Boundary Value Problems
Volume 2009, Article ID 654539,19pages 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, 2Ling Guo,
1and Shengmao Fu
11Department of Mathematics, Northwest Normal University, Lanzhou 730070, China 2Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China
Correspondence should be addressed to Shengmao Fu,[email protected]
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.
Copyrightq2009 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 I1. 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 theL∞ estimate for the population densityw of predator species, there is not cross-diffusion forwin 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 systems2–17.
In this paper, we consider the following cross-diffusion system:
ut
duα11u2α12uvα13uw
xxβv−au−bu
2−cu3− u2w 1u2,
vt
dvα21uvα22v2α23vw
wt
d3wα31uwα32vwα33w2
xx−kw−γw
2 αu2w 1u2,
uxx, t vxx, t wxx, t 0, x0,1, t >0,
ux,0 u0x, vx,0 v0x, wx,0 w0x, 0< x <1,
1.1
whered, d3, αiji, j 1,2,3, α, β, γ, a, b, c,andkare positive constants. Also,d, d3are linear diffusion coefficients ofu, v, w, respectively, whileαiii1,2,3are referred as self-diffusion
pressures, andαiji /j, i, j 1,2,3are cross-diffusion pressures. Ifα12 α21 α23 α31 α320, then1.1reduces to the system1.4of 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 · Wk
p0,1,| · |p · Lp0,1. 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 u0, v0, w0 ∈ Wp10,1, p > 1, then 1.1 has a unique nonnegative
solutionu, v, w ∈ C0, T, W1
p0,1
C∞0, T, C∞0,1, where T ≤ ∞ is the maximal existence time for the solution. Ifu, v, wsatisfies
sup|u·, t|1,p,|v·, t|1,p,|w·, t|1,p : 0< t < T<∞, 2.1
thenT ∞. If, in addition,u0, v0, w0∈Wp20,1, thenu, v, w∈C0,∞, Wp20,1.
The main result in this section is as follows.
Theorem 2.1. Letu0, v0, w0 ∈W220,1,u, v, wis the unique nonnegative solution of 1.1in its maximal existence interval0, T. Assume that
8α11α21α31> α21α213α212α31,
8α12α22α32> α32α221α223α12,
8α13α23α33> α23α231α232α13.
Then there existst0 > 0 and positive constantsM, M which depend ond, d3, αiji, 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∈t0, T
≤M, 2.3
max{ux, t, vx, t, wx, t: 0≤x≤1, t0≤t < T} ≤M, 2.4
andT ∞. In particular, ifd, d3 ≥ 1, d3/d ∈d, d, whered ≤1anddare positive constants, thenM, Mdepend ond, d, but do not depend ond, d3 ≥1.
The following Gagliardo-Nirenberg-type inequalities and corresponding corollary play an importance role in the proof ofTheorem 2.1.
Theorem 2.2see18. LetΩ ⊂ Rn be a bounded domain with ∂Ω ∈ Cm. For every function
u∈Wm
r Ω, 1≤q, r≤ ∞, the derivativeDju0≤j < msatisfies the inequality
Dju p≤C
|Dmu|ar|u|q1−a|u|q, 2.5
provided one of the following three conditions is satisfied:1r ≤q,20 < nr −q/mrq < 1, or
3nr−q/mrq1, andm−n/qis not a nonnegative integer, where1/pj/na1/r−m/n 1−a/q, for alla∈j/m,1, and the positive constantCdepends onn, m, j, q, r, a.
Corollary 2.3. There exists a positive constantCsuch that
|u|2≤C|ux|21/3|u|21/3|u|1
, ∀u∈W210,1, 2.6
|u|4≤C|ux|21/2|u|11/2|u|1
, ∀u∈W1
20,1, 2.7
|u|7/2 ≤C|ux|210/21|u|111/21|u|1
, ∀u∈W210,1, 2.8
|ux|2≤C
|uxx|32/5|u|21/5|u|1
, ∀u∈W220,1. 2.9
For simplicity, denote thatCis Sobolev embedding constant or other kind of absolute constant.
