Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 183285, 25 pages
doi:10.1155/2012/183285
Research Article
Global Behavior for a Strongly Coupled
Predator-Prey Model with One Resource and Two
Consumers
Yujuan Jiao1, 2 and Shengmao Fu1
1
2
College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
College of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou 730124,
China
Correspondence should be addressed to Shengmao Fu, fusm@nwnu.edu.cn
Received 23 March 2012; Accepted 15 May 2012
Academic Editor: Michiel Bertsch
Copyright q 2012 Y. Jiao and S. Fu. 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.
We consider a strongly coupled predator-prey model with one resource and two consumers, in
which the first consumer species feeds on the resource according to the Holling II functional
response, while the second consumer species feeds on the resource following the BeddingtonDeAngelis functional response, and they compete for the common resource. Using the energy
estimates and Gagliardo-Nirenberg-type inequalities, the existence and uniform boundedness
of global solutions for the model are proved. Meanwhile, the sufficient conditions for global
asymptotic stability of the positive equilibrium for this model are given by constructing a
Lyapunov function.
1. Introduction
The principle of competitive exclusion asserts that two or more consumer species cannot
coexist indefinitely upon a single limiting resource, which dates back to the pioneering work
of Volterra 1 in the 1920s. Subsequently, Ayala 2 in 1969 demonstrated experimentally that
two species of Drosophila can coexist upon a single limiting resource. Ayala’s experiments
have received much attention see the comprehensive survey by Cantrell and Cosner
3. Schoener 4 in 1976 found that intraspecific interference among consumers may
lead to coexistence of multiple consumer species upon a single resource. To examine
more closely the implications of feeding interference among conspecific consumers on
2
Abstract and Applied Analysis
consumer-resource dynamics, Cantrell et al. 5 in 2004 proposed the following predator-prey
system:
u
auv
Auw
du
ru 1 −
−
−
,
dt
K
1 bu 1 Bu Dw
dv
eu v −l ,
dt
1 bu
dw
Eu
w −L ,
dt
1 Bu Dw
1.1
where r, a, b, e, l, A, B, D, E, L, and K are positive constants, ut represents the density of the
limiting resource at time t, vt and wt denote two consumers species. The first consumer
species feeds upon the resource according to the Holling II functional response, while
the second consumer species feeds upon the resource following the Beddington-DeAngelis
functional response, and they compete for the common resource. For more details on the
backgrounds about this system see 5.
The system 1.1 has a positive equilibrium E u, v , w
under the suitable conditions,
where
l
,
e − bl
u
v e
u
A
ru
1−
−
u − L ,
E − BL
al
K
DE
w
u−L
E − BL
. 1.2
DL
The Jacobian matrix of the system 1.1 at E can be written as
⎞
⎛
a11 a12 a13
J E ⎝a21 a22 a23 ⎠,
a31 a32 a33
1.3
where
abv
ru
ABw
,
u
−
K 1 b
u2 1 Bu
Dw
2
a11
a13 −
a33 −
A
u1 Bu
Dw
1 Bu
2
DE
uw
Dw
2
1 Bu
a12 −
ev
< 0,
a21 < 0,
a22 a23 a32 0.
2
u
1 b
> 0,
a
u
< 0,
1 b
u
a31 Ew1
Dw
Dw
2
1 Bu
> 0,
1.4
The following results were proved in 5:
1 the system 1.1 is dissipative;
2 the positive equilibrium E of 1.1 is locally stable if −a11 a33 > 0 and a11 a11 a33 −
a13 a31 − a12 a21 − a33 a11 a33 − a31 a31 < 0; and
3 the positive equilibrium E of 1.1 is globally stable if max{b, B}K − u
< 1.
Abstract and Applied Analysis
3
BK −→ B,
–
–
–
–
–
–
L
−→ L,
r
A
−→ A,
r
–
eK
−→ e,
l
bK −→ b,
–
l
−→ l,
r
a
−→ a,
r
EK
−→ E
L
–
D −→ D,
rt −→ t,
–
w −→ w,
–
v −→ v,
–
u
−→ u,
K
–
Rescaling the system 1.1 such that
1.5
yields
du
auv
Auw
u1 − u −
−
,
dt
1 bu 1 Bu Dw
dv
eu lv −1 ,
dt
1 bu
dw
Eu
Lw −1 .
dt
1 Bu Dw
1.6
The corresponding weakly coupled reaction-diffusion system for 1.6 is as follows:
Auw
auv
−
, x ∈ Ω, t > 0,
1 bu 1 Bu Dw
eu , x ∈ Ω, t > 0,
vt d2 Δv lv −1 1 bu
Eu
wt d3 Δw Lw −1 , x ∈ Ω, t > 0,
1 Bu Dw
ut d1 Δu u1 − u −
∂ν u ∂ν v ∂ν w 0,
ux, 0 u0 x,
1.7
x ∈ ∂Ω, t > 0,
vx, 0 v0 x,
wx, 0 w0 x,
x ∈ Ω,
where Ω ⊂ RN is a bounded smooth domain, ν is the outward unit normal vector of the
boundary ∂Ω, ∂ν ∂/∂ν. The constants d1 , d2 , and d3 , called diffusion coefficients, are
positive, and u0 x, v0 x, and w0 x are nonnegative functions which are not identically
zero. The system 1.7 has a constant positive steady-state solution E∗ u∗ , v∗ , w∗ if and
only if
e > b,
E − B > e − b,
DEe − b − 1 − Ae − bE − B − e − b > 0,
1.8
where
u∗ 1
,
e−b
v∗ e{DEe − b − 1 − Ae − bE − B − e − b}
E − B − e − b
w∗ .
De − b
aDEe − b2
,
1.9
4
Abstract and Applied Analysis
In 6, Hei and Yu proved the following main results.
1 The equilibrium u∗ , v∗ , w∗ of 1.7 is locally asymptotically stable if 1.8 and
DE2 be − b − 1 − e Ae − bBe − bEE − B − e − b < 0
1.10
hold.
2 Let d2∗ and d3∗ be fixed positive constants which satisfy d2∗ μ1 ≥ le − b − 1/1 b
≥ LE − B − 1/1 B.
and
Then there exists a positive constant D1 , such that 1.7 has no nonconstant positive
solution if d1 > D1 , d2 ≥ d2∗ and d3 ≥ d3∗ , where 0 μ0 < μ1 < μ2 < · · · are the eigenvalues of
the operator −Δ on Ω with the homogeneous Neumann boundary condition.
