Volume 2009, Article ID 378763,26pages doi:10.1155/2009/378763
Research Article
Global Behavior for a Diffusive Predator-Prey
Model with Stage Structure and Nonlinear Density
Restriction-I: The Case in
R
n
Rui Zhang,
1Ling Guo,
2and Shengmao Fu
21Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China 2Department of Mathematics, Northwest Normal 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
This paper deals with a Holling type III diffusive predator-prey model with stage structure and nonlinear density restriction in the spaceRn. We first consider the asymptotical stability of equilibrium points for the model of ODE type. Then, the existence and uniform boundedness of global solutions and stability of the equilibrium points for the model of weakly coupled reaction-diffusion type are discussed. Finally, the global existence and the convergence of solutions for the model of cross-diffusion type are investigated when the space dimension is less than 6.
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
Population models with stage structure have been investigated by many researchers, and various methods and techniques have been used to study the existence and qualitative properties of solutions1–9. However, most of the discussions in these works are devoted to either systems of ODE or weakly coupled systems of reaction-diffusion equations. In this paper we investigate the global existence and convergence of solutions for a strongly coupled cross-diffusion predator-prey model with stage structure and nonlinear density restriction. Nonlinear problems of this kind are quite difficult to deal with since the usual idea to apply maximum principle arguments to get priori estimates cannot be used here10.
Consider the following predator-prey model with stage-structure:
X1 BX2−r1X1−CX1−η1X12−η2X13−
X2 CX1−r2X2,
X3 −r3X3−η3X23AX3 EX12 1FX2
1 ,
1.1
whereX1t ,X2t denote the density of the immature and mature population of the prey, respectively, X3t is the density of the predator. For the prey, the immature population is nonlinear density restriction.X3is assumed to consumeX1with Holling type III functional responseEX12/1FX21 and contributes to its growth with rateAEX21/1FX21 . For more details on the backgrounds of this model see references11,12.
Using the scaling u √FX1, v r2
√
F/C X2, w E/r2
√
F X3, dτ r2dtand redenotingτbyt, we can reduce the system1.1 to
uβv−au−bu2−cu3− u2w 1u2 ≡f1, vu−v≡f2,
w−kw−γw2 αu 2w
1u2 ≡f3,
1.2
whereβ BC/r2
2, a r1C /r2, b η1/r2
√
F, c η2/r2F, k r3/r2, α AE/r2F, γ η3√F/E.
To take into account the natural tendency of each species to diffuse, we are led to the following PDE system of reaction-diffusion type:
ut−d1Δuβv−au−bu2−cu3− u
2w
1u2, x∈Ω, t >0, vt−d2Δvu−v, x∈Ω, t >0,
wt−d3Δw−kw−γw2 αu
2w
1u2, x∈Ω, t >0, ∂ηu∂ηv∂ηw0, x∈∂Ω, t >0,
ux,0 u0x ≥0, vx,0 v0x ≥0, wx,0 w0x ≥0, x∈Ω,
1.3
whereΩis a bounded domain inRnwith smooth boundary∂Ω,ηis the outward unit normal vector on ∂Ω, and∂η ∂/∂η.u0x , v0x , w0x are nonnegative smooth functions on Ω. The diffusion coefficientsdi i1,2,3 are positive constants. The homogeneous Neumann
boundary condition indicates that system 1.3 is self-contained with zero population flux across the boundary. The knowledge for system1.3 is limitedsee13–17 .
account the effect of self-as well as cross-diffusion18–26. In this paper we are led to the following cross-diffusion system:
ut Δd1α11uα13w u βv−au−bu2−cu3− u2w
1u2, x∈Ω, t >0, vt Δd2α22v v u−v, x∈Ω, t >0,
wt Δd3α33w w−kw−γw2 αu
2w
1u2, x∈Ω, t >0, ∂u
∂ν ∂v ∂ν
∂w
∂ν 0, x∈∂Ω, t >0,
ux,0 u0x ≥0, vx,0 v0x ≥0, wx,0 w0x ≥0, x∈Ω,
1.4
whered1, d2, d3 are the diffusion rates of the three species, respectively. αii i 1,2,3 are
referred as self-diffusion pressures, andα13is cross-diffusion pressure. The term self-diffusion implies the movement of individuals from a higher to a lower concentration region. Cross-diffusion expresses the population fluxes of one species due to the presence of the other species. The value of the cross-diffusion coefficient may be positive, negative, or zero. The term 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
27. Forαij/0, problem1.4 becomes strongly coupled with a full diffusion matrix. As far
as the authors are aware, very few results are known for cross-diffusion systems with stage-structure.
The main purpose of this paper is to study the asymptotic behavior of the solutions for the reaction-diffusion system1.3 , the global existence, and the convergence of solutions for the cross-diffusion system1.4 . The paper will be organized as follows. InSection 2a linear stability analysis of equilibrium points for the ODE system 1.2 is given. In Section 3the uniform bound of the solution and stability of the equilibrium points to the weakly coupled system 1.3 are proved. Section 4 deals with the existence and the convergence of global solutions for the strongly coupled system1.4 .
