Based on the permanence result, sufficient conditions are established for the existence and uniqueness of globally attractive almost periodic solution

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

EXISTENCE AND PERMANENCE OF ALMOST PERIODIC SOLUTIONS FOR LESLIE-GOWER PREDATOR-PREY MODEL

WITH VARIABLE DELAYS

TIANWEI ZHANG, XIAORONG GAN

Abstract. By constructing a suitable Lyapunov functional and using almost periodic functional hull theory, we study the almost periodic dynamic behavior of a discrete Leslie-Gower predator-prey model with constant and variable delays. Based on the permanence result, sufficient conditions are established for the existence and uniqueness of globally attractive almost periodic solution.

A example and a numerical simulation are given to illustrate the our results.

1. Introduction

Leslie [12, 13] introduced a predator-prey model where the “carrying capacity” of the predator’s environment is proportional to the number of prey. Leslie stresses the fact that there are upper limits to the rates of increase of both prey and predator, which are not recognized in the Lotka-Volterra model. These upper limits can be approached under favorable conditions: for the predator, when the number of prey per predator is large; for the prey, when the number of predators (and perhaps the number of prey also) is small. In the case of continuous time, these considerations lead to the model

x⁰₁=x1 r1−b1x1−a1x2

, x⁰₂=x₂ r₂−a₂x₂

x1

, (1.1)

which are known as Leslie-Gower predator-prey model [20]. System (1.1) is one of the simplest having maximum growth rates which each population approaches under favorable conditions.

It is well known that time delays of one type or another have been incorporated into mathematical models of population dynamics due to maturation time, cap- turing time or other reasons. In general, delay differential equations exhibit much more complicated dynamics than ordinary differential equations since a time delay could cause a stable equilibrium to become unstable and cause the populations to fluctuate. We refer to the monographs of Cushing [4], Gopalsamy [7], Kuang [11] for general delayed biological systems and to Beretta and Kuang [1, 2], Faria and Mag- alhaes [6], Gopalsamy [8, 9], May [19], Song and Wei [23], Xiao and Ruan [24], Liu

2000Mathematics Subject Classification. 34K14, 34K20, 39A11.

Key words and phrases. Variable delay; permanence; almost periodicity; Leslie-Gower model.

c

2013 Texas State University - San Marcos.

Submitted July 23, 2012. Published April 24, 2013.

1

and Yuan [17] and the references cited therein for studies on delayed predator-prey systems. Some scholars have explored the dynamics of the following Leslie-Gower models with constant delays [2, 18, 25, 26, 27]:

x⁰₁(t) =x₁(t) r₁−b₁x₁(t−τ)−a₁x₂(t) , x⁰₂(t) =x2(t) r2−a2

x2(t) x1(t)

; (1.2)

x⁰₁(t) =x1(t) r1−b1x1(t−τ)−a1x2(t) , x⁰₂(t) =x2(t) r2−a2

x2(t−τ) x₁(t)

; (1.3)

x⁰₁(t) =x1(t) r1−b1x1(t)−a1x2(t) , x⁰₂(t) =x2(t) r2−a2

x2(t−τ) x₁(t−τ)

; (1.4)

x⁰₁(t) =x1(t) r1−b1x1(t)−a1x2(t−τ1) , x⁰₂(t) =x₂(t) r₂−a₂ x2(t)

x₁(t−τ₂)

; (1.5)

whereτ, τ1 andτ2 are nonnegative constants.

Firstly, in the real world, the delays in differential equations of biological phe- nomena are usually time-varying. Thus, it is worthwhile continuing to discuss the Leslie-Gower predator-prey model with time-varying delays. Secondly, many au- thors [3, 12, 14, 15, 16, 21, 22, 28, 30] have argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations. Discrete time models can also provide efficient computational models of continuous models for numerical simulations. Thirdly, Leslie-Gower predator-prey models have not been well stud- ied yet in the sense that most results are for models with constant environment [2, 18, 25, 26, 27]. This means that the models have been assumed to be au- tonomous, that is, all biological or environmental parameters have been assumed to be constants in time. However, this is rarely the ease in real life, because many biological and environmental parameters do vary in time (e.g., naturally subject to seasonal fluctuations). When this is taken into account, a model must be nonau- tonomous, which is more difficult to analyze in general. But, in doing so, one should also take advantage of the properties of those varying parameters. For example, one may assume the parameters are periodic or almost periodic for seasonal rea- sons. Based on the above points, we consider the following discrete Leslie-Gower predator-prey model with pure and variable delays:

x1(n+ 1) =x1(n) exp

r1(n)−b1(n)x1(n− bc1(n)c)−a1(n)x2(n− bc3(n)c) , x2(n+ 1) =x2(n) expn

r2(n)−a2(n)x2(n− bc2(n)c) x₁(n− bc₄(n)c) o

,

(1.6) where{ri(n)},{b1(n)},{ai(n)}and{cj(n)} are bounded nonnegative almost peri- odic sequences, i= 1,2, j= 1,2,3,4,bacdenotes the algebraically largest integer which does not exceed a. Under the assumptions of almost periodicity of the co- efficients of (1.6), our purpose of this paper is to establish sufficient conditions for the existence and uniqueness of globally attractive almost periodic solution of (1.6)

by constructing a suitable Lyapunov functional and almost periodic functional hull theory. Obviously, systems (1.2)-(1.5) are special cases of (1.6).

