Global dynamics of two systems of exponential difference equations by Lyapunov function

Transcription

1 Khan Advances in Difference Equations :297 R E S E A R C H Open Access Global dynamics of two systems of exponential difference equations by Lyapunov function Abdul Qadeer Khan * * Correspondence: abdulqadeerkhan1@gmailcom Department of Mathematics University of Azad Jammu and Kashmir Muzaffarabad Pakistan Abstract In this paper we study the boundedness character and persistence existence and uniqueness of the positive equilibrium local and global behavior and rate of convergence of positive solutions of two systems of exponential difference equations Furthermore by constructing a discrete Lyapunov function we obtain the global asymptotic stability of the unique positive equilibrium point Some numerical examples are given to verify our theoretical results MSC: 39A10; 40A05 Keywords: difference equations; boundedness; persistence; asymptotic behavior; Lyapunov function; rate of convergence 1 Introduction and preliminaries Since difference equations and systems of difference equations containing exponential terms have many potential applications in biology there are many papers dealing with such equations See for example the following El-Metwally et al [1] have investigated the boundedness character asymptotic behavior periodicity nature of the positive solutions and stability of the equilibrium point of the following population model: x n+1 = α + βx n 1 e x n where the parameters α β are positive numbers and the initial conditions are arbitrary non-negative real numbers Ozturk et al [2] have investigated the boundedness asymptotic behavior periodicity and stability of the positive solutions of the following difference equation: y n+1 = α + βe y n γ + y n 1 wherethe parameters α β γ are positive numbers and the initial conditions are arbitrary non-negative numbers 2014 Khan; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

2 Khan Advances in Difference Equations :297 Page 2 of 21 Bozkurt [3] has investigated the local and global behavior of positive solutions of the following difference equation: y n+1 = αe y n + βe y n 1 wherethe parameters α β γ and the initial conditions are arbitrary positive numbers Papaschinopoulos et al [4] have investigated the boundedness persistence and asymptotic behavior of positive solutions of the following two directional interactive and invasive species models: x n+1 = a + bx n 1 e y n y n+1 = c + dy n 1 e x n wherethe parameters a b c d and the initial conditions are arbitrary positive numbers Papaschinopoulos et al [5] have investigated the asymptotic behavior of the solutions of the following three systems of difference equations of exponential form: x n+1 = α + βe y n γ + y n 1 x n+1 = α + βe y n γ + x n 1 x n+1 = α + βe x n γ + y n 1 y n+1 = δ + ɛe yn ζ + x n 1 y n+1 = δ + ɛe xn ζ + y n 1 y n+1 = δ + ɛe yn ζ + x n 1 where the parameters α β γ δ ɛ δ are positive numbers and the initial conditions are arbitrary non-negative numbers Papaschinopoulos and Schinas [6] have investigated the asymptotic behavior of the positive solutions of the systems of the two difference equations: x n+1 = a + by n 1 e y n y n+1 = c + dx n 1 e x n x n+1 = a + by n 1 e x n y n+1 = c + dx n 1 e y n wherethe parameters a b c d and the initial conditions are arbitrary positive numbers Recently Khan and Qureshi [7] have investigated the qualitative behavior of the following exponential type system of rational difference equations: x n+1 = αe y n + βe y n 1 γ + αx n + βx n 1 y n+1 = α 1e xn + β 1 e xn 1 γ 1 + α 1 y n + β 1 y n 1 n =01 where α β γ α 1 β 1 γ 1 and the initial conditions x 0 x 1 y 0 y 1 are positive real numbers Motivated by the above studies our aim in this paper is to investigate the qualitative behavior of positive solutions of the following two systems of exponential rational difference equations: x n+1 = αe y n + βe y n 1 y n+1 = α 1e xn + β 1 e xn 1 γ 1 + α 1 x n + β 1 x n 1 n =01 1

3 Khan Advances in Difference Equations :297 Page 3 of 21 and x n+1 = αe x n + βe x n 1 y n+1 = α 1e yn + β 1 e yn 1 γ 1 + α 1 x n + β 1 x n 1 n =01 2 where the parameters α β γ α 1 β 1 γ 1 and the initial conditions 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 the unique positive equilibrium point and rate of convergence of positive solutions of systems 1 and 2 which converge to its unique positive equilibrium point For basic theory and applications of difference equations we refer the reader [8 16] and references therein Let us consider the four-dimensional discrete dynamical system of the form x n+1 = f x n x n 1 y n y n 1 y n+1 = gx n x n 1 y n y n 1 n =01 3 where f : I 2 J 2 I and g : I 2 J 2 J are continuously differentiable functions and I J are some intervals of real numbers Furthermore a solution {x n y n } n= 1 of system 3 is uniquely determined by the initial conditions x i y i I J for i { 1 0} Along with system 3 we consider the corresponding vector map F =f x n g y n An equilibrium point of 3isapoint x ȳ that satisfies x = f x x ȳ ȳ ȳ = g x x ȳ ȳ The point x ȳ isalsocalledafixedpointofthevectormap F Definition 1 Let x ȳ be an equilibrium point of system 3 i An equilibrium point x ȳ is said to be stable if for every ε >0thereexistsδ >0 such that for every initial condition x i y i i { 1 0} 0 i= 1 x i y i x ȳ < δ implies x n y n x ȳ < ε for all n >0where is the usual Euclidean norm in R 2 ii An equilibrium point x ȳ is said to be unstable if it is not stable iii An equilibrium point x ȳ is said to be asymptotically stable if there exists η >0 such that 0 i= 1 x i y i x ȳ < η and x n y n x ȳ as n iv An equilibrium point x ȳ is called a global attractor if x n y n x ȳ as n v An equilibrium point x ȳ is called an asymptotic global attractor if it is a global attractor and stable Definition 2 Let x ȳ be an equilibrium point of the map F =f x n g y n where f and g are continuously differentiable functions at x ȳ The linearized system of 3 about the equilibrium point x ȳ is X n+1 = FX n =F J X n

