ON THE STABILITY OF SOME SYSTEMS OF EXPONENTIAL DIFFERENCE EQUATIONS. N. Psarros, G. Papaschinopoulos, and C.J. Schinas

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1 Opuscula Math. 38, no , Opuscula Mathematica ON THE STABILITY OF SOME SYSTEMS OF EXPONENTIAL DIFFERENCE EQUATIONS N. Psarros, G. Papaschinopoulos, C.J. Schinas Communicated by Stevo Stević Abstract. In this paper we prove the stability of the zero equilibria of two systems of difference equations of exponential type, which are some extensions of an one-dimensional biological model. The stability of these systems is investigated in the special case when one of the eigenvalues is equal to -1 the other eigenvalue has absolute value less than 1, using centre manifold theory. In addition, we study the existence uniqueness of positive equilibria, the attractivity the global asymptotic stability of these equilibria of some related systems of difference equations. Keywords: difference equations, asymptotic behaviour, global stability, centre manifold, biological dynamics. Mathematics Subject Classification: 39A INTRODUCTION Difference equations systems of difference equations containing exponential terms have numerous potential applications in biology. A large number of papers dealing with such or related equations have been published see, e.g, [6, 16, 22 24]. In [34], the following model incorporating litter inhibition is discussed: ea blt ea blt B t+1 = cn 1 + e, L a blt t+1 = L2 t L t + d + ckn 1 + e, a blt B is the living biomass, L the litter mass, N the total soil nitrogen, t the time measured in years constants a, b, c, d > 0 0 < k < 1. In this model, the living biomass B is reduced below its equilibrium, by litter. Litter decay is determined by d, while litter production is k times the living biomass. The complexity of the grassl ecosystem makes its study interesting but complicated. In addition, in [18], the authors studied the boundedness the persistence of the positive solutions, the existence, c Wydawnictwa AGH, Krakow

2 96 N. Psarros, G. Papaschinopoulos, C.J. Schinas the attractivity the global asymptotic stability of the unique positive equilibrium, as well as the existence of periodic solutions of the following equation: x2 n e k dxn x n+1 = a + c b + x n 1 + e, k dxn a 0, 1, a, b, c, d, k are positive constants x 0 is a positive real number. Motivated by this discrete time model recent studies of symmetric close to symmetric systems of difference equations see, e.g, [9, 11, 19, 25, 26], in this paper, we will study the stability of the zero equilibria of the following systems: y n x n e k1 d1xn x n+1 = a 1 + c 1 b 1 + y n 1 + e, y x n y n e k2 d2yn k1 d1xn n+1 = a 2 + c b 2 + x n 1 + e k2 d2yn x n y n e k1 d1yn x n+1 = a 1 + c 1 b 1 + x n 1 + e, y y n x n e k2 d2xn k1 d1yn n+1 = a 2 + c 2, 1.2 k2 d2xn b 2 + y n 1 + e a 1, a 2, b 1, b 2, c 1, c 2, d 1, d 2, k 1, k 2, are real constants the initial values x 0 y 0 are real numbers. In addition, we will investigate the asymptotic behaviour of the positive solutions of the following systems of difference equations: 2 y n e d1 k1xn x n+1 = a 1 + c 1 b 1 + y n 1 + e, y e b2 k2xn d1 k1xn n+1 = a e + c e d2 k3yn b2 k2xn e, d2 k3yn y n e d1 k1xn 2 x n+1 = a 1 + c 1 b 1 + y n 1 + e, y x n e d2 k2yn d1 k1xn n+1 = a 2 + c 2, 1.4 d2 k2yn b 2 + x n 1 + e 2 x n e d1 k1yn x n+1 = a 1 + c 1 b 1 + x n 1 + e, y e b2 k2xn d1 k1yn n+1 = a e + c e d2 k3yn b2 k2xn e, d2 k3yn x n e d1 k1yn 2 x n+1 = a 1 + c 1 b 1 + x n 1 + e, y y n e d2 k2xn d1 k1yn n+1 = a 2 + c 2, 1.6 d2 k2xn b 2 + y n 1 + e a 1, a 2, b 1, b 2, c 1, c 2, d 1, d 2, k 1, k 2, k 3 are positive constants the initial values x 0 y 0 are positive. The results of this paper could be used to create more elaborate biological models to facilitate understing the underlying ecological mechanisms. The results obtained for the systems provide conditions for stability of the zero equilibria of those systems. Those equilibria correspond to the physical situation both quantities x y vanish. The asymptotic behaviour of positive solutions of scalar equations related to the previous systems is studied in [22]. For some related cyclic systems of difference equations see [9, 28, 31] [32], as well as some three-dimensional systems see, e.g., [19, 27, 33]. Finally we note that, since difference equations have several applications in applied sciences, there exists a rich bibliography concerning theory applications of difference equations see [1 34].

