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Boundary Value Problems

Volume 2010, Article ID 285961,24pages doi:10.1155/2010/285961

Research Article

Global Existence and Convergence of Solutions to

a Cross-Diffusion Cubic Predator-Prey System with

Stage Structure for the Prey

Huaihuo Cao

1

and Shengmao Fu

2

1Department of Mathematics and Computer Science, Chizhou College, Chizhou 247000, China

2College of Mathematics and Information Science, Northwest Normal University, Lanzhou 730070, China

Correspondence should be addressed to Huaihuo Cao,caguhh@yahoo.com.cn Received 3 December 2009; Accepted 30 March 2010

Academic Editor: Wenming Zou

Copyrightq2010 H. Cao 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 study a cubic predator-prey system with stage structure for the prey. This system is a generalization of the two-species Lotka-Volterra predator-prey model. Firstly, we consider the asymptotical stability of equilibrium points to the system of ordinary differential equations type. Then, the global existence of solutions and the stability of equilibrium points to the system of weakly coupled reaction-diffusion type are discussed. Finally, the existence of nonnegative classical global solutions to the system of strongly coupled reaction-diffusion type is investigated when the space dimension is less than 6, and the global asymptotic stability of unique positive equilibrium point of the system is proved by constructing Lyapunov functions.

1. Introduction and Mathematical Model

The predator-prey model as, which follows, the ordinary differential equation system

du dt

b1b2ub3u2

ub4uv,

dv

dtcv

αuβvv

1.1

is said to be the general Lotka-Volterra predator-prey model in 1–3, and to be cubic predator-prey system in 4, where u, v are the population densities of prey and predator species at time t, respectively. b3, b4, c, α, β are positive constants, b1 is nonnegative as the

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mortality rate of predator population, and the survival of predator species is dependent on the survival state of prey species, and b2ub3u2, βv are the respective density restriction

terms of prey and predator species.b4uis the predation rate of the predator, andαuis the

conversion rate of the predator. In4, three questions about system1.1are discussed: the stability of nonnegative equilibrium points, and the existence, as well as numbers of limit cycle.

Referring to5, we establish cubic predator-prey system with stage structure for the prey as follows:

dx1

dt η1x2−r1x1−η2x1b2x

2

1−b3x13−b4x1x3,

dx2

dt η2x1−r2x2,

dx3

dtcx3

αx1−βx3

x3,

1.2

where x1 and x2 are the population densities of the immature and mature prey species,

respectively, andx3 denotes the density of the predator species. The predators live only on

the immature prey species, as well as the survival of the predator species is dependent on the survival state of the immature prey species.η1, η2, r1, r2, b3, b4, c, α, βare positive constants,

and the sign ofb2is undetermined.η1 andr1are the birth rate and the mortality rate of the

immature prey species, respectively.r2 and care the net mortality rate of the mature prey

population and the predator population, andη2is the conversion rate of the immature prey

to the mature prey species.b2x1−b3x21andβx3are the respective density restriction terms of

the immature prey species and predator species.b4x1is the predation rate of the predator to

the immature prey population, andαx1is the conversion rate of the predator.

Using the scaling

u1

α r2

x1, u2

α η2

x2, u3

β r2

x3, dτr2dt, 1.3

and redenotingτbyt, system1.2reduces to

du1

dt a0u2−a1u1a2u

2

1−a3u31−ku1u3,

du2

dt u1−u2,

du3

dtbu1−u3u3,

1.4

wherea0η1η2/r22, a1 r1η2/r2, a2b2/α, a3 b3/r2, kb4/β,andbc/r2are positive

constants, anda2b2is undetermined to the sign.

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we derive the following PDE system of reaction-diffusion type:

u1tdu1a0u2−a1u1a2u21−a3u31ku1u3, x∈Ω, t >0,

u2tdu2u1−u2, x∈Ω, t >0,

u3tdu3 −bu1−u3u3, x∈Ω, t >0,

∂ηui 0, i1,2,3, xΩ, t >0,

uix,0 ui0x≥0, i1,2,3, x∈Ω,

1.5

where∂η ∂/∂η,η is the unit outward normal vector of the boundaryΩwhich we will assume to be smooth. The homogeneous Neumann boundary condition indicates that the above system is self-contained with zero population flux across the boundary. The positive constantsd1,d2, andd3are said to be the diffusion coefficients, and the initial valuesui0x

i1,2,3are nonnegative smooth functions.

Note that, in recent years, there has been considerable interest to investigate the global behavior of a system of interacting populations by taking into account the effect of self as well as cross-diffusion. According to the ideas in6–13, especially to8,9, the cross-diffusion term will be only included in the third equation, that is, the following cross-diffusion system:

ut Δ

d111u2

a0va1ua2u2−a3u3−kuw, x∈Ω, t >0,

vt Δ

d222v2

uv, x∈Ω, t >0,

wt Δ

d331uwα32vwα33w2

buww, x∈Ω, t >0,

uηx, t vηx, t wηx, t 0, xΩ, t >0,

ux,0 u0x≥0, vx,0 v0x≥0, wx,0 w0x≥0, x∈Ω.

1.6

In the above, di, αii i 1,2,3, α31, and α32 are positive constants. d1, d2 and d3 are the

diffusion rates of the three species, respectively. αii i 1,2,3 are referred to as self-diffusion pressures.α31andα32are cross-diffusion pressures. 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. Generally, 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

9.