Aj, Bj, Cj are some positive constants which depend onαiji, j 1,2,3,β, a, b, c, k, γ, α. Also,Kj
are positive constants which depend onαiji, j1,2,3,β, a, b, c, k, γ, α, d, d3. Whend, d3 ≥1,Kj
do not depend ond, d3, but ond, d.
Proof ofTheorem 2.1
linearly, we have
d dt
1
0
uaβvdx≤ −a 1
0 vdx
1
0
βu−bu2dx. 2.10
From Young inequality and H ¨older inequality, we can see
d dt
1
0
uaβvdx≤C1− a aβ
1
0
uaβvdx, 2.11
whereC1 1/4bβa/aβ2.From which it follows that there exists a constantτ0 >0, such that
1
0 udx,
1
0
vdx≤M0, t≥τ0, 2.12
whereM0 2C1aβ/amax{aβ−1,1}.
Secondly, taking integration of the third equations in2.7over domain0,1, 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 constantτ0>0, such that
1
0
wdx≤ 2|α−k|
γ , t≥τ0. 2.14
LetM1max{M0,2|α−k|/γ},τ1max{τ0,τ0}. Then
1
0 udx,
1
0 vdx,
1
0
wdx≤M1, t≥τ1. 2.15
Moreover, there exists a positive constantM1 which depends onβ, a, b, c, k, γ, αand theL1 -norm ofu0, v0, w0, such that
1
0 udx,
1
0 vdx,
1
0
Step 2. estimate|u|2,|v|2 and|w|2. Multiplying the first three inequalities ofCorollary 2.3by u, v, w, respectively, and integrating over0,1, we have
1 2
d dt
1
0 u2dx
≤ −d 1
0 u2xdx−
1
0
2α11uα12vα13wux2 α12uuxvxα13uuxwx
dxβ
1
0 uvdx,
1 2
d dt
1
0 v2dx
≤ −d 1
0 vx2dx−
1
0
α21u2α22vα23wv2xα21vuxvxα23vvxwx
dx
1
0 uvdx,
1 2
d dt
1
0 w2dx
≤ −d3 1
0
w2xdx− 1
0
α31uα32v2α33wwx2α31wuxwxα32wvxwx
dx.
2.16
Letdmin{d, d3}. By the above three inequalities and Young inequality, we have
1 2
d dt
1
0
u2v2w2dx
≤ −d 1
0
u2xvx2w2xdx− 1
0
qux, vx, wxdx
β1
2 α
1
0
u2v2w2dx,
2.17
where
qux, vx, wx 2α11uα12vα13wu2x α21u2α22vα23wvx2 α31uα32v2α33ww2x
α12uα21vuxvx α13uα31wuxwx α23vα32wvxwx
2.18
is quadratic form ofux, vx, wx. It is not hard to verify thatqux, vx, wxis positive definite if
2.2holds. Moreover, if2.2holds, then
1 2
d dt
1
0
u2v2w2d≤ −d 1
0
u2xv2xw2xdx β1
2 α
1
0
u2v2w2dx.
2.19
Now we proceed in the following two cases.
iIt holds thatt≥τ1. By2.6and2.15, we have 1
0u2xdx≥1/CM41 1
0u2dx 3−M2
and
−d 1
0
u2xv2xw2xdx≤3dM21−C2d 1
0
u2v2w2dx 3
. 2.20
By2.19and2.20, we can see that
1 2
d dt
1
0
u2v2w2dx
≤ −C2d 1
0
u2v2w2dx 3
β1
2 α
1
0
u2v2w2dx3dM21.