3 On the stability of some systems of exponential difference equations STABILITY OF ZERO EQUILIBRIUM OF SYSTEM 1.1 In the following, we prove the stability of the zero equilibrium of System 1.1, using Centre Manifold Theory. Proposition 2.1. Consider System 1.1 a 1, b 1, b 2, c 2, k 1, are real positive constants a 2, c 1, k 2 are real negative constants such that c 2 < 1 + e k2, 1 + ek1 e k1 < c 1 < c 2e k2 1 + e k1 e k1 1 + e k2, c 1e k1 1 + c 2e k2 = a 1a 2, e k1 1 + e k2 b 1 b 2 } d 2 > max {0, A 1, A 2, A 3, 2.3 A 1 = c ek2 1 d 1 Γe k1 c 2 Γ 2 e k2 1 + e k1 2 + a 1Γ 3 A 2 = ek2 c 2 Γ e k2 A 3 = b 2 1 a 2 b 2 2 c1 d 1 e k1 1 + e k1 2 + a 1 2 Γ b 2 a 2 1 b e k2 1 + c e k2 2 b 2 1 c. 2e k2 1 e k2,, Then the zero equilibrium of 1.1 is stable. Proof. The Jacobian matrix J 0 at the zero equilibrium for 1.1 is ] J 0 = [ c1e k 1 1+e k 1 a 1 b 1 a 2 b 2 c 2e k 2 1+e k 2 Calculating the eigenvalues of J 0, using we obtain λ 1 = 1, λ 2 = 1 + c 1e k1 1 + e k1 + c 2e k2 Now, the initial system can be written as [ xn+1 y n+1 ] [ ] xn = J 0 + y n. 1 + e k2 so λ 2 < 1. fx, y = a 1y a 1y + c 1xe k1 d1x y + b 1 b e k1 d1x c 1e k1 1 + e gx, y = a 2x a 2x + c 2ye k2 d2y x + b 2 b e k2 d2y c 2e k2 1 + e [ ] fxn, y n, 2.4 gx n, y n k1 x, k2 y.

4 98 N. Psarros, G. Papaschinopoulos, C.J. Schinas We let now [ xn y n ] = T [ un T is the matrix that diagonalizes J 0 defined by [ ] 1 1 T =, Γ Γ = b 11 + e k1 1 + c 1 a e k1 Then, 2.4 can be written as [ ] [ ] [ un un = v n+1 0 λ 2 v n ],, = b 11 + e k2 1 + c a e k2 v n ] + [ ] ˆfun, v n, 2.6 ĝu n, v n ˆfu, v a 1 Γu + v = R b 1 + Γu + v a 1Γu + v + c 1u + ve k1 d1u+v c 1u + ve k1 b e k1 d1u+v 1 + e k1 a 2 u + v b 2 + u + v a 2u + v + c 2Γu + ve k2 d2γu+ v c 2Γu + ve k2, b e k2 d2γu+ v 1 + e k2 ĝu, v a 1 Γu + v = R Γ b 1 + Γu + v a 1Γu + v + c 1u + ve k1 d1u+v c 1u + ve k1 b e k1 d1u+v 1 + e k1 + a 2u + v b 2 + u + v a 2u + v + c 2Γu + ve k2 d2γu+ v c 2Γu + ve k2 b e k2 d2γu+ v 1 + e k2 2.7 R = 1 Γ. We now let v = hu with hu = ψu + Ou 4, ψu = ηu 2 + θu 3, η, θ R. The use of this approximation is justified by Theorem 7 of [1]. Consequently, according to Theorem 8 of [1] using 2.6, the study of the stability of the zero equilibrium of System 1.1 reduces to the study of the stability of the zero equilibrium of the equation: u n+1 = u n + ˆfu n, ψu n = Gu n. 2.8 We now need to determine η the coefficient in the Taylor expansion. From 2.6, we conclude that map h must satisfy the centre manifold equation see [1, p. 34], [3, p. 243], [15], [16, p. 642] [19]: h u + ˆfu, hu λ 2 hu ĝu, hu = 0.

5 On the stability of some systems of exponential difference equations 99 Keeping the terms up to u 3 using 2.7, we obtain η = R c 1 d 1 Γe k1 1 λ e k1 2 + a 1Γ 3 b 2 a 2 1 b 2 c 2d 2 Γ 2 e k e k From we obtain G 0 = 1 G 0 = R 6c 1d 2 1e 2k1 1 + e k c 1d 2 1e k1 1 + e k1 2 12c 1d 1 ηe k1 1 + e k1 2 12a 1 Γη b 2 + 6a 1Γ 3 1 b 3 1 6a2 b 3 12a 2η 2 b 2 12c 2d 2 Γe k2 η e k2 2 6c 2d 2 2Γ 3 e 2k2 1 + e k c 2d 2 2Γ 3 e k2 1 + e k From 2.10 since R > 0, we deduce that if the following inequalities hold, then, G 0 > 0: η c 1d 1 e k1 1 + e k1 2 a 1 2 Γ b 2 + a 2 1 b 2 + c 2d 2 Γe k e k2 2 > 0, 2.11 Γ 3 2a1 b 3 + c 2d 2 2e k2 e k e k2 3 > 0, 2.12 c 1 d 2 1 e k1 1 e k1 1 + e k1 3 2a 2 b 3 > Now, from 2.3, we have that d 2 > A 1 so from 2.9 we conclude that η < 0. Inequality 2.11 holds, since η < 0 from 2.3 we obtain d 2 > A 2. Moreover, from 2.1, we obtain c 1 > 1+ek 1 therefore from 2.5 we get Γ < 0. Hence, 2.12 e k 1 holds, since from 2.3 we have d 2 > A 3. Finally, 2.13 is always true, since c 1, a 2 < 0, k 1 > 0. So, we have shown, that if the conditions in the proposition hold, then G 0 > 0. Hence, for the Schwarzian derivative see [3], [13], we have Sf0 < 0. Therefore, from Theorem 8 of [1], the zero equilibrium of 1.1 is stable. 3. STABILITY OF ZERO EQUILIBRIUM OF SYSTEM 1.2 In the following, we prove the stability of the zero equilibrium of System 1.2, using Centre Manifold Theory. Proposition 3.1. Consider System 1.2 b 1, b 2, c 1, c 2, k 1, k 2 are real positive constants a 1, a 2 are real negative constants such that 2 < a 1 b 1 + a 2 b 2 < 0, b 1 < a 1 < 0, b 2 < a 2 <

