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
1Rui 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−bu2−cu3− u2w 1u2, vt
dvα21uvα22v2α23vw
xxu−v, 0< x <1, t >0,
wt
d3wα31uwα32vwα33w2
xx−kw−γw2 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 · Wpk0,1,| · |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, Wp10,1
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.
2.2
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 ≤1 anddare 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∈WrmΩ, 1≤q, r≤ ∞, the derivativeDju0≤j < msatisfies the inequality
Dju
p≤C
|Dmu|ar|u|1−aq |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, where 1/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|1/32 |u|2/31 |u|1
, ∀u∈W210,1, 2.6
|u|4≤C
|ux|1/22 |u|1/21 |u|1
, ∀u∈W210,1, 2.7
|u|7/2 ≤C
|ux|10/212 |u|11/211 |u|1
, ∀u∈W210,1, 2.8
|ux|2≤C
|uxx|3/52 |u|2/51 |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
Step 1. Estimate |u|1,|v|1,|w|1. Firstly, taking integration of the first and second equations in 2.7 over the domain0,1, respectively, and combining the two integration equalities
linearly, we have
d dt
1
0
u aβ
v
dx≤ −a 1
0
vdx 1
0
βu−bu2
dx. 2.10
From Young inequality and H ¨older inequality, we can see
d dt
1
0
u aβ
v
dx≤C1− a aβ
1
0
u aβ
v
dx, 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
wdx≤M1, t≥0. 2.15
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
2α11uα12vα13wu2xα12uuxvxα13uuxwx
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
u2v2w2 dx
≤ −d 1
0
u2xvx2w2x dx−
1
0
qux, vx, wxdxβ1 2 α
1 0
u2v2w2 dx,
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
u2v2w2 d≤ −d
1
0
u2xv2xw2x dx
β1 2 α
1
0
u2v2w2 dx.
2.19 Now we proceed in the following two cases.
iIt holds thatt≥τ1. By2.6and2.15, we have1
0u2xdx≥1/CM411
0u2dx3−M21,
and
−d 1
0
u2xv2xw2x
dx≤3dM21−C2d 1
0
u2v2w2 dx
3
. 2.20
By2.19and2.20, we can see that 1
2 d dt
1
0
u2v2w2 dx
≤ −C2d 1
0
u2v2w2 dx
3
β1 2 α
1 0
u2v2w2
dx3dM21.
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≤M2, 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,
2.24
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
2yt≤ − 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 1
0Px2 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
2yt≤ − 1
0
u2tdx− 1
0
v2tdx−η 1
0
w2tdx−C3
1
0
uvw
u2t v2tw2t dx
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.
2.28
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
≤M4/52 d−8/5 1
0
u7dx 1/5
, 1
0
u4dx≤ 1
0
u7dx
2/51
0
u2dx 3/5
≤M3/52 d−6/5 1
0
u7dx 2/5
, 1
0
u5dx≤ 1
0
u7dx
3/51
0
u2dx 2/5
≤M2/52 d−4/5 1
0
u7dx 3/5
, 1
0
u6dx≤ 1
0
u7dx
4/51
0
u2dx 1/5
≤M1/52 d−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
≤M4/52 d−8/5 1
0
u7dx 1/5
, 1
0
u3vdx≤ 1
0
u7dx
2/51
0
u2dx
1/101
0
v2dx 1/2
≤M3/52 d−6/5 1
0
u7dx 2/5
, 1
0
u6vdx≤ 6 7
1
0
u7dx1 7
1
0
v7dx≤ 6 7
1
0
u7v7 dx, 1
0
u4vdx≤ 1 2
1
0
u2vdx1 2
1
0
u6vdx≤ 1
2M24/5d−8/5 1
0
u7dx 1/5
3 7
1
0
u7v7 dx, 1
0
uutdx≤ 1 2
1
0
udx 2
1
0
uu2tdx≤ 1
2M1d−1 2
1
0
uu2tdx, 1
0
u2utdx≤ 1 2
1
0
u3dx 2
1
0
uu2tdx≤ 1
2M4/52 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
2M2/52 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
2M4/52 d−8/5 1
0
u7dx 1/5
2
1
0
vu2tdx, 1
0
u2vutdx≤ 1 2
1
0
u4vdx 2
1
0
vu2tdx
≤ 1
4M4/52 d−8/5 1
0
u7dx 1/5
3 14
1
0
u7v7 dx
2 1
0
vu2tdx, 1
0
u3vutdx≤ 1 2
1
0
u6vdx 2
1
0
vu2tdx≤ 3 14
1
0
u7v7 dx
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
u2tdx≤ −1 2
1
0
Pxx2 dx 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
≤
a2β12a
M2d−42ba1M3/52 d−13/5 1
0
u7dx 1/5
2acb22
M3/52 d−6/5 1
0
u7dx 2/5
2bcM2/52 d1/5 1
0
u7dx 3/5
c2M1/52 d8/5 1
0
u7dx 4/5
, 1
0
g2dx≤d−2 1
0
u2v2
dx≤2M2d−4,
η 1
0
h2dx≤d−2η
k2α21
0
w2dx2kd−1γη 1
0
w3dxγ2η 1
0
w4dx
≤η
k2α2
M2d−42kγηM4/52 d−13/51 1
0
w7dx 1/5
γ2ηM3/52 d−6/5 1
0
w7dx 2/5
.
