GLOBAL DYNAMICS OF THE SYSTEM OF TWO EXPONENTIAL DIFFERENCE EQUATIONS

Transcription

1 Electronic Journal of Mathematical Analysis an Applications Vol. 7(2) July 209, pp ISSN: X(online) GLOBAL DYNAMICS OF THE SYSTEM OF TWO EXPONENTIAL DIFFERENCE EQUATIONS MAI NAM PHONG Abstract. The purpose of this paper is to investigate the bouneness an persistence of the solutions, the global stability of the unique positive equilibrium point an the rate of convergence of solutions of the system of two ifference equations which contains exponential terms: x n+ = a + e (bxn+cyn), y n+ = α + e (βxn+γyn), + bx n + cy n + βx n + γy n where the parameters a, b, c,, α, β, γ, an the initial values x 0, y 0 are positive real numbers. Furthermore, we give some numerical examples to illustrate our theoretical results.. Introuction Difference equations arise in the situations in which the iscrete values of the inepenent variable involve. Many practical phenomena are moele with the help of ifference equations [, 3, 8]. In engineering, ifference equations arise in control engineering, igital signal processing, electrical networks, etc. In social sciences, ifference equations arise to stuy the national income of a country an then its variation with time, Cobweb phenomenon in economics, etc. Recently, there has been a great interest in stuying the qualitative properties of ifference equations an systems of ifference equations of exponential form [4, 6,, 2, 3, 4, 5, 9]. In [4], the authors examine the bouneness, the asymptotic behavior, the perioic character of the solutions an the stability character of the positive equilibrium of the ifference equation: x n+ = a + bx n e xn, () where a, b are positive constants an the initial values x, x 0 are positive numbers. Furthermore, in () the authors use a as the immigration rate an b as the growth rate in the population moel. In fact, this was a moel suggeste by the people from the Harvar School of Public Health; stuying the population ynamics of one species x n. 200 Mathematics Subject Classification. 39A0. Key wors an phrases. Equilibrium points, local stability, global behavior, rate of convergence, positive solutions. Submitte Oct. 26,

2 EJMAA-209/7(2) GLOBAL DYNAMICS OF THE SYSTEM 257 In [], the authors stuie the bouneness, the asymptotic behavior, the perioicity an the stability of the positive solutions of the ifference equation: y n+ = α + βe yn γ + y n, (2) where α, β, γ are positive constants an the initial values y, y 0 are positive numbers. In [7], the authors explore the bouneness, the asymptotic behavior an the rate of convergence of the positive solutions of the system of two ifference equations: x n+ = α + βe xn, y n+ = + ζe yn, (3) γ + y n η + x n where α, β, γ,, ζ, η are positive constants an the initial values x 0, y 0 are positive real values. Motivate by these above papers, in this paper, we will investigate the bouneness, the persistence an the asymptotic behavior of the positive solutions of the following system of exponential form: x n+ = a + e (bxn+cyn) + bx n + cy n, y n+ = α + e (βxn+γyn) + βx n + γy n, (4) where a, b, c,, α, β, γ, are positive constants an the initial values x 0, y 0 are positive real values. Moreover, we establish the rate of convergence of a solution that converges to the equilibrium E = ( x, ȳ) of (4). 2. Global behavior of solutions of system (4) The following lemma shows that every positive solution {(x n, y n )} n=0 of (4) is boune an persists. Lemma 2.. Every positive solution of system (4) is boune an persists. Proof. Let (x n, y n ) be an arbitrary solution of (4). From (4) we can see that x n a +, y n α +, n =, 2,... (5) In aition, from (4) an (5) we get x n b(a+) a + e c(α+), y n β(a+) α + e γ(α+) + b(a+) + c(α+) + β(a+) + γ(α+) Therefore, from (5) an (6) the proof of lemma is complete. The next lemma establishes an invariant set for the system (4), n = 2, 3,... (6) Lemma 2.2. Let {(x n, y n )} n=0 be a positive solution of the system (4). Then [ a + e b(a+) c(α+), a + ] + b(a+) + c(α+) is an invariant set for the system (4). Proof. It follows from inuction. [ α + e β(a+) γ(α+) + β(a+) + γ(α+), α + ] The following result will be useful in establishing the global attractivity character of the equilibrium of system (4).