6 100 N. Psarros, G. Papaschinopoulos, C.J. Schinas Let 2a e A 1 = k1 2 a 1 + b 1 4 b 2 2 a 2 + b 2 1 e k1 a 2, 1 e2k1 A 2 = a2 2a 1 + b e k e k2 e k2 1 2a 1 b 1 b 3, 2 e3k1 e k2 a 2 + b 2 A 3 = 1 + a a e k e k2, b 1 b 2 e k1+k2 K 1 = 1 + ek1 b 2 2a 2 a 2 + b 2 1 e k1, K 2 = a 1c 1 e k1 a 1 + b 1 2, 2a e K 3 = k e k1 a 2 + b 2 b 2 c 1 c 2 b 3, 1 e ek1 k2 e k2 1 K 4 = a 1 + b 1 2 a e k e k2 2 b 2 1 b2 2 c2 1 c 2e 2k1 e k2. Suppose also that the following relations hold: 4a 1 + b 1 a 2 + b 2 < a 1 a 2 e k1 1e k2 1, A 1 < c 1 < Then the zero equilibrium of 1.2 is stable. A2 A 3, c 2 A 3 c K 1 < d 1 < K 2, 3.3 K 3 < d 2 < K Proof. Firstly, we note that 4a 1 + b 1 a 2 + b 2 < a 1 a 2 e k1 1e k2 1 implies that A A 1 < 2 A 3. Next, we can easily see that K 1 < K 2 is true, since from 3.2, we have c 1 > A 1. Moreover, using c 1 c 2 < A 3 c 2 1 < A2 A 3 from 3.2, we have c 3 1c 2 < A 2 therefore K 3 < K 4 is true. Now, the Jacobian matrix J 0 at the zero equilibrium for 1.2 is ] J 0 = [ a1 c 1e k 1 b 1 1+e k 1 c 2e k 2 a 2 1+e k 2 b 2 Calculating the eigenvalues of J 0, using we obtain λ 1 = 1, λ 2 = 1 + a 1 b 1 + a 2 b 2 so λ 2 < 1..

7 On the stability of some systems of exponential difference equations 101 Now, the initial system can be written as [ ] [ ] xn+1 xn = J y 0 + n+1 y n We let now [ ] fxn, y n, 3.5 gx n, y n fx, y = a 1x a 1x + c 1ye k1 d1y x + b 1 b e k1 d1y c 1e k1 y, k1 1 + e gx, y = a 2y a 2y + c 2xe k2 d2x y + b 2 b e k2 d2x c 2e k2 x. k2 1 + e [ xn y n ] = T [ un T is the matrix that diagonalizes J 0 defined by [ ] Γ T =, 1 1 Then, 3.5 can be written as v n ], b 1 c 1 e k1 Γ = b 1 + a e k1, = b 2 c 1 e k1 a 2 + b e k [ un+1 v n+1 ] [ ] [ 1 0 un = 0 λ 2 ˆfu, v a 1 Γu + v = R b 1 + Γu + v a 1Γu + v b 1 v n ] + [ ] ˆfun, v n, 3.7 ĝu n, v n + c 1u + ve k1 d1u+v c 1u + ve k1 1 + e k1 d1u+v 1 + e k1 a 2 u + v b 2 + u + v a 2u + v + c 2Γu + ve k2 d2γu+ v c 2Γu + ve k2 b e k2 d2γu+ v 1 + e k2 ĝu, v a 1 Γu + v = R b 1 + Γu + v a 1Γu + v b 1 Γ + c 1u + ve k1 d1u+v c 1u + ve k1 1 + e k1 d1u+v 1 + e k1 a 2 u + v b 2 + u + v a 2u + v + c 2Γu + ve k2 d2γu+ v c 2Γu + ve k2 b e k2 d2γu+ v 1 + e k2 R = 1 Γ. 3.8,