For any bounded sequence {f(n)} defined on Z, let f^u = sup_n∈Z{f(n)}, f^l = inf_n∈Z{f(n)}.

Throughout this paper, we assume that

(H1) 0< r_i^l≤ri(n)≤r^u_i, 0< a^l_i≤ai(n)≤a^u_i and 0< b^l₁≤b1(n)≤b^u₁,∀n∈Z, i= 1,2.

Let ¯ci := sup_n∈Zbci(n)c, c_i := inf_n∈Zbci(n)c, i = 1,2,3,4, c0 = P4

i=1¯ci. We consider system (1.6) together with the initial conditions

x_i(θ) =ϕ_i(θ)≥0, θ∈[−c0,0]_Z, ϕ_i(0)>0, i= 1,2. (1.7) One can easily show that the solutions of (1.6) with initial conditions (1.7) are defined and remain positive forn∈Z⁺:= [0,+∞)_Z.

The organization of this article is as follows. In Section 2, we give some basic definitions and necessary lemmas which will be used in later sections. In Section 3, permanence of (1.6) is considered. In Section 4, global attractivity of (1.6) is investigated by constructing a suitable Lyapunov functional. In Section 5, some sufficient conditions are established for the existence and uniqueness of almost pe- riodic solution of (1.6) by using almost periodic functional hull theory. An example and numerical simulation are given in Section 6.

2. Preliminaries

Let us state the following definitions and lemmas, which will be useful in proving our main result.

Definition 2.1([29]). A sequencex:Z→Ris called analmost periodic sequence if the-translation set ofx,

E{, x}={τ∈Z:|x(n+τ)−x(n)|< ,∀n∈Z}

is a relatively dense set inZfor all >0; that is, for any given >0, there exists an integerl()>0 such that each interval of lengthl() contains an integerτ ∈E{, x}

such that

|x(n+τ)−x(n)|< , ∀n∈Z.

The valueτ is called the-translation number or-almost period.

Definition 2.2 ([29]). Let f : Z×D →R, where D is an open set inC :={φ : [−τ,0]_Z →R}. f(n, φ) is said to be almost periodic innuniformly for φ∈D, or uniformly almost periodic for short, if for any >0 and any compact setS in D, there exists a positive integerl(,S) such that any interval of lengthl(,S) contains a integerτ for which

|f(n+τ, φ)−f(n, φ)|< , ∀n∈Z, φ∈S. The valueτ is called the-translation number off(n, φ).

Definition 2.3 ([29]). The hull off, denoted by H(f), is defined by H(f) ={g(n, x) : lim

k→∞f(n+τk, x) =g(n, x) uniformly onZ×S} for some sequence{τk}, whereS is any compact set inD.

Definition 2.4. Suppose that (x1, x2) is any solution of (1.6). (x1, x2) is said to be a strictly positive solution onZif forn∈Z,

0< inf

n∈Z

xi(n)≤sup

n∈Z

xi(n)<∞, i= 1,2.

Lemma 2.5 ([29]). A sequence {x(n)} is almost periodic if and only if for any sequence {h⁰_k} ⊂ Z there exists a subsequence {hk} ⊂ {h⁰_k} such that x(n+hk) converges uniformly onn∈Zask→+∞. Furthermore, the limit sequence is also an almost periodic sequence.

3. Permanence

In this section, we obtain the following permanence result of (1.6).

Lemma 3.1. Assume that(H1)holds, then every solution(x1, x2)of (1.6)satisfies lim sup

n→∞

x_i(n)≤M_i, i= 1,2, where

M₁:= min (r1

b₁)^uexp{r^u₁(¯c₁+ 1)},exp{r₁^u(¯c1+ 1)−1}

b^l₁

, M₂:= min

(r2

a₂)^uM₁exp{r^u₂(¯c₂+ 1)},M1exp{r₁^u(¯c1+ 1)−1}

a^l₂

.

Proof. Let (x₁, x₂) be any positive solution of (1.6) with initial conditions (1.7).

From the first equation of (1.6) it follows that

x1(n+ 1)≤x1(n) exp{r1(n)} ≤x1(n) exp{r^u₁}, which yields

x1(n− bc1(n)c)≥x1(n) exp{−r^u₁¯c1}, which implies

x₁(n+ 1)≤x₁(n) exp

r₁(n)−b₁(n) exp{−r^u₁¯c₁}x1(n)

. (3.1)

First, we present two cases to prove that lim sup

n→∞

x1(n)≤M1.

