(−1 + δ) + (−1 + α))(β + γη)3 , (β + γ(L + η))2 (γδη + (β + (−1 + α)η))2 Lα(β + γη)3 , (β + γ(L + η))2 (γδη + (β + (−1 + α)η))2 Lβ(γ(−1 + δ) + (−1 + α))(β + γη)2 , (β + γ(L + η))2 (γδη + (β + (−1 + α)η))2 C=

D= E= F = G= and

H=

(β + γη)(L + βδ + γδη) . (γδη + (β + (−1 + α)η))2

Let S(λ) = λ2 , T (λ) = (A − B + C)λ − D + E − F + G − H. Assume that Ω < (β(γ − γδ + ) + αγη)2 (γδη + (β + (−1 + α)η)). Then one has |T (λ)|

≤ A+B+C +D+E+F +G+H Ω < 1. < (β(γ − γδ + ) + αγη)2 (γδη + (β + (−1 + α)η))

Then, by Rouche’s theorem, S(λ) and S(λ) − T (λ) have the same number of zeroes in an open unit disk |λ|< 1. Hence, the unique positive equilibrium point P is locally asymptotically stable. 3.1. Global character Theorem 3.4. Let I = [a, b] and J = [c, d] be real intervals, and let f : I ×J → I and g : I × J → J be continuous functions. Consider the system (2.1) with initial conditions (x0 , y0 ) ∈ I × J. Suppose that the following statements are true. (i) f (x, y) is non-decreasing in x and non-increasing in y. (ii) g(x, y) is non-decreasing in both arguments. (iii) If (m1 , M1 , m2 , M2 ) ∈ I 2 × J 2 is a solution of the system m1 = f (m1 , M2 ), M1 = f (M1 , m2 ) m2 = g(m1 , m2 ), M2 = g(M1 , M2 )

ADE_231_edited

[04/05 13:56]

6

6/15

Q. Din

such that m1 = M1 and m2 = M2 , then there exists exactly one equilibrium point (¯ x, y¯) of the system (2.1) such that lim (xn , yn ) = (¯ x, y¯). n→∞

Proof. According to the Brouwer fixed point theorem, the function F : I × J → I × J defined by F (x, y) = F (f (x, y), g(x, y)) has a fixed point (¯ x, y¯), which is a fixed point of the system (2.1). Assume that m01 = a, M10 = b, m02 = c, M20 = d such that mi+1 = f (mi1 , M2i ), M1i+1 = f (M1i , mi2 ), 1 and mi+1 = g(mi1 , mi2 ), M2i+1 = g(M1i , M2i ). 2 Then m01 = a ≤ f (m01 , M20 ) ≤ f (M10 , m02 ) ≤ b = M10 , and m02 = c ≤ g(m01 , m02 ) ≤ g(M10 , M20 ) ≤ d = M20 . Moreover, one has m01 ≤ m11 ≤ M11 ≤ M10 , and m02 ≤ m12 ≤ M21 ≤ M20 . We similarly have m11 = f (m01 , M20 ) ≤ f (m11 , M21 ) ≤ f (M11 , m12 ) ≤ f (M10 , m02 ) ≤ M11 , and m12 = g(m01 , m02 ) ≤ g(m11 , m12 ) ≤ g(M11 , M21 ) ≤ g(M10 , M20 ) ≤ M21 . Now observe that for each i ≥ 0, a = m01 ≤ m11 ≤ · · · ≤ mi1 ≤ M1i ≤ M1i−1 ≤ · · · ≤ M10 = b, and c = m02 ≤ m12 ≤ · · · ≤ mi2 ≤ M2i ≤ M2i−1 ≤ · · · ≤ M20 = d. Hence, mi1 ≤ xn ≤ M1i , and mi2 ≤ yn ≤ M2i for n ≥ 2i+1. Let m1 = lim mi1 , n→∞

M1 = lim M1i , m2 = lim mi2 , and M2 = lim M2i . Then a ≤ m1 ≤ M1 ≤ b n→∞ n→∞ n→∞ and c ≤ m2 ≤ M2 ≤ d. By the continuity of f and g, one has m1 = f (m1 , M2 ), M1 = f (M1 , m2 ), m2 = g(m1 , m2 ), M2 = g(M1 , M2 ). Hence, m1 = M1 , m2 = M2 .

Theorem 3.5. Assume that ηγ − β 6= 0, then the unique positive equilibrium point P of the system (1.1) is a global attractor.