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The paper will be organized as follows. In Section 2, we analyze the asymptotical stability of equilibrium points for the ODE system1.4via linearization and the Lyapunov method. In Section 3, we prove the global existence of solutions and the stability of the equilibrium points to the diffusion system1.5. InSection 4, we investigate the existence of nonnegative classical global solutions by assuminga0,a1,a2,a3,k,bto be positive constants

only for the simplicity of calculation, and the global asymptotic stability of unique positive equilibrium point to the cross-diffusion system1.6.

2. Equilibrium Solution of the ODE System

In this section we discuss the stability of unique positive equilibrium point for system1.4. The following theorem shows that the solution of system1.4is bounded.

Theorem 2.1. Let u1t, u2t, u3t be the solution of system 1.4with initial values ui0 > 0i 1,2,3, and let0, T be the maximal existence interval of the solution. Then0 < uit

Mi i1,2,3, t∈0, T, where

M1max

u10 a0a1u20, C0

1 a0

a1 ,

M2max

u10

a0a1

,C0 a1

,

M3max{u30, M1−b}.

2.1

The aboveC0is a positive constant depending only ona0, a1, a2, a3, and furtherT.

Proof. It is easy to see that1.4has a unique positive local solutionu1t, u2t, u3t. Let

T ∈0,∞be the maximal existence time of the solution, and combinu1andu2linearly, that

is,u1 a0a1u2, it follows from1.4that

d

dtu1 a0a1u2≤ −a1u2a0u1a2u

2

1−a3u31. 2.2

Using Young inequality, we can check that there exists a positive constantC0depending only

ona0, a1, a2,anda3such that

a0u1a2u21a3u31C0− a1

a0a1

u1. 2.3

It follows that

d

dtu1 a0a1u2≤C0− a1

a0a1

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which implies that there existM1andM2referring to2.1such that 0 < u1 ≤M1, 0 < u2 ≤

M2, andt∈0, T.

Finally, we note that du3/dtb u1 − u3u3 ≤ −b M1 − u3u3. Let M3

max{u30, M1−b}, then

du3−M3

dt u3−M3u3≤0 u30−M3≤0. 2.5

From the comparison inequality for the ODE, we haveu3−M3≤0,t∈0, T.

Thus the solutions for system1.4are bounded. Further, from the extension theorem of solutions, we haveT ∞.

By the simple calculation, the sufficient conditions for system1.4having a unique positive equilibrium point as follows:

ia2−k 2

a3a1−a0−kb > 2a3b;iimax{a1−a0,a2−2a3bb} ≤ bka2 −

a3bb, where the left equal sign holds if and only ifa1−a0 < a2 −2a3bb;iiia1 −a0 ≤

ba2 −a3bkb;iv a1 −a0 < kba2 −2a3bb;v kb < a1 −a0 ≤ a2 −a3bband

a2−k ≥ max{2a3b,2

a3a1−a0−kb}, where the second equal sign holds if and only if

a3b2> a1−a0−kb;vi2

a3a1−a0−kb< a2−k≤2a3bandkb < a1−a0<a2−a3bb.

If one of the above conditions holds, then system 1.4 has the unique positive equilibrium pointu1, u2, u3, where

u1u2

a2−k

a2−k24a3kba0−a1

2a3 , u3u1−b.

2.6

Theorem 2.2. System1.4has the unique positive equilibrium point u1, u2, u3when one of the

above conditions (i), (ii), (iii), (iv), (v), and (vi) holds. Ifa1−a0<a22−2ka2/4a3 kbholds, then

u1, u2, u3is locally asymptotically stable.

Theorem 2.2is easy to be obtained by using linearization; therefore, we omit its proof.

The objective of this section is to prove the following result.

Theorem 2.3. System1.4has the unique positive equilibrium point u1, u2, u3when one of the

above conditions (i), (ii), (iii), (iv), (v), and (vi) holds. Ifa1−a0<min{a22−2ka2/4a3 kb, kb

a2k/a3}holds, thenu1, u2, u3is globally asymptotically stable.

Proof. We make use of the general Lyapunov function

Vut

3

i1

αi

uiuiuiln

ui

ui

, 2.7

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dV dt

3

i1

αi

uiui

ui

dui

dt

α1

u1−u1

u1

×

a0

u1

u1u2−u2−u2u1−u1−u1u1−u1−a2a3u1a3u1−ku1u3−u3

α2

u2−u2

u2

1

u2

u2u1−u1−u1u2−u2 α3u3u1−u1−u3−u3

u3−u3

u3

α1−a2a3u1a3u1u1−u12−α3u3−u32

α3−1u1−u1u3−u3 a0α1u1−u1

×

u2

u1u1

u1−u1 1

u1

u2−u2

α2u2−u2u1−u1u2−u2−u2u1

u2u2

.

2.8

Letα2a0α1andα31. Then

dV

dtα1−a2a3u1a3u1u1−u1

2α

3u3−u32

a0α1

1

u1

u2

u1

u1−u1−

u1

u2

u2−u2 2

.

2.9

We observe that

a1−a0< kb

a2k

a3

2.10

is a sufficient condition of−a2a3u1a3u1 >0. So, when condition2.10holds, we have

dV

dt ≤0. 2.11

Set D {u ∈ IntR3

: dV/dt 0} {u1, u2, u3}. According to the Lyapunov-LaSalle

invariance principle 14, u1, u2, u3 is global asymptotic stability if inequality 2.10and

all conditions ofTheorem 2.2are satisfied.Theorem 2.3is, thus, proved.

