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Hindawi Publishing Corporation
Boundary Value Problems
Volume 2009, Article ID 654539, 19 pages
doi:10.1155/2009/654539
Research Article
Global Behavior for a Diffusive Predator-Prey
Model with Stage Structure and Nonlinear Density
Restriction-II: The Case in
R
1
Rui Zhang,
1, 2
Ling Guo,
1
and Shengmao Fu
1
1
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
2
Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China
Correspondence should be addressed to Shengmao Fu,
Received 2 April 2009; Accepted 31 August 2009
Recommended by Wenming Zou
A Holling type III predator-prey model with self- and cross-population pressure is considered.
Using the energy estimate and Gagliardo-Nirenberg-type inequalities, the existence and uniform
boundedness of global solutions to the model are dicussed. In addition, global asymptotic stability
of the positive equilibrium point for the model is proved by Lyapunov function.
Copyright q 2009 Rui Zhang et al. 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
This paper is a continuation of Part I 1. In Section 3 of Part I, using the energy estimate and
bootstrap arguments, the global existence of solutions for a Holling type III cross-diffusion
predator-prey model with stage-structure has been discussed when the space dimension be
less than 6. However, to obtain the L
∞
estimate for the population density w of predator
species, there is not cross-diffusion for w in Part I.
All diffusive predator-prey systems behave, more or less, in the same way, for both
semilinear and cross-diffusive models, at least for small values of the cross diffusivities.
Consequently, all the available information for linear diffusive models is essential to realize
the behavior of the most complicated cross-diffusive systems 2–17.
In this paper, we consider the following cross-diffusion system:
u
t
du α
11
u
2
α
12
uv α
13
uw
xx
βv − au − bu
2
− cu
3
−
u
2
w
1 u
2
,
v
t
dv α
21
uv α
22
v
2
α
23
vw
xx
u − v, 0 <x<1,t>0,
2 Boundary Value Problems
w
t
d
3
w α
31
uw α
32
vw α
33
w
2
xx
− kw − γw
2
αu
2
w
1 u
2
,
u
x
x, t
v
x
x, t
w
x
x, t
0,x 0, 1,t>0,
u
x, 0
u
0
x
,v
x, 0
v
0
x
,w
x, 0
w
0
x
, 0 <x<1,
1.1
where d, d
3
,α
ij
i, j 1, 2, 3,α,β,γ,a,b,c,and k are positive constants. Also, d,d
3
are linear
diffusion coefficients of u, v, w, respectively, while α
ii
i 1, 2, 3 are referred as self-diffusion
pressures, and α
ij
i
/
j,i, j 1, 2, 3 are cross-diffusion pressures. If α
12
α
21
α
23
α
31
α
32
0, then 1.1 reduces to the system 1.4 of Part I.
Recently, the work in 18–20 studied the existence, uniform boundedness, and
uniform convergence of global solutions for the Lotka-Volterra cross-diffusion models
without stage-structure in the case that the space dimension n 1. In this paper, we
consider mainly the existence and uniform boundedness of global solutions for the model
1.1 with nonlinear density restriction and stage-structure. Moreover, global asymptotic
stability of the positive equilibrium point for 1.1 is proved by an important lemma
of 21. The proof is complete and complement the uniform convergence theorem in
18–20.
2. Global Existence and Uniform Boundedness
For simplicity, denote |·|
k,p
·
W
k
p
0,1
, |·|
p
·
L
p
0,1
. The local existence result of
solutions to 1.1 is an immediate consequence of a series of papers 22, 23 by Amann.
Roughly speaking, if u
0
,v
0
,w
0
∈ W
1
p
0, 1,p > 1, then 1.1 has a unique nonnegative
solution u, v, w ∈ C0,T,W
1
p
0, 1
C
∞
0,T,C
∞
0, 1, where T ≤ ∞ is the maximal
existence time for the solution. If u, v, w satisfies
sup
|
u
·,t
|
1,p
,
|
v
·,t
|
1,p
,
|
w
·,t
|
1,p
:0<t<T
< ∞, 2.1
then T ∞. If, in addition, u
0
,v
0
,w
0
∈ W
2
p
0, 1, then u, v, w ∈ C0, ∞,W
2
p
0, 1.
The main result in this section is as follows.
Theorem 2.1. Let u
0
,v
0
,w
0
∈ W
2
2
0, 1, u, v, w is the unique nonnegative solution of 1.1 in its
maximal existence interval 0,T. Assume that
8α
11
α
21
α
31
>α
21
α
2
13
α
2
12
α
31
,
8α
12
α
22
α
32
>α
32
α
2
21
α
2
23
α
12
,
8α
13
α
23
α
33
>α
23
α
2
31
α
2
32
α
13
.
2.2
Boundary Value Problems 3
Then there exists t
0
> 0 and positive constants M, M
which depend on d, d
3
,α
ij
i, j 1, 2, 3,
β, a, b, c, k, γ, α, such that
sup
|
u
·,t
|
1,2
,
|
v
·,t
|
1,2
,
|
w
·,t
|
1,2
: t ∈
t
0
,T
≤ M
, 2.3
max
{
u
x, t
,v
x, t
,w
x, t
:0≤ x ≤ 1,t
0
≤ t<T
}
≤ M, 2.4
and T ∞. In particular, if d, d
3
≥ 1,d
3
/d ∈ d, d,whered ≤ 1 and d are positive constants,
then M
,Mdepend on d, d, but do not depend on d, d
3
≥ 1.