8 102 N. Psarros, G. Papaschinopoulos, C.J. Schinas We now let v = hu with hu = ψu + Ou 4, ψu = ηu 2 + θu 3, η, θ R. Using this approximation is justified by Theorem 7 of [1]. Consequently, according to Theorem 8 of [1] using 3.7, the study of the stability of the zero equilibrium of System 1.2 reduces to the study of the stability of the zero equilibrium of the equation u n+1 = u n + ˆfu n, ψu n = Gu n. 3.9 We now need to determine η the coefficient in the Taylor expansion. From 3.7, we conclude that map h must satisfy the centre manifold equation see [1, p. 34], [3, p. 243], [15], [16, p. 642] [19]: h u + ˆfu, hu λ 2 hu ĝu, hu = 0. Keeping the terms up to u 3 using 3.8, we obtain η = R c1 d 1 e k1 1 λ e k1 2 + a 1Γ 2 a2 b 2 Γ 1 b 2 + c 2d 2 Γ 2 e k e k From we obtain G 0 = 1 G 0 = R 6c 1d 2 1e 2k1 1 + e k c 1d 2 1e k1 1 + e k1 2 12c 1d 1 ηe k1 1 + e k1 2 12a 1 Γη 6a2 b a 2η b 2 2 b a 1Γ 3 b c 2d 2 Γe k2 η 1 + e k2 2 6c 2d 2 2Γ 3 e 2k2 1 + e k c 2d 2 2Γ 3 e k2 1 + e k From 3.11 since R < 0, we deduce that if the following inequalities hold, then, G 0 > 0: η c 1d 1 e k1 1 + e k1 2 a 1 Γ b 2 + a 2 1 b 2 + c 2d 2 2 Γe k e k2 2 < 0, 3.12 Γ 3 2a1 b 3 + 2c 2d 2 2 e 2k e k2 3 c 2d 2 2 e k2 1 + e k2 2 < 0, c 1d 2 1e 2k1 1 + e k1 3 + c 1d 2 1e k1 1 + e k1 2 2 a 2 b 3 < Now, from the second inequalites of , we obtain from 3.10 that η > 0. In addition, from conditions 3.1 equation 3.6 we obtain Γ < 0. Hence, 3.12 holds, since η > 0, a 1, a 2, Γ < 0 c 1, c 2, d 1, d 2 > 0. Moreover, from the first inequality of 3.4 since Γ < 0, we deduce that 3.13 is true. Finally, from the first inequality of 3.3, we obtain that 3.14 is also true. So, we have shown, that if the conditions in the proposition hold, then G 0 > 0. Hence, for the Schwarzian derivative, we have Sf0 < 0. Therefore, from Theorem 8 of [1], the zero equilibrium of 1.2 is stable.

9 On the stability of some systems of exponential difference equations GLOBAL BEHAVIOUR OF SOLUTIONS OF SYSTEM 1.3 In this section, we will investigate the asymptotic behaviour of the positive solutions of 1.3. We will use the following theorem, which is essentially a slight modification of Theorem 1.16 of [5]. The proof of Theorem 1.16 of [5] can be easily adapted to this case. Theorem 4.1. Let f, g, with f : R + R + R +, g : R + R + R + be continuous functions, R + = 0,. Let a, A, b, B be positive numbers, such that a < A, b < B f : [a, A] [b, B] [a, A], g : [a, A] [b, B] [b, B]. Consider the system of difference equations x n+1 = fx n, y n, y n+1 = gx n, y n, n = 0, 1, Suppose that fx, y is a non-increasing function with respect to x a non-decreasing with respect to y. Moreover, suppose that gx, y is a non-increasing function with respect to x a non-increasing function with respect to y. Finally suppose that, if m, M, r, R are real numbers such that if M = fm, R, m = fm, r, R = gm, r, r = gm, R, then m = M r = R. Then the system of difference equations 4.1 has a unique positive equilibrium x, ȳ every positive solution of System 4.1 which satisfies x n0 [a, A], y n0 [b, B] tends to the unique equilibrium of 4.1. We now prove the following result. Proposition 4.2. Consider System 1.3. Suppose that the following relations hold true: k 1 c 1 < 1, k 3 c 2 < a 1 a 2 k 2 < 1 k 1 c 1 1 k 3 c Then System 1.3 has a unique positive equilibrium every positive solution of System 1.3 tends to the unique equilibrium of 1.3 as n. Proof. System 1.3 can be written as x n+1 = fx n, y n, fx, y = a 1 y 2 e d1 k1x y n+1 = gx n, y n, b 1 + y + c e d1 k1x 4.4 e b2 k2x e d2 k3y gx, y = a e b2 k2x + c ed2 k3y We can now easily see that fx, y is non-increasing in x non-decreasing in y, while gx, y is non-increasing in x non-increasing in y. We now show that f : [a, A] [b, B] [a, A] g : [a, A] [b, B] [b, B], B = a 2 + c 2, A = a 1 B 2 b 1 + B + c 1,

10 104 N. Psarros, G. Papaschinopoulos, C.J. Schinas e b2 k2a e d2 k3b b = a e b2 k2a + c e d2 k3b a = a 1 b 1 + b + c e d1 k1a. b 2 e d1 k1a Indeed, from 4.5, we can easily obtain that gx, y a 2 + c 2 = B. Now, for y B, B from 4.4, we get fx, y a 2 1 b + c 1+B 1 = A. Moreover, for x A y B from 4.5, we obtain e b2 k2a e d2 k3b gx, y a e b2 k2a + c 2 = b. 1 + ed2 k3b Finally, taking x A, y b from 4.4, we conclude that fx, y a 1 b 2 e d1 k1a b 1 + b + c 1 = a. 1 + ed1 k1a Now, let m, M, r, R be positive real numbers such that M = fm, R, m = fm, r, R = gm, r r = gm, R. Therefore, we obtain M m = fm, R fm, r R r = gm, r gm, R. So, we can write M m = fm, R fm, R + fm, R fm, r R r = gm, r gm, r + gm, r gm, R. Using the Mean Value Theorem, we obtain M m = f x ξ 1, Rm M + f y M, ξ 2 R r R r = g x ξ 3, rm M + g y M, ξ 4 r R for some ξ 1, ξ 3 m, M ξ 2, ξ 4 r, R. Therefore, we can write However, we have M m f x ξ 1, R m M + f y M, ξ 2 R r 4.6 R r g x ξ 3, r m M + g y M, ξ 4 r R. 4.7 f x ξ 1, R = c 1k 1 e d1 k1ξ1 1 + e d1 k1ξ1 2, f ym, ξ 2 = a 1 2b 1 ξ 2 + ξ 2 2 b 1 + ξ 2 2, 4.8 g x ξ 3, r = a 2k 2 e b2 k2ξ3 1 + e b2 k2ξ3 2, g ym, ξ 4 = c 2k 3 e d2 k3ξ4 1 + e d2 k3ξ From 4.6, 4.7, 4.8, 4.9, we obtain M m 1 k 1 c 1 a 1 R r, R r 1 k 3 c 2 a 2 k 2 M m. 4.10