Case I.There exists al₀∈Z⁺ such thatx₁(l₀+ 1)≥x₁(l₀). Then by (3.1), r1(l0)−b1(l0) exp{−r^u₁¯c1}x1(l0)≥0,

which implies

x1(l0)≤(r1

b₁)^uexp{r^u₁¯c1} ≤M1. On the one hand, from (3.1),

x1(l0+ 1)≤x1(l0) exp{r^u₁} ≤(r₁ b1

)^uexp{r₁^u(¯c1+ 1)}; (3.2) on the other hand, from (3.1),

x(l0+ 1)≤ b1(l0) exp{−r^u₁¯c1}x1(l0) b₁(l₀) exp{−r^u₁¯c₁} exp

r₁^u−b1(l0) exp{−r^u₁¯c1}x1(l0)

≤ exp{r^u₁(¯c1+ 1)−1}

b^l₁ ,

(3.3)

here we used

maxx>0 xexp{r^u₁−x}= exp{r₁^u−1}.

Together with (3.2)-(3.3), we have x1(l0+ 1)≤M1:= min (r1

b₁)^uexp{r^u₁(¯c1+ 1)},exp{r^u₁(¯c1+ 1)−1}

b^l₁

. (3.4) We claim that

x1(n)≤M1, ∀n≥l0.

In fact if there exists an integer k0 ≥l0+ 2 such thatx1(k0)> M1, and letting1

be the least integer betweenl0andk0 such thatx1(1) = maxl₀≤n≤k0{x1(n)}, then

1≥l0+ 2 andx1(1)> x1(1−1), which implies from the argument as that in (3.4) that

x1(1)≤M1< x1(k0).

This is impossible. This proves the claim.

Case II.x1(n)≥x1(n+ 1),∀n∈Z⁺. In particular, lim_n→∞x1(n) exists, denoted by ¯x1. Taking limit in the first equation of (1.6) gives

n→∞lim

r1(n)−b1(n)x1(n− bc1(n)c)−a1(n)x2(n− bc2(n)c)

= 0.

Hence ¯x1≤(^r_b¹

1)^u≤M1. This proves the claim.

From the two claims above, lim sup_n→∞x1(n)≤M1. For arbitrary >0, there existsn0∈Z⁺ such that

x1(n)≤M1+ forn≥n0. Forn > n₀+ 2c₀, from the second equation in (1.6), we have

x₂(n+ 1)≤x₂(n) exp

r₂(n)−a2(n)x2(n− bc2(n)c)

M₁+ ≤x₂(n) exp{r₂^u}, which yields

x2(n− bc2(n)c)≥x2(n) exp{−r^u₂¯c2}, which implies

x2(n+ 1)≤x2(n) exp

r2(n)−a2(n) exp{−r₂^u¯c2}x2(n) M₁+

. Similar to the above argument asx1, we can easily obtain that

lim sup

n→∞

x2(n)≤M2:= min (r2

a2

)^uM1exp{r^u₂(¯c2+ 1)},M1exp{r₁^u(¯c1+ 1)−1}

a^l₂

.

This completes the proof.

Lemma 3.2. Assume that(H1) and the following condition hold:

(H2) r^l₁> a^u₁M2.

Then every solution(x1, x2)of (1.6)satisfies lim inf

n→∞xi(n)≥mi, i= 1,2, where

α:= exp

b^u₁M1+a^u₁M2−r^l₁

¯ c1

, β:= exp{a^u₂M2

m₁ −r₂^l

¯ c2}, m₁:=r^l₁−a^u₁M2

b^u₁α exp{r₁^l−a^u₁M₂−b^u₁αM₁}, m₂:=r₂^lm1

βa^u₂ exp{r₂^l−a^u₂βM2

m₁ }.

Proof. From the definition ofM1, we obtain

r^l₁−b^u₁M1−a^u₁M2≤r^l₁−b^u₁M1≤r₁^l−b^u₁(r1

b1

)^u≤r₁^l−r₁^u≤0, which impliesα≥1.

By Lemma 3.1 and (H2), for an arbitrary >0, there existsn₁∈Z⁺such that x_i(n)≤M_i+, r₁^l > a^u₁(M₂+), ∀n≥n₁, i= 1,2.

Forn > n1+c0, from the first equation of (1.6), we have

x1(n+ 1)≥x1(n) exp{r^l₁−b^u₁(M1+)−a^u₁(M2+)}.

So

x1(n− bc1(n)c)≤x1(n) exp

[b^u₁(M1+) +a^u₁(M2+)−r^l₁]¯c1 :=x1(n)α(), where

α() := exp

[b^u₁(M₁+) +a^u₁(M₂+)−r₁^l]¯c₁ ≥1.

From the first equation of (1.6), we have

x₁(n+ 1)≥x₁(n) exp{r^l₁−a^u₁(M₂+)−b^u₁α()x₁(n)}, ∀n≥n₀+c₀. (3.5) Next, we present two cases to prove that

lim inf

n→∞x1(n)≥m1.