4 Khan Advances in Difference Equations :297 Page 4 of 21 where X n = x n x n 1 y n y n 1 and F J is the Jacobian matrix of system 3 about the equilibrium point x ȳ Lemma 1 [17] Consider the system X n+1 = FX n n =01where X isafixedpointoff If all eigenvalues of the Jacobian matrix J F about X lieinsidetheopenunitdisk λ <1then X is locally asymptotically stable If any of the eigenvalue has a modulus greater than one then X is unstable The following result gives the rate of convergence of solutions of a system of difference equations: X n+1 = A + Bn X n 4 where X n is an m-dimensional vector A C m m is a constant matrix and B : Z + C m m isamatrixfunctionsatisfying Bn 0 5 as n where denotes any matrix norm which is associated with the vector norm x y = x2 + y 2 Proposition 1 Perron s theorem [18] Suppose that condition 5 holds If X n is a solution of 4 then either X n =0for all large n or or ρ = lim Xn 1/n n X n+1 ρ = lim n X n exists and is equal to the modulus of one of the eigenvalues of matrix A 2 On the system x n+1 = αe yn +βe y n 1 γ +αy n y +βy n+1 = α 1e xn +β 1 e x n 1 n 1 γ 1 +α 1 x n +β 1 x n 1 In this section we shall investigate the asymptotic behavior of system 1 Let x ȳ bethe equilibrium point of system 1then x = α + βe ȳ γ +α + βȳ ȳ = α 1 + β 1 e x γ 1 +α 1 + β 1 x To construct the corresponding linearized form of system 1 we consider the following transformation: x n x n 1 y n y n 1 f f 1 g g 1 6

5 Khan Advances in Difference Equations :297 Page 5 of 21 where f = αe y n + βe y n 1 f 1 = x n g = α 1e xn + β 1 e xn 1 γ 1 + α 1 x n + β 1 x n 1 g 1 = y n The Jacobian matrix about the fixed point x ȳ under the transformation 6 is given by F J x ȳ= 0 0 αe ȳ + x γ +α+βȳ α 1e x +ȳ γ 1 +α 1 +β 1 x βe ȳ + x γ +α+βȳ β 1e x +ȳ γ 1 +α 1 +β 1 x Boundedness and persistence The following theorem shows that every positive solution {x n y n } of system 1 is bounded and persists Theorem 1 Every positive solution {x n y n } of system 1 is bounded and persists Proof Let {x n y n } be an arbitrary solution of 1 From 1 we have x n α + β γ = U 1 y n α 1 + β 1 γ 1 = U 2 n =012 7 In addition from 1and7 we have α 1 +β 1 α + βe γ 1 x n γ +α + β α = L 1 y n α 1 + β 1 e α+β γ 1+β 1 γ 1 γ 1 +α 1 + β 1 α+β γ = L 2 n =23 8 Hence from 7 and8we get L 1 x n U 1 L 2 y n U 2 n =34 So the proof is complete 22 Existence of invariant set for solutions Theorem 2 Let {x n y n } be a positive solution of system 1 Then [L 1 U 1 ] [L 2 U 2 ] is an invariant set for system 1 Proof For any positive solution {x n y n } of system 1 with initial conditions x 0 x 1 [L 1 U 1 ] and y 0 y 1 [L 2 U 2 ] we have and x 1 = αe y 0 + βe y 1 α + β γ + αy 0 + βy 1 γ x 1 = αe y 0 + βe y 1 γ + αy 0 + βy 1 α 1 +β 1 α + βe γ 1 γ +α + β α 1+β 1 γ 1