The following Gagliardo-Nirenberg-type inequalities and corresponding corollary
play an importance role in the proof of Theorem 2.1.
Theorem 2.2 see 18. Let Ω ⊂ R
n
be a bounded domain with ∂Ω ∈ C
m
. For every function
u ∈ W
m
r
Ω, 1 ≤ q, r ≤∞, the derivative D
j
u 0 ≤ j<m satisfies the inequality
D
j
u
p
≤ C
|
D
m
u
|
a
r
|
u
|
1−a
q
|
u
|
q
, 2.5
provided one of the following three conditions is satisfied: 1 r ≤ q, 2 0 <nr − q/mrq < 1,or
3 nr −q/mrq 1, and m −n/q is not a nonnegative integer, where 1/p j/na1/r −m/n
1 − a/q, for all a ∈ j/m,1, and the positive constant C depends on n, m, j, q, r, a.
Corollary 2.3. There exists a positive constant C such that
|
u
|
2
≤ C
|
u
x
|
1/3
2
|
u
|
2/3
1
|
u
|
1
, ∀u ∈ W
1
2
0, 1
, 2.6
|
u
|
4
≤ C
|
u
x
|
1/2
2
|
u
|
1/2
1
|
u
|
1
, ∀u ∈ W
1
2
0, 1
, 2.7
|
u
|
7/2
≤ C
|
u
x
|
10/21
2
|
u
|
11/21
1
|
u
|
1
, ∀u ∈ W
1
2
0, 1
, 2.8
|
u
x
|
2
≤ C
|
u
xx
|
3/5
2
|
u
|
2/5
1
|
u
|
1
, ∀u ∈ W
2
2
0, 1
. 2.9
For simplicity, denote that C is Sobolev embedding constant or other kind of absolute constant.
A
j
,B
j
,C
j
are some positive constants which depend on α
ij
i, j 1, 2, 3, β, a, b, c,k, γ, α.Also,K
j
are positive constants which depend on α
ij
i, j 1, 2, 3, β, a, b, c, k, γ, α, d, d
3
.Whend, d
3
≥ 1, K
j
do not depend on d, d
3
, but on d, d.
Proof of Theorem 2.1
Step 1. Estimate |u|
1
, |v|
1
,|w|
1
. Firstly, taking integration of the first and second equations
in 2.7 over the domain 0, 1, respectively, and combining the two integration equalities
4 Boundary Value Problems
linearly, we have
d
dt
1
0
u
a β
v
dx ≤−a
1
0
vdx
1
0
βu − bu
2
dx. 2.10
From Young inequality and H
¨
older inequality, we can see
d
dt
1
0
u
a β
v
dx ≤ C
1
−
a
a β
1
0
u
a β
v
dx, 2.11
where C
1
1/4bβ a/a β
2
. From which it follows that there exists a constant τ
0
> 0,
such that
1
0
udx,
1
0
vdx ≤ M
0
,t≥ τ
0
, 2.12
where M
0
2C
1
a β/a max{a β
−1
, 1}.
Secondly, taking integration of the third equations in 2.7 over domain 0, 1, we have
d
dt
1
0
wdx ≤
α − k
1
0
wdx − γ
1
0
wdx
2
. 2.13
This implies that there exists a constant τ
0
> 0, such that
1
0
wdx ≤
2
|
α − k
|
γ
,t≥ τ
0
. 2.14
Let M
1
max{M
0
, 2|α − k|/γ}, τ
1
max{τ
0
, τ
0
}. Then
1
0
udx,
1
0
vdx,
1
0
wdx ≤ M
1
,t≥ τ
1
. 2.15
Moreover, there exists a positive constant M
1
which depends on β, a, b, c, k, γ, α and the L
1
-
norm of u
0
,v
0
,w
0
, such that
1
0
udx,
1
0
vdx,
1
0
wdx ≤ M
1
,t≥ 0. 2.15
Boundary Value Problems 5
Step 2. estimate |u|
2
, |v|
2
and |w|
2
. Multiplying the first three inequalities of Corollary 2.3 by
u, v, w, respectively, and integrating over 0, 1, we have
1
2
d
dt
1
0
u
2
dx
≤−d
1
0
u
2
x
dx −
1
0
2α
11
u α
12
v α
13
w
u
2
x
α
12
uu
x
v
x
α
13
uu
x
w
x
dx β
1
0
uvdx,
1
2
d
dt
1
0
v
2
dx
≤−d
1
0
v
2
x
dx −
1
0
α
21
u 2α
22
v α
23
w
v
2
x
α
21
vu
x
v
x
α
23
vv
x
w
x
dx
1
0
uvdx,
1
2
d
dt
1
0
w
2
dx
≤−d
3
1
0
w
2
x
dx −
1
0
α
31
u α
32
v 2α
33
w
w
2
x
α
31
wu
x
w
x
α
32
wv
x
w
x
dx.
2.16
Let d
min{d, d
3
}. By the above three inequalities and Young inequality, we have
1
2
d
dt
1
0
u
2
v
2
w
2
dx
≤−d
1
0
u
2
x
v
2
x
w
2
x
dx −
1
0
q
u
x
,v
x
,w
x
dx
β 1
2
α
1
0
u
2
v
2
w
2
dx,
2.17
where
q
u
x
,v
x
,w
x
2α
11
u α
12
v α
13
w
u
2
x
α
21
u 2α
22
v α
23
w
v
2
x
α
31
u α
32
v 2α
33
w
w
2
x
α
12
u α
21
v
u
x
v
x
α
13
u α
31
w
u
x
w
x
α
23
v α
32
w
v
x
w
x
2.18
is quadratic form of u
x
,v
x
,w
x
. It is not hard to verify that qu
x
,v
x
,w
x
is positive definite if
2.2 holds. Moreover, if 2.2 holds, then
1
2
d
dt
1
0
u
2
v
2
w
2
d ≤−d
1
0
u
2
x
v
2
x
w
2
x
dx
β 1
2
α
1
0
u
2
v
2
w
2
dx.
