Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2008, Article ID 706154, 13 pages
doi:10.1155/2008/706154
Research Article
Almost Periodic Solution of a Diffusive
Mixed System with Time Delay
and Type III Functional Response
Qiong Liu
Department of Mathematics and Computer Science, Guangxi Qinzhou University,
Qinzhou, Guangxi 535000, China
Correspondence should be addressed to Qiong Liu, gqlq08@163.com
Received 21 February 2008; Accepted 2 June 2008
Recommended by Manuel De La Sen
A delayed predator-prey model with diffusion and competition is proposed. Some sufficient
conditions on uniform persistence of the model have been obtained. By applying LiapunovRazumikhin technique, we will point out, under almost periodic circumstances, a set of sufficient
conditions that assure the existence and uniqueness of the positive almost periodic solution which
is globally asymptotically stable.
Copyright q 2008 Qiong Liu. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium,
provided the original work is properly cited.
1. Introduction
In the nature world, diffusion often occurs in an ecological environment; that is, species can
diffuse between patches. The works about autonomous systems in this field were pioneered
by Levin, after Levin 1, Kishimoto 2, and Takeuchi 3 studied this kind of model. But all
the coefficients in the system they studied are constants. Since biological and environmental
parameters are naturally subject to fluctuation in time, the effects of a varying environment
are considered as important selective forces on systems in a fluctuating environment. More
realistic and interesting models should take into account both the seasonality of the changing
environment and the effects of time delays 4–7. This motivated Chen et al. 8–11, and
others to consider nonautonomous predator-prey models with almost periodic coefficients
and diffusion. In this paper, we study the almost periodic solution of the delayed predatorprey model with diffusion and competition so as to obtain some conditions under which three
species are uniformly persistent. In addition, we obtain that for the almost periodic system
there exists a unique almost positive periodic solution which is globally asymptotically stable.
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Discrete Dynamics in Nature and Society
The organization of this paper is as follows. In the next section, we develop our model,
establish its important properties, and give several lemmas, which will be a key for our proofs
and discussions. In Section 3, sufficient conditions are given for uniform persistence of three
species. In Section 4, by applying Liapunov-Razumikhin technique, we prove the existence and
uniqueness of the positive almost periodic solution which is globally asymptotically stable.
Finally, we give a discussion of our results.
2. Model and preliminaries
It is assumed that the ecosystem is composed of two isolated patches, and the prey population
can disperse among the patches instantaneously. The state variables of the models, xi xi t i 1, 2, describe the densities of the prey population in Patch 1 and Patch 2, respectively.
We suppose that the net exchange of the prey population from Patch j to Patch i is proportional
to the difference of the concentration between xj − xi with Di t, i, j 1, 2. The state variables
of the models, xi xi t i 3, 4, describe the densities of the predator population in Patch 1
with competition.
Let us consider the following delayed diffusive predator-prey system with competition
and functional response:
α2 tx12 x4
α1 tx12 x3
x1 x1 a10 t − a11 tx1 −
−
D1 t x2 − x1 ,
2
2
1 β1 tx1 1 β2 tx1
x2 x2 a20 t − a21 tx2 D2 t x1 − x2 ,
α1 tx12 t − τ1
,
−
a
x3 x3 − a30 t a31 t
tx
−
a
tx
32
3
34
4
1 β1 tx12 t − τ1
α2 tx12 t − τ2
x4 x4 − a40 t a41 t
− a42 tx4 − a43 tx3 ,
1 β2 tx12 t − τ2
with the initial condition
x1 s φ1 s ∈ C −τ, 0, R ,
2.1
s ∈ −τ, 0, φ1 0 ≥ 0, xi 0 φi ≥ 0 constants, i 2, 3, 4.
2.2
Here, ai0 t and ai1 t i 1, 2 represent the intrinsic growth rate and the intraspecific
interference coefficient of the prey population xi i 1, 2, respectively. We then assume
that the death rate of the predator population xi i 3, 4 in Patch 1 is proportional to
both the existing predator population with the proportional functions a30 t and, respectively,
a40 t and to its square with the proportional functions a32 t and, respectively, a42 t. The
predator consumes the prey according to Holling type III functional response 12, 13, that is,
α1 tx12 x3 /1 β1 tx12 and α2 tx12 x4 /1 β2 tx12 . τi i 1, 2 is the time to digest food in the
.
predator organism. τ max{τ1 , τ2 }. R {z : z ≥ 0}.
We introduce some notations and definitions, and state some preliminary lemmas which
will be useful for establishing our main results.
