Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 15 (2007), pp. 229–238. ISSN: 1072-6691. URL: http://ejde.mathMississippi State Universityṫxstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

GLOBAL ATTRACTIVITY IN A NONLINEAR DIFFERENCE EQUATION

CHUANXI QIAN, YIJUN SUN

Abstract. In this paper, we study the asymptotic behavior of positive solutions of the nonlinear difference equation

x_{n+1} = x_{n}f(x_{n−k}), where f : [0, ∞) → (0, ∞) is a unimodal function, and k is a nonnegative integer. Sufficient conditions for the positive equilibrium to be a global attractor of all positive solutions are established. Our results can be applied to to some difference equations derived from mathematical biology.

1. Introduction

Our aim in this paper is to study the global attractivity of the difference equation

x_{n+1} = x_{n}f(x_{n−k}), n = 0, 1, …  (1.1)

where k ∈ {0, 1, 2, … }, and f : [0, ∞) → (0, ∞) is a unimodal function; i.e, f is first increasing, and then decreasing. Global attractivity of (1.1) under different assumptions on f has been studied by several authors, see, for example, Kocic and Ladas [5], Qian [10] and Graef and Qian [2]. However, few results can be found under the assumption that f is a unimodal function. Besides its theoretic interest, our motivation to study (1.1) comes from the following observation. Consider the delay logistic equation

x′(t) = x(t)[a + bx(t − τ) − cx^{2}(t − τ)], t ≥ 0,  (1.2)

where

a, c, τ ∈ (0, ∞) and b ∈ (−∞, ∞).  (1.3)

This equation is a model of single species with a quadratic per-capita growth rate (see [1] for details). One may see [4] and [7] also for some extensions of (1.2). The following difference equation with piecewise constant arguments

x′(t) = x(t)[a + bx([t − k]) − cx^{2}([t − k])], t ≥ 0,  (1.4)

2000 Mathematics Subject Classification. 39A10.

Key words and phrases. Nonlinear difference equation; global attractor;

unimodal function; positive equilibrium.

\begin{matrix}c \\ ◯ \end{matrix}2007 Texas State University - San Marcos.

Published February 28, 2007.

where [·] denotes the greatest integer function, may be viewed as a semidiscretization of (1.2). By an argument similar to that in [3, Section 8.2], one can see that (1.4) leads to the following equation

x_{n+1} = x_{n}e^{a+bx_{n−k}−cx_{n−k}^{2}}, n = 0, 1, …  (1.5)

whose oscillatory and stability properties completely characterize the same properties for (1.4), and so this leads us to study (1.5), which is a special case of (1.1).

In the following, we only consider the positive solutions of (1.1). If we let

x_{−k}, x_{−k+1}, …, x_{0}  (1.6)

be k + 1 given nonnegative numbers with x_{0} > 0, and x̄ be the unique solution of f(x) = 1, then (1.1) has a unique positive solution with initial condition (1.6). Clearly, x̄ is the only positive equilibrium. In the following section, we establish some sufficient conditions such that x̄ attracts all the positive solutions of (1.1). Then, in Section 3, we apply our results about (1.1) to (1.5) to establish some sufficient conditions for the global attractivity of the positive equilibrium of (1.5).

2. Global attractivity of (1.1)

Consider the difference equation

x_{n+1} = g(x_{n}), n = 1, 2, …,  (2.1)

where g ∈ C[R, R]. Let G be any set in R. We say that V is a Liapunov function for (2.1) on G if

(i) V is continuous on G, and

(ii) V̇ (x) = V (g(x)) − V (x) ≤ 0 for all x ∈ G.

The following lemma on the asymptotic behavior of (2.1) is taken from [8] and will be needed later.

Lemma 2.1. Let G be a bounded open positively invariant set. If (i) V is a Liapunov function for (2.1) on G, (ii) M ⊂ G, where M is the largest invariant set of E = {x : V̇ (x) = 0, x ∈ Ḡ}, (iii) V is constant on M.

Then M is globally asymptotically stable relative to G.

The following theorem is our main result in this section.

Theorem 2.2. Let f : [0, ∞) → (0, ∞) satisfy the following assumptions: (i) f is a unimodal function obtaining its maximum at x = x^{∗}; (ii) x̄ > x^{∗} is the unique equilibrium point; (iii) [ln f(x)]′′ ≤ 0 on (x^{∗}, m_{0}); (iv) x̄f(m_{0})^{k+1} ≥ x^{∗};

(v) f(m_{0}) ≥ \frac{1}{f(x^{∗})},

where m_{0} = x̄f(x^{∗})^{k+1}. Then x̄ is the global attractor of all positive solutions of (1.1).