3. Stability of the PDE System without Cross-Diffusion

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Denote thatFu f1, f2, f3T, whereu u1, u2, u3, f1a0u2−a1u1a2u21a3u31

ku1u3, f2 u1−u2,and f3 −bu1 −u3u3. It is easy to see thatf1, f2, f3 ∈ C1R 3

with

R3 {u1, u2, u3 | ui ≥ 0, i 1,2,3}. The standard PDE theory15shows that1.5has the unique solution u1, u2, u3 ∈ CΩ×0, TC2,1Ω×0, T

3

, where T ≤ ∞is the maximal existence time. The following theorem shows that the solution of1.5is uniformly bounded, and thusT ∞.

Theorem 3.1. Let u1, u2, u3 ∈ CΩ×0, TC2,1Ω×0, T 3

be the solution of system

1.5 with initial values ui0x ≥ 0 i 1,2,3, and let T be the maximal existence time.

Then 0 ≤ u1x, t, u2x, tM1, 0 ≤ u3x, tM2, and t ∈ 0, T, where M1 is a

positive constant depending only on Ω and all coefficients of 1.5 and ui0LΩi 1,2,

M2 max{u30LΩ, M1b}. Furthermore,Tand uix, t > 0on Ωfor any t > 0if

ui0≥/≡0 i1,2,3.

Proof. Letu1, u2, u3be the solution of1.5with initial valuesui0x≥ 0i 1,2,3. From

the maximum principle for parabolic equations16, it is not hard to verify thatuix, t ≥0 for x, t ∈ Ω×0, T i 1,2,3, whereT is the maximal existence time of the solution

u1, u2, u3. Furthermore, we know by the strong maximum principle thatuix, t > 0 onΩ for allt >0 ifui0 ≥/≡0i1,2,3. Next we prove that the solutionu1, u2, u3is bounded

onΩ×0, T.

Integrating the first two equations of1.5overΩand adding the results linearly, we have that, by Young inequality,

d dt

Ωu1 a0a1u2dx

Ωa0u1−a1u2dx

Ω

a2u21−a3u31

dx

≤ −a1

Ωu2dx

Ω

a0

a2 2

a3

u1−a2u21

dx

Ca1 a0a1

Ωu1 a0a1u2dx

3.1

for some positive constant C depending only on the coefficients of 1.5. Therefore,

u1,2tL1Ω is bounded in 0,∞. Using 17, Exercise 5 of Section 3.5, we obtain that

u1,2tLΩ is also bounded in 0,∞. Now note that supΩ×0,u1,2x, tM1. The maximum principle givesu3x, t≤max{u30L∞Ω, M1−b}:M2. The proof ofTheorem 3.1

is completed.

In order to prove the global stability of unique positive equilibrium solution for system

1.5, we first recall the following lemma which can be found in7,17.

Lemma 3.2. Leta and bbe positive constants. Assume that φ, ϕC1a,,ϕt > 0,and

φ is bounded from below. Ifφt ≤ −bϕt, andϕtK in a,for some constantK, then

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Let 0 μ1 < μ2 < μ3 < · · · be the eigenvalues of the operator−Δ onΩ with the

homogeneous Neumann boundary condition, and letEμibe the eigenspace corresponding toμiinC1Ω. Denote thatX{uC

3

∂ηu0, xΩ},{φij;j1,2, . . . ,dimEμi}is an orthonormal basis ofEμi,andXij{c·φij|cR3}. Then

X

i1

Xi, Xi

dimEμi

j1

Xij. 3.2

Next we present the clear proof of the the global stability by two steps:

Step 1Local Stability. LetDdiagd1, d2, d3andLDFuuDΔ {aij}, where

Fuu ⎛ ⎜ ⎜ ⎝

−2a3u21a2u1−a0 a0 −ku1

1 −1 0

u3 0 −u3 ⎞ ⎟ ⎟

⎛ ⎜ ⎜ ⎝

a11 a12 a13

a21 a22 a23

a31 a32 a33 ⎞ ⎟ ⎟

. 3.3

The linearization of1.5atuis

utLu. 3.4

For eachi≥1,Xiis invariant under the operatorL, andλis an eigenvalue ofLonXiif and only if it is an eigenvalue of the matrix−μiDFuu.

The characteristic polynomial of−μiDFuuis given by

ϕiλ λ3Aiλ2BiλCi, 3.5

where

Aiμid1d2d3−a11−a331,

Biμ2id1d2d1d3d2d3 μid11−a33−d2a11a33 d31−a11

a11a33−a13a31−a33−a11a0,

Ciμ3id1d2d3μ2id1d3−a11d2d3−a33d1d2

μid1a33−d2a11a33−a13a31 d3a11a0 a33a11a0−a13a31.

3.6

Thus

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whereP0, P1, P2,andP3are given by

P0 a11a33a13a31a33−a11a33 a11a0−a331a0−a11a0,

P1d1a11a33−a13a31−a11a0−d2a11a0 a33 d3a11a33−a33−a13a31 −a11a33−1d11−a33−d2a11a33 d31−a11,

P2 d1d2d3d11−a33−d2a11a33 d31−a11 −a11d1d2d3 d2d1d3−a33d3d1d2,

P3 d1d2d1d2d1d3d2d3 d23d1d2.