2. Global Stability for System
1.2
Let E0 0,0,0 . If β > a, then1.2 has semitrivial equilibriaE1m0, m0,0 , where m0
b24cβ−a −b /2c. To discuss the existence of the positive equilibrium point of1.2 , we give the following assumptions:
α > k, β > a,
k
α−k < m0,
β−a−c
2
b2 8c ≤
b√p1 24c
24β−ac2
3b24cβ−a−c−b√p1,
2.1
thatβ−a−c u2−2bu3−3cu4−βaattains its maximum atu √p1−3b /12c, thus when
β−a−c /2b2/8c≤b√p
1/24c24β−a c2/3b24cβ−a−c −b√p1 ,g1u <00< u < m0 . l3 has the asymptotew α−k/γ and passes the point k/α−k,0 . In this case, l1 and l3 have unique intersection u∗, w∗ , as shown in Figure 1. E∗ u∗, v∗, w∗ is the unique positive equilibrium point of1.2 , wherev∗ u∗,w∗ 1u∗2 /u∗ β−a−bu∗−cu∗2 , kγw∗αu∗2/1u∗2 . In addition, the restriction of the existence of the positive equilibrium can be removed, ifβ < ac.
The Jacobian matrix of the equilibriumE0is
JE0
⎛ ⎜ ⎜ ⎝
−a β 0 1 −1 0
0 0 −k
⎞ ⎟ ⎟
⎠. 2.2
The characteristic equation ofJE0 isλk λ2 1aλa−β 0.E0 is a saddle for β > a. In addition, the dimensions of the local unstable and stable manifold ofE0are 1 and 2, respectively.E0is locally asymptotically stable forβ < a.
The Jacobian matrix of the equilibriumE1is
JE1
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
a11 β − m2
0 1m2
0
1 −1 0
0 0 a33
⎞ ⎟ ⎟ ⎟ ⎟
⎠, 2.3
wherea11 −a−2bm0−3cm20, a33 −kαm20/1m20 . The characteristic equation ofJE1 isλ3A1λ2B1λC10, where
A1−a11−a331,
B1a11a33−a33−
a11β
,
C1a33a11β,
H1 A1B1−C1 a11a33
a33−a11a33
a11β
−a33
1β−a11β
.
2.4
According to Routh-Hurwitz criterion,E1is locally asymptotically stable fora11β <0 and a33<0, that is,m20α−k < kandm0>
b23cβ−a −b /3c. The Jacobian matrix of the equilibriumE∗is
JE∗
⎛ ⎜ ⎜ ⎝
a11 β a13
1 −1 0
a31 0 a33
⎞ ⎟ ⎟
k α−k
m0
O u
α−k γ
w
l3
E∗ l1
Figure1
where
a11−a−2bu∗−3cu∗2− 2u ∗w∗
1u∗2 2, a13 −
u∗2 1u∗2,
a31 2αu ∗w∗
1u∗2 2, a33−γw
∗.
2.6
The characteristic equation ofJE∗ isλ3A2λ2B2λC20, where
A2−a11−a331,
B2 a11a33−a13a31−a33−a11β,
C2a33
a11β
−a13a31,
H2A2B2−C2 a11a33 a13a31a33−a11a33a11β−a331β−a11β.
2.7
According to Routh-Hurwitz criterion, E∗ is locally asymptotically stable for a11 β < 0. Obviously,a11β <0 can be checked by2.1 .
Now we discuss the global stability of equilibrium points for1.2 .
Theorem 2.1. (i) Assume that2.1 ,
bcu∗−u
∗β−a−bu∗
22√1u∗2 >
√
u∗21u∗2
8u∗21 2 1 8,
γ α>
1 2,
2.8
(ii) Assume thatβ > a, m20α−k < k, and
b23cβ−a −b/3c < m0 < 2k/αhold, then the equilibrium pointE1of 1.2 is globally asymptotically stable.
(iii) Assume thatβ≤aholds, then the equilibrium pointE0of 1.2 is globally asymptotically stable.
Proof. i Define the Lyapunov function
Et u−u∗−u∗ln u u∗
βv−v∗−v∗ln v v∗
1
α
w−w∗−w∗ln w w∗
.
2.9
Calculating the derivative ofEt along the positive solution of1.2 , we have
Et −β u∗
v uu−u
∗ −
u vv−v
∗
2
−u−u∗ 2
bcucu∗ w
∗1−u∗u
1u∗2 1u2
− c
αw−w
∗ 2 u−u∗ w−w∗ u∗u
1u∗2 1u2 −
u 1u2
≤ −u−u∗ 2
bcucu∗ w∗1−u∗u
1u∗2 1u2 −
u2 21u2 2 −
uu∗ 2 21u∗2 21u2 2
uuu∗
1u∗2 1u2 2
− γ α− 1 2
w−w∗ 2.
2.10
When u ∈ 0,∞ , the minimum of 1−u∗u/1u2 anduuu∗ /1u2 2 is−u∗2/2 2√1u∗2 and 0, respectively; the maximum ofuu∗ /1u2 is u/1u2 are u∗
√
1u∗2 /2 and 1/2, respectively. Thus, when 2.8 hold, Et ≤ 0. According to the Lyapunov-LaSalle invariance principle28,E∗is globally asymptotically stable if2.1 –2.3 hold.
ii Let
Et
u−m0−m0ln u m0
β
v−m0−m0ln v m0
1
αw. 2.11
Then
Et − β m0
v
uu−m0 −
u
vv−m0
2
−
bcucm0 u−m0 2
c αw
2−w
m0u 1u2 −
k α
.
2.12
iii Let
Et uβv 1
αw, 2.13
then
Et β−au−bu2−cu3−k αw−
γ αw
2. 2.14
Thus,Et ≤0 forβ≤a. This completes the proof ofTheorem 2.1.