2.21
Thus, there exists positive constantsτ2 > τ1andM2depending ond, d3, β, a, b, c, k, γ, α, such that
1
0 u2dx,
1
0 v2dx,
1
0
w2dx≤M
2, t≥τ2. 2.22
Since the zero point of the right-hand side in2.21can be estimated by positive constants independent ofd, whend≥1. ThusM2do not depend ond≥1.
iit≥0. Repeating estimates iniby2.9, we can obtain that there exists a positive constantM2depending ond, d3, β, a, b, c, k, γ, αand theL1,L2-norm ofu0, v0, w0, such that
1
0 u2dx,
1
0 v2dx,
1
0
w2dx≤M2, t≥0, 2.22
whend≥1,M1is independent ofd.
Step 3. Estimate|ux|2,|vx|2,|wx|2. Introduce the scaling that
u u
d1
, v v
d1
, w w
d1
, td1t, 2.23
denoteηd3/d, and redenoteu, v, w, tbyu, v, w, t,respectively. Then2.7reduces to
utPxxfu, v, w, 0< x <1, t >0,
vtQxxgu, v, w, 0< x <1, t >0,
wtRxxhu, v, w, 0< x <1, t >0,
uxx, t vxx, t wxx, t 0, x0,1, t >0,
ux,0 u0x, vx,0 v0x, wx,0 w0x, 0< x <1,
whereP uα11u2α12uvα13uw,Qvα21uvα22v2α23vw,Rηwα31uwα32vw α33w2,fu, v, w βd−1v−ad−1u−bu2−cdu3−du2w/1d2u2,gu, v, w d−1u−v, hu, v, w −kd−1w−rw2 αdu2w/1d2u2. We still proceed in following two cases.
iIt holds thatt≥τ2∗dτ2. From2.15and2.22, we can easily obtain that
1
0 udx,
1
0 vdx,
1
0
wdx≤M1d−1,
1
0 u2dx,
1
0 v2dx,
1
0
w2dx≤M2d−2,
|P|1,|Q|1,|R|1 ≤DK1d−1,
2.25
whereK1 2η M2d−2,Dmax{M1, α11α12α13, α21α22α23, α31α32α33}. Multiply the first three equations in2.24byPt, Qt, Rtand integrate them over0,1,
respectively, then adding up the three new equations, we have
1 2y
t≤ −1 0
u2tdx− 1
0
v2tdx−η 1
0 wt2dx−
1
0
qut, vt, wtdx
1
0
12α11uα12vα13wutfα12uvtfα13uwtf
dx
1
0
α21vutg 1α21u2α22vα23wvtgα23vwtg
dx
1
0
α31wuthα32wvth
ηα31uα32v2α33w
wth
dx,
2.26
wherey 10P2
x Qx2 R2xdx. It is not hard to verify by2.4 that there exists a positive
constantC3depending only onαiji, j 1,2,3, such that
qut, vt, wt≥C3uvw
u2tvt2w2t. 2.27
Thus,
1 2y
t≤ −1 0
u2tdx− 1
0
v2tdx−η 1
0
w2tdx−C3 1
0
uvwu2t v2tw2tdx
1
0
12α11uα12vα13wutfdx
1
0
1α21u2α22vα23wvtgdx
1
0
ηα31uα32v2α33w
wthdx
1
0
α12uvtfdx
1
0
α13uwtfdx
1
0
α21vutgdx
1
0
α23vwtgdx
1
0
α31wuthdx
1
0
α32wvthdx.