3 Let di i 1, 3 be fixed positive constants. Assume that a∗11 > 0,
d3∗ μ1
4d2 − 4eld1 − ed2 DE − B − AE − B − 1
min
,
4d2
DE − B
>
1
e−b
1.11
and 1.8 hold. Furthermore, assume that one of the following conditions is satisfied:
i a∗11 a∗33 − a∗31 a∗13 < 0, μ̌ ∈ μn , μn1 for some n ≥ 1, and the sum σn ni1 dim Eμi is odd;
∈ μk , μk1 , μ̌ ∈ μn , μn1 for some n ≥ k ≥ 1, and the sum
ii a∗11 a∗33 − a∗31 a∗13 > 0, μ
σn nik1 dim Eμi is odd.
Then there exists a positive constant D2 , such that 1.7 has at least one nonconstant positive
solution if d2 ≥ D2 , where
a∗11 −u∗ a∗33 −
a∗13 −
a∗31 μ̌ μ
abv∗ ABw∗
2 ∗ ,
e 2 u∗
E u
DELu∗ w∗
1 Bu∗ Dw∗ 2
,
Aw∗
,
Eu∗
ELw∗ 1 Dw∗ 1 Bu∗
Dw∗ 2
a∗11 d3 a∗33 d1 a∗11 d3 a∗33 d1 −
1.12
,
a∗11 d3 − a∗33 d1
2
4a∗31 a∗13 d1 d3
2d1 d3
a∗11 d3 − a∗33 d1
2
,
4a∗31 a∗13 d1 d3
2d1 d3
and Eμi is the eigenspace corresponding to μi in H 1 Ω.
4 The bifurcation of nonconstant positive solutions for 1.7 was studied.
In recent years, the SKT type cross-diffusion systems have attracted the attention of
a great number of investigators and have been successfully developed on the theoretical
Abstract and Applied Analysis
5
backgrounds. The above work mainly concentrate on 1 the instability and stability induced
by cross-diffusion, and the existence of nonconstant positive steady-state solutions 7–14;
2 the global existence of strong solutions 15–23; 3 the global existence of weak solutions
based on semidiscretization or finite element approximation 24–30; and 4 the dynamical
behaviors 18, 19, 31, 32, and so forth. The corresponding SKT type cross-diffusion system
for 1.7 is as follows:
Auw
auv
−
, x ∈ Ω, t > 0,
ut Δ d1 u α11 u2 α12 uv α13 uw u1 − u −
1 bu 1 Bu Dw
eu , x ∈ Ω, t > 0,
vt Δ d2 v α21 uv α22 v2 α23 vw lv −1 1 bu
Eu
wt Δ d3 w α31 uw α32 vw α33 w2 Lw −1 , x ∈ Ω, t > 0,
1 Bu Dw
∂ν u ∂ν v ∂ν w 0,
ux, 0 u0 x,
x ∈ ∂Ω, t > 0,
vx, 0 v0 x,
wx, 0 w0 x,
x ∈ Ω,
1.13
where αij i, j 1, 2, 3 are positive constants, αii i 1, 2, 3 are referred as self-diffusion
pressures, and αij i, j 1, 2, 3, i / j are cross-diffusion pressures. The self-diffusion implies
the movement of individuals from a higher to lower concentration region. Cross-diffusion
expresses the population fluxes of one species due to the presence of the other species.
The value of cross-diffusion coefficient may be positive, negative, or zero. The positive
cross-diffusion coefficient denotes the movement of the species in the direction of lower
concentration of another species and negative cross-diffusion coefficient denotes that one
species tends to diffuse in the direction of higher concentration of another species e.g., 33.
The local existence of solutions for the system 1.13 is an immediate consequence of a
series of important papers 34–36 by Amann. Roughly speaking, if u0 x, v0 x, and w0 x in
Wp1 Ω with p > n, then 1.13 has a unique nonnegative solution u, v, w ∈ C0, T , Wp1 Ω ∩
C∞ 0, T , C∞ Ω, where T ∈ 0, ∞ is the maximal existence time for the solution. If the
solution u, v, w satisfies the estimate
sup u·, tWp1 Ω , v·, tWp1 Ω , w·, tWp1 Ω : 0 < t < T < ∞,
1.14
then T ∞. Moreover, if u0 x, v0 x, w0 x ∈ Wp2 Ω, then u, v, w ∈ C0, ∞, Wp2 Ω.
For the following SKT system
ut d1 Δ 1 αv γu u au1 − u − cv,
vt d2 Δ1 δvv bv1 − du − v,
∂ν u ∂ν v 0,
ux, 0 u0 x,
x ∈ Ω, t > 0,
x ∈ ∂Ω, t > 0,
vx, 0 v0 x,
x ∈ Ω, t > 0,
x ∈ Ω,
P
6
Abstract and Applied Analysis
Yamada in 23 proposed four open problems:
1 the global existence of solutions of P in the case δ > 0 and the space dimension
N ≥ 6;
2 the global existence in the case γ 0;
3 in order to study the asymptotic behavior of u, v as t → ∞ need to establish the
uniform boundedness of global solutions; and
4 the global existence of solutions for the following full SKT system:
ut d1 Δ 1 αv γu u au1 − u − cv,
vt d2 Δ 1 βu δv v bv1 − du − v,
∂ν u ∂ν v 0,
x ∈ Ω, t > 0,
x ∈ Ω, t > 0,
x ∈ ∂Ω, t > 0,
vx, 0 v0 x,
ux, 0 u0 x,
1.15
x∈Ω
with α, γ, β, δ > 0.
Very few global existence results for 1.13 are known. The main purpose of this paper
is to establish the uniform boundedness of global solutions for the system 1.13 in one space
dimension. For convenience, we consider the following system:
ut d1 u α11 u2 α12 uv α13 uw
Auw
auv
−
, 0 < x < 1, t > 0,
xx
1 bu 1 Bu Dw
eu , 0 < x < 1, t > 0,
vt d2 v α21 uv α22 v2 α23 vw
lv −1 xx
1 bu
Eu
, 0 < x < 1, t > 0,
wt d3 w α31 uw α32 vw α33 w2
Lw −1 xx
1 Bu Dw
ux x, t vx x, t wx x, t 0,
ux, 0 u0 x,
u1 − u −
x 0, 1, t > 0,
vx, 0 v0 x,
wx, 0 w0 x,
0 < x < 1.