3.8

According to the Routh-Hurwitz criterion18, for eachi≥ 1, the three rootsλi,1, λi,2, λi,3 of

ϕiλ 0 all have negative real parts if and only if Ai > 0,Ci > 0, and Hi > 0. Noting thata33 < 0 anda13a31 < 0, the three roots have negative real parts ifa11a0 < 0. A direct

calculation shows thata11a0is negative if

a1−a0 <

a2 2−2a2k

4a3

kb. 3.9

Now we can conclude that there exists a positive constantδsuch that

Re{λi,1},Re{λi,2},Re{λi,3} ≤ −δ, i≥1. 3.10

In fact, letλμiξ, then

ϕiλ μ33Aiμ22BiμiξCi ϕiξ. 3.11

Sinceμi → ∞asi → ∞, it follows that

lim i→ ∞

ϕiξ

μ3

i

ξ3 d1d2d3ξ2 d1d2d2d3d1d3ξd1d2d3ϕξ. 3.12

It is easy to see thatd1, d2, d3 are the three roots ofϕξ 0. Thus, there exists a positive

constantδsuch that

Re{−d1},Re{−d2},Re{−d3} ≤ −δ. 3.13

By continuity, we see that there existsi0 such that the three rootsξi,1, ξi,2, ξi,3 ofϕiξ 0 satisfy

Re{ξi,1},Re{ξi,2},Re{ξi,3} ≤ −

δ

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So

Re{λi,1},Re{λi,2},Re{λi,3} ≤ −

μiδ 2 ≤ −

δ

2, ii0. 3.15

Let

δmax

1≤ii0

{Re{λi,1},Re{λi,2},Re{λi,3}}, 3.16

thenδ > 0, and3.10holds forδmin{δ, δ/ 2}.

Consequently, the spectrum ofL, consisting only of eigenvalues, lies in{Reλ≤ −δ}if

3.9holds, and the local stability ofufollows19, Theorem 5.1.1.

Step 2Global Stability. In the following,Cdenotes a generic positive constant which does not depend onx ∈Ωandt ≥0. Letube the unique positive solution. Then it follows from

Theorem 3.1thatu·, tis bounded uniformly onΩ, that is,ui·, t∞≤Cfor allt≥0. By20, TheoremA2,

ui·, tC2ΩC,t≥1. 3.17

Define the Lyapunov function

Et

Ω

u1−u1−u1lnu1

u1

dxa0

Ω

u2−u2−u2lnu2

u2 dx k Ω

u3−u3−u3lnu3

u3

dx.

3.18

ThenEt≥0 for allt >0. Using1.5and integrating by parts, we have

Et

Ω

u1−u1

u1 u1ta0

u2−u2

u2 u2tk

u3−u3

u3 u3t dx − Ω

d1u1

u2 1

|∇u1|2a0

d2u2

u2 2

|∇u2|2kd3u3 u2

3 |∇u3|2

dx Ω

u1−u1

u1

f1u a0

u2−u2

u2

f2u k

u3−u3

u3

f3u dx ≤ Ω

−−a2a3u1a3u1u1−u12−ku3−u32

a0u1−u1

u2

u1u1

u1−u1

1

u1

u2−u2

a0u2−u2

u1−u1u2−u2−u2u1

u2u2

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Ω

−−a2a3u1a3u1u1−u12−ku3−u32

dx

a0

u1 Ω u2 u1

u1−u1−

u1

u2

u2−u2 2

dx

≤ −

Ω−a2a3u1a3u1u1−u1

2dxk

Ωu3−u3

2dx.

3.19

Takingl1−a2a3u1andl3k, we have that

Et≤ −l1

Ωu1−u1

2dxl 3

Ωu3−u3

2dx, 3.20

wherel1>0 holds for

a1−a0< a3ba2k

a3

. 3.21

FromTheorem 3.1the solutionuiof1.5is bounded, and so are the derivatives ofu1−u12

andu3−u32by equations in1.5. ApplyingLemma 3.2, we obtain

lim t→ ∞

Ωu1−u1

2dx0, lim

t→ ∞

Ωu3−u3

2dx0. 3.22

Asu2

iC, it follows that

Et≤ −

Ω

d1u1

u2 1

|∇u1|2a0

d2u2

u2 2

|∇u2|2kd3u3 u2

3 |∇u3|2

dx

≤ −C

Ω

|∇u1|2|∇u2|2|∇u3|2dx

ϕt.

3.23

Using inequality3.17and system1.5, the derivative ofϕtis bounded in1,∞. From

Lemma 3.2, we conclude thatϕt → 0 ast → ∞. Therefore

lim t→ ∞

Ω

|∇u1|2|∇u2|2|∇u3|2dx0. 3.24

Using the Poincar´e inequality yields

lim t→ ∞

Ωu1−u1

2dx lim

t→ ∞

Ωu2−u2

2dx lim

t→ ∞

Ωu3−u3

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whereuit 1/|Ω| Ωuidx, i1,2,3.Thus, it follows from3.22and3.25that

|Ω|u1tu12

Ωu1−u1

2dx2

Ωu1−u1

2dx2

Ωu1−u1

2dx−→0 3.26

ast → ∞. So we haveu1tu1 ast → ∞. Similarly,u3tu3 ast → ∞. Therefore,

there exists a sequence{tm}withtm → ∞such thatu1tm → 0. Asu2tmis bounded, there exists a subsequence of{tm}, still denoted by the same notation, and nonnegative constantu2

such that

u2tm−→u2. 3.27

Atttm, from the first equation of1.5, we have

|Ω|u1tm

Ωu1tdx !! !! tm

Ω

dΔu1f1u

dx!!!!

tm

Ω

a0

u1

u1u2−u2−u2u1−u1dx !! !! tm

Ω−u1u1−u1−a2a3u1u1dx !! !! tm

Ω−ku1u3−u3dx !! !! tm

−→0.