3. Global Behavior of System
1.3
In this section we discuss the existence, uniform boundedness of global solutions, and the stability of constant equilibrium solutions for the weakly coupled reaction-diffusion system
1.3 . In particular, the unstability results in Section 2 also hold for system 1.3 because solutions of1.2 are also solutions of1.3 .
Theorem 3.1. Letu0x , v0x , w0x be nonnegative smooth functions onΩ. Then system1.3 has a unique nonnegative solutionux, t , vx, t , wx, t ∈CΩ×0,∞ C2,1Ω×0,∞ 3, and
0≤u≤M1 max
⎧ ⎪ ⎨ ⎪
⎩supΩ u0,supΩ v0,
b24cβ−a−b 2c
⎫ ⎪ ⎬ ⎪ ⎭,
0≤v≤M2M1,
0≤w≤M3max
⎧ ⎪ ⎨ ⎪ ⎩supΩ w0,
αM12
γ1M12
− k
γ
⎫ ⎪ ⎬ ⎪ ⎭
3.1
onΩ×0,∞ . In particular, ifu0, v0, w0≥/≡ 0, thenu, v, w >0for allt >0, x∈Ω.
Proof. It is easily seen that f1, f2, f3 is sufficiently smooth in R3 and possesses a mixed quasimonotone property inR3
Remark 3.2. When c 0 namely η2 0 , system 1.3 reduces to a system in which the immature population of the prey is linear density restriction. Similar to the proof of
Theorem 3.1, we have
M1M2max
!
sup
Ω u0,supΩ v0,
β−a b
"
,
M3max
⎧ ⎪ ⎨ ⎪ ⎩supΩ
w0,
αM1 2
γ1M1 2−
k γ
⎫ ⎪ ⎬ ⎪ ⎭.
3.2
Now we show the local and global stability of constant equilibrium solutionsE0, E1, E∗ for1.3 , respectively.
Theorem 3.3. i Assume that 2.1 holds, then the equilibrium point E∗ of 1.3 is locally asymptotically stable.
ii Assume thatβ > a,m20α−k < k, andm0 >
b23cβ−a −b/3chold, then the equilibrium pointE1of1.3 is locally asymptotically stable.
iii Assume thatβ < aholds, then the equilibrium pointE0of 1.3 is locally asymptotically stable.
Proof. Let 0μ1 < μ2 < μ3 <· · · be the eigenvalues of the operator−ΔonΩwith Neumann boundary condition, and letEμi be the eigenspace corresponding toμiinC1Ω . Let
X
#
U∈$C1Ω%3, ∂ηU0, x∈∂Ω
&
, Xij
'
c·φij :c∈R3
(
, 3.3
where{φij;j1, . . . ,dimEμi }is an orthonormal basis ofEμi , then
X⊕∞i1Xi, Xi⊕
dimEμi
j1 Xij. 3.4
i LetDdiagd1, d2, d3 ,LDΔ FUE∗ DΔ {aij}, where
a11 −a−2bu∗−3cu∗2−
2u∗w∗
1u∗2 2, a12β, a13 −
u∗2 1u∗2,
a21 1, a22−1, a230,
a31
2αu∗w∗
1u∗2 2, a320, a33−γw
∗.
The linearization of1.3 isUt LUat E∗. For eachi ≥ 1,Xi is invariant under the
operator L, andλis an eigenvalue of L onXi, if and only ifλis an eigenvalue of the matrix
−μiDFUE∗ . The characteristic equation isϕiλ λ3Aiλ2BiλCi0, where
Aiμid1d2d3 −a11−a331,
Biμ2id1d2d1d3d2d3
μid11−a33 −d2a11a33 d31−a11
a11a33−a13a31−a33−
a11β
,
Ciμ3id1d2d3μ2id1d3−a33d1d2−a11d2d3
−μi
d1a33−d2a11a33−a13a31 d3
a11β
a33a11β−a13a31,
HiAiBi−CiP3μi3P2μ2i P1μiP0,
P3 d1d2 d1d2d1d3d2d3 d23d1d2 ,
P2 d1d2d3 d11−a33 −d2a11a33 d31−a11
−a11d1d2d3 d2d1d3 −a33d3d1d2 ,
P1d1a11a33−a13a31−a11β−d2a11βa33
d3a11a33−a33−a13a31
−a11a33−1 d11−a33 −d2a11a33 d31−a11 ,
P0 a11a33
a13a31a33−a11a33
a11β
−a33
1β−a11β
.
3.6
From Routh-Hurwitz criterion, we can see that three eigenvaluesdenoted byλi,1,λi,2,λi,3 all have negative real parts if and only ifAi>0, Ci>0, Hi>0. Noting thata11, a13, a33 <0, a31 > 0, we must havea11β <0. It is easy to check thata11β <0 ifg1u1 <0seeSection 2 .
We can conclude that there exists a positive constantδ, such that
Re{λi,1},Re{λi,2},Re{λi,3} ≤ −δ, i≥1. 3.7
In fact, letλμiξ, then
ϕiλ μ3iξ3i Aiμ2iξi2BiμiξCi ϕ)iξ . 3.8
Sinceμi → ∞asi → ∞, it follows that
lim
i→ ∞
)
ϕiξ μ3
i
Clearly, ϕ)ξ has the three roots−d1,−d2,−d3. Letd min{d1, d2, d3}. By continuity, there existsi0such that the three rootsξi1, ξi2, ξi3ofϕ)iξ 0 satisfy
Re{ξi1},Re{ξi2},Re{ξi3} ≤ − d
2, i≥i0. 3.10
Let −δ) max0≤i≤i0{Re{λi1},Re{λi2},Re{λi3}}, then δ >) 0. Let δ min{δ, d/) 2}, then3.7 holds. According to30, Theorem 5.1.1, we have the locally asymptotically stability ofE∗.
ii The linearization of1.4 isUtLUatE1, whereLDΔ FUE1 DΔ {aij},
and
a11 −a−2bm0−3cm20, a12β, a13 − m 2 0 1m20,
a21 1, a22−1, a230,
a310, a320, a33 −k αm2
0 1m2
0 .