Case I.There exists al₀≥n₀+c₀such thatx₁(l₀+ 1)≤x₁(l₀). Then from (3.5), r^l₁−a^u₁(M₂+)−b^u₁α()x₁(l₀)≤0,

which implies

x1(l0)≥ r₁^l−a^u₁(M2+) b^u₁α() . In view of (3.5), we can easily obtain that

x₁(l₀+ 1)≥ r₁^l−a^u₁(M2+)

b^u₁α() exp{r^l₁−a^u₁(M₂+)−b^u₁α()(M₁+)}

:=m₁()≤r^l₁−a^u₁(M2+) b^u₁α() . We claim that

x1(n)≥m1() forn≥l0.

By way of contradiction, assume that there exists a p0 ≥ l0 such that x1(p0) <

m1(). Then p0 ≥ l0 + 2. Let p1 ≥ l0 + 2 be the smallest integer such that x1(p0) < m1(). Then x1(p1−1) > x1(p1). The above argument produces that x₁(p₁)≥m₁(), a contradiction. This proves the claim.

Case II.We assume thatx₁(n)< x₁(n+1), for alln≥n₀+c₀. Then lim_n→∞x₁(n) exists, denoted byx₁. Taking limit in the first equation of (1.6) gives

n→∞lim

r1(n)−b1(n)x1(n− bc1(n)c)−a1(n)x2(n− bc2(n)c)

= 0.

Hencex₁≥^rl1−a_bu^u1M2 1

≥m₁() and lim_→0m₁() =m₁. This proves the claim.

From the two claims above, lim inf_n→∞x₁(n)≥m₁. There exists two positive constants₀ andn₂ such that

x_i(n)≤M_i+₀, x₁(n)> m₁−₀>0, ∀n > n2+c₀, i= 1,2.

From the second equation of (1.6), we have x2(n+ 1)≥x2(n) exp

r^l₂−a^u₂(M2+0)

m₁−₀ , ∀n > n2+c0. So

x₂(n− bc2(n)c)≤x₂(n) exp

[a^u₂(M₂+₀) m1−0

−r^l₂]¯c₂ :=x₂(n)β(₀) for alln > n2+c0, where

β(0) := exp{a^u₂(M2+0) m₁−₀ −r^l₂

¯ c2} ≥1.

From the second equation of (1.6), we have x2(n+ 1)≥x2(n) exp

r^l₂−a^u₂β(₀)x₂(n) m1−0

, ∀n≥n0+c0. (3.6) Similar to the above analysis forx1, we can obtain

lim inf

n→∞x2(n)≥m2:=r^l₂m1

βa^u₂ exp

r^l₂−a^u₂βM2

m1

.

The proof is complete.

By Lemmas 3.1 and 3.2, we can easily show the following result.

Theorem 3.3. Assume that(H1)–(H2) hold, then every solution (x1, x2) of (1.6) satisfies

m_i≤lim inf

n→∞N_i(n)≤lim sup

n→∞

N_i(n)≤M_i, i= 1,2.

That is,(1.6)is permanent.

4. Global attractivity

In this section, we investigate the global attractivity of (1.6). Define a function χ: [0,∞)_Z→ {0,1}as follows:

χ(s) :=

(0, ifs= 0, 1, ifs∈[1,∞)_Z. Let

µ₁:= exp{r₁^u−b^l₁m₁−a^l₁m₂}, µ₂:= exp{r₂^u−a^l₂m₂ M1

}, ν1:= max{µ1,1}, ν2:= max{µ2,1},

δ1:= max{r₁^u, b^u₁M1+a^u₁M2}, δ2:= max{r₂^u,a^u₂M₂ m1

}.

Theorem 4.1. Assume that (H1)—(H2)hold. Suppose further that

(H3) there exist two positive constants λ₁ and λ₂ such that min{Θ1,Θ₂} > 0, where

Θ₁:=λ₁min[b^l₁, 2 M1

−b^u₁]−λ₁M₁µ₁(b^u₁)²χ(¯c₁)¯c₁(¯c₁−c₁+ 1)−λ₁ν₁δ₁b^u₁χ(¯c₁)¯c₁

−λ2χ(¯c2)¯c2(¯c4−c₄+ 1)M₂²µ2(a^u₂)²

m³₁ −λ2a^u₂(¯c4−c₄+ 1)M2

m²₁

and

Θ2:=λ2min[a^l₂ M1

, 2 M2

− a^u₂ m1

]−λ₂a^u₂χ(¯c₂)¯c₂ν₂δ₂ m1

−λ₂χ(¯c₂)¯c₂(¯c₂−c₂+ 1)M₂µ₂(a^u₂)²

m²₁ −λ₁M₁µ₁a^u₁b^u₁χ(¯c₁)¯c₁(¯c₃−c₃+ 1)

−λ1a^u₁(¯c3−c₃+ 1).

Then (1.6)is globally attractive.