3 258 M. N. PHONG EJMAA-209/7(2) Theorem 2.3. [2] Let R = [a, b ] [c, ] an f : R [a, b ], g : R [c, ] be a continuous functions such that: (a) f(x, y) is ecreasing in both variables an g(x, y) is ecreasing in both variables for each (x, y) R; (b) If (m, M, m 2, M 2 ) R 2 is a solution of { M = f(m, m 2 ), m = f(m, M 2 ) (7) M 2 = g(m, m 2 ), m 2 = g(m, M 2 ) then m = M an m 2 = M 2. Then the following system of ifference equations: x n+ = f(x n, y n ), y n+ = g(x n, y n ) (8) has a unique equilibrium ( x, ȳ) an every solution (x n, y n ) of the system (8) with (x 0, y 0 ) R converges to the unique equilibrium ( x, ȳ). In aition, the equilibrium ( x, ȳ) is globally asymptotically stable. Now we state the main theorem of this section. Theorem 2.4. Consier system (4). Suppose that the following relations hol true: > b + c, > β + γ. (9) Then system (4) has a unique positive equilibrium ( x, ȳ) an every positive solution of system (4) tens to the unique positive equilibrium ( x, ȳ) as n. In aition, the equilibrium ( x, ȳ) is globally asymptotically stable. Proof. We consier the functions where u, v I J = f(u, v) = a + e (bu+cv) + bu + cv, [ a + e b(a+) c(α+) + b(a+) + c(α+) α + e (βu+γv) g(u, v) = + βu + γv, a + ] [ α + e β(a+) γ(α+) + β(a+) + γ(α+) (0), α + ]. () It is easy to see that f(u, v), g(u, v) are ecreasing in both variables for each (u, v) I J. In aition, from (0) an () we have f(u, v) I, g(u, v) J as (u, v) I J an so f : I J I, g : I J J. Now let m, M, m 2, M 2 be positive real numbers such that an M = a + e (bm+cm2) + bm + cm 2, m = a + e (bm+cm2) + bm + cm 2, (2) M 2 = α + e (βm+γm2), m 2 = α + e (βm+γm2). (3) + βm + γm 2 + βm + γm 2 Moreover arguing as in the proof of Theorem.2.3, it suffices to assume that From (2) we get m M, m 2 M 2. (4) M + bm M + cm 2 M = a + e (bm+cm2), m + bm M + cm M 2 = a + e (bm+cm2), (5)

4 EJMAA-209/7(2) GLOBAL DYNAMICS OF THE SYSTEM 259 which implies that (M m ) + cm (m 2 M 2 ) + cm 2 (M m ) = e (bm+cm2) e (bm+cm2). (6) From (6) we have (M m ) + cm (m 2 M 2 ) + cm 2 (M m ) = e (bm+cm2+bm+cm2)+θ [b(m m ) + c(m 2 m 2 )], where bm + cm 2 θ bm + cm 2. From (7) we get from which we have (M m )[ + cm 2 be (bm+cm2+bm+cm2)+θ ] = (M 2 m 2 )[cm + ce (bm+cm2+bm+cm2)+θ ], (7) (8) (M 2 m 2 ) = + cm 2 be (bm+cm2+bm+cm2)+θ cm + ce (bm+cm2+bm+cm2)+θ (M m ). (9) From (3) we imply that From (20) we obtain M 2 + βm M 2 + γm 2 M 2 = α + e (βm+γm2), m 2 + βm 2 M + γm 2 M 2 = α + e (βm+γm2). (20) (M 2 m 2 ) + βm 2 (m M ) + βm (M 2 m 2 ) = e (βm+γm2) e (βm+γm2). (2) From (2) we have (M 2 m 2 ) + βm 2 (m M ) + βm (M 2 m 2 ) = e (βm+γm2+βm+γm2)+θ2 [β(m m ) + γ(m 2 m 2 )], where βm + γm 2 θ 2 βm + γm 2. From (22) we get (M m )[βm 2 + βe (βm+γm2+βm+γm2)+θ2 ] (M 2 m 2 )[ + βm γe (βm+γm2+βm+γm2)+θ2 ] = 0. From two relations (9) an (23) we obtain (M m ) [ ( + cm2 be (bm+cm2+bm+cm2)+θ )( + βm γe (βm+γm2+βm+γm2)+θ2 ) cβ(m + e (bm+cm2+bm+cm2)+θ )(M 2 + βe (βm+γm2+βm+γm2)+θ2 ) ] = 0. (24) By using inequality (9), we have ( + cm 2 be (bm+cm2+bm+cm2)+θ )( + βm γe (βm+γm2+βm+γm2)+θ2 ) > ( + cm 2 b)( + βm γ) > cβ( + M )( + M 2 ). Moreover, we have cβ(m + e (bm+cm2+bm+cm2)+θ )(M 2 + βe (βm+γm2+βm+γm2)+θ2 ) < cβ(m + )(M 2 + ). (22) (23) (25) (26)