3.11
The characteristic equation of−μiDFUE1 isϕiλ λ3Aiλ2BiλCi0, where
Aiμid1d2d3 −a11−a331,
Biμ2id1d2d1d3d2d3
μid11−a33 −d2a11a33 d31−a11
a11a33−a33−a11β,
Ciμ3id1d2d3μ2id1d3−a33d1d2−a11d2d3
−μi
d1a33−d2a11a33d3a11βa33a11β,
HiAiBi−CiP3μi3P2μ2i P1μiP0,
P3 d1d2 d1d2d1d3d2d3 d23d1d2 ,
P2 d1d2d3 d11−a33 −d2a11a33 d31−a11
−a11d1d2d3 d2d1d3 −a33d3d1d2 ,
P1 d1
a11a33−
a11β
−d2
a11β
a33
d3a11a33−a33
−a11a33−1 d11−a33 −d2a11a33 d31−a11 ,
P0 a11a33 a33−a11a33a11β−a331β−a11β.
3.12
The three roots ofϕiλ 0 all have negative real parts fora11β <0 anda33<0. Namely,E1 is the locally asymptotically stable, ifm2
0α−k < kandm0>
iii The linearization of1.3 isUtLUatE0, whereLDΔ FUE0 DΔ {aij},
and
a11 −a, a12 β, a130,
a21 1, a22−1, a230,
a310, a32 0, a33−k.
3.13
Similar toi ,E1is locally asymptotically stable, whenβ < a.
Remark 3.4. When c 0, denote E0 0,0,0 . If β > a, then 1.3 has the semitrivial equilibrium pointE1 m0, m0,0 , wherem0 β−a/b. Ifα > k, β > a, kb2<α−k β−a 2< 27b2α−k , then1.3 has a unique positive equilibrium pointE∗ u∗, v∗, w∗ . Similar as
Theorem 3.3, we have the following.
i Ifβ > a,α > k, and kb2 < α−k β−a 2 < 27b2α−k namely,α > k,β > a,
k/α−k <β−a /b <3√3 , thenE∗is locally asymptotically stable.
ii Ifβ > aandα−k β−a2< kb2, thenE1is locally asymptotically stable.
iii Ifβ < a, thenE0is locally asymptotically stable.
Before discussing the global stability, we give an important lemma which has been proved in31, Lemma 4.1or in32, Lemma 2.5.3.
Lemma 3.5. Let a, b be positive constants. Assume that φ, ψ ∈ C1a,∞ ,ψt ≥ 0, and φ is bounded from below. Ifφt ≤ −bψt andψt ≤ K∀t≥ a for some positive constantK, then limt→ ∞ψt 0.
Theorem 3.6. i Assume that2.1 ,
bcu∗−u
∗β−a−bu∗
22√1u∗2 >
√
u∗21u∗2
8u∗21 2 1 8,
γ α>
1 2,
3.14
hold, then the equilibrium pointE∗of system1.3 is globally asymptotically stable.
ii Assume thatβ > a, m2
0α−k < k, and
b23cβ−a −b /3c < m
0 <2k/αhold, then the equilibrium pointE1of system1.3 is globally asymptotically stable.
Proof. Let u, v, w be the unique positive solution of 1.3 . ByTheorem 3.1, there exists a positive constant C which is independent ofx∈ Ωandt ≥ 0 such thatu·, t ∞,v·, t ∞,
w·, t ∞≤C, fort≥0. By33, TheoremA2,
u·, t C2αΩ ,v·, t C2αΩ ,w·, t C2αΩ ≤C, ∀t≥t0,∀t0>0. 3.15
i Define the Lyapunov function
Et
*
Ω
u−u∗−u∗ln u u∗
dxβ
*
Ω
v−v∗−v∗ln v v∗ dx 1 α * Ω
w−w∗−w∗ln w w∗
dx.
3.16
ByTheorem 3.1,Et t >0 is defined well for all solutions of1.3 with the initial functions u0, v0, w0≥/≡ 0. It is easily see thatEt ≥0 andEt 0 if and only ifuu∗.
Calculating the derivative ofEt along positive solution of1.3 by integration by parts and the Cauchy inequality, we have
Et −
*
Ω
d1u∗ u2 |∇u|
2βd2v∗ v2 |∇v|
2d3w∗ αw2|∇w|
2 dx * Ω
u−u∗ f1u, v, w
u βv−v
∗ f2u, v, w v
1 αw−w
∗ f3u, v, w w
dx
≤ −
*
Ωu−u
∗ 2
bcucu∗ w
∗1−u∗u
1u∗2 1u2 −
u2 21u2 2 −
uu∗ 2 21u∗2 21u2 2
uuu∗
1u∗2 1u2 2
dx−
γ
α− 1 2
*
Ωw−w
∗ 2dx.