1.16
We firstly investigate the global existence and the uniform boundedness of the solutions
for 1.16, then prove the global asymptotic stability of the positive equilibrium u∗ , v∗ , w∗ of 1.16 by an important lemma from 37. The proof is complete and complement to the
uniform convergence theorems in papers 38–40.
It is obvious that u∗ , v∗ , w∗ is the unique positive equilibrium of the system 1.16 if
1.8 holds.
For simplicity, we denote · Wpk 0,1 by | · |k,p and · Lp 0,1 by | · |p . Our main results are
as follows.
Abstract and Applied Analysis
7
Theorem 1.1. Let u0 x, v0 x, w0 x ∈ W22 0, 1, u, v, w be the unique nonnegative solution of
the system 1.16 in the maximal existence interval 0, T . Assume that
8α11 α21 α31 > α21 α213 α212 α31 ,
8α12 α22 α32 > α32 α221 α223 α12 ,
1.17
8α13 α23 α33 > α23 α231 α232 α13 .
Then there exist t0 > 0 and positive constants M
, M which depend on di , αij i, j
1, 2, 3, a, b, e, l, A, B, D, E and L, such that
sup |u·, t|1,2 , |v·, t|1,2 , |w·, t|1,2 : t ∈ t0 , T ≤ M
,
1.18
max{ux, t, vx, t, wx, t : x, t ∈ 0, 1 × t0 , T } ≤ M,
1.19
and T ∞. Moreover, in the case that d1 , d2 , d3 ≥ 1, d2 /d1 , d3 /d1 ∈ d, d, where d and d are
positive constants, M
, M depend on d, d, but do not depend on d1 , d2 , d3 .
Remark 1.2. Since the continuous embedding H 1 Ω → L∞ Ω holds only in one space
dimension, we can only establish the uniform maximum-norn estimates about time for the
solution in one space dimension.
Theorem 1.3. Assume that all conditions in Theorem 1.1 are satisfied. Assume further that
2
4αβu∗ v∗ w∗ d1 d2 d3 > u∗ M2 αα23 v∗ βα32 w∗ d1 2α11 M α12 M α13 M
2
αv∗ M2 α13 u∗ βα31 w∗ d2 α21 M 2α22 M α23 M
1.20
βw∗ M2 α12 u∗ αα21 v∗ 2 d3 α31 M α32 M 2α33 M,
DEe − b − be − b − 1 > Ae − bE − B − e − bB − b,
1.21
and 1.8 hold, where α a1 bu∗ /el, β A1 Bu∗ /EL1 Dw∗ , M is given by 1.19.
Then the unique positive equilibrium u∗ , v∗ , w∗ of 1.16 is globally asymptotically stable.
Remark 1.4. The system 1.16 has no nonconstant positive steady-state solution if all
conditions of Theorem 1.3 hold.
8
Abstract and Applied Analysis
Examples. The following two examples satisfy all conditions of Theorem 1.3:
α11 1
,
4
α12 1,
α13 1,
α21 5
,
2
α22 1,
α23 α31 1,
α32 1,
α33 2,
b 1,
B
e 3,
1
,
2
D
1
,
4
A 1,
2
,
3
E 3,
d1 , d2 , d3 1;
α11 2,
α21 3
,
2
α31 2,
α12 1,
α22 1
,
2
α32 2,
5
,
2
b
1
,
2
e
B
1
,
2
D
3
,
4
α13 2,
1.22
α23 1,
α33 3,
A
2
,
3
E 4,
d1 , d2 , d3 1.
2. Global Solutions
In order to establish the uniform W21 -estimates of the solutions for the system 1.16,
the following Gagliardo-Nirenberg-type inequalities and the corresponding corollary play
important roles see 38, 41.
Theorem 2.1. Let Ω ⊂ Rn be a bounded domain with ∂Ω ∈ Cm . For every function u ∈ Wrm Ω, 1 ≤
q, r ≤ ∞, the derivative Dj u 0 ≤ j < m satisfies the inequality
j a
|u|q ,
D u ≤ C |Dm u|r |u|1−a
q
p
2.1
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/n a1/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.
Abstract and Applied Analysis
9
Corollary 2.2. There exists a universal constant C such that
2/3
|u|2 ≤ C |ux |1/3
2 |u|1 |u|1 ,
∀u ∈ W21 0, 1,
2.2
∀u ∈ W21 0, 1,
2.3
1/2
|u|4 ≤ C |ux |1/2
2 |u|1 |u|1 ,
3/5
|u|5/2 ≤ C |ux |2/5
2 |u|1 |u|1 ,
∀u ∈ W21 0, 1,
2.4
2/5
|ux |2 ≤ C |uxx |3/5
2 |u|1 |u|1 ,
∀u ∈ W22 0, 1.
2.5
Throughout this paper, we always denote that C is a Sobolev embedding constant or
other kind of universal constant, Aj , Bj , Cj are some positive constants which depend only on
αij i, j 1, 2, 3, a, b, e, l, A, B, D, E and L, Kj are positive constants depending on di , αij i, j 1, 2, 3, a, b, e, l, A, B, D, E and L. When d1 , d2 , d3 ≥ 1, d2 /d1 , d3 /d1 ∈ d, d, Kj depend on d, d,
but do not depend on d1 , d2 , d3 .
Proof of Theorem 1.1. Taking integration of the three equations in 1.16 over 0, 1, respectively, and combining the three integration equalities linearly, we have
d
dt
1
el ELu av Awdx ≤
1
0
el ELu1 − u − alv − ALwdx.
2.6
0
It follows from the Young inequality and the Hölder inequality that
d
dt
1
el ELu av Awdx ≤ C1 − k
0
1
el ELu av Awdx,
2.7
0
where k min{l, L}, C1 k 12 el EL/4. So there exist positive constants M0 and τ0
depending on a, b, e, l, A, B, D, E, and L, such that
1
1
u dx,
0
1
v dx,
0
w dx ≤ M0 ,
t ≥ τ0 .
2.8
0
Moreover, there exists a positive constant M0
which depends on a, b, e, l, A, B, D, E, L and
L1 -norm of u0 , v0 , w0 , such that
1
1
u dx,
0
1
v dx,
0
0
w dx ≤ M0
,
t ≥ 0.