3.28

In view of3.22and3.27, it follows from3.28thatu2u2, thus

lim

m→ ∞u2tm u2. 3.29

According to 3.17, there exists a subsequence of {tm}, denoted still by {tm}, and nonnegative functionswiC2Ω, such that

lim

m→ ∞ui·, tmwi·C2Ω0, i1,2,3. 3.30

In view of3.29and noting that in factu1tu1andu3tu3, we know thatwiui, i 1,2,3. Therefore,

lim

m→ ∞ui·, tmuiC2Ω0, i1,2,3. 3.31

The global asymptotic stability ofufollows from3.31and the local stability ofu.

Theorem 3.3. System1.5has the unique positive equilibrium point u1, u2, u3when one of the

conditions (i), (ii), (iii), (iv), (v), and (vi) inSection 2holds. If 3.9and3.21hold, thenu1, u2, u3

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4. Global Existence of Classical Solutions and Convergence

In this section, we discuss the existence of nonnegative classical global solutions and the global asymptotic stability of unique positive equilibrium point of system1.6.

Some notations throughout this section are as follows: QT Ω × 0, T, u

WpkΩ means that DαuLpΩ for any |α| ≤ k with α α1, α2, . . . , αn, uWk pΩ

Ω"|α|≤k|Dαu|pdx

1/p

, uWp2,1QT means that u, uxi, uxixji, j 1,2, . . . , n and ut

are in LpQ

T, uW2,1

p QT QT|u|

p|Du|p|D2u|p|u

t|pdx dt

1/p

, and uV2QT

sup0≤tTu·, tL2ΩuL2Q

TwithV2QT L

0, T, L2ΩW1,0 2 QT. To obtainC2α,1α/2Q

Tnormal estimates of the solution for1.6, we present a series of lemmas in the following.

Lemma 4.1. Letu, v, wbe the solution of 1.6. Then there exists a positive constantM0(1) such

that

0≤u, vM0, 0≤w,t≥0. 4.1

Proof. By applying the comparison principle20to system1.6, we haveu≥0, v≥0,and

w≥0 inQT. To prove thatu, vM0in the following, we consider the auxiliary problem

utd12α11uΔu2α11

n

i1

uxiuxi f1, x∈Ω, t >0,

vtd22α22vΔv2α22

n

i1

vxivxi f2, x∈Ω, t >0,

∂ηux, t ∂ηvx, t 0, xΩ, t >0,

ux,0 u0x/≡0, vx,0 v0x/≡0, x∈Ω.

4.2

Notice that the functionsf1andf2are sufficiently smooth inR2, and are quasimonotone in

R2

. Let0,0andM, Nbe a pair of upper-lower solutions for4.2, whereMandNare positive constants. Direct calculation with inequalities

a0Na1Ma2M2−a3M3≤0,

MN≤0, u0≤M, v0≤N

4.3

yieldsM max{a2

a2

24a3|a0−a1|/2a3,u0L∞Ω}and N max{M,v0LΩ}. It follows that there existsM0 Kmax{M, N} ≥ 1 for anyt ≥ 0, whereK is a big enough

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Lemma 4.2. LetX d1α11uu, anduLQTfor the solution to following equation:

ut Δd1α11uu f1, x, t∈Ω×0, T,

∂ηu0, x, tΩ×0, T,

ux,0 u0x≥0, x∈Ω,

4.4

whered111are positive constants and0≤wL2QT. Then there exists a positive constantCT,

depending onu0W1

andu0L∞Ω, such that

XW2,1

2 QTCT. 4.5

Furthermore,

XV2QT,uL2n2/nQT. 4.6

Proof. It is easy to check, fromX d1α11uu, that

Xt d12α11uΔXC1−C2w, 4.7

whereC1 d12α11ua0va1ua2u2 −a3u3andC2 kd12α11uu.C1and C2 are

bounded inQTfrom4.1. Multiplying4.7by−ΔX, and integrating by parts overQt, yields

1 2

Ω|∇Xx, t|

2dx1

2

Ω|∇Xx,0|

2dxd 1

Qt

X|2dx dt

QT

|C1C2wΔX|dx dt.

4.8

Using H ¨older inequality and Young inequality to estimate the right side of4.8, we have

C1C2wL2Q

TΔXL2QTm1

1wL2Q

T

ΔXL2Q

T

m211M3 2

2d1

d1

2 ΔX

2

L2QT

4.9

with somem1>0. Substituting4.9into4.8yields

sup

0≤tT

Ω|∇Xx, t|

2dxd 1

Qt

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where m2 depends on u0W1

2Ω and u0L∞Ω. Since XL

2Q

T, the elliptic regularity estimate10, Lemma 2.3yields

QT |Xxixj|

2dx dtm

3, i, j 1, . . . , n. 4.11

From 4.7, we have XtL2QT. Hence, XW22,1QTCT. Moreover, the Sobolev

embedding theorem shows that4.6holds.