3.17
It is not hard to verify that
Et ≤ −l1
*
Ωu−u
∗ 2dx−l 3
*
Ωw−w
∗ 2dx, 3.18
if3.14 hold. ApplyingLemma 3.5, we can obtain
lim
t→ ∞
*
Ωu−u
∗ 2dx0, lim
t→ ∞
*
Ωw−w
RecomputingEt , we find
Et ≤ −
*
Ω
d1u∗ u2 |∇u|
2βd2v∗ v2 |∇v|
2d3w∗ αw2 |∇w|
2dx
≤ −C
*
Ω
|∇u|2|∇v|2|∇w|2dx−gt .
3.20
From3.15 , we can see thatgt is bounded int0,∞ ,t0>0. It follows fromLemma 3.5and
3.15 thatgt → 0 ast → ∞. Namely,
lim
t→ ∞
*
Ω
|∇u|2|∇v|2|∇w|2dx0. 3.21
Using the Pioncar´e inequality, we have
lim
t→ ∞
*
Ωu−u
2dx lim
t→ ∞
*
Ωv−v
2dx lim
t→ ∞
*
Ωw−w
2dx0, 3.22
whereut 1/|Ω| +Ωudx, vt 1/|Ω|+Ωvdx, wt 1/|Ω| +Ωw dx.Noting that
|Ω||ut −u∗|2
*
Ωu−u
∗ 2dx≤2
*
Ωu−u
2dx2
*
Ωu−u
∗ 2dx,
|Ω||wt −w∗|2
*
Ωw−w
∗ 2dx≤2
*
Ωw−w
2dx2
*
Ωw−w
∗ 2dx,
3.23
according to3.19 and3.22 , we can see
ut → u∗, wt → w∗ t → ∞ . 3.24
Thus, there exists{tm}, utm → 0 astm → ∞. Applying the boundness of{vtm }, there
exists a subsequence of{vtm }, denoted still by{vtm }, such thatvtm → v., On the one hand
*
Ωutdx
--tm
|Ω|utm −→0, tm−→ ∞. 3.25
On the other hand
*
Ωutdx
--tm
*
Ω
d1Δuf1u, v, w dx--
--tm
*
Ωf1u, v, w dx
--tm
*
Ω
$
βv−v∗ −abuu∗ cu2uu∗u∗2u−u∗ −duw−w∗ %dx--
--tm
.
According to3.19 to compute the limit of the previous equation and using the uniqueness of the limit, we havev,v∗, and
lim
tm→ ∞
vtm v∗. 3.27
It follows from 3.15 that there exists a subsequence of {tm}, denoted still by {tm}, and
nonnegative functionsgi ∈C2Ω , i1,2,3, such that
u·, tm −→g1· , v·, tm −→g2·, w·, tm −→g3· inC2
Ω. 3.28
Applying3.19 –3.27 , we obtain thatg1u∗, g2 v∗, g3w∗, and
u·, tm −→u∗, v·, tm −→v∗, w·, tm −→w∗ inC2Ω. 3.29
In view ofTheorem 3.3, we can conclude thatE∗is globally asymptotically stable.
ii Let
Et
*
Ω
u−m0−m0ln u m0
dxβ
*
Ω
v−m0−m0ln v m0
dx 1 α
*
Ωwdx. 3.30
Then
Et −m0
*
Ω
d1 u2|∇u|
2βd2 v2|∇v|
2
dx
*
Ω
u−u∗ f1u, v, w
u βv−v
∗ f2u, v, w v
1
αf3u, v, w
dx
≤ −
*
Ω
β m0
v
uu−m0 −
u vv−m0
2
−
*
Ω
bcucm0 u−m0 2 γ
αw 2−w
m0u 1u2 −
k α
dx.
3.31
Therefore,Et ≤ −bcm0
+
Ωu−m0 2dx− γ α
+
Ωw2dx.It follows that the equilibrium point
E1of1.3 is globally asymptotically stable.
iii Define
Et 1 2
*
Ω
Then
Et −
*
Ω
d1|∇u|2βd2|∇v|2d3|∇w|2dx
*
Ω
uf1u, v, w βvf2u, v, w wf3u, v, w
dx.
3.33
Whena > β, k > α,
Et ≤ −
*
Ω
$
au2βv2 k−α w2%dx. 3.34
The following proof is similar toi .
Remark 3.7. Whenc0,Theorem 3.6shows the following.
i Assume thatβ > a, α > k,k/α−k <β−a /b <3√3,
b−u
∗β−a−bu∗
22√1u∗2 >
√
u∗21u∗22
8u∗21 2 1 8,
γ α >
1
2, 3.35
hold, then the equilibrium pointE∗of1.3 is globally asymptotically stable.
ii Assume thatβ > aand b2k/β−a > max{α−k β−a , bα/2}hold, then the equilibrium pointE1of1.3 is globally asymptotically stable.
iii Assume thatβ < aandk > αhold, then the equilibrium pointE0of1.3 is globally asymptotically stable.
Example 3.8. Consider the following system:
X1t−D1ΔX15X2−0.6X1−1.4X1−2X12−6X31−X3
2X2 1
12X12, x∈Ω, t >0,
X2t−D2ΔX21.4X1−X2, x∈Ω, t >0,
X3t−D3ΔX3−X3−
√
2X23X3 2X2
1 12X2
1
, x∈Ω, t >0,
∂ηXi0, i1,2,3, x∈∂Ω, t >0,
Xix,0 Xi0x ≥0, i1,2,3, x∈Ω.
3.36
4. Global Existence and Stability of Solutions for the System
1.4
By34–36, we have the following result.