ADE_231_edited

[04/05 13:56]


7/15

7

δy+xy Proof. Let f (x, y) = αx−βxy 1+γx and g(x, y) = 1+ηy . Then it is easy to see that f (x, y) is non-decreasing in x and non-increasing in y. Moreover, g(x, y) is non-decreasing in both x and y. Let (m1 , M1 , m2 , M2 ) be a positive solution of the system

m1 = f (m1 , M2 ), M1 = f (M1 , m2 ), m2 = g(m1 , m2 ), M2 = g(M1 , M2 ). Then one has m1 =

αm1 − βm1 M2 αM1 − βM1 m2 , M1 = , 1 + γm1 1 + γM1

(3.1)

δM2 + M1 M2 δm2 + m1 m2 , M2 = . 1 + ηm2 1 + ηM2

(3.2)

and m2 = From (3.1), one has 1 + γm1 = α − βM2 , 1 + γM1 = α − βm2 .

(3.3)

On subtraction, (3.3) implies that γ(m1 − M1 ) = β(m2 − M2 ).

(3.4)

Similarly, from (3.2), one has 1 + ηm2 = δ + m1 , 1 + ηM2 = δ + M1 .

(3.5)

On subtraction, (3.5) implies that η(m2 − M2 ) = (m1 − M1 ).

(3.6)

Comparing (3.4) and (3.6), one has (ηγ − β)(m1 − M1 ) = 0. Then one has m1 = M1 and m2 = M2 . Hence, from Theorem 3.4 the equilibβ−βδ+(−1+α)η γ(−1+δ)+(−1+α) rium point , of the system (1.1) is a global β+γη β+γη attractor. Theorem 3.6. Assume that α > 1, δ >1, and ηγ − β 6= 0. Then the γ(−1+δ)+(−1+α) unique positive equilibrium point (¯ x, y¯) = β−βδ+(−1+α)η , β+γη β+γη is globally asymptotically stable. Proof. The proof follows from Theorem 3.3 and Theorem 3.5.

3.2. Rate of convergence In this section we determine the rate of convergence of a solution that converges to the unique positive equilibrium point of the system (1.1). The following result gives the rate of convergence of solutions of a system of difference equations: Xn+1 = (A + B(n)) Xn ,

(3.7)

ADE_231_edited

[04/05 13:56]

8

8/15

Q. Din

where Xn is an m-dimensional vector, A ∈ C m×m is a constant matrix, and B : Z+ → C m×m is a matrix function satisfying kB(n)k→ 0

(3.8)

as n → ∞ ,where k·k denotes any matrix norm which is associated with the vector norm p k(x, y)k= x2 + y 2 . Proposition 3.7. (Perron’s theorem)[15] Suppose that condition (3.8) holds. If Xn is a solution of (3.7), then either Xn = 0 for all large n or ρ = lim (kXn k)1/n n→∞

(3.9)

exists and is equal to the modulus of one of the eigenvalues of matrix A. Proposition 3.8. [15] Suppose that condition (3.8) holds. If Xn is a solution of (3.7), then either Xn = 0 for all large n or ρ = lim

n→∞

kXn+1 k kXn k

(3.10)

exists and is equal to the modulus of one of the eigenvalues of matrix A. Let {(xn , yn )} be any solution of the system (1.1) such that lim xn = x ¯, n→∞ , γ(−1+δ)+(−1+α) and lim yn = y¯, where (¯ x, y¯) = β−βδ+(−1+α)η . To find β+γη β+γη n→∞

the error terms, one has from the system (1.1) xn+1 − x ¯ = =

α¯ x − βx ¯y¯ αxn − βxn yn − 1 + γxn 1 + γx ¯ βx ¯ (α − βyn ) (xn − x ¯) − (yn − y¯), (1 + γxn )(1 + γ x ¯) 1 + γx ¯

and yn+1 − y¯ = =

δyn + xn yn δ y¯ + ¯ xy¯ − 1 + ηyn 1 + η y¯ ¯ y δ + xn (xn − x ¯) + (yn − x ¯) . 1 + η y¯ (1 + ηyn )(1 + ¯ y)

Let e1n = xn − x ¯ and e2n = yn − y¯, then one has e1n+1 = an e1n + bn e2n , and e2n+1 = cn e1n + dn e2n , where

(α − βyn ) βx ¯ , bn = − , (1 + γxn )(1 + γ x ¯) 1 + γx ¯ ¯ y δ + xn cn = , dn = . 1 + η y¯ (1 + ηyn )(1 + ¯ y)

an =

ADE_231_edited

[04/05 13:56]