6 Khan Advances in Difference Equations :297 Page 6 of 21 Moreover and y 1 = α 1e x 0 + β 1 e x 1 γ 1 + α 1 x 0 + β 1 x 1 α 1 + β 1 γ 1 y 1 = α 1e x 0 + β 1 e x 1 α 1 + β 1 e α+β γ γ 1 + α 1 x 0 + β 1 x 1 γ 1 +α 1 + β 1 α+β γ Hence x 1 [L 1 U 1 ]andy 1 [L 2 U 2 ] Similarly one can show that if x k [L 1 U 1 ]and y k [L 2 U 2 ] then x k+1 [L 1 U 1 ]andy k+1 [L 2 U 2 ] 23 Existence and uniqueness of the positive equilibrium and local stability Theorem 3 Suppose that η < γ γ 1 +α 1 + β 1 L 1 +α + βα1 + β 1 e L 1 2 γ1 +α 1 + β 1 L 1 9 where η =α + βα 1 + β 1 e α 1 +β 1 e L 1 γ 1 +α 1 +β 1 L 1 L 1 γ + α + β γ1 +α 1 + β 1 U 1 +α + βα 1 + β 1 e L 1 γ 1 +1+U 1 α 1 + β 1 Then system 1 has a unique positive equilibrium point x ȳ in [L 1 U 1 ] [L 2 U 2 ] Proof Consider the following system of equations: x = α + βe y γ +α + βy y = α 1 + β 1 e x γ 1 +α 1 + β 1 x α+βe f x Let Fx = γ +α+βf x x wheref x = α 1+β 1 e x γ 1 +α 1 +β 1 x and x [L 1 U 1 ] Then it follows that FL 1 = α+βe f L 1 γ +α+βf L 1 L 1NowFL 1 >0ifandonlyif α + βe α 1 +β 1 e L 1 γ 1 +α 1 +β 1 L 1 > L 1 γ + α + βα 1 + β 1 e L 1 γ 1 +α 1 + β 1 L 1 Furthermore we have FU 1 = α+βe f U 1 γ +α+βf U 1 U 1 where f U 1 = α 1+β 1 e U 1 γ 1 +α 1 +β 1 U 1 Itiseasytosee that FU 1 <0ifandonlyif α + βe α 1 +β 1 e U 1 γ 1 +α 1 +β 1 U 1 < U 1 γ + α + βα 1 + β 1 e U 1 γ 1 +α 1 + β 1 U 1 Hence Fx has at least one positive solution in [L 1 U 1 ] Furthermore assume that condition 9issatisfiedthenonehas dfx dx < 1+ η γ γ 1 +α 1 + β 1 L 1 +α + βα 1 + β 1 e L 1 2 γ 1 +α 1 + β 1 L 1 <0 Hence Fx = 0 has a unique positive solution in [L 1 U 1 ] This completes the proof

7 Khan Advances in Difference Equations :297 Page 7 of 21 Theorem 4 Assume that α + βα 1 + β 1 e L 2 + U 1 e L 1 + U 2 < γ +α + βl2 γ1 +α 1 + β 1 L 1 Then the unique positive equilibrium point x ȳ in [L 1 U 1 ] [L 2 U 2 ] of system 1 is locally asymptotically stable Proof The characteristic polynomial of the Jacobian matrix F J x ȳ about the equilibrium point x ȳisgivenby Pλ=λ 4 Let λ=λ 2 and λ= αα 1 e ȳ + xe x + ȳ γ +α + βȳγ 1 +α 1 + β 1 x λ2 αβ 1 + βα 1 e ȳ + xe x + ȳ γ +α + βȳγ 1 +α 1 + β 1 x λ ββ 1 e ȳ + xe x + ȳ γ +α + βȳγ 1 +α 1 + β 1 x αα 1 e ȳ + xe x + ȳ γ +α + βȳγ 1 +α 1 + β 1 x λ2 + αβ 1 + βα 1 e ȳ + xe x + ȳ γ +α + βȳγ 1 +α 1 + β 1 x λ + ββ 1 e ȳ + xe x + ȳ γ +α + βȳγ 1 +α 1 + β 1 x Assume that α + βα 1 + β 1 e L 2 + U 1 e L 1 + U 2 <γ +α + βl 2 γ 1 +α 1 + β 1 L 1 Then one has λ αα 1 e ȳ + xe x + ȳ γ +α + βȳγ 1 +α 1 + β 1 x + αβ 1 + βα 1 e ȳ + xe x + ȳ γ +α + βȳγ 1 +α 1 + β 1 x ββ 1 e ȳ + xe x + ȳ + γ +α + βȳγ 1 +α 1 + β 1 x e ȳ + xe x + ȳ =αα 1 + αβ 1 + βα 1 + ββ 1 γ +α + βȳγ 1 +α 1 + β 1 x < α + βα 1 + β 1 e L 2 + U 1 e L 1 + U 2 γ +α + βl 2 γ 1 +α 1 + β 1 L 1 Then by Rouche s theorem λ and λ λ have the same number of zeroes in an open unit disk λ < 1 Hence the unique positive equilibrium point x ȳ in[l 1 U 1 ] [L 2 U 2 ]ofsystem1 is locally asymptotically stable 24 Global character Theorem 5 If <1 α + βe L 2 < x γ +α + βl 2 and α 1 + β 1 e L 1 < ȳ γ 1 +α 1 + β 1 L 1 10 then the unique positive equilibrium point x ȳ of system 1 is globally asymptotically stable Proof Arranging as in [19] we consider the following discrete time analogof the Lyapunov function:

8 Khan Advances in Difference Equations :297 Page 8 of 21 xn V n = x x 1 ln x n yn + ȳ x ȳ 1 ln y n ȳ The nonnegativity of V n follows from the following inequality: x 1 ln x 0 x >0 Furthermore we have ln ln xn+1 x n yn+1 y n = ln 1 1 x n x n+1 = ln 1 1 y n y n+1 x n+1 x n x n+1 y n+1 y n y n+1 Assume that 10 holdstrue then it followsthat xn+1 V n+1 V n = x x 1 ln x n+1 x yn ȳ ȳ 1 ln y n ȳ x n+1 x n 1 x x n+1 yn+1 + ȳ ȳ +y n+1 y n 1 ȳ 1 ln y n+1 xn x ȳ x 1 ln x n x y n+1 αe y n + βe y n 1 x =x n+1 x n αe y n + βe y n 1 α1 e x n + β 1 e x n 1 ȳγ 1 + α 1 x n + β 1 x n 1 +y n+1 y n α 1 e x n + β 1 e x n 1 α + βe L 2 xγ +α + βl 2 U 1 L 1 α + βe L 2 α1 + β 1 e L 1 ȳγ 1 +α 1 + β 1 L 1 +U 2 L 2 0 α 1 + β 1 e L 1 for all n 0 Thus V n is a non-increasing non-negative sequence It follows that lim n V n 0 Hence we obtain lim n V n+1 V n = 0 Then it follows that lim n x n+1 = x and lim n y n+1 = ȳ FurthermoreV n V 0 for all n 0 which shows that x ȳ [L 1 U 1 ] [L 2 U 2 ] is uniformly stable Hence the unique positive equilibrium point x ȳ [L 1 U 1 ] [L 2 U 2 ]ofsystem1 is globally asymptotically stable 25 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 system 1 Let {x n y n } be any solution of system 1 suchthatlim n x n = x andlim n y n = ȳ To find the error terms note that x n+1 x = αe y n + βe y n 1 α + βe ȳ γ +α + βȳ = αe y n e y n ȳ 1 βe yn 1e yn 1 ȳ 1

9 Khan Advances in Difference Equations :297 Page 9 of 21 So α xy n ȳ β xy n 1 ȳ = αe y n y n ȳ + O 1 y n ȳ 2 βe yn 1y n 1 ȳ + O 2 y n 1 ȳ 2 α xy n ȳ β xy n 1 ȳ x n+1 x = αe y n + x y n ȳ βe yn 1 + x y n 1 ȳ+o 1 yn ȳ 2 Similarly + O 2 yn 1 ȳ 2 11 α 1 e x n + ȳ β 1 e x n 1 + ȳ y n+1 ȳ = x n x x n 1 x+o 3 xn x 2 γ 1 + α 1 x n + βx n 1 γ 1 + α 1 x n + β 1 x n 1 From 11and12 we have + O 4 xn 1 x 2 12 x n+1 x αe y n + x y n ȳ βe yn 1 + x y n 1 ȳ α 1 e x n + ȳ β 1 e x n 1 + ȳ y n+1 ȳ x n x x n 1 x γ 1 + α 1 x n + βx n 1 γ 1 + α 1 x n + β 1 x n 1 13 Let e 1 n = x n x and e 2 n = y n ȳthensystem13canberepresentedas where e 1 n+1 a ne 2 n + b ne 2 n 1 e2 n+1 c ne 1 n + d ne 1 n 1 a n = αe y n + x b n = βe yn 1 + x α 1 e x n + ȳ c n = d n = β 1e xn 1 + ȳ γ 1 + α 1 x n + βx n 1 γ 1 + α 1 x n + β 1 x n 1 Moreover lim a n = αe ȳ + x n γ +α + βȳ lim c n = α 1e x + ȳ n γ 1 +α 1 + β 1 x lim b n = βe ȳ + x n γ +α + βȳ lim d n = β 1e x + ȳ n γ 1 +α 1 + β 1 x So the limiting system of the error terms can be written as e 1 n+1 e 1 n e 2 n+1 = e 2 n 0 0 αe ȳ + x γ +α+βȳ α 1e x +ȳ γ 1 +α 1 +β 1 x βe ȳ + x γ +α+βȳ β 1e x +ȳ γ 1 +α 1 +β 1 x e 1 n e 1 n 1 e 2 n e 2 n 2