2.8
10
Abstract and Applied Analysis
Multiplying the first three equations in the system 1.16 by u, v, w, respectively, and
integrating over 0, 1, we have
1 d
2 dt
1
u2 dx ≤ −d1
1
0
0
1
u2x dx −
1
0
2α11 u α12 v α13 wu2x α12 vx α13 wx uux dx
u2 dx,
0
1 d
2 dt
1
v2 dx ≤ −d2
0
0
1 d
2 dt
1
1
el
b
w2 dx ≤ −d3
1
1
0
α21 u 2α22 v α23 wvx2 α21 ux α23 wx vvx dx
2.9
v2 dx,
0
1
0
0
vx2 dx −
EL
B
wx2 dx −
1
1
α31 u α32 v 2α33 wwx2 α31 ux α32 vx wwx dx
0
w2 dx,
0
from which it follows that
1 d
2 dt
1
u2 v2 w2 dx
0
1
1
1
EL 1 2
dx ≤ −d
v dx w dx − qux , vx , wx dx
B 0
0
0
0
0
1
1
1
el EL
u2x vx2 wx2 dx 1 u2 v2 w2 dx − qux , vx , wx dx,
≤ −d
b
B
0
0
0
2.10
u2x
vx2
wx2
el
u dx b
2
1
2
where d min{d1 , d2 , d3 },
qux , vx , wx 2α11 u α12 v α13 wu2x α12 u α21 vux vx α21 u 2α22 v α23 wvx2
α13 u α31 wux wx α31 u α32 v 2α33 wwx2 α23 v α32 wvx wx .
2.11
It is obvious that qux , vx , wx is a positive definite quadratic form of ux , vx , wx if 1.17 holds.
So 1.17 implies that
1 d
2 dt
1
0
1
1
el EL
u2 v2 w2 dx ≤ −d
u2x vx2 wx2 dx 1 u2 v2 w2 dx.
b
B
0
0
2.12
Abstract and Applied Analysis
11
Now, we proceed in the following two cases.
i One has t ≥ τ0 . The inequality 2.2 implies that |u|62 ≤ C|ux |22 |u|41 |u|61 ≤
1
1
3
CM04 |ux |22 M02 . So we have 0 u2x dx ≥ 1/CM04 0 u2 dx − M02 , and
−
1
0
u2x
vx2
wx2
1
dx ≤ −
9CM04
1
u v w
2
2
2
3
dx
0
3M02 .
2.13
It follows from 2.12 and 2.13 that
1 d
2 dt
1
0
u2 v2 w2 dx
⎧
⎨
≤ d −C2
⎩
1
3
u2 v2 w2 dx
0
⎫
1
⎬
EL
1
el
u2 v2 w2 dx .
3M02 1 ⎭
d
b
B
0
2.14
This means that there exist positive constants τ1 and M1 depending on di i
1, 2, 3, a, b, e, l, A, B, D, E, and L, such that
1
u2 dx,
1
0
v2 dx,
1
0
w2 dx ≤ M1 ,
t ≥ τ1 .
2.15
0
When d ≥ 1, M1 is independent of d because the zero point of the right-hand side in 2.14
can be estimated by positive constants independent of d.
ii One has t ≥ 0. Repeating estimates in i by 2.8
, we can obtain that there exists
a positive constant M1
depending on di i 1, 2, 3, a, b, e, l, A, B, D, E, L and the L1 , L2 -norm
of u0 , v0 , w0 , such that
1
u2 dx,
0
1
v2 dx,
1
0
0
w2 dx ≤ M1
,
t ≥ 0.
2.15
When d ≥ 1, M1
is independent of d.
To estimate |ux |2 , |vx |2 , |wx |2 , we introduce the following scaling:
u
u
,
d1
v v
,
d1
w
w
,
d1
t d1 t.
2.16
12
Abstract and Applied Analysis
, v , w,
t, respectively, then the
Denoting ξ d2 /d1 , η d3 /d1 , and using u, v, w, t instead of u
system 1.16 reduces to
ut Pxx fu, v, w,
0 < x < 1, t > 0,
vt Qxx gu, v, w,
0 < x < 1, t > 0,
wt Rxx hu, v, w,
0 < x < 1, t > 0,
ux x, t vx x, t wx x, t 0,
x 0, 1, t > 0,
vx, 0 v0 x,
ux, 0 u
0 x,
2.17
wx, 0 w
0 x,
0 < x < 1,
where P uα11 u2 α12 uv α13 uw, Q ξv α21 uv α22 v2 α23 vw, R ηw α31 uw α32 vw α33 w2 , fu, v, w d1−1 u − u2 − auv/1 bd1 u − Auw/1 Bd1 u Dd1 w, gu, v, w lv−d1−1 eu/1 bd1 u, hu, v, w Lw−d1−1 Eu/1 Bd1 u Dd1 w.
We still proceed in the following two cases.
i One has t ≥ τ1∗ d1 τ1 . It is not hard to verify that
1
1
u dx,
0
1
1
v dx,
0
u2 dx,
0
1
0
v2 dx,
0
w dx ≤ M0 d1−1 ,
1
0
w2 dx ≤ M1 d1−2 ,
2.18
|P |1 , |Q|1 , |R|1 ≤ A1 K1 d1−1 ,
where K1 1 ξ η M1 d1−1 , A1 max{M0 , α11 α12 α13 , α21 α22 α23 , α31 α32 α33 }.
Multiplying the first three equations in 2.17 by Pt , Qt , Rt , integrating them over the
domain 0, 1, respectively, and then adding up the three integration equalities, we have
1 y t −
2
1
0
1
u2t dx
−ξ
1
0
vt2 dx
−η
1
0
wt2 dx
−
1
qut , vt , wt dx
0
1 2α11 u α12 v α13 wut f α12 uvt f α13 uwt f dx
0
1
ξ α21 u 2α22 v α23 wvt g α21 vut g α23 vwt g dx
2.19
0
1
η α31 u α32 v 2α33 w wt h α31 wut h α32 wvt h dx,
0
1
where yt 0 Px2 Qx2 R2x dx. Notice by 1.17 that there exists a positive constant C3
depending only on αij i, j 1, 2, 3, such that
qut , vt , wt ≥ C3 u v w u2t vt2 wt2 .