Let R4 {X ∈ R4 : X x1 , x2 , x3 , x4 , xi > 0, i 1, 2, 3, 4}. C C−τ, ∞ × R × R ×
4
R , R . Assume Ω is a subset of R ×C−τ, 0, R ×R ×R ×R . Denote by f f1 , f2 , f3 , f4 T :
Ω → R4 the map defined by the right-hand side of system 2.1. If V : R × C → R is a
Qiong Liu
3
continuous function, then the upper right derivative of V t, x with respect to system 2.1 is
defined as
D V t, X lim sup
h→0
1 V t h, X hft, X − V t, X .
h
2.3
Obviously, the global existence and uniqueness of solutions of system 2.1 are guaranteed by
the smoothness properties of f see 14, 15 for details on fundamental properties of retarded
functional differential equations.
For convenience, we introduce the following notations:
ψ sup ψt ,
ψ inf ψt .
t≥0
t≥0
2.4
In this paper, we need all the coefficients to satisfy
min
i1,2,3,4,
j0,1,2,3,
aij , αi , βi , Di > 0,
max aij , αi , βi , Di < ∞.
2.5
i1,2,3,4,
j0,1,2,3.
Definition 2.1. System 2.1 is said to be uniformly persistent if there exists a compact region
D ⊂ R4 such that every solution x1 , x2 , x3 , x4 of system 2.1 with initial conditions 2.2
eventually enters and remains in the region D.
For convenience, the set CIP {u : 0, ∞ → 0, ∞ | us is positive and nondecreasing
for s > 0, u0 0}.
Lemma 2.2 see 16, 17. Consider the following almost periodic equation:
x t g t, xt .
2.6
Let CH ∗ {xt ∈ C : xt supθ∈−τ,0 |xt θ| < H ∗ }, SH ∗ {x ∈ Rn : |x| < H ∗ }, H ∗ ∈ R
or H ∗ ∞, g : R × CH ∗ → Rn , and g is uniformly almost periodic with respect to t. Let V :
R × SH ∗ × SH ∗ → R . Assume that the following conditions hold:
i ax − y ≤ V t, x, y ≤ bx − y, a·, b· ∈ CIP, b0 > 0;
ii |V t, x1 , y1 − V t, x2 , y2 | ≤ Lx1 − x2 y1 − y2 , where L is a positive constant;
iii there exists a continuous nondecreasing function P S such that
P S > S if S > 0,
D V t, x1 t, x2 t ≤ −CV t, x1 t, x2 t , C ∈ R ,
if P V t, x1 t, x2 t ≥ V t θ, x1 t θ, x2 t θ , θ ∈ −τ, 0.
2.7
If system 2.6 has a solution ξt : ξt ≤ H < H ∗ , t ≥ t0 , then system 2.6 has a unique
positive almost periodic solution ηt which is uniformly asymptotically stable, and modη ⊂ modg.
Furthermore, if g is ω-periodic with respect to t, then system 2.6 has a positive ω-periodic solution
which is globally asymptotically stable.
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Discrete Dynamics in Nature and Society
Here, modφ denotes the module of φt which is the set consisting of all real numbers
which are finite linear combinations of elements of the set
Λ
1
α ∈ R | lim
T →∞T
T
φt exp−iαtdt /
0
2.8
0
with integer coefficients.
Lemma 2.3. R4 {x1 , x2 , x3 , x4 | xi > 0, i 1, 2, 3, 4} is a positive invariant set of system 2.1.
Proof. Let x1 , x2 , x3 , x4 be a solution of system 2.1 with initial conditions 2.2. Hence, for
t ∈ R and x1 , x2 , x3 , x4 ∈ R4 , we can derive
x1 |x1 0 D1 tx2 > 0 for x2 > 0,
x2 |x2 0 D2 tx1 > 0 for x1 > 0,
t
x3 > x3 0 exp
− a30 s − a32 sx3 s − a34 sx4 s ds > 0,
0
x4 > x4 0 exp
t
− a40 s − a42 sx4 s − a43 sx3 s ds
2.9
> 0.
0
Therefore, we obtain the positive invariance of R4 . This completes the proof.
We will focus our discussion on R4 with respect to a biological meaning. This also
ensures the solution with positive initial value to be positive all the time.
3. Uniform persistence
In what follows, we want to construct an ultimately bounded region of system 2.1.