Lemma 4.3Lemma 4.3 can be presented by combining Lemmas 2.3 and 2.4 in11. Letp >

1,p24p/nq1, and letwsatisfy

sup

0≤tT

wL2p/p1ΩwL2QT<, 4.12

and there exist positive constantsβ ∈0,1andCT such that Ω|w·, t|βdxCTtT. Then

there exists a positive constantMindependent ofw but possibly depending onn,Ω,p,β,andCT

such that

wLqQ TM

⎧ ⎨ ⎩1

sup

0≤tT

wtL2p/p1Ω

4p/np1p

w2L/2pQ

T ⎫ ⎬

. 4.13

Finally, one proposes some standard embedding results which are important to obtain the C2α,1α/2Q

Tnormal estimates of the solution for1.6.

Lemma 4.4. LetΩ⊂RNbe a fixed bounded domain and∂Ω⊂C2. Then for alluWq2,1QTwith

q≥1, one has

1∇uLpQ

TCuWq2,1QT, for all1≤pn2q/n2−q, q < n2,

2∇uLpQ

TCuWq2,1QT, for all1≤p≤ ∞, qn2,

3∇uCα,α/2QTCuW2,1

q QT for all1−n2/qα≤1, q > n2,

whereCis a positive constant dependent onq, n, Ω,andT.

The main result about the global existence of nonnegative classical solution for the cross-diffusion system1.6is given as follows.

Theorem 4.5. Assume thatu0 ≥0, v0 ≥ 0,andw0 ≥0satisfy homogeneous Neumann boundary

conditions and belong toCΩfor someλ 0,1. Then system1.6has a unique nonnegative

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Proof.

Step 1. L1-, L2-Estimates andLq-Estimates of w. Firstly, integrating the third equation of1.6 overΩ, we have

d dt

Ωw dx≤ 1 2

Ωu

2dx1

2

Ωw

2dx

≤ |Ω|

2 M

2 0−

1 2|Ω|

Ωw dx 2

.

4.14

Thus

Ωw dx≤max

M0|Ω|,

Ωw0dx :M1,t≥0. 4.15

Furthermore

wL1QT

T

0

M1dt:M2. 4.16

Integrating4.14in0, Tand moving terms yield

wL2QT

M20|Ω|T2w0L

1/2

:M3. 4.17

Secondly, multiplying the third equation of1.6byqwq−1 q >1and integrating over

Ω, we have

d dt

Ωw

qdx≤ −4

q−1d3

q

Ω !!

!∇wq/2!!!2dx−8q

q−1α33

q12

Ω !!

!∇wq1/2!!!2dx

q−1α31

Ω∇u· ∇w

qdxq1α

32

Ω∇v· ∇w qdx

q

Ωw

qbuwdx.

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Integrating the above expression in0, t tTyields

Ωw

qx, tdx4

q−1d3

q

Qt !!

!∇wq/2!!!2dx dt8q

q−1α33

q12

Qt !!

!∇wq1/2!!!2dx dt

Ωw q

0xdx

q−1α31

Qt

u· ∇wqdx dtq−1α32

Qt

v· ∇wqdx dt

q

Qt

wqbuwdx dt.

4.19

Since∇uL2n2/nQ

TfromLemma 4.2, and using H ¨older inequality and Young inequality,

we have

q−1α31

Qt

u· ∇wqdx dtq−1α32

Qt

v· ∇wqdx dt

≤ 2q

q−1α31

q1

!! !! !

Qt

wq−1/2∇wq1/2· ∇u dx dt!!!!!

2q

q−1α32

q1

!! !! !

Qt

wq−1/2∇wq1/2· ∇v dx dt!!!!!

≤ 2q

q−1

q1

α31∇uL2n2/nQ

32∇vL2n2/nQT

× wq−1/2

Ln2QTwq1/2L2QT

C3)))∇

wq1/2)))

L2QTw q−1/2

Ln2QT

C3ε)))∇wq1/2))) 2

L2Q

T

C3

4εw

q−1/22

Ln2QT.

4.20

From4.1andαpβp/p1αp1 1/p1βp1 α, β >0, it holds that

q

Qt

wqbuwdx dtqbM0

Qt

wqdx dtq

Qt

wq1dx

qbM0|QT|1/q1wqLq1Q

tqw q1

Lq1Q

t

≤−bM02|QT|C4.

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Taking w wq1/2 and selecting a proper εsuch that C

3ε ≤ 4qq−1α33/q12, then

applying4.20and4.21to4.19yields

Ωw

2q/q1x, tdx4

q−1d3

q

Qt !!

!∇wq/q1!!!2dx dt4q

q−1α33

q12

Qt

|∇w|2dx dt

w0qLqΩ

C3

4εw

2q−1/q1

Lq−1n2/q1QTC4 ≤C5

1wL2qq−−11/qn2/q11Q T

.