Theorem 4.1. Ifu0, v0, w0 ∈Wp1Ω , p > n, then1.4 has a unique nonnegative solutionu, v, w∈
C0, T , W1
pΩ
C∞0, T , C∞Ω , whereT ≤∞is the maximal existence time of the solution. If the solutionu, v, w satisfies the estimate
sup'u·, t Wp1Ω ,v·, t Wp1Ω ,w·, t Wp1Ω : 0< t < T(<∞, 4.1
thenT ∞. If, in addition,u0, v0, w0∈Wp2Ω , thenu, v, w∈C0,∞ , Wp2Ω .
In this section, we consider the existence and the convergence of global solutions to the system1.4 .
Theorem 4.2. Let α11, α22 > 0 and the space dimension n < 6. Suppose that u0, v0, w0 ∈ C2λΩ 0 < λ < 1 are nonnegative functions and satisfy zero Neumann boundary conditions.
Then1.4 has a unique nonnegative solutionu, v, w∈C2λ,1λ/2Ω×0,∞ .
In order to proveTheorem 4.2, some preparations are collected firstly.
Lemma 4.3. Letu, v, w be a solution of 1.4 . Then
u, v≥0, 0≤w≤M1, in QT≡Ω×0, T ,
sup 0<t<T
u·, t L1Ω ,sup 0<t<T
v·, t L1Ω ≤C1T ,
uL2QT ,vL2QT ≤C2T ,
4.2
whereM1max{α/γ,w0L∞Ω }.
Proof. From the maximum principle for parabolic equations, it is not hard to verify that u, v, w≥0 andwis bounded.
Multiplying the second equation of1.4 byaβ , adding up the first equation of
1.4 , and integrating the result overΩ, we obtain
d dt
*
Ω
uaβvdx≤ −a
*
Ωv dx
*
Ω
βu−bu2dx. 4.3
Using Young inequality and H ¨older inequality, we have
*
Ω
βu−bu2dx≤C2,1− a aβ
*
whereC2,1 1/4b βa/aβ 2|Ω|.It follows from4.3 and4.4 that
d dt
*
Ω
uaβvdx≤C2,1− a aβ
*
Ω
uaβvdx. 4.5
Thus,
u·, t L1Ω ,v·, t L1Ω ≤C2,2, 4.6
whereC2,2depends onv0L1Ω ,u0L1Ω and coefficients of1.4 . In addition, there exists a positive constantC1T , such that
sup 0<t<T
u·, t L1Ω ,sup 0<t<T
v·, t L1Ω ≤C1T . 4.7
Integrating the first equation of1.4 overΩ, we have
d dt
*
Ωu dx≤β
*
Ωv dx−b
*
Ωu
2dx. 4.8
Integrating4.8 from 0 toT, we have
*
Ωux, T dx−
*
Ωux,0 dx≤β
*T
0
*
Ωv dx dt−b
*T
0
*
Ωu
2dx dt. 4.9
According to4.7 , there exists a positive constantC2T , such that
uL2QT ≤C2T . 4.10
Multiplying the second equation of1.4 byvand integrating it overΩ, we obtain
1 2
d dt
*
Ωv
2dx−
*
Ωd22α22v |∇v|
2dx
*
Ω
uv−v2dx
≤ 1
2
*
Ωu
2dx− 1 2
*
Ωv
2dx.
4.11
Integrating the previous inequation from 0 toT, we have
vL2QT ≤C2T . 4.12
Lemma 4.4. Letu, v, w be a solution of1.4 ,w1 d3α33w w, andτ < T. Then there exists a positive constantC3τ depending onw0W1
2Ω andw0L∞Ω , such that
w1W22,1Qτ ≤C3τ . 4.13
Proof. w1satisfies the equation
w1t d32α33w Δw1c1c2 u
2
1u2, 4.14
wherec1, c2are functions ofwand so are bounded because ofLemma 4.3.
Multiply the second equation of1.4 by−Δw1and integrate it overQτ to obtain
1 2
*
Ω|∇w1|
2x, τ dx−1 2
*
Ω|∇w1|
2x,0 dxd 3
*
Qτ
|∇w1|2dx ds
≤
*
Qτ |Δw1|
---c1c2
u2 1u2
---dx ds
≤m1Δw1L2Q
τ ≤ d3
2 Δw1 2
L2Qτ m1 2d3.
4.15
Then
*
Ω|∇w1|
2x, τ dxd3
*
Qτ
|∇w1|2dx ds
≤
*
Ω|∇w1|
2x,0 dx m1 2d3 ,
4.16
and w1 ∈ W22,1QT . From a disposal similar to the proof of Lemma 2.2 in23, we have
∇w1∈V2Qτ . Using a standard embedding result, we obtain∇w1∈L2n2/nQ τ .
Lemma 4.5see23, Lemmas 2.3 and 2.4 . Letq >1,q) 24q/nq1 ,β)∈0,1 , and let CT>0be any number which may depend onT. Then there is a constantM2depending onn, q,Ω,β), andCTsuch that
..g..
L)q
QT ≤M2
⎧ ⎨ ⎩1
/
sup 0≤t≤T
..g·, t ..
L2q/q1 Ω
04q/nq1 q)
..∇g..2/q) L2Q
T
⎫ ⎬
⎭, 4.17
for anyg ∈C0, T , W1
2Ω with
+
Ω|g·, t |
)
βdx 1/β)≤C
T for allt∈0, T.
To obtain L∞-estimates of u, v, we establish Lq-estimates of u, v in the following
lemma.