Theorem 3.1. There exist three constants Mi > Mi∗ i 1, 2, 3 such that xj t ≤ M1 j 1, 2,
x3 t ≤ M2 , and x4 t ≤ M3 for each positive solution x1 t, x2 t, x3 t, x4 t of system 2.1 with
t large enough, where
M1∗
max
a10 a20
,
,
a11 a21
α1
.
A a31
− a30 > 0,
β1
M2∗ A
,
a32
M3∗ B
,
a42
α2
.
B a41
− a40 > 0.
β2
3.1
3.2
Proof. Suppose that x1 t, x2 t, x3 t, x4 t is a solution of system 2.1 with initial conditions
2.2. According to the first two equations of 2.1, we have
x1 ≤ a10 x1 − a11 x12 D1 t x2 − x1 ,
x2 ≤ a20 x2 − a21 x22 D2 t x1 − x2 .
3.3
Qiong Liu
5
We define the following lines in x1 -x2 plane:
Line L1 : x1 M1 ,
0 ≤ x2 ≤ M1 ,
Line L2 : x2 M1 ,
0 ≤ x1 ≤ M1 .
3.4
Then, we have
x1 |L1 < 0,
x2 |L2 < 0.
3.5
Hence, it follows from
max x1 0, x2 0 ≤ M1
3.6
that
max x1 t, x2 t ≤ M1
for t ≥ 0.
3.7
If
max x1 0, x2 0 > M1 ,
3.8
we only consider what follows. If xi > M1 , i 1, 2, from the given condition we get
ai0 xi − ai1 xi2 < M1 ai0 − ai1 M1 < 0,
i 1, 2.
3.9
Let
.
−α max M1 ai0 − ai1 M1 ,
i1,2
gt max x1 t, x2 t .
3.10
Next, we consider the following three cases.
Case 1. x1 0 > x2 0, g0 x1 0 > M1 . Then, there exists ε > 0 such that gt x1 t > M1
for t ∈ 0, ε. We also derive that
x1 ≤ a10 x1 − a11 x12 < −α < 0.
3.11
Hence, if t2 > t1 and t1 , t2 ∈ 0, ε, we get
g t2 − g t1 < −α t2 − t1 .
3.12
Case 2. x2 0 > x1 0, g0 x2 0 > M1 . Similarly, we could obtain that there exists 0, ε. If
t2 > t1 and t1 , t2 ∈ 0, ε, we get
g t2 − g t1 < −α t2 − t1 .
3.13
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Discrete Dynamics in Nature and Society
Case 3. x2 0 x1 0 g0 > M1 . We can also find an interval 0, ε such that gt x1 t >
M1 or gt x2 t > M1 . In the same way, if t2 > t1 and t1 , t2 ∈ 0, ε, we can obtain
g t2 − g t1 < −α t2 − t1 .
3.14
Now, we can know that if g0 > M1 , gt will monotonously decrease by speed α. So, there
exists T1 > 0. If t ≥ T1 , we have
gt < M1 .
According to the third equation of 2.1, we have
α1 tx12 t − τ1
x3 ≤ x3 − a30 a31
− a32 x3
1 β1 tx12 t − τ1
α1
− a32 x3 ,
< x3 − a30 a31
β1
α1
− a32 x3 .
x3 x3 M2 < x3 − a30 a31
β1
3.15
3.16
Hence, it follows from x3 0 ≤ M2 that x3 t ≤ M2 for t ≥ 0.
If
x3 0 > M2 ,
we only consider what follows. If x3 > M2 , from the given condition we obtain
α1
α1
− a32 x3 < M2 − a30 a31
− a32 M2 < 0.
x3 − a30 a31
β1
β1
Let
We also derive that
3.17
3.18
α1
−β M2 − a30 a31
− a32 M2 .
β1
3.19
α1
− a32 M2 −β < 0.
x3 < M2 − a30 a31
β1
3.20
Hence, if t2 > t1 and t1 , t2 ∈ 0, ε, we get
x3 t2 − x3 t1 < −β t2 − t1 .
3.21
Now, we can know that if x3 0 > M2 , x3 t will monotonously decrease by speed β. So, there
exists T2 such that x3 t < M2 for t ≥ T2 . Similarly, we also get
α2 tx12 t − τ2
x4 ≤ x4 − a40 a41
−
a
x
42 4
1 β2 tx12 t − τ2
3.22
α2
< x4 − a40 a41
− a42 x4 .
β2
We can also choose the same M3 . There exists T3 > 0 such that x4 t < M3 for t > T3 . This
completes the proof.