10 Khan Advances in Difference Equations :297 Page 10 of 21 which is similar to the linearized system of 1 about the equilibrium point x ȳ Using Proposition 1 one has the following result Theorem 6 Assume that {x n y n } be a positive solution of system 1 such that lim n x n = x and lim n y n = ȳ where xin[l 1 U 1 ] and ȳin[l 2 U 2 ] Then the error vector e n = e 1 n e 1 n 1 e 2 n e 2 n 2 of every solution of 1 satisfies both of the following asymptotic relations: lim en 1 n = λ 1234 F J x ȳ e n+1 lim n n e n = λ 1234 F J x ȳ where λ 1234 F J x ȳ are the characteristic roots of Jacobian matrix F J x ȳ 3 On the system x n+1 = αe xn +βe x n 1 γ +αy n +βy n 1 y n+1 = α 1e yn +β 1 e y n 1 γ 1 +α 1 x n +β 1 x n 1 In this section we shall investigate the asymptotic behavior of system 2 Let x ȳ bethe equilibrium point of system 2 then x = α + βe x γ +α + βȳ ȳ = α 1 + β 1 e ȳ γ 1 +α 1 + β 1 x To construct the corresponding linearized form of system 2 we consider the following transformation: x n x n 1 y n y n 1 f f 1 g g 1 14 where f = αe x n + βe x n 1 f 1 = x n g = α 1e yn + β 1 e yn 1 γ 1 + α 1 x n + β 1 x n 1 g 1 = y n The Jacobian matrix about the fixed point x ȳ under transformation 14 is given by F J x ȳ= A B C D A 1 B 1 C 1 D where A = γ +α+βȳ B = γ +α+βȳ C = γ +α+βȳ D = γ +α+βȳ A 1 = γ 1 +α 1 +β 1 x B 1 = β 1 ȳ γ 1 +α 1 +β 1 x C 1 = α 1e ȳ γ 1 +α 1 +β 1 x D 1 = β 1e ȳ γ 1 +α 1 +β 1 x αe x βe x 31 Boundedness and persistence Theorem 7 Every positive solution {x n y n } of system 2 is bounded and persists α x β x α 1 ȳ

11 Khan Advances in Difference Equations :297 Page 11 of 21 Proof Let {x n y n } be an arbitrary solution of 2 then x n α + β γ From 2 and15 we have = U 1 y n α 1 + β 1 γ 1 = U 2 n =01 15 α+β α + βe γ x n γ +α + β α 1+β 1 = L 1 y n α 1 + β 1 e α 1 +β 1 γ 1 γ 1 γ 1 +α 1 + β 1 α+β γ = L 2 n =23 16 Hence from 15and16we get L 1 x n U 1 L 2 y n U 2 n =34 This proves the statement Theorem 8 Let {x n y n } be a positive solution of system 2 Then [L 1 U 1 ] [L 2 U 2 ] is an invariant set for system 2 Proof Follows by induction 32 Existence and uniqueness and local stability The following theorem shows the existence and uniqueness of the positive equilibrium point of system 2 Theorem 9 If U 1 +1e L 1+ e L 1 L γ 1 α+β e L 1 γ e L 1 α + β +1 < L 2 L 1 1 γ 2 17 L 1 α + β then system 2 has a unique positive equilibrium point x ȳ in [L 1 U 1 ] [L 2 U 2 ] Proof Consider the following system of algebraic equations: x = α + βe x γ +α + βy y = α 1 + β 1 e y γ 1 +α 1 + β 1 x 18 Assume that x y [L 1 U 1 ] [L 2 U 2 ] then it follows from 18that y = e x x γ α + β x = e y y γ 1 α 1 + β 1 Defining Fx= e hx hx γ 1 α 1 + β 1 x where hx= e x γ x α+β x [L 1 U 1 ] It is easy to see that FL 1 = e hl 1 hl 1 γ 1 α 1 +β 1 L 1 >0if and only if e e L 1 L γ 1 α+β > e L 1 L 1 γ α+β L 1 + γ 1 α 1 +β 1 Also FU 1 = e hu 1 hu 1 γ 1 α 1 +β 1 U 1 <0ifand

12 Khan Advances in Difference Equations :297 Page 12 of 21 only if e e U 1 U γ 1 α+β < e U 1 U 1 γ α+β U 1 + γ 1 α 1 +β 1 Hence Fx has at least one positive solution in [L 1 U 1 ] Furthermore assume that condition 17issatisfiedthenonehas F x = x +1e x+ e x x γ α+β e x x γ α+β +1 x 2 e x x γ α+β 2 1 U 1 +1e L 1+ e L 1 L γ 1 α+β e L 1 L 1 γ α+β +1 L 2 1 e L 1 L 1 γ α+β 2 1<0 Hence Fx = 0 has a unique positive solution in [L 1 U 1 ] This completes the proof Theorem 10 If α + βα 1 + β 1 e L 1 L 2 + U 1 U 2 <1 U 1 U 2 γ +α + βl 2 γ1 +α 1 + β 1 L 1 19 then the unique positive equilibrium point x ȳ of system 2 is locally asymptotically stable Proof The characteristic equation of the Jacobian matrix F J x ȳ about the equilibrium point x ȳisgivenby λ 4 p 4 λ 3 + p 3 λ 2 + p 2 λ + p 1 =0 where p 4 = A + C 1 p 3 = AC 1 B A 1 C D 1 p 2 = AD 1 A 1 D + BC 1 B 1 C p 1 = BD 1 B 1 D Assuming condition 19one has 4 α + βe x p i = γ +α + βȳ + α 1 + β 1 e ȳ γ 1 +α 1 + β 1 x i=1 + αα 1 + αβ 1 + α 1 β + ββ 1 e x ȳ +αα 1 + αβ 1 + α 1 β + ββ 1 xȳ γ +α + βȳγ 1 +α 1 + β 1 x = x + ȳ + α + βα 1 + β 1 e x ȳ + xȳ γ +α + βȳγ 1 +α 1 + β 1 x < U 1 + U 2 + α + βα 1 + β 1 e L 1 L 2 + U 1 U 2 <1 20 γ +α + βl 2 γ 1 +α 1 + β 1 L 1 Therefore inequality 20 and Remark 131 of reference [20] implies that the unique positive equilibrium point x ȳ ofsystem2 is locally asymptotically stable This completes the proof 33 Global character Theorem 11 If α + βe L 1 < x γ +α + βl 2 and α 1 + β 1 e L 2 < ȳ γ 1 +α 1 + β 1 L 1 21 then the unique positive equilibrium point x ȳ of system 2 is globally asymptotically stable