5 260 M. N. PHONG EJMAA-209/7(2) Then from (24), (25) an (26) imply m = M, so from (9) we have m 2 = M 2. Hence from Theorem 2.3 system (4) has a unique positive equilibrium ( x, ȳ) an every positive solution of system (4) tens to the unique positive equilibrium ( x, ȳ) as n. In aition, the equilibrium ( x, ȳ) is globally asymptotically stable. This completes the proof of the theorem. 3. Rate of convergence In this section we give the rate of convergence of a solution that converges to the equilibrium E = ( x, ȳ) of the systems (4) for all values of parameters. The rate of convergence of solutions that converge to an equilibrium has been obtaine for some two-imensional systems in [9] an [0]. The following results give the rate of convergence of solutions of a system of ifference equations x n+ = [A + B(n)]x n (27) where x n is a k-imensional vector, A C k k is a constant matrix, an B : Z + C k k is a matrix function satisfying B(n) 0 when n, (28) where enotes any matrix norm which is associate with the vector norm; also enotes the Eucliean norm in R 2 given by x = (x, y) = x 2 + y 2. (29) Theorem 3.. ([8]) Assume that conition (28) hols. system (27), then either x n = 0 for all large n or If x n is a solution of n ρ = lim n xn (30) exists an is equal to the moulus of one of the eigenvalues of matrix A. Theorem 3.2. ([8]) Assume that conition (28) hols. system (27), then either x n = 0 for all large n or x n+ ρ = lim n x n If x n is a solution of exists an is equal to the moulus of one of the eigenvalues of matrix A. The equilibrium point of the system (4) satisfies the following system of equations x = a + e (b x+cȳ) + b x + cȳ ȳ = α + e (β x+γȳ) + β x + γȳ The map T associate to the system (4) is ( ) f(x, y) T (x, y) = = g(x, y) (3). (32) a + e (bx+cy) + bx + cy α + e (βx+γy) + βx + γy. (33)