2.19
Now we proceed in the following two cases.
i It holds that t ≥ τ
1
.By2.6 and 2.15, we have
1
0
u
2
x
dx ≥ 1/CM
4
1
1
0
u
2
dx
3
−M
2
1
,
6 Boundary Value Problems
and
−d
1
0
u
2
x
v
2
x
w
2
x
dx ≤ 3d
M
2
1
− C
2
d
1
0
u
2
v
2
w
2
dx
3
. 2.20
By 2.19 and 2.20, we can see that
1
2
d
dt
1
0
u
2
v
2
w
2
dx
≤−C
2
d
1
0
u
2
v
2
w
2
dx
3
β 1
2
α
1
0
u
2
v
2
w
2
dx 3d
M
2
1
.
2.21
Thus, there exists positive constants τ
2
>τ
1
and M
2
depending on d, d
3
,β,a,b,c,k,γ,α, such
that
1
0
u
2
dx,
1
0
v
2
dx,
1
0
w
2
dx ≤ M
2
,t≥ τ
2
. 2.22
Since the zero point of the right-hand side in 2.21 can be estimated by positive constants
independent of d
, when d
≥ 1. Thus M
2
do not depend on d
≥ 1.
ii t ≥ 0. Repeating estimates in i by 2.9
, we can obtain that there exists a positive
constant M
2
depending on d, d
3
,β,a,b,c,k,γ,αand the L
1
, L
2
-norm of u
0
,v
0
,w
0
, such that
1
0
u
2
dx,
1
0
v
2
dx,
1
0
w
2
dx ≤ M
2
,t≥ 0, 2.22
when d
≥ 1, M
1
is independent of d
.
Step 3. Estimate |u
x
|
2
, |v
x
|
2
,|w
x
|
2
. Introduce the scaling that
u
u
d
1
, v
v
d
1
, w
w
d
1
,
t d
1
t, 2.23
denote η d
3
/d, and redenote u, v, w,
t by u, v, w, t, respectively. Then 2.7 reduces to
u
t
P
xx
f
u, v, w
, 0 <x<1,t>0,
v
t
Q
xx
g
u, v, w
, 0 <x<1,t>0,
w
t
R
xx
h
u, v, w
, 0 <x<1,t>0,
u
x
x, t
v
x
x, t
w
x
x, t
0,x 0, 1,t>0,
u
x, 0
u
0
x
,v
x, 0
v
0
x
,w
x, 0
w
0
x
, 0 <x<1,
2.24
Boundary Value Problems 7
where P uα
11
u
2
α
12
uv α
13
uw, Q v α
21
uv α
22
v
2
α
23
vw, R ηw α
31
uw α
32
vw
α
33
w
2
, fu, v, wβd
−1
v − ad
−1
u − bu
2
− cdu
3
− du
2
w/1 d
2
u
2
, gu, v, wd
−1
u − v,
hu, v, w−kd
−1
w − rw
2
αdu
2
w/1 d
2
u
2
. We still proceed in following two cases.
i It holds that t ≥ τ
∗
2
dτ
2
.From2.15 and 2.22, we can easily obtain that
1
0
udx,
1
0
vdx,
1
0
wdx ≤ M
1
d
−1
,
1
0
u
2
dx,
1
0
v
2
dx,
1
0
w
2
dx ≤ M
2
d
−2
,
|
P
|
1
,
|
Q
|
1
,
|
R
|
1
≤ DK
1
d
−1
,
2.25
where K
1
2 ηM
2
d
−2
, D max {M
1
,α
11
α
12
α
13
,α
21
α
22
α
23
,α
31
α
32
α
33
}.
Multiply the first three equations in 2.24 by P
t
,Q
t
,R
t
and integrate them over 0, 1,
respectively, then adding up the three new equations, we have
1
2
y
t
≤−
1
0
u
2
t
dx −
1
0
v
2
t
dx − η
1
0
w
2
t
dx −
1
0
q
u
t
,v
t
,w
t
dx
1
0
1 2α
11
u α
12
v α
13
w
u
t
f α
12
uv
t
f α
13
uw
t
f
dx
1
0
α
21
vu
t
g
1 α
21
u 2α
22
v α
23
w
v
t
g α
23
vw
t
g
dx
1
0
α
31
wu
t
h α
32
wv
t
h
η α
31
u α
32
v 2α
33
w
w
t
h
dx,
2.26
where
y
1
0
P
2
x
Q
2
x
R
2
x
dx. It is not hard to verify by 2.4 that there exists a positive
constant C
3
depending only on α
ij
i, j 1, 2, 3, such that
q
u
t
,v
t
,w
t
≥ C
3
u v w
u
2
t
v
2
t
w
2
t
. 2.27
Thus,
1
2
y
t
≤−
1
0
u
2
t
dx −
1
0
v
2
t
dx − η
1
0
w
2
t
dx − C
3
1
0
u v w
u
2
t
v
2
t
w
2
t
dx
1
0
1 2α
11
u α
12
v α
13
w
u
t
fdx
1
0
1 α
21
u 2α
22
v α
23
w
v
t
gdx
1
0
η α
31
u α
32
v 2α
33
w
w
t
hdx
1
0
α
12
uv
t
fdx
1
0
α
13
uw
t
fdx
1
0
α
21
vu
t
gdx
1
0
α
23
vw
t
gdx
1
0
α
31
wu
t
hdx
1
0
α
32
wv
t
hdx.