Lemma 4.6. Letα11, α22 >0,1 < q <2n1 /n−2 , then there exist positive constantsCq, T andCT , such that
uLqQ
T ,vLqQT ≤C
Proof. Multiply the first equation of1.4 byquq−1 forq >1 and integrate by parts overΩto obtain d dt * Ωu
qdx≤ −qq−1* Ωu
q−2d
12α11u |∇u|2dx
−α13q−1* Ω∇u
q · ∇w dxqβ
*
Ωu
q−1v dx.
4.19
Integrating4.19 from 0 tot, we have
*
Ωu
qx, t dx−
*
Ωu q
0x dxq
q−1*
Qt
uq−2d12α11u |∇u|2dx ds
≤ −α13
q−1*
Qt
∇uq · ∇w dx dsqβ
*
Qt
uq−1v dx ds.
4.20
Then substitution ofuq−2|∇u|2 4/q2 |∇uq/2 |2,uq−1|∇u|2 4/q1 2 |∇uq1 /2 |2 into
4.20 leads to
*
Ωu
qx, t dx4
q−1d1 q
*
Qt
---∇uq/2---2dx ds8α11q
q−1
q12
*
Qt
---∇uq1 /2---2dx ds
≤
*
Ωu q
0xdx−α13
q−1*
Qt
∇uq · ∇w dx dsqβ
*
Qt
uq−1v.
4.21
It follows from H ¨older inequality andLemma 4.3that
qβ
*
Qt
uq−1v≤qβuq−1 /2Ln2QT uq−1 /2L2n2/nQT vL2QT
≤C3,1uLq−q1−1 n2/2Q
T
.
4.22
Note that 1/21/n2 n/2n2 1, andn2 ≥2n2 /nforn ≥ 2. From H ¨older inequality, Young inequality, andLemma 4.4, we have
-*
QT
∇uq · ∇w dx dt----- 2q q1
-*
QT
uq−1 /2∇w· ∇uq1 /2dx dt----
-≤ 2q
q1u
q−1 /2
Lq−1 n2/2Q
T ∇wL2n2/nQT
..
.∇uq1 /2...
L2Q
T ≤C3,2u
q−1 /2
Lq−1 n2/2Q
T
..
.∇uq1 /2...
L2QT
≤ C3,21 2
..
.∇uq1 /2...2
L2Q
T
C3,2 21
uqL−q1−1 n2/2Q
T .
Substitution of4.22 and4.23 into4.21 leads to
*
Ωu
2q/q1
1 x, t dx
4q−1d1 q
*
Qt
---∇uq/2---2dx dt8α11q
q−1
q12
*
Qt
|∇u1|2dx dt
≤
*
Ωu q
0x dxC3,3∇u1 2
L2QT C3,4
u1
2q−1/q1
Lq−1 n2/q1 QT
,
4.24
where >0 is arbitrary andu1uq1 /2. Choosesuch that
α13C3,3< 4α11q
q12, 4.25
then it follows from4.24 that
sup 0<t<T
*
Ωu
2q/q1
1 x, t dx
*
QT
|∇u1|2dx dt≤C3,5
1u1
2q−1 /q1
Lq−1 n2/q1 Q
T
. 4.26
Let
E≡ sup 0<t<T
*
Ωu
2q/q1
1 x, t dx
*
QT
|∇u1|2dx dt. 4.27
Thenq−1 n2 /q1 <q)for
1< q < nn4
n2−4 . 4.28
According toLemma 4.5and the definition ofE, we can see
u1L)qQ
T ≤M3
1E2/nq)E1/q). 4.29
Combining4.26 and4.29 , we have
E≤C3,5
1u1
2q−1/q1
Lq)Q T
≤C3,5
#
1$M3
1E2/nq)E1/q)%2q−1/q1
&
≤C3,61Eμ ,
4.30
whereμ 2q−1 /q)q1 2/n1 < 1/q)4q/nq1 2 1. ThereforeEis bounded
From4.29 , we haveu1∈Lq)QT . Namely,u∈LrQ
T ,r q)q1 /2q12q/n.
Combining4.28 , we haveu∈LrQT , wherer <2n1 /n−2 .
Settingq2 in4.20 it is easily checked thatq2< nn4 /n2−4 , i.e.,n2,3,4,5 , we haveuV2QT ≤CT.
Multiplying the second equation of1.4 byqvq−1and integrating it overΩ, we have
d dt
*
Ωv
qdx−qq−1* Ωv
q−2d22α22v |∇v|2dxq
*
Ωv
q−1u−v dx. 4.31
The result ofvcan be obtained from an analogue of the previous proof ofu’s.
Lemma 4.7. Letn2,3,4,5, then there exists a positive constantM4such that
uL∞QT ,vL∞QT ≤M4. 4.32
Proof. We will prove this lemma by37, Theorem 7.1, page 181. At first, we rewrite the first two equations of1.4 as
∂u ∂t −
n
1
i,j1 ∂ ∂xi
/
aij
∂u ∂xj
0
−1n
i1 ∂ ∂xi
aiu u
abucu2 uw 1u2
βv,
∂v ∂t −
n
1
i,j1 ∂ ∂xi
/
bij
∂v ∂xj
0
vu,
4.33
whereaijx, t d12α11uα13w δij,aix, t α13∂w/∂xi ,bijx, t d22α22v δij,δijis
Kroneckersymbol. It follows fromLemma 4.6thatu∈LqQ
T , 1< q <2n1 /n−2 .
By the third equation of1.4 , we have
wt∇d32α33w ∇w−kw−γw2
αu2w
1u2. 4.34
It follows fromLemma 4.3thatd32α33w,−kw−γw2αu2w/1u2 is bounded inQT.