Using Young inequality, H ¨older inequality and2.24, we can obtain the following estimates:
1
0 u3dx≤
1
0 u7dx
1/51
0 u2dx
4/5
≤M42/5d−8/5 1
0 u7dx
1/5 ,
1
0 u4dx≤
1
0 u7dx
2/51
0 u2dx
3/5
≤M32/5d−6/5 1
0 u7dx
2/5 ,
1
0 u5dx≤
1
0 u7dx
3/51
0 u2dx
2/5
≤M22/5d−4/5 1
0 u7dx
3/5 ,
1
0 u6dx≤
1
0 u7dx
4/51
0 u2dx
1/5
≤M12/5d−2/5 1
0 u7dx
4/5 , 1 0 uvdx≤ 1 0 u2dx
1/21
0 v2dx
1/2
≤M2d−2,
1
0
u2vdx≤ 1
0 u7dx
1/51
0 u2dx
3/101
0 v2dx
1/2
≤M42/5d−8/5 1
0 u7dx
1/5 ,
1
0
u3vdx≤ 1
0 u7dx
2/51
0 u2dx
1/101
0 v2dx
1/2
≤M32/5d−6/5 1
0 u7dx
2/5 ,
1
0
u6vdx≤ 6
7 1
0
u7dx1
7 1
0
v7dx≤ 6
7 1
0
u7v7dx,
1
0
u4vdx≤ 1 2
1
0
u2vdx1 2
1
0
u6vdx≤ 1
2M
4/5 2 d−8/5
1
0 u7dx
1/5 3
7 1
0
u7v7dx,
1
0
uutdx≤
1 2 1 0 udx 2 1 0 uu2
tdx≤
1 2M1d
−1 2 1 0 uu2 tdx, 1 0
u2utdx≤
1 2
1
0
u3dx
2 1
0
uu2tdx≤ 1
2M 4/5 2 d−8/5
1
0 u7dx
1/5
2 1
0 uu2tdx,
1
0
u3utdx≤
1 2
1
0
u5dx
2 1
0
uu2tdx≤ 1
2M 2/5 2 d−4/5
1
0 u7dx
3/5
2 1
0 uu2tdx,
1
0
u4utdx≤
1 2
1
0
u7dx
2 1
0 uu2tdx,
1
0
uvutdx≤
1 2
1
0
u2vdx
2 1
0
vu2tdx≤ 1
2M 4/5 2 d−8
/5 1
0 u7dx
1/5
2 1
0 vu2tdx,
1
0 u2vu
tdx≤
1 2
1
0
u4vdx 2
1
0 vu2
≤ 1 4M
4/5 2 d−8/5
1
0 u7dx
1/5 3
14 1
0
u7v7dx 2
1
0 vu2
tdx,
1
0
u3vutdx≤
1 2
1
0
u6vdx
2 1
0
vu2tdx≤ 3
14 1
0
u7v7dx
2 1
0
vu2tdx. 2.29
Applying the above estimates and Gagliardo-Nirenberg-type inequalities to the terms on the right-hand side of2.28, we have
− 1
0 u2
tdx≤ −
1 2
1
0 P2
xxdx
1
0 f2dx,
− 1
0
v2tdx≤ −1
2 1
0
Q2xxdx 1
0 g2dx,
−η 1
0
w2tdx≤ −η
2 1
0
R2xxdxη 1
0 h2dx,
1
0
f2dx≤β2d−2 1
0
v2dxa2d−2 1
0
u2dxb2 1
0
u4dx2bcd2 1
0
u5dxc2d4 1
0 u6dx
2abd−1 1
0
u3dx2ac 1
0
u4dxd−2 1
0 w2dx
2ad−2 1
0
uwdx2bd−1 1
0
u2wdx2 1
0 u3wdx
≤a2β12aM
2d−42ba1M32/5d−13/5 1
0 u7dx
1/5
2acb22M32/5d−6/5 1
0 u7dx
2/5
2bcM22/5d1/5 1
0 u7dx
3/5
c2M12/5d8/5 1
0 u7dx
4/5 ,
1
0
g2dx≤d−2 1
0
u2v2dx≤2M2d−4,
η 1
0
h2dx≤d−2ηk2α2 1
0
w2dx2kd−1γη 1
0
w3dxγ2η 1
0 w4dx
≤ηk2α2M2d−42kγηM42/5d−131 /5 1
0 w7dx
1/5
γ2ηM32/5d−6/5 1
0 w7dx
2/5 .