2.28
8 Boundary Value Problems
Using Young inequality, H
¨
older inequality and 2.24, we can obtain the following estimates:
1
0
u
3
dx ≤
1
0
u
7
dx
1/5
1
0
u
2
dx
4/5
≤ M
4/5
2
d
−8/5
1
0
u
7
dx
1/5
,
1
0
u
4
dx ≤
1
0
u
7
dx
2/5
1
0
u
2
dx
3/5
≤ M
3/5
2
d
−6/5
1
0
u
7
dx
2/5
,
1
0
u
5
dx ≤
1
0
u
7
dx
3/5
1
0
u
2
dx
2/5
≤ M
2/5
2
d
−4/5
1
0
u
7
dx
3/5
,
1
0
u
6
dx ≤
1
0
u
7
dx
4/5
1
0
u
2
dx
1/5
≤ M
1/5
2
d
−2/5
1
0
u
7
dx
4/5
,
1
0
uvdx ≤
1
0
u
2
dx
1/2
1
0
v
2
dx
1/2
≤ M
2
d
−2
,
1
0
u
2
vdx ≤
1
0
u
7
dx
1/5
1
0
u
2
dx
3/10
1
0
v
2
dx
1/2
≤ M
4/5
2
d
−8/5
1
0
u
7
dx
1/5
,
1
0
u
3
vdx ≤
1
0
u
7
dx
2/5
1
0
u
2
dx
1/10
1
0
v
2
dx
1/2
≤ M
3/5
2
d
−6/5
1
0
u
7
dx
2/5
,
1
0
u
6
vdx ≤
6
7
1
0
u
7
dx
1
7
1
0
v
7
dx ≤
6
7
1
0
u
7
v
7
dx,
1
0
u
4
vdx ≤
1
2
1
0
u
2
vdx
1
2
1
0
u
6
vdx ≤
1
2
M
4/5
2
d
−8/5
1
0
u
7
dx
1/5
3
7
1
0
u
7
v
7
dx,
1
0
uu
t
dx ≤
1
2
1
0
udx
2
1
0
uu
2
t
dx ≤
1
2
M
1
d
−1
2
1
0
uu
2
t
dx,
1
0
u
2
u
t
dx ≤
1
2
1
0
u
3
dx
2
1
0
uu
2
t
dx ≤
1
2
M
4/5
2
d
−8/5
1
0
u
7
dx
1/5
2
1
0
uu
2
t
dx,
1
0
u
3
u
t
dx ≤
1
2
1
0
u
5
dx
2
1
0
uu
2
t
dx ≤
1
2
M
2/5
2
d
−4/5
1
0
u
7
dx
3/5
2
1
0
uu
2
t
dx,
1
0
u
4
u
t
dx ≤
1
2
1
0
u
7
dx
2
1
0
uu
2
t
dx,
1
0
uvu
t
dx ≤
1
2
1
0
u
2
vdx
2
1
0
vu
2
t
dx ≤
1
2
M
4/5
2
d
−8/5
1
0
u
7
dx
1/5
2
1
0
vu
2
t
dx,
1
0
u
2
vu
t
dx ≤
1
2
1
0
u
4
vdx
2
1
0
vu
2
t
dx
Boundary Value Problems 9
≤
1
4
M
4/5
2
d
−8/5
1
0
u
7
dx
1/5
3
14
1
0
u
7
v
7
dx
2
1
0
vu
2
t
dx,
1
0
u
3
vu
t
dx ≤
1
2
1
0
u
6
vdx
2
1
0
vu
2
t
dx ≤
3
14
1
0
u
7
v
7
dx
2
1
0
vu
2
t
dx. 2.29
Applying the above estimates and Gagliardo-Nirenberg-type inequalities to the terms
on the right-hand side of 2.28, we have
−
1
0
u
2
t
dx ≤−
1
2
1
0
P
2
xx
dx
1
0
f
2
dx,
−
1
0
v
2
t
dx ≤−
1
2
1
0
Q
2
xx
dx
1
0
g
2
dx,
−η
1
0
w
2
t
dx ≤−
η
2
1
0
R
2
xx
dx η
1
0
h
2
dx,
1
0
f
2
dx ≤ β
2
d
−2
1
0
v
2
dx a
2
d
−2
1
0
u
2
dx b
2
1
0
u
4
dx 2bcd
2
1
0
u
5
dx c
2
d
4
1
0
u
6
dx
2abd
−1
1
0
u
3
dx 2ac
1
0
u
4
dx d
−2
1
0
w
2
dx
2ad
−2
1
0
uwdx 2bd
−1
1
0
u
2
wdx 2
1
0
u
3
wdx
≤
a
2
β 1 2a
M
2
d
−4
2b
a 1
M
3/5
2
d
−13/5
1
0
u
7
dx
1/5
2ac b
2
2
M
3/5
2
d
−6/5
1
0
u
7
dx
2/5
2bcM
2/5
2
d
1/5
1
0
u
7
dx
3/5
c
2
M
1/5
2
d
8/5
1
0
u
7
dx
4/5
,
1
0
g
2
dx ≤ d
−2
1
0
u
2
v
2
dx ≤ 2M
2
d
−4
,
η
1
0
h
2
dx ≤ d
−2
η
k
2
α
2
1
0
w
2
dx 2kd
−1
γη
1
0
w
3
dx γ
2
η
1
0
w
4
dx
≤ η
k
2
α
2
M
2
d
−4
2kγηM
4/5
2
d
−13/5
1
1
0
w
7
dx
1/5
γ
2
ηM
3/5
2
d
−6/5
1
0
w
7
dx
2/5
.