11 On the stability of some systems of exponential difference equations 105 Therefore, from 4.10, we conclude that M m Hence, from 4.3, , we obtain a 1 a 2 k 2 M m k 1 c 1 1 k 3 c 2 M = m, R = r. Let now x n, y n be an arbitrary solution of 1.3. From the discussion above, it is obvious that y 1 B. Then we can see that x 2 A. In addition, we also have y 2 B so we can get y 3 b. Finally, since we also have x 3 A, we obtain x 4 a. Hence, we have shown that x n [a, A], y n [b, B], for all n 4. Therefore, from Theorem 4.1, System 1.3 has a unique positive equilibrium x, ỹ every positive solution of System 1.3 tends to the unique positive equilibrium as n. This completes the proof of the proposition. Proposition 4.3. Consider System 1.3, the conditions in Proposition 4.2 hold. In addition, suppose that the following relation holds true: k 1 k 3 c 1 c 2 + k 2 a 1 a 2 < Then the unique positive equilibrium x, ỹ of System 1.3 is globally asymptotically stable. Proof. First, we will prove that x, ỹ is locally asymptotically stable. The linearised system of 1.3 is [ ] [ ] α β xn w n+1 = Aw n, A =, w γ δ n =. y n α = c d1 k1 x 1k 1 e 1 + e d1 k1 x 2, β = a 1ỹ 2 + 2b 1 ỹ b 1 + ỹ 2, γ = a b2 k2 x 2k 2 e 1 + e b2 k2 x 2, δ = c 2k 3 e d2 k3ỹ 1 + e d2 k3ỹ 2. The characteristic equation of A is λ 2 α+δλ+αδ βγ = 0. From relation 4.12, we obtain αδ βγ < 1. Moreover, form 4.2, we obtain α + δ < 1 + αδ βγ. Therefore, from Theorem of [13], we deduce that x, ỹ is locally asymptotically stable. Using Proposition 4.2, we conclude that x, ỹ is globally asymptotically stable. This completes the proof of the proposition. 5. GLOBAL BEHAVIOUR OF SOLUTIONS OF SYSTEM 1.4 In this section, we will investigate the asymptotic behaviour of the positive solutions of 1.4. We will use the following theorem, which is essentially a slight modification of Theorem 1.16 of [5]. The proof of Theorem 1.16 of [5] can be easily adapted to this case.

12 106 N. Psarros, G. Papaschinopoulos, C.J. Schinas Theorem 5.1. Let f, g, with f : R + R + R +, g : R + R + R + be continuous functions, R + = 0,. Let a, A, b, B be positive numbers, such that a < A, b < B f : [a, A] [b, B] [a, A], g : [a, A] [b, B] [b, B]. Consider the system of difference equations x n+1 = fx n, y n, y n+1 = gx n, y n, n = 0, 1, Suppose that fx, y is a non-increasing function with respect to x a non-decreasing with respect to y. Moreover, suppose that gx, y is a non-decreasing function with respect to x a non-increasing function with respect to y. Finally suppose that, if m, M, r, R are real numbers such that if m = fm, r, M = fm, R, r = gm, R, R = gm, r, then m = M r = R. Then the system of difference equations 5.1 has a unique positive equilibrium x, ȳ every positive solution of system 5.1 which satisfies x n0 [a, A], y n0 [b, B] tends to the unique equilibrium of 5.1. We now prove the following result. Proposition 5.2. Consider system 1.4. Suppose that the following relations hold true: k 1 c 1 < 1, k 2 c 2 < 1, a 1, a 2 < a 1 a 2 < 1 k 1 c 1 1 k 2 c Then System 1.4 has a unique positive equilibrium every positive solution of the System 1.4 tends to the unique equilibrium of 1.4 as n. Proof. System 1.4 can be written as x n+1 = fx n, y n, y n+1 = gx n, y n, fx, y = a 1 y 2 gx, y = a 2 x 2 e d1 k1x b 1 + y + c e d1 k1x 5.4 e d2 k2y b 2 + x + c ed2 k2y We can now easily see that fx, y is non-increasing in x non-decreasing in y, while gx, y is non-decreasing in x non-increasing in y. We now show that f : [a, A] [b, B] [a, A] g : [a, A] [b, B] [b, B], A = a 1c 2 + c 1 1 a 1 a 2 + θ, B = a 2c 1 + c 2 1 a 1 a 2 + θ, e d1 k1a e d2 k2b a = c e d1 k1a b = c e d2 k2b for some θ 0,. If y B, from 5.4, we have a2 c 1 + c 2 fx, y a 1 y + c 1 a 1 + θ + c 1 A. 1 a 1 a 2