Proof. First, we show that every non-oscillatory (about x̄) solution {x_{n}} tends to x̄. We assume that x_{n} ≥ x̄ eventually. The proof for the case that x_{n} ≤ x̄ eventually is similar, and so is omitted. By (1.1),

\frac{x_{n+1}}{x_{n}} = f(x_{n−k}).

Since for large n, x_{n−k} ≥ x̄, and x̄ > x^{∗}, we have \frac{x_{n+1}}{x_{n}} ≤ f(x̄) = 1. Hence, \{x_{n}\} is non-increasing and lim_{n→∞} x_{n} = l exists. Obviously, l = x̄.

Next, we show that every oscillatory solution tends to x̄. Suppose that {x_{n}} is an oscillatory solution of (1.1). We say that x_{s} is a local maximum of {x_{n}}, if

x_{s} ≥ x̄, x_{s} ≥ x_{s−1}, x_{s} ≥ x_{s+1}.

Similarly, we say that x_{r} is a local minimum of {x_{n}}, if

x_{r} ≤ x̄, x_{r} ≤ x_{r−1}, x_{r} ≤ x_{r+1}.

By (1.1),

x_{s} = x_{s−1}f(x_{s−k−1}) ≥ x_{s−1},

so we have f(x_{s−k−1}) ≥ 1, and hence x_{s−k−1} ≤ x̄. Thus,

x_{s} = x_{s−k−1} ∏_{i=s−k}^{s} f(x_{i−k−1}) ≤ x̄[f(x^{∗})]^{k+1} = m_{0}.

So, m_{0} is an upper bound of {x_{n}}. By a similar argument, we have

x_{r} ≥ x̄[f(m_{0})]^{k+1} = m_{1}.

Thus, there exists some N_{0} > 0, such that

m_{1} ≤ x_{n} ≤ m_{0}, for n ≥ N_{0}.

Notice that under assumption (iv), m_{1} = x̄f(m_{0})^{k+1} ≥ x^{∗}, by induction, we can prove that

m_{2s+1} ≤ x_{n} ≤ m_{2s}, for n ≥ N_{s}

where {m_{s}} is defined by

\begin{matrix}m_{s+1} = x̄[f(m_{s})]^{k+1} \\ m_{0} = x̄[f(x^{∗})]^{k+1}. \end{matrix}  (2.2)

To prove x_{n} tends to x̄, it suffices to show that m_{s} tends to x̄. Let G = (0, m_{0}). Clearly, G is a positively invariant set of IVP (2.2). Define

V (x) = ( ln \frac{x}{x̄})^{2}, x ∈ G.

Then

V̇ (x) = [(k + 1) ln f(x)]^{2} − ( ln \frac{x}{x̄})^{2}.

Let

g(x) = (k + 1) ln f(x).

To get V̇ (x) < 0 on G for x ≠ x̄, we need |g(x)| < | ln \frac{x}{x̄}|, which is equivalent to

g(x) < − ln \frac{x}{x̄} = ln \frac{x̄}{x} for x < x̄;

g(x) > − ln \frac{x}{x̄} = ln \frac{x̄}{x} for x > x̄.

(2.3)

Let h(x) = ln \frac{x̄}{x}. Observe that

g′(x) = \frac{(k + 1)f′(x)}{f(x)} < 0, g′′(x) = (k + 1)(ln f(x))′′ ≤ 0 on (x^{∗}, m_{0}); (2.4)

and

(h(x))′ = −\frac{1}{x} < 0, (h(x))′′ = \frac{1}{x^{2}} > 0 for x > 0.  (2.5)

So, g and h look as in the following graph

Figure 1. g(x^{∗}) = max g(x), g(x̄) = h(x̄), by the concavity of f
and g, h(x) > g(x) for x < x̄, h(x) < g(x) for x > x̄

We first show that (2.3) holds on (x̄, m_{0}). Since g(x̄) = ln \frac{x̄}{x̄} = 0, by the monotonicity and concavity of g and ln \frac{x̄}{x}, it’s enough to show that g(m_{0}) ≥ ln(x̄/m_{0}); i.e,

[f(m_{0})]^{k+1} ≥ \frac{x̄}{m_{0}} = \frac{x̄}{x̄f(x^{∗})^{k+1}} = \frac{1}{f(x^{∗})^{k+1}} .