2.30
10 Boundary Value Problems
Thus
−
1
0
u
2
t
dx −
1
0
v
2
t
dx − η
1
0
w
2
t
dx
≤−
1
2
1
0
P
2
xx
dx −
1
2
1
0
Q
2
xx
dx −
η
2
1
0
R
2
xx
dx C
4
2 η
M
2
d
−4
C
5
d
−1
1 η
M
4/5
2
d
−8/5
1
0
u
7
w
7
dx
1/5
C
6
1 η
M
3/5
2
d
−6/5
1
0
u
5
w
7
dx
2/5
C
7
M
2/5
2
d
1/5
1
0
u
7
dx
3/5
C
8
M
1/5
2
d
8/5
1
0
u
7
dx
4/5
.
2.31
For the other terms on the right-hand side of 2.28, we have
1
0
u
t
fdx ≤ βd
−1
1
0
u
t
vdx
ad
−1
1
0
uu
t
dx
b
1
0
u
2
u
t
dx
cd
1
0
u
3
u
t
dx
d
−1
1
0
wu
t
dx
≤
β
2
a
2
1
2
M
1
d
−3
b
2
2
M
4/5
2
d
−8/5
1
0
u
7
dx
1/5
c
2
2
M
2/5
2
d
6/5
1
0
u
7
dx
3/5
3
2
1
0
uu
2
t
dx
βd
−1
2
1
0
vu
2
t
dx
1
2
1
0
wu
2
t
dx,
2α
11
1
0
uu
t
fdx ≤ 2α
11
βd
−1
1
0
uu
t
vdx
2α
11
ad
−1
1
0
u
2
u
t
dx
2α
2
11
b
1
0
u
3
u
t
dx
2α
11
dc
1
0
u
4
u
t
dx
2α
11
d
−1
1
0
uu
t
wdx
≤
α
2
11
β
2
a
2
1
M
4/5
2
d
−18/5
1
0
u
7
dx
1/5
α
2
11
b
2
3
M
2/5
2
d
−4/5
1
0
u
7
dx
3/5
α
2
11
c
2
d
2
1
0
u
7
dx 3
1
0
uu
2
t
dx
1
0
vu
2
t
dx
1
0
wu
2
t
dx,
Boundary Value Problems 11
α
12
1
0
vu
t
fdx ≤ α
12
βd
−1
1
0
u
t
v
2
dx
α
12
ad
−1
1
0
uvu
t
dx
α
2
12
b
1
0
u
2
vu
t
dx
α
12
dc
1
0
u
3
vu
t
dx
α
12
d
−1
1
0
vwu
t
dx
≤
α
2
12
2
a
2
d
−2
b
2
2
M
4/5
2
d
−8/5
1
0
u
7
dx
1/5
α
2
12
β
2
1
2
M
4/5
2
d
−18/5
1
0
v
7
dx
1/5
3α
2
12
7
b
2
2
c
2
d
2
1
0
u
7
v
7
dx
5
2
1
0
vu
2
t
dx,
α
13
1
0
wu
t
fdx ≤ α
13
βd
−1
1
0
vwu
t
dx
α
13
ad
−1
1
0
uwu
t
dx
α
2
13
b
1
0
u
2
wu
t
dx
α
13
dc
1
0
u
3
wu
t
dx
α
13
d
−1
1
0
w
2
u
t
dx
≤
α
2
13
a
2
d
−2
b
2
/2
2
M
4/5
2
d
−8/5
1
0
u
7
dx
1/5
α
2
13
2
M
4/5
2
d
−18/5
⎡
⎣
β
1
0
v
7
dx
1/5
1
0
w
7
dx
1/5
⎤
⎦
3α
2
13
7
b
2
2
c
2
d
2
1
0
u
7
w
7
dx
5
2
1
0
wu
2
t
dx,
1
0
v
t
gdx ≤ d
−1
1
0
uv
t
dx
d
−1
1
0
vv
t
dx
≤
M
1
d
−3
2
1
0
uv
2
t
vv
2
t
dx,
α
21
1
0
uv
t
gdx ≤ α
21
d
−1
1
0
u
2
v
t
dx
α
21
d
−1
1
0
uvv
t
dx
≤
α
2
21
M
4/5
2
d
−18/5
1
0
u
7
dx
1/5
2
1
0
uv
2
t
vv
2
t
dx,
2α
22
1
0
vv
t
gdx ≤ 2α
22
d
−1
1
0
uvv
t
dx
2α
22
d
−1
1
0
v
2
v
t
dx
≤
α
22
1
0
vv
2
t
dx
M
4/5
2
d
−18/5
⎡
⎣
1
0
u
7
dx
1/5
1
0
v
7
dx
1/5
⎤
⎦
1
0
vv
2
t
dx,
α
23
1
0
wv
t
gdx ≤ α
23
d
−1
1
0
uwv
t
dx
α
23
d
−1
1
0
vwv
t
dx
12 Boundary Value Problems
≤
α
23
1
0
vv
2
t
dx
2
M
4/5
2
d
−18/5
⎡
⎣
1
0
u
7
dx
1/5
1
0
v
7
dx
1/5
⎤
⎦
1
0
wv
2
t
dx,
η
1
0
w
t
hdx ≤ d
−1
η
α k
1
0
ww
t
dx
γη
1
0
w
2
w
t
dx
≤
α k
2
2
η
2
d
−3
M
1
γ
2
2
η
2
M
4/5
2
d
−8/5
1
0
w
7
dx
1/5
η
1
0
ww
2
t
dx,
α
31
1
0
uw
t
hdx ≤
α k
d
−1
α
31
1
0
uww
t
dx