Thus
− 1
0 u2tdx−
1
0
v2tdx−η 1
0 wt2dx
≤ −1 2
1
0
Pxx2 dx−1
2 1
0
Qxx2 dx−η
2 1
0
R2xxdxC4
2ηM2d−4
C5d−1
1ηM24/5d−8/5 1
0
u7w7dx 1/5
C6
1ηM32/5d−6/5 1
0
u5w7dx 2/5
C7M22/5d1/5 1
0 u7dx
3/5
C8M12/5d8/5 1
0 u7dx
4/5 .
2.31
For the other terms on the right-hand side of2.28, we have
1
0
utfdx≤βd−1
1
0 utvdx
ad−1
1
0 uutdx
b
1
0 u2utdx
cd 1
0 u3utdx
d−1
1
0 wutdx
≤ β2a21
2 M1d
−3 b2 2M
4/5 2 d−8/5
1
0 u7dx
1/5
c2
2M 2/5 2 d6/5
1
0 u7dx
3/5 3
2 1
0
uu2tdxβd −1
2
1
0
vu2tdx1
2 1
0
wu2tdx,
2α11 1
0
uutfdx≤2α11βd−1 1
0 uutvdx
2α11ad−1 1
0 u2utdx
2α211b 1
0 u3utdx
2α11dc 1
0 u4utdx
2α11d−1 1
0
uutwdx
≤α211
β2a21
M
4/5 2 d−18/5
1
0 u7dx
1/5
α211b2
3
M22/5d−4/5 1
0 u7dx
3/5
α211c2
d
2 1
0
α12 1
0
vutfdx≤α12βd−1 1
0 utv2dx
α12ad−1 1
0 uvutdx
α212b
1
0
u2vutdx
α12dc 1
0
u3vutdx
α12d−1 1
0
vwutdx
≤α212 2
a2d−2b 2
2
M42/5d−8/5 1
0 u7dx
1/5
α212
β21
2 M
4/5 2 d−18/5
1
0 v7dx
1/5
3α212 7
b2
2 c 2d2
1
0
u7v7dx 5
2 1
0 vu2tdx,
α13 1
0
wutfdx≤α13βd−1 1
0
vwutdx
α13ad−1 1
0
uwutdx
α213b
1
0
u2wutdx
α13dc 1
0
u3wutdx
α13d−1 1
0
w2utdx
≤ α213
a2d−2b2/2
2 M
4/5 2 d−8/5
1
0 u7dx
1/5
α213 2M
4/5 2 d−18
/5 ⎡ ⎣β
1
0 v7dx
1/5
1
0 w7dx
1/5⎤ ⎦
3α213 7
b2
2 c 2d2
1
0
u7w7dx5
2 1
0
wu2tdx,
1
0
vtgdx≤d−1
1
0 uvtdx
d−1
1
0 vvtdx
≤ M1d−3
2 1 0
uv2tvv2tdx,
α21 1
0
uvtgdx≤α21d−1 1
0 u2vtdx
α21d−1 1
0 uvvtdx
≤ α221
M
4/5 2 d−18/5
1
0 u7dx
1/5
2 1
0
uvt2vv2tdx,
2α22 1
0
vvtgdx≤2α22d−1 1
0 uvvtdx
2α22d−1 1
0 v2vtdx
≤ α22 1
0vvt2dx
M
4/5 2 d−18/5
⎡ ⎣1
0 u7dx
1/5
1
0 v7dx
1/5⎤ ⎦
1
0 vv2tdx,
α23 1
0
wvtgdx≤α23d−1 1
0
uwvtdx
α23d−1 1
0
vwvtdx
≤ α23 1
0vvt2dx
2 M
4/5 2 d−18
/5 ⎡ ⎣1
0 u7dx
1/5
1
0 v7dx
1/5⎤ ⎦
1
0
wv2tdx,
η 1
0
wthdx≤d−1ηαk
1
0 wwtdx
γη
1
0
w2wtdx
≤ αk2
2 η
2d−3M 1
γ2
2η 2M4/5
2 d−8/5 1
0 w7dx
1/5 η 1 0 ww2 tdx, α31 1 0
uwthdx≤αkd−1α31 1
0
uwwtdx
α31γ
1
0
uw2wtdx