From (v), we can see that (2.3) holds immediately on (x̄, m_{0}). Furthermore, we have g′(x) > ( ln \frac{x̄}{x})′ at x = x̄. So, for x ∈ (x̄, x^{∗}), again by (2.4) and (2.5), we know that g(x) > ln \frac{x̄}{x}. For x ≤ x^{∗}, since f is increasing and ln \frac{x̄}{x} is decreasing, (2.3) is satisfied automatically. Thus, we have

V̇ (x) < 0 for x ∈ G and x ≠ x̄,

and

E = {x : V̇ (x) = 0, x ∈ G} = {x̄}.

Hence, by Lemma 2.1, x̄ is a global attractor relative to G, and so every solution {m_{s}} of IVP (2.2) tends to x̄. Then, it follows that {x_{n}} tends to x̄. The proof of Theorem 2.2 is complete. □

We can get a linearized stability result by using the following lemma.

Lemma 2.3 ([9]). Assume that q ∈ R and k ∈ {0, 1, 2, … }. Then the delay difference equation

x_{n+1} − x_{n} + qx_{n−k} = 0, n = 0, 1, …  (2.6)

is asymptotically stable if and only if

0 < q < 2 cos \frac{kπ}{2k + 1}.  (2.7)

Corollary 2.4. If the assumptions of Theorem 2.2 hold, and

−f′(x̄)x̄ < 2 cos \frac{kπ}{2k + 1},  (2.8)

then (1.1) is globally asymptotically stable.

Proof. Let y_{n} = ln x_{n}, (1.1) becomes

y_{n+1} − y_{n} − ln f(e^{y_{n−k}}) = 0, n = 0, 1, …  (2.9)

The linearized form of (2.9) is

y_{n+1} − y_{n} − f′(x̄)x̄y_{n−k} = 0, n = 0, 1, …  (2.10)

Let p = −f′(x̄)x̄. We have p > 0 since f′(x̄) < 0. By (2.7), we have that (2.10) is stable if (2.8) holds. Hence, (1.1) is locally stable. Then combining with the global attractivity from Theorem 2.2, we get the global stability result. □

Although we can not prove it now, we believe that if the conditions in Theorem 2.2 hold, then (2.8) holds, and so the conditions in Theorem 2.2 imply the global stability of (1.1). In Section 3, we will show that this is true for (1.5).

3. Global attractivity of (1.5)

In this section, we apply our results in Section 2 to establish some sufficient conditions for the global attarctivity of (1.5). Two cases of (1.5) with b ≤ 0 and b > 0 are considered. For (1.5), x̄ = \frac{b+\sqrt{b^{2}+4ac}}{2c} is the only positive equilibrium.

Theorem 3.1. Assume that b ≤ 0, and

(k + 1)a ≤ ln \frac{b + \sqrt{b^{2} + 8ac}}{b + \sqrt{b^{2} + 4ac}}.  (3.1)

Then x̄ is a global attractor of all positive solutions of (1.5).

Proof. To apply Theorem 2.2, we need to show that all assumptions of Theorem 2.2 are satisfied. Here, f(x) = exp (a + bx − cx^{2}), and clearly, f(x) is decreasing on (0, ∞), obtaining its maximum at x^{∗} = 0. Note that

m_{0} = x̄f(0)^{k+1} = x̄e^{(k+1)a},  (3.2)

and it is easy to see that assumption (ii), (iii) and (iv) of Theorem 2.2 are satisfied. Now, wee only need to check (v), which is

e^{a+bm_{0}−cm_{0}^{2}} ≥ \frac{1}{e^{a}} = e^{−a},

that is,

a + bm_{0} − cm_{0}^{2} ≥ −a.  (3.3)

Let g(x) = cx^{2} − bx − 2a. Then x_{1} = \frac{b+\sqrt{b^{2}+8ac}}{2c} is the only positive solution of g(x) = 0. Since g is increasing on (0, ∞), (3.3) is equivalent to m_{0} ≤ x_{1}, which is (3.1). Thus, all assumptions of Theorem 2.2 are satisfied, and so x̄ is a global attractor of (1.5). □

The following result is a consequence of the above theorem and Corollary 2.4.

Corollary 3.2. If b ≤ 0 and (3.1) hold, then (1.5) is globally asymptotically stable.