2.20
Abstract and Applied Analysis
13
Thus,
1 y t ≤ −
2
1
u2t dx − ξ
0
1
1
vt2 dx − η
0
1
0
wt2 dx − C3
1
0
u v w u2t vt2 wt2 dx
1 2α11 u α12 v α13 wut f α12 uvt f α13 uwt f dx
0
1
2.21
ξ α21 u 2α22 v α23 wvt g α21 vut g α23 vwt g dx
0
1
η α31 u α32 v 2α33 w wt h α31 wut h α32 wvt h dx.
0
Using the Young inequality, Hölder inequality, and 2.18, we can obtain the following
estimates:
1
u dx ≤
4
% 1
u dx
0
1
uv dx ≤
u dx
u dx ≤
% 1
4
5
0
&1/6 % 1
2
v dx
2
&2/3 % 1
u5 dx 0
1
5
1
u dx
≤
4
5
v5 dx ≤
0
,
&1/3
≤
5
&1/3
5
&2/3
5
0
v dx
0
1
% 1
0
u dx
0
u4 v dx ≤
M11/3 d1−2/3
0
u dx
0
1
&1/2 % 1
2
0
3
≤
0
% 1
2
&2/3
5
u dx
0
0
1
&1/3 % 1
2
M12/3 d1−4/3
% 1
M12/3 d1−4/3
% 1
&1/3
5
v dx
,
0
&1/3
5
u dx
,
0
1
u5 v5 dx.
0
2.22
Applying the above estimates and Gagliardo-Nirenberg-type inequalities to the terms
on the right-hand side of 2.21, we have
−
1
0
−ξ
u2t dx ≤ −
1
0
−η
1
0
1
2
vt2 dx ≤ −
wt2 dx
1
0
ξ
2
2
Pxx
dx 1
η
≤−
2
0
1
f 2 dx,
0
2
Qxx
dx ξ
1
0
R2xx dx
1
η
g 2 dx,
0
1
0
h2 dx,
14
Abstract and Applied Analysis
&
1
1%
a2 v2
A2 w 2
au2 v
Au2 w
aAvw
2
−2 2
4
d1 u u dx
f dx ≤
2
2
2
bd1
Bd1
bBd12
bd1 2 Bd1 2
0
0
&2/3
% 1
2 a 2
A
1/3 −2/3
−4
5
≤ 12
M1 d1 M1 d1
2
u dx
b
B
0
% 1
&1/3
a A
2/3 −7/3
5
M1 d 1
2
u dx
,
b B
0
&
1
1%
'
e 2 (
e2 l2 v 2
2
2 −2 2
2
l d1 v dx
≤
ξl
ξ g dx ≤ ξ
1
M1 d1−4 ,
b
bd1 2
0
0
&
1
1%
2 E
E 2 L2 w 2
2
2 −2 2
2
L d1 w dx ≤ ηL 1 M1 d1−4 ,
η h dx ≤ η
2
B
Bd
0
0
1
−
1
0
u2t dx − ξ
1
0
0
2.23
wt2 dx
η 1 2
−
R dx
2 0 xx
0
0
2 *
'
a 2 A 2
e 2 (
E
2
2
12
ξl 1 M1 d1−4
ηL 1 b
B
b
B
1
≤ −
2
)
1
vt2 dx − η
1
2
Pxx
dx
M11/3 d1−2/3
ξ
−
2
% 1
1
2
Qxx
dx
&2/3
5
u dx
0
% 1
&1/3
a A
2/3 −7/3
5
M1 d 1
2
u dx
.
b B
0
Similarly, we can obtain
1
0
1
a
A
ut f dx ≤
ut
u v
w dx
bd1
Bd1
0
d−2 1
ε 1 2
1 1 3
ε 1 2
u dx uut dx u dx uu dx
≤ 1
2ε 0
2 0
2ε 0
2 0 t
ε 1 2
ε 1
a/bd1 2 1
A/Bd1 2 1
v dx vut dx w dx wu2t dx
2ε
2 0
2ε
2 0
0
0
% 1
&1/3
a 2 A 2
1
1 2/3 −4/3
−3
5
M0 d1 M1 d1
≤
1
u dx
2ε
b
B
2ε
0
ε
1
0
d1−1 u
uu2t dx
2
ε
2
1
0
vu2t dx
ε
2
1
0
wu2t dx,
2.24
Abstract and Applied Analysis
1
2α11
0
1
a
A
uut d1−1 u u2 v
w dx
bd1
Bd1
0
1
1
α211 1 5
α211 d1−2 1 3
2
u dx ε uut dx u dx ε uu2t dx
≤
ε
ε 0
0
0
0
1
1
2 2 1
2
2 1
α11 A
α11 a
2
2
2
u v dx ε vut dx u w dx ε wu2t dx
εb2 d12 0
εB2 d12 0
0
0
%
&
1/3
1
a 2 A 2
α2
α2 1 5
M12/3 d1−10/3
≤ 11 1 u5 dx
11
u dx
ε
b
B
ε 0
0
1
1
1
2
2
2ε uut dx ε vut dx ε wu2t dx,
uut f dx ≤ 2α11
0
1
α12
0
15
0
0
1
a
A
vut d1−1 u u2 v
w dx
bd1
Bd1
0
α212 1 4
α212 d1−2 1 2
ε 1 2
ε 1 2
uv dx uu dx u v dx vut dx
≤
2ε
2 0 t
2ε 0
2 0
0
α212 A2 1 2
α212 a2 1 3
ε 1 2
ε 1
v dx vut dx v w dx wu2t dx
2 0
2 0
2εb2 d12 0
2εB2 d12 0
vut f dx ≤ α12
% 1
&1/3
a 2 A 2
2α212 1 5
α212
2/3 −10/3
5
M1 d 1
1
u v5 dx
v dx
≤
2ε
b
B
5ε 0
0
1
ε 1 2
ε 1
uut dx ε vu2t dx wu2t dx,
2 0
2
0
0
1
α13
0
1
a
A
wut d1−1 u u2 v
w dx
bd1
Bd1
0
−2
1
α2 1 4
α2 d
ε 1 2
ε 1
uw2 dx uut dx 13
u w dx wu2t dx
≤ 13 1
2ε
2
2ε
2
0
0
0
0
1
2 2 1
2
2 1
α A
α a
ε
ε 1
w2 v dx vu2t dx 13 2
w3 dx wu2t dx
13 2
2 0
2 0
2εb2 d1 0
2εB2 d1 0
% 1
&1/3
a 2 A 2