Proof. From condition (H3), there exist small positive constants <min{m1, m2} andλsuch that

Θ₁() :=λ₁min b^l₁, 2

M1+−b^u₁

−λ₁(M₁+)µ₁()(b^u₁)²χ(¯c₁)¯c₁(¯c₁−c₁+ 1)

−λ1ν1()δ1()b^u₁χ(¯c1)¯c1−λ2

χ(¯c2)¯c2(¯c4−c₄+ 1)(M2+)²µ2()(a^u₂)² (m₁−)³

−λ₂a^u₂(¯c₄−c₄+ 1)(M₂+) (m1−)² > λ, Θ2() :=λ2min a^l₂

M₁+, 2

M₂+ − a^u₂ m₁−

−λ2

a^u₂χ(¯c2)¯c2ν2()δ2() m₁−

−λ2

χ(¯c2)¯c2(¯c2−c₂+ 1)(M2+)µ2()(a^u₂)² (m1−)²

−λ1(M1+)µ1()a^u₁b^u₁χ(¯c1)¯c1(¯c3−c₃+ 1)−λ1a^u₁(¯c3−c₃+ 1)> λ, where

µ1() := exp{r₁^u−b^l₁(m1−)−a^l₁(m2−)}, µ2() := exp{r^u₂ −a^l₂(m₂−) M1+ }, ν₁() := max{µ₁(),1}, ν₂() := max{µ₂(),1},

δ1() := max{r^u₁, b^u₁(M1+) +a^u₁(M2+)}, δ2() := max

r₂^u,a^u₂(M2+) m1− . Suppose that (x1, x2) and (y1, y2) are two positive solutions of (1.6). By Theorem 3.3, there exists a constantN₀>0 such that

mi−≤xi(n), yi(n)≤Mi+, n≥N0, i= 1,2.

Let

V11(n) =|lnx1(n)−lny1(n)|.

In view of (1.6), we have

V11(n+ 1) =|lnx1(n+ 1)−lny1(n+ 1)|

=

[lnx₁(n)−lny₁(n)]−b₁(n)[x₁(n− bc1(n)c)−y₁(n− bc1(n)c)]

−a₁(n)[x₂(n− bc₃(n)c)−y₂(n− bc₃(n)c)]

=

[lnx1(n)−lny1(n)]−b1(n)[x1(n)−y1(n)]

+b1(n)χ(¯c1) [x1(n)−x1(n− bc1(n)c)] + [y1(n)−y1(n− bc1(n)c)]

−a1(n)[x2(n− bc3(n)c)−y2(n− bc3(n)c)]

.

(4.1)

Define

P1(n) :=r1(n)−b1(n)x1(n− bc1(n)c)−a1(n)x2(n− bc3(n)c), ∀n∈Z, Q1(n) :=r1(n)−b1(n)y1(n− bc1(n)c)−a1(n)y2(n− bc3(n)c), ∀n∈Z. In view of (1.6), we obtain

[x₁(n)−x₁(n− bc₁(n)c)] + [y₁(n)−y₁(n− bc₁(n)c)]

=

n−1

X

s=n−bc1(n)c

x₁(s+ 1)−y₁(s+ 1)

−

n−1

X

s=n−bc1(n)c

x₁(s)−y₁(s)

=

n−1

X

s=n−bc1(n)c

x1(s)e^P1(s)−y1(s)e^Q1(s)

−

n−1

X

s=n−bc1(n)c

x1(s)−y1(s)

=

n−1

X

s=n−bc₁(n)c

x₁(s)

e^P1(s)−e^Q1(s) +

n−1

X

s=n−bc₁(n)c

x₁(s)−y₁(s)

e^Q1(s)−1

≤

n−1

X

s=n−¯c₁

x1(s)ξ1(s) b1(s)

x1(s− bc1(s)c)−y1(s− bc1(s)c)

+a1(s)

x2(s− bc3(s)c)−y2(s− bc3(s)c)

+

n−1

X

s=n−¯c₁

ξ2(s)

r1(s)−b1(s)y1(s− bc1(s)c)

−a1(s)y2(s− bc3(s)c)

x1(s)−y1(s)

≤

n−1

X

s=n−¯c₁

¯ c1

X

k=c₁

(M1+)µ1()b^u₁

x1(s−k)−y1(s−k)

+

n−1

X

s=n−¯c1

¯ c3

X

k=c₃

(M₁+)µ₁()a^u₁

x₂(s−k)−y₂(s−k)

+

n−1

X

s=n−¯c₁

ν1()δ1()

x1(s)−y1(s)

≤

n−k−1

X

s=n−k−¯c1

¯ c1

X

k=c₁

(M1+)µ1()b^u₁

x1(s)−y1(s)

+

n−k−1

X

s=n−k−¯c1

¯ c₃

X

k=c₃

(M1+)µ1()a^u₁

x2(s)−y2(s)

+

n−1

X

s=n−¯c1

ν₁()δ₁()

x₁(s)−y₁(s)

, (4.2)

where ξ₁(s) lies between e^P1(s) and e^Q1(s), ξ₂(s) lies between e^Q1(s) and 1, s = n− bc1(n)c, . . . , n−1, forn > N₀+c₀.