13 On the stability of some systems of exponential difference equations 107 Next, if x A, from 5.5, we get a1 c 2 + c 1 gx, y a 2 + θ 1 a 1 a 2 Moreover, if x A, from 5.4, we obtain Finally, for y B, from 5.5 we obtain e d1 k1a + c 2 B. fx, y c 1 = a. 1 + ed1 k1a e d2 k2b gx, y c 2 = b. 1 + ed2 k2b In addition, working in the same way as in the proof of Proposition 4.2, we can show that there exist real numbers m, M, r, R such that if m = fm, r, M = fm, R, r = gm, R, R = gm, r, then m = M r = R. Moreover, from 1.4, we can see that x n+1 a 1 y n + c 1, y n+1 a 2 x n + c 2. So, we get x n+2 a 1 a 2 x n + a 1 c 2 + c 1, y n+2 a 1 a 2 y n + a 2 c 1 + c 2. We now let w n+2 = a 1 a 2 w n + a 1 c 2 + c 1 z n+2 = a 1 a 2 z n + a 2 c 1 + c 2 with x 0 = w 0, y 0 = z 0, x 1 = w 1 y 1 = z 1. These two equations are essentially linear difference equations of first order, so solvable ones many recent equations systems are solved by using the equation such as [25 27, 29, 33]. Solving these equations, we obtain w n = A 1 a 1 a 2 n 2 + A2 1 n a 1 a 2 n 2 + a 1 c 2 + c 1 1 a 1 a 2 z n = B 1 a 1 a 2 n 2 + B2 1 n a 1 a 2 n a 2 c 1 + c 2 2 +, 1 a 1 a 2 respectively, A 1, A 2, B 1, B 2 depend on the initial conditions. Now, since x 0 = w 0, y 0 = z 0, x 1 = w 1, y 1 = z 1, working inductively, we obtain that x n w n, y n z n, n N, therefore we conclude that for some n 0 N + θ 0,, we have x n a1c2+c1 1 a 1a 2 + θ = A y n a2c1+c2 1 a 1a 2 + θ = B, n n 0. Hence, we can easily see that x n a, n > n 0 y n b, n > n 0. So, we have obtained that x n [a, A] y n [b, B], n > n 0. Therefore, using Theorem 5.1, we deduce that System 1.4 has a unique positive equilibrium x, ỹ, every positive solution of System 1.4 tends to the unique positive equilibrium as n. This completes the proof of the proposition. Proposition 5.3. Consider System 1.4. If the conditions in Proposition 5.2 hold, then the unique positive equilibrium x, ỹ of System 1.4 is globally asymptotically stable. Proof. Firstly, we will prove that x, ỹ is locally asymptotically stable. The linearised system of 1.4 is [ ] [ ] α β xn w n+1 = Aw n, A =, w γ δ n =, y n

14 108 N. Psarros, G. Papaschinopoulos, C.J. Schinas α = c d1 k1 x 1k 1 e 1 + e d1 k1 x 2, β = a 1ỹ 2 + 2b 1 ỹ b 1 + ỹ 2, γ = a 2 x 2 + 2b 2 x b 2 + x 2, δ = c 2k 2 e d2 k2ỹ 1 + e d2 k2ỹ2. The characteristic equation of A is λ 2 α + δλ + αδ βγ = 0. Therefore, from relation 5.2, we obtain αδ βγ < 1 from 5.3, we obtain α + δ < 1 + αδ βγ. Therefore, from Theorem of [13], we deduce that x, ỹ is locally asymptotically stable. Using Proposition 5.2, we conclude that x, ỹ is globally asymptotically stable. This completes the proof of the proposition. 6. GLOBAL BEHAVIOUR OF SOLUTIONS OF SYSTEM 1.5 In this section, we will investigate the asymptotic behaviour of the positive solutions of 1.5. We will use the following theorem, which is essentially a slight modification of Theorem 1.16 of [5]. The proof of Theorem 1.16 of [5] can be easily adapted to this case. Theorem 6.1. Let f, g, with f : R + R + R +, g : R + R + R + be continuous functions, R + = 0,. Let a, A, b, B be positive numbers, such that a < A, b < B f : [a, A] [b, B] [a, A], g : [a, A] [b, B] [b, B]. Consider the system of difference equations x n+1 = fx n, y n, y n+1 = gx n, y n, n = 0, 1, Suppose that fx, y is a non-decreasing function with respect to x a non-increasing with respect to y. Moreover, suppose that gx, y is a non-increasing function with respect to x a non-increasing function with respect to y. Finally suppose that, if m, M, r, R are real numbers such that if M = fm, r, m = fm, R, R = gm, r, r = gm, R, then m = M r = R. Then the system of difference equations 6.1 has a unique positive equilibrium x, ȳ every positive solution of system 6.1 which satisfies x n0 [a, A], y n0 [b, B] tends to the unique equilibrium of 6.1. We now prove the following result. Proposition 6.2. Consider System 1.5. Suppose that the following relations hold true: k 3 c 2 < 1, a 1 < k 1 c 1 a 2 k 2 < 1 k 3 c 2 1 a Then System 1.5 has a unique positive equilibrium every positive solution of System 1.5 tends to the unique equilibrium of 1.5 as n.