α
31
γ
1
0
uw
2
w
t
dx
≤
α
2
31
2
α k
2
d
−2
γ
2
2
M
4/5
2
d
−8/5
⎡
⎣
1
0
u
7
dx
1/5
1
0
w
7
dx
1/5
⎤
⎦
3α
2
31
γ
2
14
1
0
u
7
w
7
dx
1
2
1
0
uw
2
t
dx
1
2
1
0
ww
2
t
dx,
α
32
1
0
vw
t
hdx ≤
α k
d
−1
α
32
1
0
vww
t
dx
α
32
γ
1
0
vw
2
w
t
dx
≤
α
2
32
2
α k
2
d
−2
γ
2
2
M
4/5
2
d
−8/5
⎡
⎣
1
0
v
7
dx
1/5
1
0
w
7
dx
1/5
⎤
⎦
3α
2
32
14
γ
2
1
0
v
7
w
7
dx
1
2
1
0
vw
2
t
dx
1
2
1
0
ww
2
t
dx,
2α
33
1
0
ww
t
hdx ≤ 2
α k
d
−1
α
33
1
0
w
2
w
t
dx
2α
33
γ
1
0
w
3
w
t
dx
≤
α k
2
α
2
33
M
4/5
2
d
−18/5
1
0
w
7
dx
1/5
α
2
33
γ
2
M
2/5
2
d
−4/5
1
0
w
7
dx
3/5
1
0
ww
2
t
dx,
α
12
1
0
uv
t
fdx ≤ α
12
βd
−1
1
0
uvv
t
dx
α
12
ad
−1
1
0
u
2
v
t
dx
α
12
b
1
0
u
3
v
t
dx
α
12
cd
1
0
u
4
v
t
dx
α
12
d
−1
1
0
uwv
t
dx
≤
α
2
12
β
2
a
2
1
2
M
4/5
2
d
−18/5
1
0
u
7
dx
1/5
α
2
12
b
2
2
M
2/5
2
d
−4/5
1
0
u
7
dx
3/5
Boundary Value Problems 13
α
2
12
c
2
2
d
2
1
0
u
7
dx
2
3
1
0
uv
2
t
dx
1
0
vv
2
t
dx
1
0
wv
2
t
dx
,
α
13
1
0
uw
t
fdx ≤ α
13
βd
−1
1
0
uvw
t
dx
α
13
ad
−1
1
0
u
2
w
t
dx
α
13
b
1
0
u
3
w
t
dx
α
3
13
cd
1
0
u
4
w
t
dx
α
13
d
−1
1
0
uww
t
dx
≤
α
2
13
β
2
a
2
1
2
M
4/5
2
d
−18/5
1
0
u
7
dx
1/5
α
2
13
b
2
2
M
2/5
2
d
−4/5
1
0
u
7
dx
3/5
α
2
13
c
2
2
d
2
1
0
u
7
dx
3
2
1
0
uw
2
t
dx
1
2
1
0
vw
2
t
dx
1
2
1
0
ww
2
t
dx,
α
21
1
0
vu
t
gdx ≤ α
21
d
−1
1
0
uvu
t
dx
α
21
d
−1
1
0
v
2
u
t
dx
≤
α
2
21
2
M
4/5
2
d
−18/5
⎡
⎣
1
0
u
7
dx
1/5
1
0
v
7
dx
1/5
⎤
⎦
1
0
vu
2
t
dx,
α
23
1
0
vw
t
gdx ≤ α
23
d
−1
1
0
uvw
t
dx
α
23
d
−1
1
0
v
2
w
t
dx
≤
α
2
23
2
M
4/5
2
d
−18/5
⎡
⎣
1
0
u
7
dx
1/5
1
0
v
7
dx
1/5
⎤
⎦
1
0
vw
2
t
dx,
α
31
1
0
wu
t
hdx ≤ α
31
α k
d
−1
1
0
w
2
u
t
dx
α
31
γ
1
0
w
3
u
t
dx
≤
α k
2
α
2
31
2
M
4/5
2
d
−18/5
1
0
w
7
dx
1/5
α
2
31
2
γ
2
M
2/5
2
d
−4/5
1
0
w
7
dx
3/5
1
0
wu
2
t
dx,
α
32
1
0
wv
t
hdx ≤ α
32
α k
d
−1
1
0
w
2
v
t
dx
α
32
γ
1
0
w
3
v
t
dx
≤
α
2
k
2
α
2
32
2
M
4/5
2
d
−18/5
1
0
w
7
dx
1/5
α
2
32
γ
2
2
M
2/5
2
d
−4/5
1
0
w
7
dx
3/5
1
0
wv
2
t
dx. 2.32
14 Boundary Value Problems
Thus
1
0
1 2α
11
α
12
v α
13
u
t
fdx
1
0
1 α
21
u 2α
22
v α
23
w
v
t
gdx
1
0
η α
31
u α
32
v 2α
33
w
w
t
hdx
1
0
α
12
uv
t
fdx
1
0
α
13
uw
t
fdx
1
0
α
21
vu
t
gdx
1
0
α
23
vw
t
gdx
1
0
α
31
wu
t
hdx
1
0
α
32
wv
t
hdx
≤ λ
1
0
u v w
u
2
t
v
2
t
w
2
t
dx
C
9
M
1
d
−3
2 η
2
C
10
M
4/5
2
d
−8/5
2 d
−2
η
2
1
0
u
7
v
7
w
7
dx
1/5
C
11
M
4/5
2
d
−8/5
1 d
2
1
0
u
7
v
7
w
7
dx
3/5
C
12
1 d
2
1
0
u
7
v
7
w
7
dx,
2.33
where λ is a positive constant.