Proof. By (2.8) in Corollary 2.4, we need

−f′(x̄)x̄ = \frac{\sqrt{b^{2} + 4ac}(b + \sqrt{b^{2} + 4ac})}{2c} < 2 cos \frac{kπ}{2k + 1}.  (3.4)

Now, we claim that (3.1) implies (3.4). First, we want to simplify the expression. Let A = (k + 1)a and C = c/(k + 1). From (3.1), it is easy to see that A ≤ ln 2, and (3.4) can be written as

\frac{\sqrt{b^{2} + 4AC}(b + \sqrt{b^{2} + 4AC})}{2C} < 2(k + 1) cos \frac{kπ}{2k + 1}.  (3.5)

Next, we let

B = \frac{|b|}{\sqrt{C}} = −\frac{b}{\sqrt{C}} > 0.

Then (3.5) can be written as

\frac{\sqrt{B^{2} + 4A}(−B + \sqrt{B^{2} + 4A})}{2} < 2(k + 1) cos \frac{kπ}{2k + 1}.

After simplification, this becomes

\frac{\sqrt{B^{2} + 4A}}{B + \sqrt{B^{2} + 4A}}A < (k + 1) cos \frac{kπ}{2k + 1}.  (3.6)

Since A ≤ ln 2, the left hand side of Inequality (3.6) is less than ln 2. On the other hand, the right hand side of (3.6) can be written as

(k + 1) sin \frac{π}{4(k + \frac{1}{2})}.

If we use s = k + \frac{1}{2}, then the right hand side of Inequality (3.6) is

g(s) = (s + \frac{1}{2}) sin \frac{π}{4s}, s ≥ \frac{1}{2}.

We claim that g is a decreasing function for s ≥ \frac{1}{2}. Observe that

g′(s) = sin \frac{π}{4s} − (\frac{π}{4s} + \frac{π}{8s^{2}} ) cos \frac{π}{4s},

and notice that \frac{π}{4s} ≥ sin \frac{π}{4s} for s ≥ \frac{1}{2},

g′′(s) = \frac{π}{4s^{3}} cos \frac{π}{4s} − (\frac{π^{2}}{16s^{3}} + \frac{π^{2}}{32s^{4}} ) sin \frac{π}{4s}

≥ \frac{π}{4s^{3}} [1 − (\frac{π}{4s})^{2} ] − (\frac{π^{2}}{16s^{3}} + \frac{π^{2}}{32s^{4}} )\frac{π}{4s}

= \frac{π}{4s^{3}} (1 − \frac{π^{2}}{16s^{2}} − \frac{π^{2}}{16s} − \frac{π^{2}}{32s^{2}} )

≥ \frac{π}{4s^{3}} (1 − \frac{π^{2}}{12} ) > 0.

So g′ is increasing, max_{s≥ \frac{1}{2}} g′(s) = lim_{s→∞} g′(s) = 0, and g is decreasing for s ≥ \frac{1}{2}. We get

min_{s≥ \frac{1}{2}} g(s) = lim_{s→∞}(s + \frac{1}{2}) sin \frac{π}{4s} = \frac{π}{4} ,

which is greater than ln 2. Thus (3.6) holds, and therefore (3.4) holds. The proof is complete. □

EJDE-2006/CONF/15  GLOBAL ATTRACTIVITY  235

Example 3.3. Consider the difference equation

x_{n+1} = x_{n}e^{0.1−x_{n−1}−x_{n−1}^{2}}.

Here k = 1, a = 0.1, b = −1, c = 1, and

(k + 1)a = .2 < ln \frac{b + \sqrt{b^{2} + 8ac}}{b + \sqrt{b^{2} + 4ac}} ≈ 0.623.

Hence, by Corollary 3.2, x̄ is globally asymptotically stable. Theorem 3.4. Assume b > 0 and

\frac{(k + 1)D}{4c} ≤ ln \frac{b + \sqrt{2D}}{b + \sqrt{D}} ,  (3.7)

where D = b^{2} + 4ac. Then all positive solutions of (1.5) tend to x̄. Proof. Let f(x) = exp (a + bx − cx^{2}). We show that f satisfies all the conditions assumed in Theorem 2.2. Clearly, f is increasing on (0, x^{∗}) and decreasing on (x^{∗}, ∞), where x^{∗} = \frac{b}{2c}, and so assumption (i) is satisfied. (ii) and (iii) are also easy to check. We see that to have x̄f(m_{0})^{k+1} ≥ x^{∗}, we need