6 EJMAA-209/7(2) GLOBAL DYNAMICS OF THE SYSTEM 26 The Jacobian matrix of T is J T (x, y) = b[a + ( + bx + cy + )e (bx+cy) ] c[a + ( + bx + cy + )e (bx+cy) ] ( + bx + cy) 2 ( + bx + cy) 2 β[α + (α + βx + γy + )e (βx+γy) ] γ[α + (α + βx + γy + )e (βx+γy) ] (α + βx + γy) 2 (α + βx + γy) 2 (34) By using the system (32), value of the Jacobian matrix of T at the equilibrium point E = ( x, ȳ) is J T ( x, ȳ) = b[a + ( + b x + cȳ + )e (b x+cȳ) ] c[a + ( + b x + cȳ + )e (b x+cȳ) ] ( + b x + cȳ) 2 ( + b x + cȳ) 2 β[α + (α + β x + γȳ + )e (β x+γȳ) ] γ[α + (α + β x + γȳ + )e (β x+γȳ) ] (α + β x + γȳ) 2 (α + β x + γȳ) 2 (35) Our goal in this section is to etermine the rate of convergence of every solution of the system (4) in the regions where the parameters a, b, c,, α, β, γ, (0, ), ( > b+c, > β +γ) an initial conitions x 0 an y 0 are arbitrary, nonnegative numbers. ( ) ( ) e Theorem 3.3. The error vector e n = n xn x e 2 = of every solution (x n y n ȳ n, y n ) ( x, ȳ) of (4) satisfies both of the following asymptotic relations: en = λ i (J T (E)) for some i =, 2, (36) an n lim n.. e n+ lim n e n = λ i(j T (E)) for some i =, 2, (37) where λ i (J T (E)) is equal to the moulus of one of the eigenvalues of the Jacobian matrix evaluate at the equilibrium J T (E). Proof. First, we will fin a system satisfie by the error terms. The error terms are given as x n+ x = a + e (bxn+cyn) + bx n + cy n a + e (b x+cȳ) + b x + cȳ = a( + b x + cȳ) a( + bx n + cy n ) ( + bx n + cy n )( + b x + cȳ) + ( + b x + cȳ)e (bxn+cyn) ( + bx n + cy n )e (b x+cȳ) ( + bx n + cy n )( + b x + cȳ) = b( x x n) + c(ȳ y n ) + [b( x x n ) + c(ȳ y n )]e (bxn+cyn) ( + bx n + cy n )( + b x + cȳ) + ( + bx n + cy n )[e (bxn+cyn) e (b x+cȳ) ] ( + bx n + cy n )( + b x + cȳ) = b(x n x) c(y n ȳ) + [b( x x n ) + c(ȳ y n )]e (bxn+cyn) ( + bx n + cy n )( + b x + cȳ)

7 262 M. N. PHONG EJMAA-209/7(2) + ( + bx n + cy n )e (b x+cȳ) [e (bxn b x+cyn cȳ) ] ( + bx n + cy n )( + b x + cȳ) = b(x n x) c(y n ȳ) + [b( x x n ) + c(ȳ y n )]e (bxn+cyn) ( + bx n + cy n )( + b x + cȳ) + ( + bx n + cy n )e (b x+cȳ) [ b(x n x) c(y n ȳ)] ( + bx n + cy n )( + b x + cȳ) + O ((x n x)) + O 2 ((y n ȳ)) ( + bx n + cy n )( + b x + cȳ) = b [ a + e (bxn+cyn) + ( + bx n + cy n )e (b x+cȳ)] (x n x) ( + bx n + cy n )( + b x + cȳ) + c [ a + e (bxn+cyn) + ( + bx n + cy n )e (b x+cȳ)] (y n ȳ) ( + bx n + cy n )( + b x + cȳ) + ( + bx n + cy n )( + b x + cȳ) O ((x n x)) + ( + bx n + cy n )( + b x + cȳ) O 2 ((y n ȳ)) By calculating similarly, we get y n+ ȳ = γ [ α + e (βxn+γyn) + ( + βx n + γy n )e (β x+γȳ)] (x n x) ( + βx n + γy n )( + β x + γȳ) + β [ α + e (βxn+γyn) + ( + βx n + γy n )e (β x+γȳ)] (y n ȳ) ( + βx n + γy n )( + β x + γȳ) + ( + βx n + γy n )( + β x + γȳ) O 3 ((x n x)) + ( + βx n + γy n )( + β x + γȳ) O 4 ((y n ȳ)) From (38) an (39) we have Set x n+ x b [ a + e (bxn+cyn) + ( + bx n + cy n )e (b x+cȳ)] (x n x) ( + bx n + cy n )( + b x + cȳ) + c [ a + e (bxn+cyn) + ( + bx n + cy n )e (b x+cȳ)] (y n ȳ) ( + bx n + cy n )( + b x + cȳ) y n+ ȳ γ [ α + e (βxn+γyn) + ( + βx n + γy n )e (β x+γȳ)] (x n x) ( + βx n + γy n )( + β x + γȳ) + β [ α + e (βxn+γyn) + ( + βx n + γy n )e (β x+γȳ)] (y n ȳ). ( + βx n + γy n )( + β x + γȳ) Then system (40) can be represente as: e n = x n x an e 2 n = y n ȳ. (38) (39) (40) e n+ a n e n + b n e 2 n e 2 n+ c n e n + n e 2 n