Note by 2.8 and 2.9 that |P|
7/2
7/2
≤ C|P
x
|
5/3
2
|P|
11/6
1
|P |
7/2
1
, |P
x
|
10/3
2
≤
B
1
K
4/3
1
d
−4/3
|P
xx
|
2
2
K
2
1
d
−2
,and
−
1
2
1
0
P
2
xx
dx −
1
2
1
0
Q
2
xx
dx −
η
2
1
0
R
2
xx
dx ≤−B
2
min
1,η
K
−4/3
1
d
4/3
y
5/3
K
2
1
d
−2
2 η
. 2.34
Choose a small enough number >0, such that λ < C
3
. According to 2.28–2.34, we have
1
2
y
t
≤−A
1
min
1,η
K
−4/3
1
y
5/3
A
2
K
2
y
1/6
A
3
K
3
y
1/3
A
4
K
4
y
1/2
A
5
K
5
y
2/3
A
6
K
6
y
5/6
A
7
K
7
,
2.35
where y
1
0
dP
x
2
dQ
x
2
dR
x
2
dx.
However, 2.35 implies that there exist positive constants τ
3
> 0and
M
3
depending
on d, d
3
,α
ij
i, j 1, 2, 3, β, a, b, c,k, γ, α, such that
1
0
dP
x
2
dx,
1
0
dQ
x
2
dx,
1
0
dR
x
2
dx ≤
M
3
,t≥ τ
3
. 2.36
Boundary Value Problems 15
When d, d
3
≥ 1,η ∈ d, d, the coefficients of 2.35 can be estimated by constants depending
on d
, d,butnotond, d
3
. Thus, when d, d
3
≥ 1,η ∈ d, d,
M
3
depends on α
ij
i, j
1, 2, 3,β,a,b,c,k,γ,α,d
, d, and is irrelevant to d, d
3
≥ 1. Since
⎛
⎜
⎜
⎝
P
x
Q
x
R
x
⎞
⎟
⎟
⎠
⎛
⎜
⎜
⎝
P
u
P
v
P
w
Q
u
Q
v
Q
w
R
u
R
v
R
w
⎞
⎟
⎟
⎠
⎛
⎜
⎜
⎝
u
x
v
x
w
x
⎞
⎟
⎟
⎠
, 2.37
similar to 2.26 in 24, we have
|
du
x
|
|
dv
x
|
|
dw
x
|
≤ D
|
dP
x
|
|
dQ
x
|
|
dR
x
|
, 0 <x<1,t>0, 2.38
where D is a positive constant only depending on η, α
ij
i, j 1, 2, 3. Scaling back with 2.22
to original variable u, v, w, t and combining 2.36,2.38, there exist positive constants τ
3
> 0
and M
3
depending on d, d
3
,α
ij
i, j 1, 2, 3, β, a, b, c,k, γ, α, such that
1
0
u
2
x
dx,
1
0
v
2
x
dx,
1
0
w
2
x
dx ≤ M
3
,t≥ τ
3
. 2.39
In addition, when d, d
3
≥ 1,η ∈ d, d, M
3
is dependent of d, d, but independent of d, d
3
≥ 1.
ii It holds that t ≥ 0. Replacing M
1
,M
2
with M
1
,M
2
in 2.24–2.34, we can obtain
that there exists a positive constant M
3
depending on d, d
3
,α
ij
i, j 1, 2, 3, β, a, b, c, k, γ, α
and the W
1
2
-norm of u
0
,v
0
,w
0
such that
1
0
u
2
x
dx,
1
0
v
2
x
dx,
1
0
w
2
x
dx ≤ M
3
,t≥ 0. 2.39
When d, d
3
≥ 1,η ∈ d, d, M
3
is dependent of d, d, but independent of d, d
3
≥ 1.
Concluding from 2.15, 2.22, 2.39,andSobolev embedding theorem, there exists
a positive constants t
0
> 0, M, M
depending on d, d
3
,α
ij
i, j 1, 2, 3, β, a, b, c, k, γ, α, such
that 2.3 and 2.4 are satisfied. Furthermore, when d, d
3
≥ 1, η ∈ d, d and the time t is
large enough, M, M
are dependent of α
ij
i, j 1, 2, 3, β, a, b, c,k, γ, α, d, d, but independent
of d, d
3
≥ 1.
Similarly, according to 2.15
, 2.22
, 2.39
, we can see that there exists a positive
constant M
depending on d, d
3
,α
ij
i, j 1, 2, 3, β, a, b, c, k, γ, α and the initial functions
u
0
,v
0
,w
0
, such that
|
u
·,t
|
1,2
,
|
v
·,t
|
1,2
,
|
w
·,t
|
1,2
≤ M
,t≥ 0. 2.40
16 Boundary Value Problems
When d,d
3
≥ 1, η ∈ d, d, M
is dependent of d, d, but independent of d, d
3
.ThusT ∞.