(a + bm_{0} − cm_{0}^{2}) ≥ \frac{1}{k + 1} ln \frac{b}{b + \sqrt{D}} .  (3.8)

By a direct but tedious calculation, (3.8) is equivalent to

\frac{(k + 1)D}{4c} ≤ ln \frac{b + \sqrt{D + \frac{4c}{k+1} ln \frac{b+\sqrt{D}}{b}}}{b + \sqrt{D}} .  (3.9)

We claim that (3.7) implies (3.9). To prove our claim, it suffices to show

\frac{b + \sqrt{2D}}{b + \sqrt{D}} ≤ \frac{b + \sqrt{D + \frac{4c}{k+1} ln \frac{b+\sqrt{D}}{b}}}{b + \sqrt{D}} ,

which is equivalent to

\sqrt{2D} ≤ \sqrt{D + \frac{4c}{k + 1} ln \frac{b + \sqrt{D}}{b}} ,

that is

\frac{(k + 1)D}{4c} ≤ ln \frac{b + \sqrt{D}}{b} .

By (3.7), it is sufficient to show that

ln \frac{b + \sqrt{2D}}{b + \sqrt{D}} ≤ ln \frac{b + \sqrt{D}}{b} .  (3.10)

It is not difficult to see that (3.10) is equivalent to

\frac{\sqrt{2} − 1}{b + \sqrt{D}} ≤ \frac{1}{b},

which is obviously true. Thus, (iv) is satisfied.

To check (v), we need exp (a + bm_{0} − cm_{0}^{2}) ≥ exp ( − (a + bx^{∗} − cx^{∗2})); i.e,

a + bm_{0} − cm_{0}^{2} ≥ −\frac{D}{4c}.  (3.11)

Let

h(x) = a + bx − cx^{2} + \frac{D}{4c}.  (3.12)

We see that the positive solution of (3.12) is x_{2} = \frac{b+\sqrt{2D}}{2c}. Since both m_{0} and x_{2} are larger than x^{∗}, and on (x^{∗}, ∞), h is decreasing, (3.11) is equivalent to m_{0} ≤ x_{2}, which is satisfied by (3.7). Hence, it follows by Theorem 2.2 that {x_{n}} tends to x̄. □

Corollary 3.5. Assume that (3.7) holds. Then (1.5) is globally asymptotically stable.

Proof. By condition (2.8) in Corollary 2.4, we need

−f′(x̄)x̄ = \frac{\sqrt{D}(b + \sqrt{D})}{2c} < 2 cos \frac{kπ}{2k + 1},

which is equivalent to

\frac{D}{4c} < cos \frac{kπ}{2k + 1} − \frac{b\sqrt{D}}{4c} .

Then, combining this with Theorem 3.4, we know that if

\frac{D}{4c} ≤ min\{\frac{1}{k + 1} ln \frac{b + \sqrt{2D}}{b + \sqrt{D}} , cos \frac{kπ}{2k + 1} − \frac{b\sqrt{D}}{4c} \}  (3.13)

holds, then (1.5) is globally asymptotically stable. Now, we show that

\frac{1}{k + 1} ln \frac{b + \sqrt{2D}}{b + \sqrt{D}} ≤ cos \frac{kπ}{2k + 1} − \frac{b\sqrt{D}}{4c} ,  (3.14)

under the assumption (3.7). Observe that

\frac{b\sqrt{D}}{4c} = \frac{b}{\sqrt{D}} \frac{D}{4c} ≤ \frac{b}{\sqrt{D}} \frac{1}{k + 1} ln \frac{b + \sqrt{2D}}{b + \sqrt{D}}

by (3.7). So it suffices to show that

(\frac{b}{\sqrt{D}} + 1)\frac{1}{k + 1} ln \frac{b + \sqrt{2D}}{b + \sqrt{D}} ≤ cos \frac{kπ}{2k + 1}.

After simplification, this becomes

(\frac{b}{\sqrt{D}} + 1) ln \frac{b + \sqrt{2D}}{b + \sqrt{D}} ≤ (k + 1) sin \frac{π}{4k + 2}.

Let \frac{b}{\sqrt{D}} = t (0 < t < 1) and s = k + \frac{1}{2} (s ≥ \frac{1}{2}). It is sufficient to show that

(t + 1) ln \frac{t + \sqrt{2}}{t + 1} ≤ (s + \frac{1}{2}) sin \frac{π}{4s}.  (3.15)

If the maximum of the function of t is less than the minimum of the function of s, then we are done. Let

f(t) = (t + 1) ln \frac{t + \sqrt{2}}{t + 1}, g(s) = (s + \frac{1}{2}) sin \frac{π}{4s}.