Applying Theorem 10.137, Page 204to4.34 , we have
w∈Cλ1,λ1/2
QT, λ1>0. 4.35
Recall thatw1 d3α33w wsatisfy4.14 inLemma 4.4, that is,
w1t d3α33w Δw1c1c2 u2
1u2, 4.36
wherec1c2u2/1u2 is bounded. Sinced32α33w ∈ Cλ1,λ1/2Q
T by4.35 , applying
Theorem 9.137, page 341-342to4.36 , we have
It follows from 37, Lemma 3.3, page 80 that ∇w1 ∈ Ln2q/n2−q QT and so ∇w
∇w1/d32α33w ∈ Ln2q/n2−q QT . Recall from Lemma 4.6thatu, v ∈ V2QT , so that
u, v∈L∞QT by applying Theorem 7.137, Page 181to4.33 .
Proof ofTheorem 4.2. Firstly,Theorem 4.2can be proved in a similar way as Theorem 2 in21,
25when the space dimensionn1.
Secondly, for 2≤n <6, applying Lemma 3.337, Page 80to4.36 , we have
w1∈C1λ2,1λ2/2Q
T
, 0< λ2<1. 4.38
Sincew −d3
d234α33w1 /2α33, we obtain
w∈C1λ2,1λ2/2
QT, 0< λ2 <1. 4.39
The first two equations can be written in the divergence form as
ut∇d12α11uα13w ∇uα13u∇w g1x, t ,
vt∇d22α22v ∇v g2x, t ,
4.40
whereg1βv−au−bu2−cu3−u2w/1u2 ∈L∞QT , g2u−v∈L∞QT . It follows from
Lemmas 4.1, 4.5, and4.39 thatu, v, w,∇ware bounded. Thus applying Theorem 10.137, Page 204to4.40 leads to
u, v∈Cλ3,λ3/2
QT, 0< λ3<1. 4.41
We rewrite the third equation of1.4 as
wt d32α33w Δwg3x, t , 4.42
whereg32α33|∇w|2−kw−γw2αu2w/1u2 ∈Cλ3,λ3/2QT . Applying Schauder estimate 29, Theorem 3.2.6, page 114to4.42 gives
w∈C2λ4,2λ4/2
QT, whereλ4min{λ, λ3}. 4.43
Let
u2 d1α11uα13w u, v2 d2α22v v, 4.44
then
u2t d12α11uα13w Δu2g4x, t ,
v2t d22α22v Δv2g5x, t ,
whereg4 d12α11uα13w βv−au−bu2−cu3−u2w/1u2 α13uw
t,g5 d22α22v u−v .
From4.41 , we haved12α11uα13w, d22α22v∈Cλ3,λ3/2QT . It follows from4.41 and 4.43 thatg4x, t , g5x, t ∈Cλ4,λ4/2Q
T . Applying Schauder estimate to4.45 gives
u2, v2 ∈C2λ4,2λ4/2
QT. 4.46
Solving equations4.44 foru, v, respectively, we have
u, v∈C2λ4,2λ4/2
QT. 4.47
In particular, to conclude u, v, w ∈ C2λ,2λ/2Q
T , we need to repeat the above
bootstrap technique. Since T is arbitrary, so the classical solution u, v, w of 1.4 exists globally in time.
Now we discuss the global stability of the positive equilibrium E∗u∗, v∗, w∗ see
Section 2 for1.4 .
Theorem 4.8. Assume that the all conditions inTheorem 4.2,2.1 , and
1 β
abu∗cu∗2>2
u∗√1u∗22
8
u∗4 2β2,
γ α>
1
21u∗2 2 4.48
hold. Letu∗, v∗, w∗ be the unique positive equilibrium point of 1.4 , and letu, v, w be a positive solution for1.4 . Then
u·, t −u∗L2Ω −→0, v·, t −v∗L2Ω −→0, w·, t −w∗L2Ω −→0 t−→ ∞ ,
4.49
provided thatd1·d2·d3is large enough.
Proof. Define the Lyapunov function
Hu, v, w 1 2β
*
Ωu−u
∗ 2dx1 2
*
Ωv−v
∗ 2dx1 α
*
Ω
w−w∗−w∗ln w w∗
Letu, v, w be a positive solution of1.4 , Then
dH dt ≤ −
*
Ω
1
βd12α11uα13w |∇u|
2 d
22α22v |∇v|2
1
αd32α33w w∗ w2|∇w|
2 1
βα13u∇u∇w
dx
−
*
Ω
⎧ ⎨
⎩u−u∗ 2
1 β
abuu∗ cu2uu∗u∗2 wuu∗
1u2 1u∗2
−2
−1
2
/
uu∗ 1u2 −
u∗2 β
02⎫⎬
⎭dx−
1 2
*
Ωv−v
∗ 2dx
−
*
Ωw−w
∗ 2
γ α−
1 21u∗2 2
dx.
4.51
The first integrand in the right hand of the previous inequality is positive definite if
4β αw
∗d12α11uα13w d22α22v d32α33w > α2
13u2w2d22α22v . 4.52
Therefore, when the all conditions inTheorem 4.8hold, there exists a positive constantδsuch that
dHu, v, w dt ≤ −δ
*
Ω
$
u−u∗ 2 v−v∗ 2 w−w∗ 2%dx. 4.53
This implies thatu·, t −u∗L2Ω , v·, t −v∗L2Ω , w·, t −w∗L2Ω → 0 as t → ∞. So the proof ofTheorem 4.8is completed.
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 NWNU-KJCXGC-03-47 Foundation.
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