By (4.1) and (4.2), we have

∆V₁₁(n) =V₁₁(n+ 1)−V₁₁(n)

≤ −|lnx1(n)−lny1(n)|+

[lnx1(n)−lny1(n)]−b1(n)[x1(n)−y1(n)]

+b1(n)χ(¯c1)

[x1(n)−x1(n− bc1(n)c)] + [y1(n)−y1(n− bc1(n)c)]

+a1(n)

x2(n− bc3(n)c)−y2(n− bc3(n)c)

≤ − 1 σ1(n)−

1

σ1(n)−b1(n)

x1(n)−y1(n)

+

n−k−1

X

s=n−k−¯c₁

¯ c₁

X

k=c₁

(M1+)µ1()(b^u₁)²χ(¯c1)

x1(s)−y1(s)

+

n−k−1

X

s=n−k−¯c₁

¯ c₃

X

k=c₃

(M1+)µ1()a^u₁b^u₁χ(¯c1)

x2(s)−y2(s)

+

n−1

X

s=n−¯c1

ν₁()δ₁()b^u₁χ(¯c₁)

x₁(s)−y₁(s) +

n−c₃

X

s=n−¯c3

a^u₁

x₂(s)−y₂(s)

≤ −min b^l₁, 2

M1+−b^u₁

x₁(n)−y₁(n)

+

n−k−1

X

s=n−k−¯c₁

¯ c1

X

k=c₁

(M₁+)µ₁()(b^u₁)²χ(¯c₁)

x₁(s)−y₁(s)

+

n−k−1

X

s=n−k−¯c1

¯ c3

X

k=c₃

(M₁+)µ₁()a^u₁b^u₁χ(¯c₁)

x₂(s)−y₂(s)

+

n−1

X

s=n−¯c₁

ν1()δ1()b^u₁χ(¯c1)

x1(s)−y1(s) +

n−c₃

X

s=n−¯c₃

a^u₁

x2(s)−y2(s) , (4.3) here we used

|x1(n)−y₁(n)|=σ₁(n)|lnx₁(n)−lny₁(n)|, whereσ₁(n) lies between x₁(n) andy₁(n),∀n > N0+c₀. Let

V12(n) =

¯ c₁−1

X

t=0

n−1

X

s=n−k−¯c₁+t

¯ c₁

X

k=c₁

(M1+)µ1()(b^u₁)²χ(¯c1)

x1(s)−y1(s) ,

V13(n) =

¯ c₁−1

X

t=0

n−1

X

s=n−k−¯c1+t

¯ c₃

X

k=c₃

(M1+)µ1()a^u₁b^u₁χ(¯c1)

x2(s)−y2(s) ,

V₁₄(n) =

¯ c1−1

X

t=0 n−1

X

s=n−¯c1+t

ν₁()δ₁()b^u₁χ(¯c₁)

x₁(s)−y₁(s) ,

V₁₅(n) =

¯ c3−c₃

X

t=0 n−1

X

s=n−¯c3+t

a^u₁

x₂(s)−y₂(s) .

Also we obtain

∆V12(n) =V12(n+ 1)−V12(n)

= (M1+)µ1()(b^u₁)²χ(¯c1)¯c1(¯c1−c₁+ 1)

x1(n)−y1(n)

−

n−k−1

X

s=n−k−¯c₁

¯ c1

X

k=c₁

(M₁+)µ₁()(b^u₁)²χ(¯c₁)

x₁(s)−y₁(s) ,

(4.4)

∆V₁₃(n) =V₁₃(n+ 1)−V₁₃(n)

= (M₁+)µ₁()a^u₁b^u₁χ(¯c₁)¯c₁(¯c₃−c₃+ 1)

x₂(n)−y₂(n)

−

n−k−1

X

s=n−k−¯c1

¯ c₃

X

k=c₃

(M1+)µ1()a^u₁b^u₁χ(¯c1)

x2(s)−y2(s) ,

(4.5)

∆V14(n) =V14(n+ 1)−V14(n)

=ν1()δ1()b^u₁χ(¯c1)¯c1

x1(n)−y1(n)

−

n−1

X

s=n−¯c1

ν1()δ1()b^u₁χ(¯c1)

x1(s)−y1(s) ,

(4.6)

∆V15(n) =V15(n+ 1)−V15(n)

=a^u₁(¯c3−c₃+ 1)

x2(n)−y2(n) −

n−c₃

X

s=n−¯c₃

a^u₁

x2(s)−y2(s)

. (4.7) Define

V₁(n) =V₁₁(n) +V₁₂(n) +V₁₃(n) +V₁₄(n) +V₁₅(n).

From (4.3)-(4.7) it follows that

∆V1(n)≤ −{min b^l₁, 2

M1+−b^u₁

−(M1+)µ1()(b^u₁)²χ(¯c1)¯c1(¯c1−c₁+ 1)

−ν1()δ1()b^u₁χ(¯c1)¯c1}

x1(n)−y1(n) +{(M1+)µ₁()a^u₁b^u₁χ(¯c₁)¯c₁(¯c₃−c₃+ 1) +a^u₁(¯c₃−c₃+ 1)}

x₂(n)−x₂(n)

, ∀n > N₀+c₀.