8 EJMAA-209/7(2) GLOBAL DYNAMICS OF THE SYSTEM 263 where a n = b [ a + e (bxn+cyn) + ( + bx n + cy n )e (b x+cȳ)], ( + bx n + cy n )( + b x + cȳ) b n = c [ a + e (bxn+cyn) + ( + bx n + cy n )e (b x+cȳ)], ( + bx n + cy n )( + b x + cȳ) c n = γ [ α + e (βxn+γyn) + ( + βx n + γy n )e (β x+γȳ)], ( + βx n + γy n )( + β x + γȳ) n = β [ α + e (βxn+γyn) + ( + βx n + γy n )e (β x+γȳ)]. ( + βx n + γy n )( + β x + γȳ) Taking the limits of a n, b n, c n an n as n, we obtain lim a n = b [ ] a + ( + b x + cȳ + )e (b x+cȳ) n ( + b x + cȳ) 2 := A, lim b n = c [ ] a + ( + b x + cȳ + )e (b x+cȳ) n ( + b x + cȳ) 2 := B, lim c n = γ [ ] α + ( + β x + γȳ + )e (β x+γȳ) n ( + β x + γȳ) 2 := C, lim n = β [ ] α + ( + β x + γȳ + )e (β x+γȳ) n ( + β x + γȳ) 2 := D, that is a n = A + α n, b n = B + β n, c n = C + γ n, n = D + n, where α n 0, β n 0, γ n 0 an n 0 as n. Now, we have system of the form (27): e n+ = (A + B(n))e n, ( ) ( ) A B where A = αn β, B(n) = n an C D n γ n B(n) 0 as n. Thus, the limiting system of error terms can be written as: ( ) ( ) e n+ e = A n e 2. n e 2 n+ The system is exactly linearize system of (4) evaluate at the equilibrium E = ( x, ȳ). Then Theorem 3. an Theorem 3.2 imply the result. 4. Examples In orer to verify our theoretical results an to support our theoretical iscussion, we consier several interesting numerical examples. These examples represent ifferent types of qualitative behavior of solutions of the systems (4). All plots in this section are rawn with Matlab.

9 264 M. N. PHONG EJMAA-209/7(2) Example 4.. Let a = 30, b = , c = 0.8, = 0.95, ( > b + c); α = 35, β = 0.85, γ = , = 0.9, ( > β + γ). Then system (4) can be written as x n+ = 30 + e (0.0007xn+0.8yn) 35 + e (0.85xn yn), y n+ =, (4) x n + 0.8y n x n y n with initial conitions x 0 = 8 an y 0 = 7. (a) Plot of x n for the system (4) (b) Plot of y n for the system (4) (c) An attractor of the system (4) Figure. Plots for the system (4) In Figure, the plot of x n is shown in Figure (a), the plot of y n is shown in Figure (b), an an attractor of the system (4) is shown in Figure (c). Example 4.2. Let a = 25, b = , c = 0.7, = 0.89, ( > b + c); α = 5, β = 0.8, γ = , = 0.85, ( > β + γ). Then system (4) can be written as x n+ = 25 + e ( xn+0.7yn) 5 + e (0.8xn yn), y n+ =, (42) x n + 0.7y n x n y n with initial conitions x 0 = 7 an y 0 = 4. In Figure 2, the plot of x n is shown in Figure 2 (a), the plot of y n is shown in Figure 2 (b), an an attractor of the system (42) is shown in Figure 2 (c). Example 4.3. Let a = 20, b = 0.0, c = 0.6, = 0.005, ( < b + c); α = 25, β = 0.8, γ = 0.02, = 0.09, ( < β + γ). Then system (4) can be written as x n+ = 20 + e (0.0xn+0.6yn) 25 + e (0.8xn+0.02yn), y n+ =, (43) x n + 0.6y n x n y n with initial conitions x 0 = 5 an y 0 = 2.

