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Qualitative behavior of a rational difference equation
Advances in Difference Equations volume 2011, Article number: 6 (2011)
Abstract
This article is concerned with the following rational difference equation y _{ n+1}= (y _{ n }+ y _{ n1})/(p + y _{ n } y _{ n1}) with the initial conditions; y _{1}, y _{0} are arbitrary positive real numbers, and p is positive constant. Locally asymptotical stability and global attractivity of the equilibrium point of the equation are investigated, and nonnegative solution with prime period two cannot be found. Moreover, simulation is shown to support the results.
Introduction
Difference equations are applied in the field of biology, engineer, physics, and so on [1]. The study of properties of rational difference equations has been an area of intense interest in the recent years [6, 7]. There has been a lot of work deal with the qualitative behavior of rational difference equation. For example, Çinar [2] has got the solutions of the following difference equation:
Karatas et al. [3] gave that the solution of the difference equation:
In this article, we consider the qualitative behavior of rational difference equation:
with initial conditions y _{1}, y _{0} ∈ (0, + ∞), p ∈ R ^{+}.
Preliminaries and notation
Let us introduce some basic definitions and some theorems that we need in what follows.
Lemma 1. Let I be some interval of real numbers and
be a continuously differentiable function. Then, for every set of initial conditions, x _{k }, x _{k+1}, ..., x _{0} ∈ I the difference equation
has a unique solution .
Definition 1 (Equilibrium point). A point is called an equilibrium point of Equation 2, if
Definition 2 (Stability).

(1)
The equilibrium point of Equation 2 is locally stable if for every ε > 0, there exists δ > 0, such that for any initial data x _{k }, x _{k+1}, ..., x _{0} ∈ I, with
we have , for all n ≥  k.

(2)
The equilibrium point of Equation 2 is locally asymptotically stable if is locally stable solution of Equation 2, and there exists γ > 0, such that for all x _{k }, x _{k+1}, ..., x _{0} ∈ I, with
we have

(3)
The equilibrium point of Equation 2 is a global attractor if for all x _{k }, x _{k+1}, ..., x _{0} ∈ I, we have .

(4)
The equilibrium point of Equation 2 is globally asymptotically stable if is locally stable and is also a global attractor of Equation 2.

(5)
The equilibrium point of Equation 2 is unstable if is not locally stable.
Definition 3 The linearized equation of (2) about the equilibrium is the linear difference equation:
Lemma 2 [4]. Assume that p _{1}, p _{2} ∈ R and k ∈ {1, 2, ...}, then
is a sufficient condition for the asymptotic stability of the difference equation
Moreover, suppose p _{2} > 0, then, p _{1} + p _{2} < 1 is also a necessary condition for the asymptotic stability of Equation 4.
Lemma 3 [5]. Let g:[p, q]^{2} → [p, q] be a continuous function, where p and q are real numbers with p < q and consider the following equation:
Suppose that g satisfies the following conditions:

(1)
g(x, y) is nondecreasing in x ∈ [p, q] for each fixed y ∈ [p, q], and g(x, y) is nonincreasing in y ∈ [p, q] for each fixed x ∈ [p, q].

(2)
If (m, M) is a solution of system
M = g(M, m) and m = g(m, M),
then M = m.
Then, there exists exactly one equilibrium of Equation 5, and every solution of Equation 5 converges to .
The main results and their proofs
In this section, we investigate the local stability character of the equilibrium point of Equation 1. Equation 1 has an equilibrium point
Let f:(0, ∞)^{2} → (0, ∞) be a function defined by
Therefore, it follows that
Theorem 1.

(1)
Assume that p > 2, then the equilibrium point of Equation 1 is locally asymptotically stable.

(2)
Assume that 0 < p < 2, then the equilibrium point of Equation 1 is locally asymptotically stable, the equilibrium point is unstable.
Proof. (1) when ,
The linearized equation of (1) about is
It follows by Lemma 2, Equation 7 is asymptotically stable, if p > 2.
(2) when ,
The linearized equation of (1) about is
It follows by Lemma 2, Equation 8 is asymptotically stable, if
Therefore,
Equilibrium point is unstable, it follows from Lemma 2. This completes the proof.
Theorem 2. Assume that , the equilibrium point and of Equation 1 is a global attractor.
Proof. Let p, q be real numbers and assume that g:[p, q]^{2} → [p, q] be a function defined by , then we can easily see that the function g(u, v) increasing in u and decreasing in v.
Suppose that (m, M) is a solution of system
M = g(M, m) and m = g(m, M).
Then, from Equation 1
Therefore,
Subtracting Equation 10 from Equation 9 gives
Since p+Mm ≠ 0, it follows that
Lemma 3 suggests that is a global attractor of Equation 1 and then, the proof is completed.
Theorem 3. (1) has no nonnegative solution with prime period two for all p ∈ R ^{+}.
Proof. Assume for the sake of contradiction that there exist distinctive nonnegative real numbers φ and ψ, such that
is a prime periodtwo solution of (1).
φ and ψ satisfy the system
Subtracting Equation 11 from Equation 12 gives
so φ = ψ, which contradicts the hypothesis φ ≠ ψ. The proof is complete.
Numerical simulation
In this section, we give some numerical simulations to support our theoretical analysis. For example, we consider the equation:
We can present the numerical solutions of Equations 1315 which are shown, respectively in Figures 1, 2 and 3. Figure 1 shows the equilibrium point of Equation 13 is locally asymptotically stable with initial data x _{0} = 1, x _{1} = 1.2. Figure 2 shows the equilibrium point of Equation 14 is locally asymptotically stable with initial data x _{0} = 1, x _{1} = 1.2. Figure 3 shows the equilibrium point of Equation 15 is locally asymptotically stable with initial data x _{0} = 1, x _{1} = 1.2.
References
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Çinar C: On the positive solutions of the difference equation x _{ n+1} = ax _{ n1} /1+ bx _{ n } x _{ n1} . Appl Math Comput 2004,158(3):809812. 10.1016/j.amc.2003.08.140
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Authors' contributions
Xiao Qian carried out the theoretical proof and drafted the manuscript. Shi Qihong participated in the design and coordination. All authors read and approved the final manuscript.
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Qian, X., Qihong, S. Qualitative behavior of a rational difference equation . Adv Differ Equ 2011, 6 (2011). https://doi.org/10.1186/1687184720116
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DOI: https://doi.org/10.1186/1687184720116
Keywords
 Global stability
 attractivity
 solution with prime period two
 numerical simulation