Asymptotic and Boundedness Behaviour of a Second Order Difference Equation

In this paper, we study the asymptotic behaviour and boundedness of the solutions of the difference equation xn+1 = α+ βxn−1λ −xn , n = 0, 1, 2, . . . (1) where λ > 1 and α > 0, β > 0 are the immigration rate and population growth respectively and initial conditions x−1, x0 are arbitrary positive numbers.


Introduction
Difference equation containing exponential terms have many applications in biology, there are many papers dealing with such equations. Evolution of a perennial grass depends on the biomass, the litter mass and the total soil nitrogen was described by the difference equations B t+1 = cN e a−bLt 1 + e a−bLt , L t+1 = L 2 t L t + d + ckN e a−bLt 1 + e a−bLt (2) where B is the living biomass, L the litter mass, N the total soil nitrogen, t the time and constants a, b, c, d > 0 and 0 < k < 1. Oscillatory and chaotic nature of (2) was discussed in [14]. Global stability, boundedness nature and periodic character of the positive solution of the difference equation x n+1 = α + βx n−1 e −xn , n = 0, 1, 2, . . .
was investigated by El-Metwally et all [9], where α > 0 and β > 0 are the immigration rate and population growth respectively and the initial conditions x −1 and x 0 are arbitrary nonnegative numbers. Existence, uniqueness and attractivity of prime period two solution for (3) was discussed by Fotiades et all [10].
Boundedness and global asymptotic behavior of the solution of the difference equations were studied by Ozturk et all [11,12], where α and β are positive numbers k ∈ {1, 2, 3, . . . } and the x −k , x −(k−1) , . . . , x −1 , x 0 are arbitrary numbers. Boundedness and the persistence of the positive solutions, the existence, the attractivity and the global asymptotic stability of the unique positive equilibrium and the existence of periodic solutions concerning the biological model  [13], where 0 < a < 1, b, c, d, k are positive constants and x 0 is a real number.
Motivated by above studies, we generalize (3) and investigate the global attractivity and boundedness of the solutions of the difference equations (1) for λ > 1.

Preliminaries
Definition 2.1. [7] Let I ∈ R and let f : I × I → I be a continuous function. Consider the difference equation for the initial conditions x 0 , x −1 ∈ I. We say thatx is an equilibrium of equation (4) That is, the constant sequence {x n } ∞ n=−1 with x n =x for all n ≥ −1 is a solution of equation (4).

Definition 2.2. [7]
(i) The equilibriumx of equation (4) is called locally stable if for every > 0, there exists δ > 0 such that (4) is called locally asymptotically stable if it is locally stable and if there exists γ > 0 such that (4) is called globally asymptotically stable if it is locally stable and a global attractor. (v) The equilibriumx of equilibrium (4) is called unstable if it is not stable.
denote the partial derivatives of f (u, v) evaluated at an equilibriumx of equation (4). Then the equation is called the linearized equation associated with the equation (4) about the equilibrium pointx.

Theorem 2.4. [7][Linearized Stability]
(a) If both roots of the quadratic equation (6) lie in the open unit disk |µ| < 1, then the equilibriumx of (4) is locally asymptotically stable. (b) If atleast one of the roots of (6) has absolute value greater than one, then the equilibriumx of (4) is unstable. (c) A necessary and sufficient condition for both roots of (6) to lie in the open unit disk |µ| < 1, is |p| < 1 − q < 2. In this case the locally asymptotically stable equilibriumx is also called a sink. (d) A necessary and sufficient condition for one root of (6) to have absolute value greater than one and for the other to have absolute value less than one is p 2 + 4q > 0 and |p| > |1 − q|. In this case unstable equilibrium pointȳ is called a saddle point. (e) A necessary and sufficient condition for a root of (6) to have absolute value equal to one is |p| = |1 − q| or q = −1 and |p| ≤ 2. In this case the equilibriumȳ is called a nonhyperbolic point. (iv) A necessary and sufficient condition for both roots of (6) to have absolute value greater than one is |q| > 1 and |p| < |1 − q|. In this caseȳ is called a repeller.

Main Results
In this section, we discuss the local, global asymptotic stability and boundedness of the solutions of equation (1).
Lemma 3.5. We assume 1 < λ < e. Suppose that . Then (1) has no positive solution of prime period two.