2.30
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 2η
M2d−4
C5d−1 1η
M24/5d−8/5 1
0
u7w7dx 1/5
C6
1η
M3/52 d−6/5 1
0
u5w7dx 2/5
C7M2/52 d1/5 1
0
u7dx 3/5
C8M1/52 d8/5 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
2M24/5d−8/5 1
0
u7dx 1/5
c2
2M2/52 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
M4/52 d−18/5 1
0
u7dx 1/5
α211b2
3
M2/52 d−4/5 1
0
u7dx 3/5
α211c2 d2
1
0
u7dx3 1
0
uu2tdx 1
0
vu2tdx 1
0
wu2tdx,
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−2b2 2
M4/52 d−8/5 1
0
u7dx 1/5
α212 β21
2 M4/52 d−18/5 1
0
v7dx 1/5
3α212 7
b2 2 c2d2
1
0
u7v7 dx 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 M4/52 d−8/5 1
0
u7dx 1/5
α213
2M4/52 d−18/5
⎡
⎣β 1
0
v7dx 1/5
1
0
w7dx 1/5⎤
⎦
3α213 7
b2 2 c2d2
1
0
u7w7 dx5
2 1
0
wu2tdx, 1
0
vtgdx≤d−1 1
0
uvtdx d−1
1
0
vvtdx
≤ M1d−3
2 1
0
uv2tvv2t dx,
α21
1
0
uvtgdx≤α21d−1 1
0
u2vtdx
α21d−1 1
0
uvvtdx
≤ α221
M4/52 d−18/5 1
0
u7dx 1/5
2
1
0
uvt2vv2t dx,
2α22
1
0
vvtgdx≤2α22d−1 1
0
uvvtdx
2α22d−1 1
0
v2vtdx
≤ α22
1
0vvt2dx
M4/52 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
≤ α231
0vvt2dx
2 M4/52 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 η2d−3M1 γ2
2η2M4/52 d−8/5 1
0
w7dx 1/5
η 1
0
wwt2dx,
α31
1
0
uwthdx≤αkd−1α31
1
0
uwwtdx α31γ
1
0
uw2wtdx
≤ α231 2
αk2d−2γ2 2
M4/52 d−8/5
⎡
⎣1 0
u7dx 1/5
1
0
w7dx 1/5⎤
⎦
3α231γ2 14
1
0
u7w7 dx1
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
M4/52 d−8/5
⎡
⎣1
0
v7dx 1/5
1
0
w7dx 1/5⎤
⎦
3α232 14γ2
1
0
v7w7 dx1
2 1
0
vwt2dx1 2
1
0
wwt2dx,
2α33 1
0
wwthdx≤2αkd−1α33 1
0
w2wtdx
2α33γ 1
0
w3wtdx
≤ αk2α233
M4/52 d−18/5 1
0
w7dx 1/5
α233γ2
M2/52 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 M4/52 d−18/5 1
0
u7dx 1/5
α212b2
2 M2/52 d−4/5 1
0
u7dx 3/5