(4.8) Let

V21(n) =|lnx2(n)−lny2(n)|.

From (1.6), we have

V₂₁(n+ 1) =|lnx₂(n+ 1)−lny₂(n+ 1)|

=

[lnx2(n)−lny2(n)]−a2(n)x2(n− bc2(n)c)

x1(n− bc4(n)c)−y2(n− bc2(n)c) y1(n− bc4(n)c)

. (4.9) Further, it follows that

x₂(n− bc₂(n)c)

x1(n− bc4(n)c)−y₂(n− bc₂(n)c) y1(n− bc4(n)c)

= x2(n− bc2(n)c)y1(n− bc4(n)c)−y2(n− bc2(n)c)x1(n− bc4(n)c) x1(n− bc4(n)c)y1(n− bc4(n)c)

=

x₂(n− bc2(n)c)−y₂(n− bc2(n)c) x1(n− bc4(n)c)

−y₂(n− bc2(n)c)

x₁(n− bc4(n)c)−y₁(n− bc4(n)c) x1(n− bc4(n)c)y1(n− bc4(n)c) , which from (4.9) implies

V21(n+ 1)≤

[lnx2(n)−lny2(n)]−a2(n)

x2(n− bc2(n)c)−y2(n− bc2(n)c) x1(n− bc4(n)c)

+a₂(n)y₂(n− bc2(n)c)

x₁(n− bc4(n)c)−y₁(n− bc4(n)c) x1(n− bc4(n)c)y1(n− bc4(n)c)

≤

[lnx₂(n)−lny₂(n)]−a₂(n)

x₂(n)−y₂(n) x1(n− bc4(n)c)

+a2(n)χ(¯c2)

[x2(n)−x2(n− bc2(n)c)] + [y2(n)−y2(n− bc2(n)c)]

x₁(n− bc₄(n)c)

+a2(n)y2(n− bc2(n)c)

x1(n− bc4(n)c)−y1(n− bc4(n)c) x1(n− bc4(n)c)y1(n− bc4(n)c) .

(4.10) Define

P₂(n) :=r₂(n)−a₂(n)x₂(n− bc2(n)c)

x1(n− bc4(n)c), ∀n∈Z, Q2(n) :=r2(n)−a2(n)y2(n− bc2(n)c)

y1(n− bc4(n)c), ∀n∈Z. By (1.6), we obtain

[x2(n)−x2(n− bc2(n)c)] + [y2(n)−y2(n− bc2(n)c)]

=

n−1

X

s=n−bc2(n)c

x2(s)

e^P2(s)−e^Q2(s) +

n−1

X

s=n−bc2(n)c

x2(s)−y2(s)

e^Q2(s)−1

≤

n−1

X

s=n−¯c2

x2(s)ξ₁⁰(s)a2(s)

x2(s− bc2(s)c)

x1(s− bc4(s)c)−y2(n− bc2(s)c) y1(s− bc4(s)c)

+

n−1

X

s=n−¯c₂

ξ⁰₂(s)

r2(s)−a2(s)y2(s− bc2(s)c) y1(s− bc4(s)c)

x2(s)−y2(s)

≤

n−1

X

s=n−¯c₂

x2(s)ξ₁⁰(s)a2(s)

x2(s− bc2(s)c)−y2(s− bc2(s)c) x1(s− bc4(s)c)

+

n−1

X

s=n−¯c₂

x2(s)ξ⁰₁(s)a2(s)y2(s− bc2(s)c)

x1(s− bc4(s)c)−y1(s− bc4(s)c) x1(s− bc4(s)c)y1(s− bc4(s)c)

+

n−1

X

s=n−¯c₂

ξ⁰₂(s)

r2(s)−a2(s)y2(s− bc2(s)c) y1(s− bc4(s)c)

x2(s)−y2(s)

≤

n−k−1

X

s=n−k−¯c2

¯ c2

X

k=c₂

(M₂+)µ₂()a^u₂ m1−

x₂(s)−y₂(s)

+

n−k−1

X

s=n−k−¯c2

¯ c4

X

k=c₄

(M2+)²µ2()a^u₂ (m1−)²

x1(s)−y1(s)

+

n−1

X

s=n−¯c2

ν2()δ2()

x2(s)−y2(s)

, (4.11)

where ξ₁⁰(s) lies between e^P2(s) and e^Q2(s), ξ⁰₂(s) lies between e^Q2(s) and 1, s = n− bc₂(n)c, . . . , n−1, forn > N₀+c₀.