≤ α231 2
αk2d−2γ2 2
M42/5d−8/5 ⎡ ⎣1
0 u7dx
1/5
1
0 w7dx
1/5⎤ ⎦
3α231γ2 14
1
0
u7w7dx1
2 1
0
uw2tdx1
2 1
0
ww2tdx,
α32 1
0
vwthdx≤αkd−1α32 1
0
vwwtdx
α32γ
1
0
vw2wtdx
≤ α232 2
αk2d−2γ2 2
M42/5d−8/5 ⎡ ⎣1
0 v7dx
1/5
1
0 w7dx
1/5⎤ ⎦
3α232 14γ
2 1
0
v7w7dx1
2 1
0
vwt2dx1
2 1
0
wwt2dx,
2α33 1
0
wwthdx≤2αkd−1α33 1
0
w2wtdx
2α33γ
1
0
w3wtdx
≤ αk
2α2 33
M
4/5 2 d−18/5
1
0 w7dx
1/5
α233γ2
M
2/5 2 d−4/5
1
0 w7dx
3/5
1
0
wwt2dx,
α12 1
0
uvtfdx≤α12βd−1 1
0 uvvtdx
α12ad−1 1
0 u2vtdx
α12b
1
0 u3vtdx
α12cd 1
0 u4vtdx
α12d−1 1
0
uwvtdx
≤ α212
β2a21
2 M
4/5 2 d−18
/5 1
0 u7dx
1/5
α212b2
2 M
2/5 2 d−4
/5 1
0 u7dx
α212c2 2 d
2 1
0
u7dx
2
3 1
0
uv2tdx 1
0
vvt2dx 1
0 wv2tdx
,
α13 1
0
uwtfdx≤α13βd−1 1
0
uvwtdx
α13ad−1 1
0
u2wtdx
α13b
1
0 u3wtdx
α313cd 1
0 u4wtdx
α13d−1 1
0
uwwtdx
≤ α213
β2a21
2 M
4/5 2 d−18/5
1
0 u7dx
1/5
α213b2
2 M
2/5 2 d−4/5
1
0 u7dx
3/5
α213c2 2 d
2 1
0
u7dx3
2 1
0
uw2tdx 1
2 1
0
vw2tdx1
2 1
0
wwt2dx,
α21 1
0
vutgdx≤α21d−1 1
0 uvutdx
α21d−1 1
0 v2utdx
≤ α221 2M
4/5 2 d−18/5
⎡ ⎣1
0 u7dx
1/5
1
0 v7dx
1/5⎤ ⎦
1
0 vu2tdx,
α23 1
0
vwtgdx≤α23d−1 1
0
uvwtdx
α23d−1 1
0
v2wtdx
≤ α223 2M
4/5 2 d−18/5
⎡ ⎣1
0 u7dx
1/5
1
0 v7dx
1/5⎤ ⎦ 1 0 vw2 tdx, α31 1 0
wuthdx≤α31αkd−1 1
0 w2utdx
α31γ
1
0 w3utdx
≤ αk
2α2 31
2 M
4/5 2 d−18/5
1
0 w7dx
1/5
α231 2γ
2M2/5 2 d−4/5
1
0 w7dx
3/5 1 0 wu2 tdx, α32 1 0
wvthdx≤α32αkd−1 1
0 w2vtdx
α32γ
1
0
w3vtdx
≤
α2k2α2 32
2 M
4/5 2 d−18/5
1
0 w7dx
1/5
α232γ2
2 M
2/5 2 d−4/5
1
0 w7dx
3/5
1
0
Thus
1
0
12α11α12vα13utfdx
1
0
1α21u2α22vα23wvtgdx
1
0
ηα31uα32v2α33w
wthdx
1
0
α12uvtfdx
1
0
α13uwtfdx
1
0
α21vutgdx
1
0
α23vwtgdx
1
0
α31wuthdx
1
0
α32wvthdx
≤λ
1
0
uvwu2tvt2w2tdxC9 M1d
−32η2
C10
M
4/5 2 d−8/5
2d−2η2 1
0
u7v7w7dx 1/5
C11
M
4/5 2 d−8/5
1d2
1
0
u7v7w7dx 3/5
C12
1d2 1
0
u7v7w7dx,
2.33
whereλis a positive constant.