15 On the stability of some systems of exponential difference equations 109 Proof. System 1.5 can be written as x n+1 = fx n, y n, y n+1 = gx n, y n, fx, y = a 1 x 2 e d1 k1y b 1 + x + c e d1 k1y 6.4 e b2 k2x e d2 k3y gx, y = a e b2 k2x + c ed2 k3y We can now easily see that fx, y is non-decreasing in x non-increasing in y, while gx, y is non-increasing in x non-increasing in y. We now show that f : [a, A] [b, B] [a, A] g : [a, A] [b, B] [b, B], e d1 k1b A = c 1 1 a 1 + θ, B = a 2 + c 2, e b2 k2a e d2 k3b a = c e d1 k1b b = a e b2 k2a + c e d2 k3b for some θ 0,. If x A, then from 6.4 we get c1 fx, y a 1 x + c 1 a 1 + θ + c 1 A. 1 a 1 Now, from 6.5, we can easily see that gx, y a 2 + c 2 = B. So, if y B, from 6.4, we obtain e d1 k1b fx, y c 1 = a. 1 + ed1 k1b Finally, if x A y B, from 6.5 we obtain e b2 k2a e d2 k3b gx, y a e b2 k2a + c 2 = b. 1 + ed2 k3b In addition, working in the same way as in the proof of Proposition 4.2, we can show that there exist real numbers m, M, r, R such that if M = fm, r, m = fm, R, R = gm, r, r = gm, R, then m = M r = R. Moreover, we have x n+1 a 1 x n + c 1, n N. We consider the difference equation z n+1 = a 1 z n + c 1, n N we let z n be the solution such that z 0 = x 0. Then, we have z n = x 0 c1 1 a 1 a n 1 + c1 1 a 1, n N. Now, since z 0 = x 0, working inductively, we can show that x n z n, n N. Therefore, since a 1 0, 1, we get that for some θ 0, n 0 N +, we have x n c1 1 a 1 + θ = A, n n 0. Finally, we can also easily see that x n a, n 2, y n B, n 1, y n b, n > n 0. Hence, we have obtained that x n [a, A] y n [b, B], n > n 0. Therefore, using Theorem 6.1, System 1.5 has a unique positive equilibrium x, ỹ, every positive solution of System 1.5 tends to the unique positive equilibrium as n. This completes the proof of the proposition. Proposition 6.3. Consider System 1.5. If the conditions in Proposition 6.2 hold, then the unique positive equilibrium x, ỹ of System 1.5 is globally asymptotically stable.

16 110 N. Psarros, G. Papaschinopoulos, C.J. Schinas Proof. Firstly, we will prove that x, ỹ is locally asymptotically stable. The linearised system of 1.5 is [ ] [ ] α β xn w n+1 = Aw n, A =, w γ δ n =, y n α = a 1 x 2 + 2b 1 x b 1 + x 2, β = c 1k 1 e d1 k1ỹ 1 + e d1 k1ỹ 2, γ = a b2 k2 x 2k 2 e, δ = c 2k 3 e d2 k3ỹ. 1 + e b2 k2 x2 1 + e d2 k3ỹ2 The characteristic equation of A is λ 2 α + δλ + αδ βγ = 0. Now, αδ βγ < 0 < 1. Moreover, from relation 6.3 we obtain α + δ < 1 + αδ βγ. Therefore, from Theorem of [13], we deduce that x, ỹ is locally asymptotically stable. Using Proposition 6.2, we conclude that x, ỹ is globally asymptotically stable. This completes the proof of the proposition. 7. GLOBAL BEHAVIOUR OF SOLUTIONS OF SYSTEM 1.6 In this section, we will investigate the asymptotic behaviour of the positive solutions of 1.6. We will use the following theorem, which is essentially a slight modification of Theorem 1.16 of [5]. The proof of Theorem 1.16 of [5] can be easily adapted to this case. Theorem 7.1. Let f, g, with f : R + R + R +, g : R + R + R + be continuous functions, R + = 0,. Let a, A, b, B be positive numbers, such that a < A, b < B f : [a, A] [b, B] [a, A], g : [a, A] [b, B] [b, B]. Consider the system of difference equations x n+1 = fx n, y n, y n+1 = gx n, y n, n = 0, 1, Suppose that fx, y is a non-decreasing function with respect to x a non-increasing with respect to y. Moreover, suppose that gx, y is a non-increasing function with respect to x a non-decreasing function with respect to y. Finally suppose that, if m, M, r, R are real numbers such that if M = fm, r, m = fm, R, R = gm, R, r = gm, r, then m = M r = R. Then the system of difference equations 7.1 has a unique positive equilibrium x, ȳ every positive solution of System 7.1 which satisfies x n0 [a, A], y n0 [b, B] tends to the unique equilibrium of 7.1. We now prove the following result. Proposition 7.2. Consider System 1.6. Suppose that the following relations hold true: a 1, a 2 < c 1 c 2 k 1 k 2 < 1 a 1 1 a