α213
2α213 1 5
2/3 −10/3
5
M1 d 1
≤
u w5 dx
1
w dx
2ε
b
B
5ε 0
0
1
ε 1 2
ε 1 2
uut dx ε wu2t dx vut dx,
2 0
2 0
0
wut f dx ≤ α13
16
Abstract and Applied Analysis
1
α12
a
A
uvt d1−1 u u2 v
w dx
bd1
Bd1
0
1
2 −2 1
α d
α2 1 5
ε
ε 1 2
≤ 12 1
u3 dx uvt2 dx 12
u dx uv dx
2ε
2 0
2ε 0
2 0 t
0
α212 a2 1 2
α212 A2 1 2
ε 1 2
ε 1
u v dx vv dx u w dx wvt2 dx
2 0 t
2 0
2εb2 d12 0
2εB2 d12 0
% 1
&1/3
a 2 A 2
α212
α2 1 5
2/3 −10/3
5
≤
M1 d 1
1
u dx
12
u dx
2ε
b
B
2ε 0
0
1
ε 1 2
ε 1
2
ε uvt dx vv dx wvt2 dx,
2 0 t
2 0
0
uvt f dx ≤ α12
0
1
α13
1
1
a
A
uwt d1−1 u u2 v
w dx
bd1
Bd1
0
α213 1 5
α213 d1−2 1 3
ε 1
ε 1
2
u dx uwt dx u dx uwt2 dx
≤
2ε
2 0
2ε 0
2 0
0
α213 a2 1 2
α213 A2 1 2
ε 1
ε 1
2
u v dx vwt dx u w dx wwt2 dx,
2 0
2 0
2εb2 d12 0
2εB2 d12 0
% 1
&1/3
a 2 A 2
α213
α2 1 5
2/3 −10/3
5
M1 d 1
≤
1
u dx
13
u dx
2ε
b
B
2ε 0
0
1
ε 1
ε 1
2
2
ε uwt dx vwt dx wwt2 dx,
2 0
2 0
0
uwt f dx ≤ α13
0
1
ξ
ξ2 l2 d1−2 1 e/b2 1
e
ε 1 2
vt lv d1−1 v dx vv dx
dx ≤
bd1
2ε
2 0 t
0
0
ε 1 2
ξ2 l2 1 e/b2
M0 d1−3 vv dx,
≤
2ε
2 0 t
vt g dx ≤ ξ
0
1
α21
0
1
1
e
uvt lv d1−1 dx
bd1
0
α2 l2 d−2 1 e/b2 1 2
ε 1 2
u v dx vv dx
≤ 21 1
2ε
2 0 t
0
% 1
&1/3
α221 l2 1 e/b2 2/3 −10/3
ε 1 2
5
≤
M1 d1
u dx
vv dx,
2ε
2 0 t
0
uvt g dx ≤ α21
Abstract and Applied Analysis
1
2α22
α23
0
1
e
−1
wvt g dx ≤ α23 wvt lv d1 dx
bd1
0
α2 l2 d−2 1 e/b2 1 2
ε 1
v w dx wvt2 dx
≤ 23 1
2ε
2 0
0
% 1
&1/3
α223 l2 1 e/b2 2/3 −10/3
ε 1
5
M1 d1
v dx
wvt2 dx,
≤
2ε
2 0
0
1
α21
α23
0
1
e
v2 vt l d1−1 dx
bd1
0
α2 l2 d−2 1 e/b2 1 3
ε 1 2
v dx vut dx
≤ 21 1
2ε
2 0
0
% 1
&1/3
α221 l2 1 e/b2 2/3 −10/3
ε 1 2
5
≤
M1 d1
v dx
vut dx,
2ε
2 0
0
vut g dx ≤ α21
0
1
1
e
v2 vt l d1−1 dx
bd1
0
1
α222 l2 d1−2 1 e/b2 1 3
≤
v dx ε vvt2 dx
ε
0
0
%
&1/3
1
1
α221 l2 1 e/b2 2/3 −10/3
5
≤
M 1 d1
v dx
ε vvt2 dx,
ε
0
0
vvt g dx ≤ 2α22
0
1
17
1
e
−1
vwt g dx ≤ α23 v wt l d1 dx
bd1
0
α2 l2 d−2 1 e/b2 1 3
ε 1
v dx vwt2 dx
≤ 23 1
2ε
2 0
0
% 1
&1/3
α223 l2 1 e/b2 2/3 −10/3
ε 1
5
M1 d1
v dx
vwt2 dx,
≤
2ε
2 0
0
1
η
0
2
1
E
−1
wt h dx ≤ η wt Lw d1 dx
Bd1
0
η2 L2 d1−2 1 E/B2 1
ε 1
w dx wwt2 dx
≤
2ε
2 0
0
18
Abstract and Applied Analysis
η2 L2 1 E/B2
ε
M0 d1−3 ≤
2ε
2
1
α31
0
1
α32
0
1
2α33
0
1
α31
0
1
0
wwt2 dx,
1
E
−1
uwt h dx ≤ α31 uwt Lw d1 dx
Bd1
0
α231 L2 d1−2 1 E/B2 1 2
ε 1
w u dx uwt2 dx
≤
2ε
2 0
0
% 1
&1/3
α231 L2 1 E/B2 2/3 −10/3
ε 1
5
≤
M1 d1
w dx
uwt2 dx,
2ε
2 0
0
1
E
vwt Lw d1−1 dx
Bd1
0
α2 L2 d1−2 1 E/B2 1 2
ε 1
≤ 32
w v dx vwt2 dx
2ε
2
0
0
%
&1/3
1
α232 L2 1 E/B2 2/3 −10/3
ε 1
5
≤
M1 d1
w dx
vwt2 dx,
2ε
2 0
0
vwt h dx ≤ α32
1
E
w2 wt L d1−1 dx
Bd1
0
1
α2 L2 d1−2 1 E/B2 1 3
≤ 33
w dx ε wwt2 dx
ε
0
0
%
&1/3
1
1
α233 L2 1 E/B2 2/3 −10/3
5
≤
M 1 d1
w dx
ε wwt2 dx,
ε
0
0
wwt h dx ≤ 2α33
1
E
w2 ut L d1−1 dx
Bd1
0
α231 L2 d1−2 1 E/B2 1 3
ε 1
≤
w dx wu2t dx
2ε
2 0
0
% 1
&1/3
α231 L2 1 E/B2 2/3 −10/3
ε 1
5
≤
M1 d1
v dx
wu2t dx,
2ε
2
0
0
wut h dx ≤ α31
Abstract and Applied Analysis
1
19
1
E
w2 vt L d1−1 dx
Bd1
0
α232 L2 d1−2 1 E/B2 1 3
ε 1
≤
w dx wvt2 dx
2ε
2 0
0
% 1
&1/3
α232 L2 1 E/B2 2/3 −10/3
ε 1
5
≤
M1 d1
w dx
wvt2 dx.