Note by 2.8 and 2.9 that |P|77//22 ≤ C|Px|52/3|P|111 /6 |P|17/2, |Px|102/3 ≤
B1K41/3d−4/3|Pxx|22K12d−2, and
−1 2
1
0
Pxx2 dx−1
2 1
0
Q2xxdx−η
2 1
0
R2xxdx≤ −B2min
1, ηK−41 /3d4/3y5/3K12d−22η. 2.34
Choose a small enough number >0, such thatλ < C3.According to2.28–2.34, we have
1 2y
t≤ −A 1min
1, ηK−41 /3y5/3A2K2y1/6A3K3y1/3A4K4y1/2
A5K5y2/3A6K6y5/6A7K7,
2.35
wherey10dPx2 dQx2 dRx2dx.
However,2.35implies that there exist positive constantsτ3 >0 andM3depending ond, d3, αiji, j1,2,3,β, a, b, c, k, γ, α, such that
1
0
dPx2dx,
1
0
dQx2dx,
1
0
Whend, d3 ≥1, η∈d, d, the coefficients of2.35can be estimated by constants depending on d, d, but not on d, d3. Thus, when d, d3 ≥ 1, η ∈ d, d, M3 depends on αiji, j
1,2,3, β, a, b, c, k, γ, α,d,d, and is irrelevant tod, d3≥1. Since
⎛ ⎜ ⎜ ⎝
Px
Qx
Rx
⎞ ⎟ ⎟ ⎠
⎛ ⎜ ⎜ ⎝
Pu Pv Pw
Qu Qv Qw
Ru Rv Rw
⎞ ⎟ ⎟ ⎠
⎛ ⎜ ⎜ ⎝
ux
vx
wx
⎞ ⎟ ⎟
⎠, 2.37
similar to2.26in24, we have
|dux||dvx||dwx| ≤D|dPx||dQx||dRx|, 0< x <1, t >0, 2.38
whereDis a positive constant only depending onη, αiji, j1,2,3. Scaling back with2.22
to original variableu, v, w, tand combining2.36,2.38, there exist positive constantsτ3>0 andM3depending ond, d3, αiji, j1,2,3,β, a, b, c, k, γ, α, such that
1
0 u2xdx,
1
0 v2xdx,
1
0
w2xdx≤M3, t≥τ3. 2.39
In addition, whend, d3≥1, η∈d, d,M3is dependent ofd, d, but independent ofd, d3≥1. iiIt holds thatt≥0. ReplacingM1, M2withM1, M2in2.24–2.34, we can obtain that there exists a positive constantM3depending ond, d3, αiji, j 1,2,3,β, a, b, c, k, γ, α
and theW21-norm ofu0, v0, w0such that
1
0 u2xdx,
1
0 v2xdx,
1
0
wx2dx≤M3, t≥0. 2.39
Whend, d3≥1, η∈d, d,M3 is dependent ofd, d, but independent ofd, d3 ≥1.
Concluding from2.15,2.22,2.39, andSobolevembedding theorem, there exists a positive constantst0 >0,M, Mdepending ond, d3, αiji, j 1,2,3,β, a, b, c, k, γ, α, such
that2.3and2.4are satisfied. Furthermore, whend, d3 ≥ 1, η ∈ d, dand the timetis large enough,M, Mare dependent ofαiji, j1,2,3,β, a, b, c, k, γ, α,d,d, but independent
ofd, d3≥1.