13 Khan Advances in Difference Equations :297 Page 13 of 21 Proof Using arrangements for the proof of Theorem 5 and assume that 21 holdstrue then α + βe L 1 xγ +α + βl 2 V n+1 V n U 1 L 1 α + βe L 1 α1 + β 1 e L 2 ȳγ 1 +α 1 + β 1 L 1 +U 2 L 2 0 α 1 + β 1 e L 2 for all n 0sothatV n 0 is a non-increasing sequence It follows that lim n V n 0 Hence we obtain lim n V n+1 V n = 0 It follows that lim n x n+1 = x and lim n y n+1 = ȳfurthermorev n V 0 for all n 0 which implies that x ȳ [L 1 U 1 ] [L 2 U 2 ]isuniform stable Hence the unique positive equilibrium point x ȳ [L 1 U 1 ] [L 2 U 2 ]of system 2 is globally asymptotically stable 34 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 system 2 Let {x n y n } be any solution of system 2 suchthatlim n x n = x andlim n y n = ȳ To find the error terms So x n+1 x = αe x n + βe x n 1 α + βe x γ +α + βȳ = αe x n e xn x 1 βe xn 1exn 1 x 1 α xy n ȳ β xy n 1 ȳ = αe x n x n x + O 1 x n x 2 βe xn 1x n 1 x + O 2 x n 1 x 2 α xy n ȳ β xy n 1 ȳ αe x n βe x n 1 x n+1 x = x n x x n 1 x α x β x y n ȳ y n 1 ȳ + O 1 xn x 2 + O 2 xn 1 x 2 22 Similarly α 1 ȳ β 1 ȳ y n+1 ȳ = x n x x n 1 x γ 1 + α 1 x n + β 1 x n 1 γ 1 + α 1 x n + β 1 x n 1 α 1 e y n β 1 e y n 1 y n ȳ y n 1 ȳ γ 1 + α 1 x n + β 1 x n 1 γ 1 + α 1 x n + β 1 x n 1 + O 3 yn ȳ 2 + O 4 yn 1 ȳ 2 23

14 Khan Advances in Difference Equations :297 Page 14 of 21 From 22and23 we have αe x n+1 x xn γ +αy n +βy n 1 x n x βe x n 1 γ +αy n +βy n 1 x n 1 x α x β x γ +αy n +βy n 1 y n ȳ γ +αy n +βy n 1 y n 1 ȳ α y n+1 ȳ 1 ȳ β γ 1 +α 1 x n +β 1 x n 1 x n x 1 ȳ γ 1 +α 1 x n +β 1 x n 1 x n 1 x α 1 e yn β γ 1 +α 1 x n +β 1 x n 1 y n ȳ 1 e y n 1 γ 1 +α 1 x n +β 1 x n 1 y n 1 ȳ 24 Let e 1 n = x n xande 2 n = y n ȳthensystem24canberepresentedas e 1 n+1 a ne 1 n + b ne 1 n 1 + c ne 2 n + d ne 2 n 1 e2 n+1 e ne 1 n + f ne 1 n 1 + g ne 2 n + h ne 2 n 1 where αe x n βe x n 1 α x a n = b n = c n = β x α 1 ȳ β 1 ȳ d n = e n = f n = γ 1 + α 1 x n + β 1 x n 1 γ 1 + α 1 x n + β 1 x n 1 α 1 e y n β 1 e y n 1 g n = h n = γ 1 + α 1 x n + β 1 x n 1 γ 1 + α 1 x n + β 1 x n 1 Moreover αe x lim a n = n γ +α + βȳ lim c α x n = n γ +α + βȳ lim e α 1 ȳ n = n γ 1 +α 1 + β 1 x lim g α 1 e ȳ n = n γ 1 +α 1 + β 1 x βe x lim b n = n γ +α + βȳ lim d β x n = n γ +α + βȳ lim f β 1 ȳ n = n γ 1 +α 1 + β 1 x lim h β 1 e ȳ n = n γ 1 +α 1 + β 1 x So the limiting system of the error terms can be written as e 1 n+1 αe x βe x α x β x γ +α+βȳ γ +α+βȳ γ +α+βȳ γ +α+βȳ e 1 n e 2 n+1 = α 1 ȳ β e 2 γ 1 +α 1 +β 1 1 ȳ x γ 1 +α 1 +β 1 α 1e ȳ x γ 1 +α 1 +β 1 β 1e ȳ x γ 1 +α 1 +β 1 x n e 1 n e 1 n 1 e 2 n e 2 n 2 which is similar to linearized system of 2 about the equilibrium point x ȳ Using Proposition 1 onehasthefollowingresult Theorem 12 Assume that {x n y n } is a positive solution of system 2 such that lim n x n = x and lim n y n = ȳ where x in[l 1 U 1 ] and ȳin[l 2 U 2 ] Then the error vector e n = e 1 n e 1 n 1 e 2 n e 2 n 2