2ε
2 0
0
wvt h dx ≤ α32
α32
0
2.25
By the above inequalities and the condition 1.17, we have
1
1 2α11 u α12 v α13 wut f α12 uvt f α13 uwt f dx
0
1
ξ α21 u 2α22 v α23 wvt g α21 vut g α23 vwt g dx
0
1
η α31 u α32 v 2α33 w wt h α31 wut h α32 wvt h dx
0
2.26
1
C4 1 ξ2 η2 M0 d1−3
u v w u2t vt2 wt2 dx ε
0
1
1/3
C5 C6 1 5
2/3 −4/3
−2
5
5
5
1 d 1 M1 d 1
u v w dx
u v5 w5 dx,
ε
ε 0
0
≤ λε
where λ is a constant depending only on εαij i, j 1, 2, 3. Choose a small enough positive
number ε which depends on αij i, j 1, 2, 3, a, b, e, l, A, B, D, E, and L, such that λε < C3 .
Substituting inequalities 2.24 and 2.26 into 2.21, one can obtain
1 1
y t ≤ −
2
2
1
0
2
Pxx
dx
ξ
−
2
1
0
2
Qxx
dx
η
−
2
1
0
R2xx dx
2.27
B1 K2 d1−3 B2 Y B3 K3 d1−2/3 Y 2/3 B4 K4 d1−4/3 Y 1/3 ,
1
where Y 0 u5 v5 w5 dx, K2 1 ξ2 η2 M0 1 ξ ηM1 d1−1 , K3 M11/3 , K4 M12/3 1 d1−2 .
Note that
P ≥ α11 u2 ,
Q ≥ α22 v2 ,
R ≥ α33 w2 .
2.28
20
Abstract and Applied Analysis
It follows from 2.18 and 2.4 to functions P, Q, R that
Y ≤ B5
1
0
Y
1/3
P 5/2 Q5/2 R5/2 dx ≤ B6 K13/2 d1−3/2 y 1/2 B6 K15/2 d1−5/2 ,
2.29
≤ B7 K11/2 d1−1/2 y1/6 B7 K15/6 d1−5/6 ,
Y 2/3 ≤ B8 K1 d1−1 y 1/3 B8 K15/3 d1−5/3 .
Moreover, one can obtain by 2.5 and 2.18 that
−
1
2
1
0
2
Pxx
dx −
ξ
2
1
0
2
Qxx
dx −
η
2
1
0
R2xx dx
≤ −B9 min 1, ξ, η K1−4/3 d14/3 y5/3 1 ξ η K12 d1−2 .
2.30
Combining 2.27,2.29, and 2.30, we have
1 y t ≤ − A1 min 1, ξ, η K1−4/3 d14/3 y 5/3
2
A2 1 ξ η K12 d1−2 K2 d1−3 K15/2 d1−5/2 K15/3 K3 d1−7/3 K15/6 K4 d1−13/6 2.31
A3 K13/2 d1−3/2 y 1/2 A4 K1 K3 d1−5/3 y1/3 A5 K11/2 K4 d1−11/6 y 1/6 .
Multiplying inequality 2.31 by d12 , we have
1 y t ≤ − A1 min 1, ξ, η K1−4/3 y5/3
2
A2 1 ξ η K12 K2 d1−1 K15/2 d1−1/2 K15/3 K3 d1−1/3 K15/6 K4 d1−1/6
2.32
A3 K13/2 d1−1/2 y1/2 A4 K1 K3 d1−1/3 y1/3 A5 K11/2 K4 d1−1/6 y1/6 ,
1
where y 0 d1 Px 2 d1 Qx 2 d1 Rx 2 dx. The inequality 2.32 implies that there exist
+2 depending on di , αij i, j 1, 2, 3, a, b, e, l, A, B, D, E, and L,
τ2 > 0 and positive constant M
such that
1
0
2
d1 Px dx,
1
0
2
d1 Qx dx,
1
+2 ,
d1 Rx 2 dx ≤ M
t ≥ τ2 .
2.33
0
In the case that d1 , d2 , d3 ≥ 1, ξ, η ∈ d, d, the coefficients of inequality 2.31 can be estimated
+2 depends
by some constants which depend on d, d, but do not depend on d1 , d2 , d3 . So M
Abstract and Applied Analysis
21
on αij i, j 1, 2, 3, a, b, e, l, A, B, D, E, L, d, and d, but it is irrelevant to d1 , d2 , d3 , when
d1 , d2 , d3 ≥ 1 and ξ, η ∈ d, d. Since
⎛
⎞ ⎛
⎞⎛ ⎞
Px
P u Pv Pw
ux
⎝Qx ⎠ ⎝Qu Qv Qw ⎠⎝ vx ⎠,
Rx
Ru Rv Rw
wx
2.34
we can transform the formulations of ux , vx , wx into fraction representations, then distribute
the denominators of the absolute value of the fractions to the numerators item and enlarge
the term concerning with ux , vx or wx to obtain
|d1 ux | |d1 vx | |d1 wx | ≤ C
|d1 Px | |d1 Qx | |d1 Rx |,
0 < x < 1, t > 0,
2.35
where C
is a constant depending only on ξ, η, αij i, j 1, 2, 3. After scaling back and
contacting estimates 2.33 and 2.35, there exist positive constant M2 depending on
di , αij i, j 1, 2, 3, a, b, e, l, A, B, D, E, L, and τ2 > 0, such that
1
0
u2x dx,
1
0
vx2 dx,
1
0
wx2 dx ≤ M2 ,
t ≥ τ2 .