Similarly, according to2.15,2.22,2.39, we can see that there exists a positive constantM depending on d, d3, αiji, j 1,2,3,β, a, b, c, k, γ,α and the initial functions
u0, v0, w0, such that
Whend, d3 ≥1,η∈d, d,Mis dependent ofd,d, but independent ofd, d3. ThusT ∞. This completes proof ofTheorem 2.1.
3. Global Stability
From1, we know that if
α > k, β > a, $
k
α−k < m0,
β−a−c
2
b2 8c ≤
b√p1 24c
24β−ac2 3b24cβ−a−c−b√p
1 ,
H
wherep1 9b224cβ−a−c ≥ 0, then1.1has the unique position equilibrium point E∗u∗, v∗, w∗.
Theorem 3.1. Assume that all conditions inTheorem 2.1andHare satisfied. Assume further that
1
β
abu∗cu∗2>2
u∗2√1u∗22
8
u∗4
2 ,
γ α >
1
21u∗22, 3.1
4
αβw ∗d2d
3> 1
β
α23M2 1
αα32w
∗2d2α
11Mα12Mα13M
1 βα13M
21 αα31w
∗2dα
21M2α22Mα23M
1 α
1
βα12α21 2
M2w∗d3α31Mα32M2α33M
3.2
hold, whereMis the positive constant in2.4. Then the unique positive equilibrium pointE∗of1.1 is globally asymptotically stable.
Remark 3.2. SinceMis independent ofd, d3in the case ofd, d3≥1,3.2is always satisfied if dandd3are big enough.
Proof. Define the Lyapunov function
Hu, v, w 1
2β 1
0
u−u∗2dx1
2 1
0
v−v∗2dx 1 α
1
0
w−w∗−w∗ln w
w∗
Letu, v, wbe any solution of1.1with initial functionsu0x, v0x, w0x≥/≡0. From the strong maximum principle for parabolic equations, it is not hard to verify that
u, v, w >0 fort >0. Thus
dH dt ≤ −
1
0 %
1
βd2α11uα12vα13wu 2
x dα21u2α22vα23wvx2
1
αd3α31uα32v2α33w w∗ w2w
2
x
1
βα12uα21v
uxvx
1
βα13u
1
αα31 w∗
w
uxwx
α23v
1
αα32 w∗
w
vxwx
& dx
− 1
0 '
u−u∗21 β %
abuu∗ cu2uu∗u∗2 wuu∗ 1u21u∗2
&
−2−1 2
uu∗
1u2 −u ∗22
( dx−1
2 1
0
v−v∗2dx
− 1
0
w−w∗2
γ α−
1 21u∗22
dx.
3.4
The first integrand in the right hand of the above inequality is positive definite if
4
αβw
∗d2α
11uα12vα13wdα21u2α22vα23wd3α31uα32v2α33w
w2
1
βα12uα21v
1
αα13u
1
αα31 w∗
w
α23vβα32 w∗
w
> 1 β
α23vw 1
αα32w
∗2d2α
11uα12vα13w
1
βα13uw
1
αα31w
∗2dα
21u2α22vα23w
1 αw
∗1
βα12uα21v 2
d3α31uα32v2α33w.
3.5
From the maximum-norm estimate in Theorem 2.1,3.2 is a sufficient condition of3.5. Thus when3.1holds, there exists a positive constantδsuch that
dHu, v, w dt ≤ −δ
1
0
By integration by parts, H ¨older inequality and the maximum-norm estimate in
Theorem 2.1, we can see thatd/dt10u−u∗2 v−v∗2 w−w∗2dxis bounded from above. According to Lemma 3.1 in1and3.6, we obtain
|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|11,/22|u|12/2. Thus
|u·, t−u∗|∞−→0, |v·, t−v∗|∞−→0, |w·, t−w∗|∞−→0, t−→ ∞. 3.8
That is,u, v, wconverges uniformly to E∗. SinceHu, v, wis decreasing fort > 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 Provinceno. 096RJZA118, the Scientific Research Fund of Gansu Provincial Education Department, and the NWNU-KJCXGC-03-47 Foundation.
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