By (4.10) and (4.11), we have

∆V21(n) =V21(n+ 1)−V21(n)

≤ −|lnx2(n)−lny2(n)|+

[lnx2(n)−lny2(n)]−a2(n)

x2(n)−y2(n) x1(n− bc4(n)c)

+a₂(n)χ(¯c₂)

[x₂(n)−x₂(n− bc2(n)c)] + [y2(n)−y₂(n− bc2(n)c)]

x1(n− bc4(n)c)

+a₂(n)y₂(n− bc₂(n)c)

x₁(n− bc₄(n)c)−y₁(n− bc₄(n)c) x₁(n− bc4(n)c)y1(n− bc4(n)c)

≤ − 1 σ2(n)−

1

σ2(n)− a₂(n) x1(n− bc4(n)c)

x₂(n)−y₂(n)

+

n−k−1

X

s=n−k−¯c₂

¯ c₂

X

k=c₂

χ(¯c2)(M2+)µ2()(a^u₂)² (m1−)²

x2(s)−y2(s)

+

n−k−1

X

s=n−k−¯c₂

¯ c₄

X

k=c₄

χ(¯c2)(M2+)²µ2()(a^u₂)² (m₁−)³

x1(s)−y1(s)

+

n−1

X

s=n−¯c₂

a^u₂χ(¯c2)ν2()δ2() m1−

x2(s)−y2(s)

+a^u₂(M₂+) (m1−)²

n−c₄

X

s=n−¯c4

x₁(s)−y₁(s)

≤ −min a^l₂

M1+, 2

M2+− a^u₂ m1−

x₂(n)−y₂(n)

+

n−k−1

X

s=n−k−¯c₂

¯ c2

X

k=c₂

χ(¯c₂)(M₂+)µ₂()(a^u₂)² (m1−)²

x₂(s)−y₂(s)

+

n−k−1

X

s=n−k−¯c2

¯ c4

X

k=c₄

χ(¯c₂)(M₂+)²µ₂()(a^u₂)² (m1−)³

x1(s)−y1(s)

+

n−1

X

s=n−¯c₂

a^u₂χ(¯c2)ν2()δ2() m₁−

x₂(s)−y₂(s)

+a^u₂(M2+) (m₁−)²

n−c₄

X

s=n−¯c₄

x1(s)−y1(s)

. (4.12)

Here we used that

|x2(n)−y₂(n)|=σ₂(n)|lnx₂(n)−lny₂(n)|,

whereσ2(n) lies between x2(n) andy2(n), forn > N0+c0. Let V2(n) =V21(n) +V22(n) +V23(n) +V24(n) +V25(n), where

V₂₂(n) =

¯ c2−1

X

t=0

n−1

X

s=n−k−¯c₂+t

¯ c2

X

k=c₂

χ(¯c₂)(M₂+)µ₂()(a^u₂)² (m1−)²

x₂(s)−y₂(s) ,

V23(n) =

¯ c₂−1

X

t=0

n−1

X

s=n−k−¯c₂+t

¯ c₄

X

k=c₄

χ(¯c2)(M2+)²µ2()(a^u₂)² (m₁−)³

x1(s)−y1(s) ,

V24(n) =

¯ c₂−1

X

t=0 n−1

X

s=n−¯c₂+t

a^u₂χ(¯c2)ν2()δ2() m₁−

x2(s)−y2(s) ,

V25(n) =a^u₂(M2+) (m1−)²

¯ c₄−c₄

X

t=0 n−1

X

s=n−¯c₄+t

x1(s)−y1(s) .

By a similar argument as that in (4.8), we obtain

∆V₂(n)≤ −{min a^l₂

M1+, 2

M2+− a^u₂ m1−

−a^u₂χ(¯c₂)¯c₂ν₂()δ₂() m1−

−χ(¯c2)¯c2(¯c2−c₂+ 1)(M2+)µ2()(a^u₂)²

(m₁−)² }

x₂(n)−y₂(n)

+{χ(¯c₂)¯c₂(¯c₄−c₄+ 1)(M₂+)²µ₂()(a^u₂)² (m1−)³

+a^u₂(¯c4−c₄+ 1)(M2+) (m1−)² }

x1(n)−y1(n)

, ∀n > N0+c0.

(4.13)

We construct a Lyapunov functional as follows:

V(n) =λ₁V₁(n) +λ₂V₂(n), which from (4.8) and (4.13) implies

∆V(n)≤ −{λ1min b^l₁, 2

M₁+−b^u₁

−λ1(M1+)µ1()(b^u₁)²χ(¯c1)¯c1(¯c1−c₁+ 1)

−λ1ν1()δ1()b^u₁χ(¯c1)¯c1−λ2

χ(¯c2)¯c2(¯c4−c₄+ 1)(M2+)²µ2()(a^u₂)² (m1−)³

−λ2

a^u₂(¯c4−c₄+ 1)(M2+) (m₁−)² }

x1(n)−y1(n)

− {λ2min a^l₂

M1+, 2

M2+− a^u₂ m1−

−λ2

a^u₂χ(¯c2)¯c2ν2()δ2() m1−

−λ2

χ(¯c₂)¯c₂(¯c₂−c₂+ 1)(M₂+)µ₂()(a^u₂)² (m1−)²

−λ1(M1+)µ1()a^u₁b^u₁χ(¯c1)¯c1(¯c3−c₃+ 1)−λ1a^u₁(¯c3−c₃+ 1)}

×

x₂(n)−y₂(n)

≤ −λ

x₁(n)−y₁(n) +

x₂(n)−y₂(n)

, ∀n > N₀+c₀. (4.14)