Qiong Liu
7
Theorem 3.2. Suppose that system 2.1 satisfies the following conditions:
a10 − D1 > 0,
a20 − D2 > 0,
2
α1 α2
Mx > 0,
E a10 − D1 − 4a11
β1 β2
a31
α1 m21
1 β1 m21
a41
α2 m21
1 β2 m21
− a30 − a34 Mx > 0,
− a40 − a43 Mx > 0
in which
m1 3.23
a10 − D1 2a11
√
E
.
3.24
Then, system 2.1 is uniformly persistent.
Proof. Suppose x1 , x2 , x3 , x4 is a solution of system 2.1 with the initial condition 2.2.
According to the first equation of 2.1, we get
α1 tx12 tx3 t α2 tx12 tx4 t
−
x1 t ≥ x1 t a10 t − D1 t −
1 β1 tx12 t
1 β2 tx12 t
≥
−a11 tx12 t
α1 tMx α2 tMx
a10 t − D1 t x1 t −
−
.
β1 t
β2 t
3.25
So,
lim inf x1 t ≥ m1 > 0.
3.26
x1 t ≥ m1
3.27
t→∞
Then, there exists a T5 > 0 such that
for t ≥ T5 .
Similarly,
. a20 − D2
> 0.
lim inf x2 t ≥ m2 t→∞
a21
3.28
Then, there exists a T6 > 0 such that
x2 t ≥ m2
for t ≥ T6 .
3.29
From the third equation of 2.1, we obtain
α1 tm21
x3 t ≥ x3 t − a30 t a31 t
−
a
tx
t
−
a
tM
32
3
34
x .
1 β1 tm21
3.30
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Discrete Dynamics in Nature and Society
So,
2
2
. a31 α1 m1 / 1 β1 m1 − a30 − a34 Mx
lim inf x3 t ≥ m3 > 0.
t→∞
a32
3.31
Then, there exists a T7 > 0 such that
x3 t ≥ m3
for t ≥ T7 .
3.32
Similarly, we also get
2
2
. a41 α2 m1 / 1 β2 m1 − a40 − a43 Mx
lim inf x4 t ≥ m4 > 0.
t→∞
a42
3.33
Then, there exists a T8 > 0 such that
x4 t ≥ m4
for t ≥ T8 .
3.34
Finally, let
D
x1 , x2 , x3 , x4 | mx < xi < Mx , i 1, 2, 3, 4 ,
3.35
where mx mini1,2,3,4 {mi } and Mx max{M1∗ , M2∗ , M3∗ }; Mi∗ i 1, 2, 3 is given in
Theorem 3.1. From Theorem 3.1 and the above analysis, we see that D is a bounded compact
region in R4 which has positive distance from coordinate hyperplanes. Let T max{Ti , i 1, . . . , 8}, then we obtain that if t > T , then every positive solution of system 2.1 with initial
conditions 2.2 eventually enters and remains in the region D. This completes the proof.
4. Almost periodic solution
In this section, we derive sufficient conditions which guarantee that the periodic solution of
periodic system 2.2 is globally attractive.
Theorem 4.1. In addition to 2.5, 3.2, and 3.23, assume further that all the coefficients of system
2.1 are continuous and positive almost periodic functions and
D1 m2
α1 m3
α2 m4
a11 m1
M12
1 β1 M12 1 β2 M12
>
2α1 β1 M12 M3 2α2 β2 M12 M4 D2
Mx2 2α1 a31 M1
2α2 a41 M1
M
,
1
2
2 m
mx 1 β1 m2 2 1 β2 m2 2
2
1 β1 m21
1 β2 m21
1
1
D2 m1 Mx2 2α1 a31 M1
D1
2α2 a41 M1
a21 m
,
>
M
2
2
m1
mx 1 β1 m2 2 1 β2 m2 2
M22
1
1
Mx2 2α1 a31 M1
α1 M1
2α2 a41 M1
a
M
,
43
3
2
2
mx
1 β1 m21
1 β1 m21
1 β2 m21
Mx2 2α1 a31 M1
α2 M1
2α2 a41 M1
a
M
.
34
3
mx 1 β1 m2 2 1 β2 m2 2
1 β2 m21
1
1
a32 m3 >
a42 m4 >
4.1
Qiong Liu
9
Then, system 2.1 has a unique positive almost periodic solution which is globally asymptotically
stable. Furthermore, if system 2.1 is an ω-periodic system, then system 2.1 has a positive ω-periodic
solution which is globally asymptotically stable.