17 On the stability of some systems of exponential difference equations 111 Then System 1.6 has a unique positive equilibrium every positive solution of System 1.6 tends to the unique equilibrium of 1.6 as n. Proof. System 1.6 can be written as x n+1 = fx n, y n, fx, y = a 1 x 2 gx, y = a 2 y 2 e d1 k1y y n+1 = gx n, y n, b 1 + x + c e d1 k1y 7.4 e d2 k2x b 2 + y + c ed2 k2x We can now easily see that fx, y is non-decreasing in x non-increasing in y, while gx, y is non-increasing in x non-decreasing in y. We now show that f : [a, A] [b, B] [a, A] g : [a, A] [b, B] [b, B], e d2 k2a A = c 1 + θ, B = c 2 e d1 k1b + θ, a = c 1 1 a 1 1 a e d1 k1b b = c e d2 k2a for some θ 0,. If x A, from 7.4 we have c1 fx, y a 1 + θ + c 1 A 1 a 1 if y B, from 7.5 we obtain c2 gx, y a 2 + θ + c 2 B. 1 a 2 Moreover, for y B, from 7.4 we get for x A, from 7.5 we obtain e d1 k1b fx, y c e d1 k1b = a e d2 k2a gx, y c 2 = b. 1 + ed2 k2a In addition, working in the same way as in the proof of Proposition 4.2, we can show that there exist real numbers m, M, r, R such that if M = fm, r, m = fm, R, R = gm, R, r = gm, r, then m = M r = R. Moreover, we have x n+1 a 1 x n + c 1, n N. We consider the difference equation z n+1 = a 1 z n + c 1, n N we let z n be the solution such that z 0 = x 0. Then, we have z n = x 0 c 1 a n 1 + c 1, n N. 1 a 1 1 a 1 Now, since z 0 = x 0, working inductively, we can show that x n z n, n N. Therefore, since a 1 0, 1 we get that for some θ 0, n 0 N +, we have x n A = c1 1 a 1 + θ, n n 0. In the same way, we can prove that for some θ 0,

18 112 N. Psarros, G. Papaschinopoulos, C.J. Schinas n 0 N +, we have y n B = c2 1 a 2 + θ, n n 0. So, we can now easily see that x n a, n > n 0, y n b, n > n 0. Hence, we have obtained that x n [a, A] y n [b, B], n > n 0. Therefore, using Theorem 7.1, System 1.6 has a unique positive equilibrium x, ỹ, every positive solution of System 1.6 tends to the unique positive equilibrium as n. This completes the proof of the proposition. Proposition 7.3. Consider System 1.6. If the conditions in Proposition 7.2 hold, then the unique positive equilibrium x, ỹ of System 1.6 is globally asymptotically stable. Proof. First, we will prove that x, ỹ is locally asymptotically stable. The linearised system of 1.6 is [ ] [ ] α β xn w n+1 = Aw n, A =, w γ δ n =, y n α = a 1 x 2 + 2b 1 x b 1 + x 2, β = c 1k 1 e d1 k1ỹ 1 + e d1 k1ỹ 2, γ = c d2 k2 x 2k 2 e 1 + e d2 k2 x 2, δ = a 2ỹ 2 + 2b 2 ỹ b 2 + ỹ 2. The characteristic equation of A is λ 2 α + δλ + αδ βγ = 0. Now, from 7.2, we obtain that αδ βγ < αδ < 1. Moreover, from relation 7.3 we obtain α + δ < 1 + αδ βγ. Therefore, from Theorem of [13], we deduce that x, ỹ is locally asymptotically stable. Using Proposition 7.2, we conclude that x, ỹ is globally asymptotically stable. This completes the proof of the proposition. Acknowledgements The authors would like to thank the referees for their helpful suggestions. REFERENCES [1] J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, New York, Heidelberg, Berlin, [2] D. Clark, M.R.S. Kulenovic, J.F. Selgrade, Global asymptotic behavior of a two- -dimensional difference equation modelling competition, Nonlinear Anal , [3] S. Elaydi, Discrete Chaos, 2nd ed., Chapman & Hall/CRC, Boca Raton, London, New York, [4] E. El-Metwally, E.A. Grove, G. Ladas, R. Levins, M. Radin, On the difference equation, x n+1 = α + βx n 1e xn, Nonlinear Anal , [5] E.A. Grove, G. Ladas, Periodicities in Nonlinear Difference Equations, Chapman & Hall/CRC, 2005.

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20 114 N. Psarros, G. Papaschinopoulos, C.J. Schinas [24] S. Stević, On a discrete epidemic model, Discrete Dyn. Nat. Soc , Article ID [25] S. Stević, On a system of difference equations, Appl. Math. Comput , [26] S. Stević, On a system of difference equations with period two coefficients, Appl. Math. Comput , [27] S. Stević, On a third-order system of difference equations, Appl. Math. Comput , [28] S. Stević, On some periodic systems of max-type difference equations, Appl. Math. Comput , [29] S. Stević, On some solvable systems of difference equations, Appl. Math. Comput , [30] S. Stević, On a solvable system of difference equations of kth order, Appl. Math. Comput , [31] S. Stević, On a cyclic system of difference equations, J. Difference Equ. Appl , [32] S. Stević, Boundedness peristence of some cyclic-type systems of difference equations, Appl. Math. Lett , [33] S. Stević, J. Diblik, B. Iričanin, Z. Šmarda, On a third-order system of difference equations with variable coefficients, Abstr. Appl. Anal , Article ID [34] D. Tilman, D. Wedin, Oscillations chaos in the dynamics of a perennial grass, Nature , N. Psarros nikosnpsarros@hotmail.com Democritus University of Thrace School of Engineering Xanthi, 67100, Greece G. Papaschinopoulos gpapas@env.duth.gr Democritus University of Thrace School of Engineering Xanthi, 67100, Greece

21 On the stability of some systems of exponential difference equations 115 C.J. Schinas Democritus University of Thrace School of Engineering Xanthi, 67100, Greece Received: March 30, Revised: April 29, Accepted: May 1, 2017.