2.36
When d1 , d2 , d3 ≥ 1 and ξ, η ∈ d, d, M2 is independent of d1 , d2 , d3 .
iiOne has t ≥ 0. Modifying the dependency of the coefficients in inequalities 2.17–
2.22, namely, replacing M0 , M1 with M0
, M1
, there exists a positive constant M2
depending
on di , αij i, j 1, 2, 3, a, b, e, l, A, B, D, E, L, and the W21 -norm of u0 , v0 , w0 , such that
1
0
u2x dx,
1
0
vx2 dx,
1
0
wx2 dx ≤ M2
,
t ≥ 0.
2.36
Furthermore, in the case that d1 , d2 , d3 ≥ 1, ξ, η ∈ d, d, M2
depends on d, d, but does not
depend on d1 , d2 , d3 .
Summarizing estimates 2.8, 2.15, 2.36 and Sobolev embedding theorem, there
exist positive constants M, M
depending only on di , αij i, j 1, 2, 3, a, b, e, l, A, B, D, E,
and L, such that 1.18 and 1.19 hold. In particular, M, M
depend only on αij i, j 1, 2, 3, a, b, e, l, A, B, D, E, L, d, and d, but do not depend on d1 , d2 , d3 , when d1 , d2 , d3 ≥ 1
and ξ, η ∈ d, d.
Similarly, according to 2.8
, 2.15
, 2.36
, there exists a positive constant M
depending on di , αij i, j 1, 2, 3,a, b, e, l, A, B, D, E, L and the initial functions u0 , v0 , w0 , such
that
|u·, t|1,2 , |v·, t|1,2 , |w·, t|1,2 ≤ M
,
t ≥ 0.
2.37
Further, in the case that d1 , d2 , d3 ≥ 1, ξ, η ∈ d, d, M
depends only on d, d, but do not
depend on d1 , d2 , d3 . Thus, T ∞. This completes the proof of Theorem 1.1.
22
Abstract and Applied Analysis
3. Global Stability
In order to obtain the uniform convergence of the solution for the system 1.16, we recall the
following result which can be found in 37, 42.
Lemma 3.1. Let a and b be positive constants. Assume that ϕ, ψ ∈ C1 a, ∞, ψt ≥ 0 and
ϕ is bounded from below. If ϕ
t ≤ −bψt and ψ t is bounded from above in a, ∞, then
limt → ∞ ψt 0.
Proof of Theorem 1.3. Let u, v, w be a solution for the system 1.16 with initial functions
≡ 0. From the strong maximum principle for parabolic equations, it
u0 x, v0 x, w0 x ≥ /
is not hard to verify that u, v, w > 0 for t > 0. Define the function
1
u
v
u − u − u ln ∗ dx α
v − v∗ − v∗ ln ∗ dx
Hu, v, w u
v
0
0
1
w
β
w − w∗ − w∗ ln ∗ dx.
w
0
1
∗
∗
3.1
Then the time derivative of Hu, v, w for the system 1.16 satisfies
1
1
1
u − u∗
v − v∗
w − w∗
ut dx α
vt dx β
wt dx
u
v
w
0
0
0
1' ∗
u∗
u
2
−
2α
u
α
v
α
wu
d
α12 vx α13 wx ux
1
11
12
13
x
2
u
0 u
dH
dt
αv∗
αv∗
2
α
u
2α
v
α
wv
d
α21 ux α23 wx vx
2
21
22
23
x
v
v2
(
βw∗
βw∗
2
α
u
α
v
2α
ww
u
α
v
d
w
α
3
31
32
33
31 x
13 x
x dx
x
w
w2
1
1
Aw
av
eu ∗
−
dx
u − u 1 − u −
dx α v − v∗ m −1 1 bu 1 Bu Dw
1 bu
0
0
1
Eu
β w − w∗ M −1 dx.
1 Bu Dw
0
3.2
The first integrand in the right hand of 3.2 is positive definite if
4αβu∗ v∗ w∗ d1 2α11 u α12 v α13 wd2 α21 u 2α22 v α23 wd3 α31 u α32 v 2α33 w
2
> u∗ αα23 v∗ w βα32 w∗ v d1 2α11 u α12 v α13 w
2
αv∗ α13 u∗ w βα31 w∗ u d2 α21 u 2α22 v α23 w
βw∗ α12 u∗ v αα21 v∗ u2 d3 α31 u α32 v 2α33 w.
3.3
Abstract and Applied Analysis
23
By the maximum-norm estimate in Theorem 1.1, the condition 1.20 implies 3.3. Therefore,
we have
dH
≤ −
dt
1'
1−
0
(
ABw∗
abv∗
−
u − u∗ 2 dx
1 bu∗ 1 bu 1 Bu∗ Dw∗ 1 Bu Dw
1
MDEu∗
w − w∗ 2 dx
∗ Dw ∗ 1 Bu Dw
Bu
1
0
1'
1
(
MDEu∗
ABw∗
abv∗
∗ 2
≤ −
−
dx
−
β
1−
−
u
u
w − w∗ 2 dx
∗
∗
∗
∗
∗
1 bu
1 Bu Dw
0
0 1 Bu Dw
1
1
∗ 2
− l1 u − u dx − βl2 w − w∗ 2 dx,
−β
0
0
3.4
where
l1 1 −
abv∗
ABw∗
−
,
∗
1 bu
1 Bu∗ Dw∗
MDEu∗
l2 > 0.
1 Bu∗ Dw∗
3.5
The condition 1.21 implies l1 > 0. Using the similar argument in the proof of Theorem 4.2 in
42, by the maximum-norm estimate in Theorem 1.1 and some tedious calculations, we can
prove
1
lim
t→∞
∗ 2
u − u dx lim
0
t→∞
1
0
∗ 2
v − v dx lim
t→∞
1
w − w∗ 2 dx 0.
3.6
0
1/2
that
It follows from 3.6 and Gagliardo-Nirenberg-type inequality |u|∞ ≤ C|u|1/2
1,2 |u|2
∗
∗
∗
u, v, w converges uniformly on u , v , w . By the fact that Hu, v, w is decreasing for t ≥
0, it is obvious that u∗ , v∗ , w∗ is globally asymptotically stable. So the proof of Theorem 1.3
is completed.
Acknowledgments
This work is supported by the China National Natural Science Foundation no. 11061031;
11161041, the Fundamental Research Funds for the Gansu University, NWNU-KJCXGC-0361 and the Fundamental Research Funds for the Central Universities no. zyz2012074.
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