Proof. Consider the product system of 2.1:
α2 tx12 x4
α1 tx12 x3
x1 x1 a10 t − a11 tx1 −
−
D1 t x2 − x1 ,
2
2
1 β1 tx1 1 β2 tx1
x2 x2 a20 t − a21 tx2 D2 t x1 − x2 ,
α1 tx12 t − τ1
x3 x3 − a30 t a31 t
− a32 tx3 − a34 tx4 ,
1 β1 tx12 t − τ1
α2 tx12 t − τ2
x4 x4 − a40 t a41 t
− a42 tx4 − a43 tx3 ,
1 β2 tx12 t − τ2
y1
y1 a10 t − a11 ty1 −
α1 ty12 y3
α2 ty12 y4
4.2
−
D1 t y2 − y1 ,
1 β1 ty12 1 β2 ty12
y2 y2 a20 t − a21 ty2 D2 t y1 − y2 ,
α1 ty12 t − τ1
−
a
y3 y3 − a30 t a31 t
ty
−
a
ty
32
3
34
4 ,
1 β1 ty12 t − τ1
α2 ty12 t − τ2
y4 y4 − a40 t a41 t
−
a
ty
−
a
ty
42
4
43
3 .
1 β2 ty12 t − τ2
It is easily noted that the existence and uniqueness of the positive almost periodic solution
of system 2.1 are equivalent to the existence and uniqueness of the positive almost periodic
solution of system 4.2. Then, choose the following function:
4 ln xi t − ln yi t.
V t V t, xi , yi 4.3
i1
Obviously, V t satisfies conditions i and ii of Lemma 2.2. Next, we will prove that V t
satisfies condition iii of Lemma 2.2. It follows that
x1 y1
α1 y1 y3
α2 y1 y4
x2 y 2
α1 x1 x3
α2 x1 x4
−
−
−
−a11 x1 − y1 − D1
−
−
−
x1 y 1
x1 y 1
1 β1 x12 1 β1 y12
1 β2 x12 1 β2 y12
4.4
in which
α1 y1 y3
α1 x1 x3
−
2
1 β1 x1 1 β1 y12
α1 β1 y1 y3 x1 y1
α1 y1 α1 x3
−
x3 − y 3 ;
x1 − y 1 2
2
2
2
1 β1 x1
1 β1 x1
1 β1 x1 1 β1 y1
4.5
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Discrete Dynamics in Nature and Society
also,
x3 y3
−
x3 y 3
x4 y4
−
x4 y 4
D2 D2 y1 x1 − y 1 ,
x2 − y 2 x2 y 2
x1
α1 x1 t − τ1 y1 t − τ1
a31 x1 t − τ1 − y1 t − τ1
2
2
1 β1 x1 t − τ1 1 β1 y1 t − τ1
− a32 x3 − y3 − a34 x4 − y4 ,
α2 x1 t − τ2 y1 t − τ2
a41 x1 t − τ2 − y1 t − τ2
2
2
1 β1 x1 t − τ2 1 β1 y1 t − τ2
− a42 x4 − y4 − a43 x3 − y3 .
x2 y2
−
x2 y 2
− a21 −
4.6
In this regard, after few computations, it is noted that
4
xi t yi t
sgn xi t − yi t
−
D V t, xi , yi xi t yi t
i1
α1 β1 y1 y3 x1 y1
D1 y2
α2 x4
α1 x3
− a11 −
−
−
x1 y 1
1 β1 x12
1 β2 x12
1 β1 x12 1 β1 y12
α2 β1 y1 y4 x1 y1
D2 y1 x2 − y 2 x1 − y1 − a21 −
2
2
x
y
1 β2 x1 1 β2 y1
2 2
D1 x2 − y2
− a32 x3 − y3 − a42 x4 − y4 sgn x1 − y1
x1
α1 y1 α2 y1 − sgn x1 − y1 x3 − y3 − sgn x1 − y1 x4 − y 4
2
2
1 β1 x1
1 β2 x1
D2 x1 − y1
sgn x2 − y2
x2
a31 α1 x1 t − τ1 y1 t − τ1
sgn x3 − y3 x1 t − τ1 − y1 t − τ1
2
2
1 β1 x1 t − τ1 1 β1 y1 t − τ1
− a34 sgn x3 − y3 x4 − y4
a41 α2 x1 t − τ2 y1 t − τ2
sgn x4 − y4 x1 t − τ2 − y1 t − τ2
2
2
1 β1 x1 t − τ2 1 β1 y1 t − τ2
− a43 sgn x4 − y4 x3 − y3
D1 m2
α1 m3
α2 m4
2α1 β1 M12 M3
−
≤ − a11 2
M12
1 β1 M12 1 β2 M12
1 β1 m21
D2 m1 D1 2α2 β2 M12 M4 D2 x2 − y2 x
−
a
− −
−
y
−
1
1
21
2
2
m2
m1
M2
1 β2 m21
Qiong Liu
11
α1 M1
α2 M1
x4 − y4 x
−
a
− a32 a
−
y
a
43
3
3
42
34
2
2
1 β1 m1
1 β2 m1
2α1 a31 M1 2α2 a41 M1 x1 t − τ1 − y1 t − τ1 x1 t − τ2 − y1 t − τ2 .