15 Khan Advances in Difference Equations :297 Page 15 of 21 of every solution of 2 satisfies both of the following asymptotic relations: lim en 1 n = λ1234 F J x ȳ e n+1 lim n n e n = λ1234 F J x ȳ where λ 1234 F J x ȳ are the roots of the characteristic polynomial of F J x ȳ 4 Examples In order to verify our theoretical results and to support our theoretical discussions we consider several interesting numerical examples These examples represent different types of qualitative behavior of solutions of the systems of nonlinear difference equations 1 and 2 All plots in this section are drawn with Mathematica Example 1 Let α = β =3024γ =1128α 1 =1005β 1 =1025γ 1 =1022Then system 1canbewrittenas x n+1 = 00005e y n +3024e y n e x n +1025e x n 1 y n+1 = y n +3024y n x n +1025x n 1 with initial conditions x 1 =18x 0 =001y 1 =1109y 0 =18 In this case the unique positive equilibrium point of system 25 isgivenby x ȳ = Moreover in Figure 1 the plot of x n is shown in Figure 1a the plot of y n isshowninfigure1b and an attractor of system 25isshowninFigure1c a Plot of x n for system 25 b Plot of y n for system 25 Figure 1 Plots for system 25 c An attractor of system 25

16 Khan Advances in Difference Equations :297 Page 16 of 21 Figure 2 Plot of system 26 a Plot of x n for system 27 b Plot of y n for system 27 Figure 3 Plots for system 27 c An attractor of system 27 Example 2 Let α =230β =132γ =500α 1 = 111 β 1 =135γ 1 =600Thensystem1 can be written as x n+1 = 230e y n +132e y n y n +132y n 1 y n+1 = 111e xn +135e xn x n +135x n 1 n =01 26 with initial conditions x 1 =109x 0 =07y 1 =178y 0 =051Theplotofsystem26 is showninfigure2 Example 3 Let α =50β =14γ =30α 1 =05β 1 =125γ 1 =08Thensystem2 can be written as x n+1 = 50e x n +14e x n y n +14y n 1 y n+1 = 05e yn +125e yn x n +125x n 1 27 with initial conditions x 1 =08x 0 =02y 1 =18y 0 =088

17 Khan Advances in Difference Equations :297 Page 17 of 21 a Plot of x n for system 28 b Plot of y n for system 28 Figure 4 Plots for system 28 c An attractor of system 28 In this case the unique positive equilibrium point of system 27 is given by x ȳ = Moreover in Figure 3 the plot of x n isshowninfigure3a the plot of y n isshowninfigure3b and an attractor of system 27isshowninFigure3c Example 4 Let α =45β =14γ =66α 1 =11β 1 =125γ 1 =05Thensystem2 can be written as x n+1 = 45e x n +14e x n y n +14y n 1 y n+1 = 11e yn +125e yn x n +125x n 1 28 with initial conditions x 1 =217x 0 =03y 1 =28y 0 =098 In this case the unique positive equilibrium point of system 28 isgivenby x ȳ = Moreover in Figure 4 the plot of x n isshowninfigure4a the plot of y n isshowninfigure4b and an attractor of system 28isshowninFigure4c Example 5 Let α =125β =45γ =112α 1 =18β 1 =32γ 1 = 009 Then system 2can be written as x n+1 = 125e x n +45e x n y n +45y n 1 y n+1 = 18e yn +32e yn x n +32x n 1 29 with initial conditions x 1 =207x 0 =03y 1 =09y 0 =02 In this case the unique positive equilibrium point of system 29 isgivenby x ȳ = Moreover in Figure 5 the plot of x n is shown in Figure 5a the plot of y n isshowninfigure5b and an attractor of system 29isshowninFigure5c

18 Khan Advances in Difference Equations :297 Page 18 of 21 a Plot of x n for system 29 b Plot of y n for system 29 Figure 5 Plots for system 29 c An attractor of system 29 Example 6 Let α =1245β = 111 γ =1266α 1 =11β 1 =32γ 1 =09Thensystem2 can be written as x n+1 = 1245e x n + 111e x n y n + 111y n 1 y n+1 = 11e yn +32e yn x n +32x n 1 30 with initial conditions x 1 = 1117 x 0 =03y 1 =08y 0 =08 In this case the unique positive equilibrium point of system 30 is unstable Moreover in Figure 6 the plot of x n isshowninfigure6a the plot of y n isshowninfigure6b and aphaseportraitofsystem30isshowninfigure6c Example 7 Let α =1145β =201γ =1266α 1 =15β 1 =232γ 1 =3Thensystem2 can be written as x n+1 = 1145e x n +201e x n y n +201y n 1 y n+1 = 15e yn +232e yn x n +232x n 1 31 with initial conditions x 1 =09x 0 =05y 1 =0001y 0 =08 In this case the unique positive equilibrium point of system 31 is unstable Moreover in Figure 7 the plot of x n is shown in Figure 7a the plot of y n is shown in Figure 7b and aphaseportraitofsystem31isshowninfigure7c Example 8 Let α =2145β =166γ =2566α 1 =16β 1 =252γ 1 =3Thensystem2 can be written as x n+1 = 2145e x n +166e x n y n +21y n 1 y n+1 = 16e yn +252e yn x n +252x n 1 32 with initial conditions x 1 =29x 0 =03y 1 =002y 0 =17