This completes proof of Theorem 2.1.
3. Global Stability
From 1, we know that if
α>k, β>a,
k
α − k
<m
0
,
β − a − c
2
b
2
8c
≤
b
√
p
1
24c
24
β − a
c
2
3b
2
4c
β − a − c
− b
√
p
1
,
H
where p
1
9b
2
24cβ − a − c ≥ 0, then 1.1 has the unique position equilibrium point
E
∗
u
∗
,v
∗
,w
∗
.
Theorem 3.1. Assume that all conditions in Theorem 2.1 and H are satisfied. Assume further that
1
β
a bu
∗
cu
∗2
> 2
u
∗2
√
1 u
∗2
2
8
u
∗4
2
,
γ
α
>
1
2
1 u
∗2
2
, 3.1
4
αβ
w
∗
d
2
d
3
>
1
β
α
23
M
2
1
α
α
32
w
∗
2
d 2α
11
M α
12
M α
13
M
1
β
α
13
M
2
1
α
α
31
w
∗
2
d α
21
M 2α
22
M α
23
M
1
α
1
β
α
12
α
21
2
M
2
w
∗
d
3
α
31
M α
32
M 2α
33
M
3.2
hold, where M is the positive constant in 2.4. Then the unique positive equilibrium point E
∗
of 1.1
is globally asymptotically stable.
Remark 3.2. Since M is independent of d, d
3
in the case of d, d
3
≥ 1, 3.2 is always satisfied if
d and d
3
are big enough.
Proof. Define the Lyapunov function
H
u, v, w
1
2β
1
0
u − u
∗
2
dx
1
2
1
0
v − v
∗
2
dx
1
α
1
0
w − w
∗
− w
∗
ln
w
w
∗
dx. 3.3
Boundary Value Problems 17
Let u, v, w be any solution of 1.1 with initial functions u
0
x,v
0
x,w
0
x ≥
/
≡0.
From the strong maximum principle for parabolic equations, it is not hard to verify that
u, v, w > 0fort>0. Thus
dH
dt
≤−
1
0
1
β
d 2α
11
u α
12
v α
13
w
u
2
x
d α
21
u 2α
22
v α
23
w
v
2
x
1
α
d
3
α
31
u α
32
v 2α
33
w
w
∗
w
2
w
2
x
1
β
α
12
u α
21
v
u
x
v
x
1
β
α
13
u
1
α
α
31
w
∗
w
u
x
w
x
α
23
v
1
α
α
32
w
∗
w
v
x
w
x
dx
−
1
0
u − u
∗
2
1
β
a b
u u
∗
c
u
2
uu
∗
u
∗2
w
u u
∗
1 u
2
1 u
∗2
−2 −
1
2
u u
∗
1 u
2
− u
∗2
2
dx −
1
2
1
0
v − v
∗
2
dx
−
1
0
w − w
∗
2
γ
α
−
1
2
1 u
∗2
2
dx.
3.4
The first integrand in the right hand of the above inequality is positive definite if
4
αβ
w
∗
d 2α
11
u α
12
v α
13
w
d α
21
u 2α
22
v α
23
w
d
3
α
31
u α
32
v 2α
33
w
w
2
1
β
α
12
u α
21
v
1
α
α
13
u
1
α
α
31
w
∗
w
α
23
v βα
32
w
∗
w
>
1
β
α
23
vw
1
α
α
32
w
∗
2
d 2α
11
u α
12
v α
13
w
1
β
α
13
uw
1
α
α
31
w
∗
2
d α
21
u 2α
22
v α
23
w
1
α
w
∗
1
β
α
12
u α
21
v
2
d
3
α
31
u α
32
v 2α
33
w
.
3.5
From the maximum-norm estimate in Theorem 2.1, 3.2 is a sufficient condition of 3.5.
Thus when 3.1 holds, there exists a positive constant δ such that
dH
u, v, w
dt
≤−δ
1
0
u − u
∗
2
v − v
∗
2
w − w
∗
2
dx. 3.6
18 Boundary Value Problems
By integration by parts, H
¨
older inequality and the maximum-norm estimate in
Theorem 2.1, we can see that d/dt
1
0
u − u
∗
2
v − v
∗
2
w − w
∗
2
dx is bounded from
above. According to Lemma 3.1in1 and 3.6,weobtain
|
u
·,t
− u
∗
|
2
−→ 0,
|
v
·,t
− v
∗
|
2
−→ 0,
|
w
·,t
− w
∗
|
2
−→ 0,
t −→ ∞
. 3.7
Using Gagliardo-Nirenberg inequalities, we have |u·,t|
∞
≤ C|u|
1/2
1,2
|u|
1/2
2
.Thus
|
u
·,t
− u
∗
|
∞
−→ 0,
|
v
·,t
− v
∗
|
∞
−→ 0,
|
w
·,t
− w
∗
|
∞
−→ 0,
t −→ ∞
. 3.8
That is, u, v, w converges uniformly to E
∗
. Since Hu, v, w is decreasing for t>0, E
∗
is
globally asymptotically stable.
Acknowledgments
This work has been partially supported by the China National Natural Science Foundation
no. 10871160, the NSF of Gansu Province no. 096RJZA118, the Scientific Research Fund of
Gansu Provincial Education Department, and the NWNU-KJCXGC-03-47 Foundation.
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