2
2
1 β1 m21
1 β2 m21
4.7
It follows from 4.1 that
4 4 1 xi − yi ≤ V t, xi , yi ≤ 1
xi − yi .
Mx i1
mx i1
Choose P s Mx /mx s > s > 0, as 1/Mx s > 0, bs 1/mx s > 0. When
P V t, xi t, yi t ≥ V t θ, xi t θ, yi t θ , θ ∈ −τ, 0, i 1, 2, 3, 4,
x1 t − τ − y1 t − τ ≤ Mx ln x1 t − τ − ln y1 t − τ
≤ Mx V t − τ, xi t − τ, yi t − τ
≤ Mx
4.8
4.9
Mx V t, xi t, yi t ;
mx
then
Mx2 2α1 a31 M1 2α1 a31 M1 ≤
x
t
−
τ
−
y
t
−
τ
1
1
1
1
2 V t, xi t, yi t ,
2 2
2
m
x 1 β1 m
1 β1 m1
1
Mx2 2α2 a41 M1 2α2 a41 M1 x1 t − τ2 − y1 t − τ2 ≤
V t, xi t, yi t .
2
2
mx 1 β2 m2
1 β2 m21
1
Hence,
D V t, xi , yi ≤
4.10
D1 m2
α1 m3
α2 m4 m1
− a11 M12
1 β1 M12 1 β2 M12
2α1 β1 M12 M3 2α2 β2 M12 M4 D2
ln x1 − ln y1 M
x
2 2
2 2
m2
1 β1 m1
1 β2 m1
D2 m1 D1
ln x2 − ln y2 m
− a21 M
2
2
2
m1
M2
α1 M1
ln x3 − ln y3 − a32 m3 a
M
43
3
2
1 β1 m1
α2 M1
ln x4 − ln y4 M
a
− a42 m4 34
4
2
1 β2 m1
Mx2
mx
2α1 a31 M1
2α2 a41 M1
V t, xi t, yi t ≤ −CV t, xi t, yi t ,
2
2
1 β1 m21
1 β2 m21
4.11
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Discrete Dynamics in Nature and Society
where
−C max
D1 m2
α1 m3
α2 m4 − a11 m1
M12
1 β1 M12 1 β2 M12
2α1 β1 M12 M3 2α2 β2 M12 M4 D2
Mx2 2α1 a31 M1
2α2 a41 M1
M
x
2 2 m
mx 1 β1 m2 2 1 β2 m2 2
2
1 β1 m21
1 β2 m21
1
1
D2 m1 Mx2 2α1 a31 M1
D1
2α2 a41 M1
M
− a21 m
2
2
m1
mx 1 β1 m2 2 1 β2 m2 2
M22
1
1
Mx2 2α1 a31 M1
α1 M1
2α2 a41 M1
a
M
43
3
2
2
mx
1 β1 m21
1 β1 m21
1 β2 m21
Mx2 2α1 a31 M1
α2 M1
2α2 a41 M1
M
.
a
34
4
mx 1 β1 m2 2 1 β2 m2 2
1 β2 m21
1
1
− a32 m3 − a42 m4 4.12
This completes the proof.
5. Discussion
In this work, we consider a nonautonomous delayed predator-prey model with competition
and diffusion. Some sufficient conditions on uniform persistence of the model have been given.
By means of the Liapunov-Razumikhin technique, it is also seen that, under almost periodic
circumstances, the existence and uniqueness of the positive almost periodic solution which is
globally asymptotically stable are governed by several inequalities.
Acknowledgments
The author is thankful to the learned referees for their valuable comments which have helped
to present a better exposition of the paper. This work is supported by the first project proposals
of Guangxi education teaching reform in the 11th five-year plan 2005240, and the project
of qualified course reform and establishment of the new century teaching reform in the 11th
five-year plan 2006072.
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