19 Khan Advances in Difference Equations :297 Page 19 of 21 a Plot of x n for system 30 b Plot of y n for system 30 Figure 6 Plots for system 30 c Phase portrait of system 30 a Plot of x n for system 31 b Plot of y n for system 31 Figure 7 Plots for system 31 c Phase portrait of system 31

20 Khan Advances in Difference Equations :297 Page 20 of 21 a Plot of x n for system 32 b Plot of y n for system 32 Figure 8 Plots for system 32 c Phase portrait of system 32 In this case the unique positive equilibrium point of system 32 is unstable Moreover in Figure 8 the plot of x n isshowninfigure8a the plot of y n isshowninfigure8b and aphaseportraitofsystem32isshowninfigure8c 5 Conclusion This work is related to the qualitative behavior of some systems of exponential rational difference equations We have investigated the existence and uniqueness of the positive steady state of system 1and2 Forallpositive values oftheparameters theboundedness and persistence of positive solutions are proved Moreover we have shown that the unique positive equilibrium point of system 1and2 is locallyas well as globallyasymptotically stable under certain parametric conditions 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 of determining the possible global behaviors of the system and determining which parametric conditions lead to these long-term behaviors By constructing a discrete Lyapunov function we have obtained the global asymptotic stability of the positive equilibrium of 1 and2 Finally some illustrative examples are provided to support our theoretical discussion First two examples show that the unique positive equilibrium point of system 1 is stable with different parametric values Meanwhile Examples 3 4and5 show that the unique positive equilibrium point of system 2 is stable whereas the last three examples show that the unique positive equilibrium point of system 2 is unstable with suitable parametric choices Competing interests The author declares that he has no competing interests Author s contributions The author carried out the proof of the main results and approved the final manuscript

21 Khan Advances in Difference Equations :297 Page 21 of 21 Acknowledgements The author thanks 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 Received: 25 September 2014 Accepted: 6 November 2014 Published: 26 Nov 2014 References 1 El-MetwallyEGroveEALadasGLevinsRRadinM:Onthedifferenceequationx n+1 = α + βx n 1 e xn Nonlinear Anal OzturkIBozkurtFOzenS:Onthedifferenceequationy n+1 = α+βe yn γ +y n 1 Appl Math Comput Bozkurt F: Stability analysis of a nonlinear difference equation Int J Mod Nonlinear Theory Appl Papaschinopoulos G Radin MA Schinas CJ: On the system of two difference equations of exponential form: x n+1 = a + bx n 1 e yn y n+1 = c + dy n 1 e xn Math Comput Model Papaschinopoulos G Radin MA Schinas CJ: Study of the asymptotic behavior of the solutions of three systems of difference equations of exponential form Appl Math Comput Papaschinopoulos G Schinas CJ: On the dynamics of two exponential type systems of difference equations Comput Math Appl Khan AQ Qureshi MN: Behavior of an exponential system of difference equations Discrete Dyn Nat Soc 2014 Article ID doi:101155/2014/ Din Q Qureshi MN Khan AQ: Dynamics of a fourth-order system of rational difference equations Adv Differ Equ 2012 Article ID Kulenović MRS Ladas G: Dynamics of Second Order Rational Difference Equations Chapman & Hall/CRC London Elsayed EM: Solutions of rational difference system of order two Math Comput Model Elsayed EM: Behavior and expression of the solutions of some rational difference equations J Comput Anal Appl Elsayed EM El-Metwally H: Stability and solutions for rational recursive sequence of order three J Comput Anal Appl Elsayed EM El-Metwally HA: On the solutions of some nonlinear systems of difference equations Adv Differ Equ 2013 Article ID Din Q Khan AQ Qureshi MN: Qualitative behavior of a host-pathogen model Adv Differ Equ 2013 Article ID Khan AQ Qureshi MN Din Q: Global dynamics of some systems of higher-order rational difference equations Adv Differ Equ 2013 Article ID Qureshi MN Khan AQ Din Q: Asymptotic behavior of a Nicholson-Bailey model Adv Differ Equ 2014 Article ID Sedaghat H: Nonlinear Difference Equations: Theory with Applications to Social Science Models Kluwer Academic Dordrecht Pituk M: More on Poincaré s and Perron s theorems for difference equations J Differ Equ Appl Enatsu Y Nakata Y Muroya Y: Global stability for a class of discrete SIR epidemic models Math Biosci Eng Kocic VL Ladas G: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications Kluwer Academic Dordrecht / Cite this article as: Khan: Global dynamics of two systems of exponential difference equations by Lyapunov function Advances in Difference Equations :297