10 EJMAA-209/7(2) GLOBAL DYNAMICS OF THE SYSTEM 265 (a) Plot of x n for the system (42) (b) Plot of y n for the system (42) (c) An attractor of the system (42) Figure 2. Plots for the system (42) (a) Plot of x n for the system (43) (b) Plot of y n for the system (43) (c) Phase portrait of the system (43) Figure 3. Plots for the system (43)

11 266 M. N. PHONG EJMAA-209/7(2) In this case, the unique positive equilibrium point of the system (43) is unstable. Moreover, in Figure 3, the plot of x n is shown in Figure 3 (a), the plot of y n is shown in Figure 3 (b), an a phase portrait of the system (43) is shown in Figure 3 (c). References [] Agarwal, R.P., Difference Equations an Inequalities, Secon E. Dekker, New York, (2000). [2] Burgić, DŽ., Nurkanović, Z., An example of globally asymptotically stable anti-monotonic system of rational ifference equations in the plane, Sarajevo Journal of Mathematics. 5(8) (2009) [3] S. Elayi, An Introuction to Difference Equations, 3r e., Springer-Verlag, New York, [4] El-Metwally, E., Grove, E.A., Laas, G., Levins, R., Rain, M., On the ifference equation, x n+ = α + βx n e xn, Nonlinear Anal. 47 (200) [5] Elsaye, E.M., Dynamics an behavior of a higher orer rational ifference equation, J. Nonlinear Sci. Appl. 9 (206), pp [6] Khuong, V. V., Phong, M. N., On the system of two ifference equations of exponential form, Int. J. Difference Equ. 8(5) (203), [7] Khuong, V. V., Phong, M. N., Global behavior an the rate of convergence of a system of two ifference equations of exponential form, PanAmer. Math. J. 27()(207), pp [8] Kocic, V. L., Laas, G., Global behavior of nonlinear ifference equations of higher orer with applications, Kluwer Acaemic, Dorrecht, (993). [9] Kulenović, M. R. S., Nurkanović, Z.,The rate of convergence of solution of a three imensional linear fractional systems of ifference equations, Zbornik raova PMF Tuzla - Svezak Matematika. 2 (2005) 6. [0] Kulenović, M.R.S., Nurkanović, M., Asymptotic behavior of a competitive system of linear fractional ifference equations, Av. Difference Equ. Art. ID 9756 (2006) 3pp. [] Ozturk, I., Bozkurt, F., Ozen, S., On the ifference equation y n+ = (α + βe yn )/(γ + y n ), Appl. Math. Comput.8 (2006) [2] Papaschinopoluos, G., Ellina, G., Papaopoulos, K.B., Asymptotic behavior of the positive solutions of an exponential type system of ifference equations, Appl. Math. Comput. 245 (204) [3] Papaschinopoluos, G., Fotiaes, N., Schinas, C.J., On a system of ifference equations incluing negative exponential terms, J. Difference Equ. Appl. 20(5-6) (204) [4] Papaschinopoluos, G., Rain, M.A., Schinas, C.J., On a system of two ifference equations of exponential form: x n+ = a + bx n e yn, y n+ = c + y n e xn, Math. Comput. Moel. 54 (20) [5] Papaschinopoluos, G., Rain, M.A., Schinas, C.J., Stuy of the asymptotic behavior of the solutions of three systems of ifference equations of exponential form, Appl. Math. Comput. 28 (202) [6] Phong, M.N.,A note on a system of two nonlinear ifference equations, Electronic Journal of Mathematical Analysis an Applications, 3() (205) [7] Phong, M.N., Khuong, V.V., Asymptotic behavior of a system of two ifference equations of exponential form, Int. J. Nonlinear Anal. Appl. 7(2)(206), pp [8] Pituk, M.,More on Poincare s an Peron s theorems for ifference equations, J. Difference Equ. Appl. 8 (2002) [9] Stefaniou, G., Papaschinopoluos, G., Schinas, C.J.,On a system of two exponential type ifference equations, Com. Appl. Nonlinear Anal. 7(2) (200) 3. Mai Nam Phong Department of Mathematical Analysis, University of Transport an Communications, Hanoi City, Vietnam aress: mnphong@utc.eu.vn