4.22

Denote thatE≡sup0<t<T Ωw2q/q1x, tdx Q

T|∇w|

2dx dt. Then it follows from4.22that

EC6

1wL2qq−−11/qn2/q11Q T

. 4.23

It is easy to see that 2q/q1<2<qandq−1n2/q1<q24q/nq1for any

q < nn4/n24; hence

EC7

1w2LqqQ−1/q1 T

. 4.24

Takeβ2/q1∈0,1. Then it follows fromL1Ω-estimates ofw,namely4.15, that

wLβΩ

Ω|w·, t|

βdx1w1 L1ΩM

1

1 ,tT. 4.25

It follows fromLemma 4.3and4.24that

EC7 ⎡ ⎣1

MMsup

0<t<T

w·, t4Lq/nq2q/q1Ω1qw

2/q

L2Q

T

2q−1/q1⎤ ⎦

C8 ⎡ ⎣1

sup

0<t<T

w·, t2Lq/q2q/q11Ω

4q−1/nq1q

w2L2Q

T

2q−1/q1q ⎤ ⎦

C8

1E2n/nq·2q−1/q1−.

4.26

Since2n/nq·2q−1/q1 ∈ 0,1,Eis bounded by contrary proof. It follows thatwLqQ

T is bounded, that is, wL

q1q/ 2Q

T. It is easy to check that q1q/ 2 ∈

1,2n1/n−2for allq < nn4/n24still denoteq1q/ 2 byq, then

wLqQT,q

1,2n1 n−2

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Finally, we observe thatq2 satisfiesn24q < n24nwithn2,3,4,5. So takeq2 for

4.17and4.19. Then there exists a positive constantM4such that

wV2QTM4. 4.28

Step 2. L∞-Estimates ofw. We rewrite the third equation of1.6as a linear parabolic equation

∂w ∂t

n

i,j1

∂xi

aijx, t∂w

∂xj

n

i1

∂xi

aix, twbuww, 4.29

whereaijx, t d3 α3132v2α33wδij,aix, t α31∂u/∂xi α32∂v/∂xi,δij are Kronecker symbols.

To apply the maximum principle15, Theorem 7.1, page 181to4.15to obtainwLQT, we need to verify that the following conditions hold:1wV2QTis bounded;2 "n

i,j1aijx, tξiξjν "n

i1ξ2i; 3 "n

i1a2ix, t, buwLq,rQ

Tμ1, where ν and μ1 are

positive constants, andqandrsatisfy

1

r n

2q 1−χ, 0< χ <1, q

n

21−χ,

r

1 1−χ,

, n≥2.

4.30

Next we verify conditions1–3in turn. From4.28, condition1is true forn≤5. One can chooseνd3such that condition2holds. To verify condition3, the first equation

of1.6is written in the divergence form

ut∇ ·d12α11uu a0va1ua2u2−a3u3−kuw, 4.31

whered12α11uis bounded inQTbyLemma 4.1, anda0va1ua2u2−a3u3−kuwLqQT forqn2/2,2n1/n−2from4.27. Application of the H ¨older continuity result

15, Theorem 10.1, page 204to4.19yields

uCβ,β/2QT, β∈0,1. 4.32

Returning to4.7, sinceC1C2wLqQTfor anyqn2/2,2n1/n−2by4.1 and4.27, andd1α11uCβ,β/2QTby4.32, then by applying the parabolic regularity theorem15, Theorem 9.1, pages 341-342to4.7we have

XWq2,1QT,q

n2 2 ,

2n1

n−2

. 4.33

Hence∇XLn2q/n2−qQ

TfromLemma 4.4, which shows that∇uLn2q/n2−qQT.

Similarly,∇vLn2q/n2−qQ

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|∇u|2,|∇v|2 ∈ Ln2q/2n2−qQ

T, which imply that "ni1a2ix, tLn2q/2n2−qQT. In addition, buw obviously belongs to LqQ

T. It follows that one can select q r

n2p/2n2−p. Now the above three conditions are satisfied, andwLQTfrom15, Theorem 7.1, page 181. RecallingLemma 4.1, thus there exists a positive constantM5for any

T >0 such that

uLQT, vLQT, wLQTM5. 4.34

Step 3. The Proof of the Classical Solution u, v, wof 1.6 in QTfor Any T > 0. Because

d111u2αt d1 2α11uΔd1u α11u2 d1 2α11uf1, we have from 4.34 that

X d111u2 ∈ Wq2,1QT for any q > 1. So ∇X/2

QT for all β∗ ∈ 0,1. It follows from15, Lemma 3.3, page 80that XC1β∗,1β∗/2Q

T. And direct calculation

X d1α11uuyieldsud1

d2

14α11X/2α11. So we have

uC1β∗,1β∗/2QT,β∗∈0,1. 4.35

The third equation of1.6can be written as

wt∇ ·d3α3132v2α33ww α31∇32∇vwbuww. 4.36

Summarizing the above conclusions that are proved, we know that−buwwLQT andu, v, w,u,vare all bounded inQT. It follows from15, Theorem 10.1,page 204that there existsσ1∈0,1such that

w11/2Q T

. 4.37

The proof ofLemma 4.2is similar. Then we have∇vL2Q

T, that is,vV2QT. Applying the13, Theorem 10.1,page 204to the second equation1.6, there existsσ2∈0,1