We claim that f is increasing on (0, 1). To this end, observe that

f′(t) = ln \frac{t + \sqrt{2}}{t + 1} − \frac{\sqrt{2} − 1}{t + \sqrt{2}}, f′′(t) = \frac{2\sqrt{2} − 3}{(t + \sqrt{2})^{2}(t + 1)} < 0.

Since f′ is decreasing and min_{0<t<1} f′(t) = f′(1) ≈ 0.0166 > 0, f is increasing for 0 < t < 1. We know from the proof of Corollary 3.2 that g is decreasing on (\frac{1}{2}, ∞). It is easy to see that max_{0<t<1} f(t) = f(1) ≈ 0.376 and min_{s≥ \frac{1}{2}} g(s) = lim_{s→∞} g(s) ≈ 0.786. Thus, we have f(t) ≤ g(s) always; i.e, (3.15) holds. The proof of Corollary 3.5 is complete. □

Example 3.6. Consider the difference equation

x_{n+1} = x_{n}e^{0.01+x_{n−3}−20x_{n−3}^{2}}.

Here a = .01, b = 1, c = 20 and k = 3. So

\frac{(k + 1)D}{4c} ≈ .09 ≤ ln \frac{b + \sqrt{2D}}{b + \sqrt{D}} ≈ .825.

By Corollary 3.5, x̄ is globally asymptotically stable.

4. Remarks

Consider the difference equation

x_{n+1} − x_{n} = x_{n}(a + bx_{n−k} − cx_{n−k}^{2}),  (4.1)

where a, c ∈ (0, ∞), b ∈ (−∞, +∞), and k ∈ {0, 1, … }. Equation (4.1) may be viewed as a discrete analogue of the delay differential equation (1.3). The global attractivity of the positive equilibrium when k = 0 was investigated by Rodrigues [11]. Kocic and Ladas [6], posted the following research projects: Obtain a global stability result for the positive equilibrium x̄ of (4.1) when k ≥ 1, and obtain explicit sufficient conditions on a, b, c and k so that all solutions of (4.1) with appropriate initial condition remain positive for all n ≥ 0. By using an argument similar to one in this paper, we can establish a sufficient condition for the positive equilibrium x̄ to be a global attractor of all positive solutions of (4.1). Then together with the linearized stability result, we can obtain a global stability result for the positive equilibrium x̄ of (4.1) when k ≥ 1. However, the problem that how to obtain explicit sufficient condition so that all solutions remain positive for all n ≥ 0 remains open.

References

[1] K. Gopalsamy, and G. Ladas, On the oscillations and asymptotic behavior of Ṅ(t) = N(t)[a+ bN(t − τ) − cN^{2}(t − τ)]. Quart. Appl. Math., 48:433-440, 1990.

[2] J. R. Graef and C. Qian, Global attractivity in a nonlinear difference equation and its applications, Dynam. Systems Appl. 15(2006), 89-96.

[3] I. Györi and G. Ladas, Oscillation Theorey of Delay Differential Equations with Applications. Clarendon Press, Oxford, 1991.

[4] X. Z. He, B. G. Zhang and K. Gopalsamy, Single species dynamics in changing environments. Dynamics and Stability of Systems, Vol.9, 1994.

[5] V. L. Kocic and G. Ladas, Global attractivity in nonlinear delay difference equations. Proc. Amer. Math. Soc., 115:1083-1088, 1992.

[6] V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Kluwer Academic Publishers, 1993.

[7] G. Ladas and C. Qian, Oscillation and global stability in a delay logistic equation. Dynamics and Stability of Systems, 9:153-162, 1994.

[8] J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1979.

[9] S. H. Levin and R. May, A note on difference-delay equations, Theor. Pop. Biol., 9:178-187, [10] C. Qian, Global attractivity in nonlinear delay difference equations. Bull. Institute Math.

Acad. Sinica, 24, 187-203, 1996.

[11] I. W. Rodrigues, Oscillation and attractivity in a discrete model with quadratic nonlinearity, Applicable Anal., 47:45-55, 1992.

Chuanxi Qian Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, USA

   E-mail address: qian@math.msstate.edu

Yijun Sun Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, USA

   E-mail address: ys101@msstate.edu