9/15


9

Moreover, lim an =

n→∞

x ¯β α − y¯β , lim bn = − , (1 + x ¯γ)2 n→∞ 1+x ¯γ

y¯ δ+x ¯ . , lim dn = n→∞ 1 + y¯η (1 + y¯η)2 Now the limiting system of error terms can be written as # 1 " α−¯yβ x ¯β − 1+¯ en+1 e1n (1+¯ xγ)2 xγ = , y¯ δ+¯ x e2n e2n+1 1+¯ yη (1+¯ y η)2 lim cn =

n→∞

which is similar to the linearized system of (1.1) about the equilibrium point (¯ x, y¯). Using proposition (3.7), one has following result. Theorem 3.9. Assume that {(xn , yn )} is a positive solution of the system (1.1) such that lim xn = x ¯ and lim yn = y¯, where n→∞ n→∞ β − βδ + (−1 + α)η γ(−1 + δ) + (−1 + α) (¯ x, y¯) = , . β + γη β + γη 1 en Then the error vector en = of every solution of (1.1) satisfies both e2n of the following asymptotic relations: 1

lim (ken k) n = |λ1,2 FJ (¯ x, y¯)|, lim

n→∞

n→∞

ken+1 k = |λ1,2 FJ (¯ x, y¯)|, ken k

where λ1,2 FJ (¯ x, y¯) are the characteristic roots of the Jacobian matrix FJ (¯ x, y¯).

4. Examples In this section, we consider some numerical examples which show that under a suitable choice of parameters α, β, γ, δ, , η, the unique positive equilibγ(−1+δ)+(−1+α) , of the system (1.1) is globally rium point β−βδ+(−1+α)η β+γη β+γη asymptotically stable. Example. Let α = 1.001, β = 0.03, γ = 0.6, δ = 1.002, = 1.7, η = 0.9. Then the system (1.1) can be written as xn+1 =

1.001xn − 0.03xn yn 1.002yn + 1.7xn yn , yn+1 = , 1 + 0.6xn 1 + 0.9yn

(4.1)

with initial conditions x0 = 0.0002, y0 = 0.0006. In this case, the unique positive equilibrium point P of the system (4.1) is given by β − βδ + (−1 + α)η γ(−1 + δ) + (−1 + α) , = (0.00142132, 0.00490694). β + γη β + γη Moreover, the plot is shown in Figure 1.

ADE_231_edited

10

[04/05 13:56]

10/15

Q. Din

Figure 1. Plot of the system (4.1). Example. Let α = 2.5, β = 0.7, γ = 2.3, δ = 2.7, = 4.2, η = 6.8. Then the system (1.1) can be written as 2.5xn − 0.7xn yn 2.7yn + 4.2xn yn xn+1 = , yn+1 = , (4.2) 1 + 2.3xn 1 + 6.8yn with initial conditions x0 = 0.84, y0 = 0.5. In this case, the unique equilibrium point P of the system (4.2) is given by β − βδ + (−1 + α)η γ(−1 + δ) + (−1 + α) , = (0.48493, 0.549516). β + γη β + γη Moreover, the plot is shown in Figure 2. Example. Let α = 22, β = 1.7, γ = 20.5, δ = 6, = 0.2, η = 2.8. Then the system (1.1) can be written as 2.5xn − 1.7xn yn 6yn + 0.2xn yn xn+1 = , yn+1 = , (4.3) 1 + 20.5xn 1 + 2.8yn with initial conditions x0 = 0.7, y0 = 0.8. In this case, the unique equilibrium point P of the system (4.3) is given by β − βδ + (−1 + α)η γ(−1 + δ) + (−1 + α) , = (0.871147, 1.84794). β + γη β + γη Moreover, the plot is shown in Figure 3. Example. Let α = 170, β = 11, γ = 2.7, δ = 50, = 1.7, η = 7. Then the system (1.1) can be written as 170xn − 11xn yn 50yn + 1.7xn yn xn+1 = , yn+1 = , (4.4) 1 + 2.7xn 1 + 7yn

ADE_231_edited

[04/05 13:56]


11/15

11

Figure 2. Plot of the system (4.2).

Figure 3. Plot of the system (4.3). with initial conditions x0 = 7, y0 = 5. In this case, the unique positive equilibrium point P of the system (4.4) is given by β − βδ + (−1 + α)η γ(−1 + δ) + (−1 + α) , = (17.1276, 11.1597). β + γη β + γη Moreover, the plot of the system (4.4) is shown in Figure 4. An attractor of the system is shown in Figure 5.

ADE_231_edited

12

[04/05 13:56]

Q. Din

Figure 4. Plot of the system (4.4).

Figure 5. An attractor of the system (4.4).

12/15

ADE_231_edited

[04/05 13:56]


13/15

13

Conclusions This work is related to the qualitative behavior of a discrete-time LotkaVolterra model. The continuous form of this model is given by dx dy = ax − bx2 − cxy, = mxy + ny − py 2 , dt dt where a, b, c, m, n, p are positive constants. Moreover, the discrete form (1.1) of the continuous model is obtained by using some nonstandard difference scheme such that the equilibrium points in both cases are conserved. We proved that the system (1.1) has four equilibrium points, which are locally asymptotically stable under certain conditions. The main contribution in this paper is to prove that the unique positive equilibrium point β − βδ + (−1 + α)η γ(−1 + δ) + (−1 + α) , P = β + γη β + γη of the system (1.1) is globally asymptotically stable. Furthermore, we have investigated the rate of convergence of the solution that converges to the unique positive equilibrium point of the system (1.1). Some numerical examples are provided to support our theoretical results. These examples are experimental verifications of theoretical discussions. Author’s contributions The author carried out the proof of the main results and approved the final manuscript. Competing interests The author declares that he has no competing interests. Acknowledgements The author would like to thank the main editor and 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.