such that

v22/2

QT. 4.38

Furthermore, applying Schauder estimate15, page 320-321yieldsvC2σ∗,1σ∗/2Q

Tfor

σ∗min{σ2, λ}. Selectingαmin{σ1, σ∗}and using Sobolev embedding theorem, we have

C2σ∗,1σ∗/2Q

TCα,α/2QT. Still applying Schauder estimate, we have

vC2σ,1σ/2Q

T

, σmin{α, λ}. 4.39

Letw d3α313233ww. Thenwsatisfies

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wherefx, t d3α3132v2α33wbuwwα31utα32vtw. By4.35–4.38, we haved3α3132v2α33w, fx, tCα,α/2QT. So applying Schauder estimate to4.40 yieldswC2σ,1σ/2QT. Sincewd3α3132v

d3α3132v24α33w/2α33,

we have

wC2σ,1σ/2Q

T

, σmin{α, λ}. 4.41

The first equation of1.6can be written as

ut d12α11uΔugx, t, 4.42

wheregx, t 2α11|∇u|2a

0va1ua2u2−a3u3−kuw. By4.35,4.39, and4.41, we

haved12α11u, gx, tCα,α/2QT. So applying Schauder estimate to4.42yields

uC2σ,1σ/2QT, σmin{α, λ}. 4.43

In particular, if λ < α, then σ λ; in other words, Theorem 4.5 is proved. For the caseα < λ, from Sobolev embedding theorem, we have C2σ,1σ/2Q

T Cλ,λ/2QT. Repeating the above bootstrap and Shauder estimate arguments, this completes the proof of

Theorem 4.5. About space dimensionn1, see21.

Theorem 4.6. System 1.6 has the unique positive equilibrium point u, v, w when one of the conditions (i), (ii), (iii), (iv), (v), and (vi) in Section 2 holds. Let the space dimension be n ≤ 5, and let the initial values u0, v0, w0 be nonnegative smooth functions and satisfy the homogenous

Neumann boundary conditions. If the following condition 4.44holds, then the solutionu, v, w of 1.6converges tou, v, winL:

4αud1d2d3> βw

αuα232d12α11M0 α31M02d22α22M0

, 4.44

whereα2/a201/2a3u,andβ ka3uα.

Proof. Define the Lyapunov function

Ht α

Ω

uuulnu

u

dx1

2

Ωvv

2dxβ

Ω

wwwlnw

w

dx, 4.45

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1.6,Htis well posed for allt ≥0 fromTheorem 4.5. According to system1.6, the time derivative ofHtsatisfies

dHt dt

Ω

αu

u2d12α11u|∇u| 2 d

22α22v|∇v|2

βw

w2d3α3132v2α33w|∇w|

2βα31w

wu α32w

wvw

dx − Ω .

αa0 ua2

a3uu

uu2 vv2βww2

−1a0α

u

uuvv βuuww/dx

≤ −

Ω

αu

u2d12α11u|∇u| 2 d

22α22v|∇v|2

βw

w2d3α3132v2α33w|∇w|

2βα31w

wu α32w

wvw

dx − Ω 0 α a0 ua2

Ma0/a21/a2a20/4a2a3u

u

uu2

a0α

u uuvv

1

uvv

2 1 dx − Ω

αa3u

2 uu

2uuvv vv2 dx − Ω

αa3u

2 uu

2βuuww βww2dx.

4.46

It is easy to check that the final three integrands on the right side of the above expression are positive definite because of the electing ofα, β, and the sufficient and necessary conditions of the first integrand being positive definite are the following:

4αβu wd12α11ud22α22vd3α3132v2α33w

> αuβα32w2d12α11u

βα31wu 2

d22α22v.

4.47

Noticing that4.44is the sufficient conditions of4.47, so there exists a positive constant

δ >0 such that

dHt dt ≤ −δ

Ω

uu2 vv2 ww2dx,

dHt

dt <0 u, v, w/ u, v, w.

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Similar to the tedious calculations of dHt/dt, using integration by parts, H ¨older inequality, and 4.34, one can verify that d/dt Ωuu2 vv2 ww2dx is bounded from above. Thus we have from4.48andLemma 3.2inSection 3that

|u·, tu|2−→0, |v·, tv|2−→0, |w·, tw|2 −→0 t−→ ∞. 4.49

In addition,Htis decreasing for t ≥ 0, so we can conclude that the solution u, v, wis globally asymptotically stable. The proof ofTheorem 4.6is completed.

Acknowledgments

The work of this author was partially supported by the Natural Science Foundation of Anhui Province Education Department KJ2009B101 and the NSF of Chizhou CollegeXK0833

caguhh@yahoo.com.cn. The work of this author was partially supported by the China National Natural Science Foundation10871160, the NSF of Gansu Province096RJZA118, and NWNU-KJCXGC-03-47, 61 Foundationsfusm@nwnu.edu.cn.

References

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2 J. B. Zheng, Z. X. Yu, and J. T. Sun, “Existence and uniqueness of limit cycle for prey-predator systems with sparse effect,”Journal of Biomathematics, vol. 16, no. 2, pp. 156–161, 2001.

3 C. Shen and B. Q. Shen, “A necessary and sufficient condition of the existence and uniqueness of the limit cycle for a class of prey-predator systems with sparse effect,”Journal of Biomathematics, vol. 18, no. 2, pp. 207–210, 2003.

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Mathematical Methods in the Applied Sciences, vol. 30, no. 5, pp. 501–511, 2007.

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11 Y. S. Choi, R. Lui, and Y. Yamada, “Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion,”Discrete and Continuous Dynamical Systems, vol. 10, no. 3, pp. 719–730, 2004.

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