References [1] W. Krebs, A General Predator-Prey Model, Math. Comp. Model. Dyn. Syst., 9 (2003), 387–401. [2] Linda J.S. Allen, An Introduction to Mathematical Biology, Prentice Hall, (2007). [3] F. Brauer, C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer, (2000). [4] L. Edelstein-Keshet, Mathematical Models in Biology, McGraw-Hill, (1988). [5] S. Ahmad,On the nonautonomous Lotka-Volterra competition equation, Proc. Amer. Math. Soc., 117(1993), 199–204. [6] X. Tang, X. Zou, On positive periodic solutions of Lotka-Volterra competition systems with deviating arguments, Pro. Amer. Math. Soc., 134(2006), 2967– 2974.

ADE_231_edited

14

[04/05 13:56]

14/15

Q. Din

[7] Z. Zhou, X. Zou, Stable periodic solutions in a discrete periodic logistic equation, Appl. Math. Lett., 16(2)(2003), 165–171. [8] X. Liu, A note on the existence of periodic solution in discrete predator-prey models, Appl. Math. Model., 34(2010), 2477–2483. [9] M. R. S. Kulenović and G. Ladas, Dynamics of Second Order Rational Difference Equations, Chapman and Hall/CRC, (2002) [10] E. Camouzis and G. Ladas, Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC Press, Boca Raton, (2007). [11] S. Elaydi, An Introduction to Difference Equations, 3rd ed., Springer-Verlag, New York, (2005). [12] R. P. Agarwal and P. Wong, Advanced Topics in Difference Equations, Kluwer Academic Publishers, (1997). [13] R. P. Agarwal, Difference Equations and Inequalities, 1st edition, Marcel Dekker, New York, (1992), 2nd edition, (2000). [14] E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, Chapman and Hall/CRC Press, Boca Raton, (2004). [15] M. Pituk, More on Poincare’s and Perron’s theorems for difference equations, J. Diff. Eq. App., 8(2002), 201–216. [16] M. Aloqeili, Dynamics of a rational difference equation, App. Math. Comput., 176(2)(2006), 768–776. [17] C. Cinar, On the positive solutions of the difference equation system xn+1 = 1 ; yn+1 = xn−1ynyn−1 , App. Math. Comput., 158(2004), 303–305. yn [18] S. Stević, On some solvable systems of difference equations, App. Math. Comput., 218(2012),5010–5018. [19] I. Bajo, E. Liz, Global behaviour of a second-order nonlinear difference equation, J. Diff. Eq. Appl., 17(10)(2011), 1471–1486. [20] S. Kalabuˆsić, M. R. S. Kulenović, E. Pilav, Dynamics of a two-dimensional system of rational difference equations of Leslie–Gower type, Ad. Diff. Eq., (2011), doi:10.1186/1687-1847-2011-29. [21] N. Touafek, E.M. Elsayed, On the solutions of systems of rational difference equations, Math. Comp. Model., 55(2012), 1987–1997. [22] N. Touafek, E.M. Elsayed, On the periodicity of some systems of nonlinear difference equations, Bull. Math. Soc. Sci. Math. Roumanie, 2(2012), 217–224. [23] Q. Din, On a system of rational difference equation, Demonstratio Mathematica, In Press. [24] Q. Din, Global character of a rational difference equation, Thai Journal of Mathematics, In Press. [25] Q. Din, M. N. Qureshi, A. Q. Khan, Dynamics of a fourth-order system of rational difference equations, Ad. Diff. Eq., doi:10.1186/1687-1847-2012-216. [26] Q. Zhang, L. Yang, J. Liu, Dynamics of a system of rational third order difference equation, Ad. Diff. Eq., doi:10.1186/1687-1847-2012-136, 8 pages. [27] M. Shojaei, R. Saadati, H. Adibi, Stability and periodic character of a rational third order difference equation, Chaos, Solitons and Fractals, 39(2009), 1203— 1209.

ADE_231_edited

[04/05 13:56]


15/15

15

[28] E. M. Elsayed, Behavior and expression of the solutions of some rational difference equations, J. Comp. Anal. Appl., 15(1)(2013), 73–81. Q. Din University of Azad Jammu and Kasmir Muzaffarabad, Pakistan e-mail: [email protected]

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5