Model with Stage Structure and Nonlinear Density Restriction-I: The Case in R

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

Rui Zhang,

Ling Guo,

and Shengmao Fu

1Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China

2Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Correspondence should be addressed to Shengmao Fu,fusm@nwnu.edu.cn 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− EX²₁X3

1FX²₁,

X₂ CX1−r2X2,

X₃ −r3X₃−η₃X²₃AX₃ EX₁² 1FX₁²,

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 αu²w 1u² ≡f3,

1.2

whereβ BC/r₂², 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− u²w

1u², x∈Ω, t >0, v_t−d₂Δvu−v, x∈Ω, t >0,

wt−d3Δw−kw−γw² αu²w

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 .

In the recent years there has been considerable interest to investigate the global behavior for models of interacting populations with linear density restriction by taking into

account the effect of self-as well as cross-diffusion18–26. In this paper we are led to the following cross-diffusion system:

u_t 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 αu²w

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. α_ii 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 24

β−a c² 3b²4c

β−a−c

−b√p1

, 2.1 wherep19b²24cβ−a−c ≥0. Let one curvel1:g1u 1u² /u β−a−bu−cu² , and the other curvel₃:g₃u kγwαu²/1u² . Obviously,l₁passes the pointm0,0 . Noting

thatβ−a−c u²−2bu³−3cu⁴−βaattains its maximum atu √p1−3b /12c, thus when β−a−c /2b²/8c≤b√p₁/24c24β−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 1a λa−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 − m²₀ 1m²₀

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 , C1a33

a11β , 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

⎞

⎟⎟

⎠, 2.5

k α−k

m0

O u

α−k γ

w

l₃ E^∗

l1

Figure 1

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−a11a33

a11β

−a33

1β

−

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

hold, then the equilibrium pointE^∗of 1.2 is globally asymptotically stable.

(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^∗

β

v−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

21u^{2 2} − uu^{∗ 2} 21u^{∗2 2}1u^{2 2} uuu^∗

1u^∗2 1u^{2 2}

− γ

α−1 2

w−w^{∗ 2}.

2.10 When u ∈ 0,∞ , the minimum of 1−u^∗u /1u² anduuu^∗ /1u^{2 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−m_{0 2}

−

bcucm₀ u−m₀ ² c αw²−w

m₀u 1u² −k

α

.

2.12

Noting that the maximum ofu/1u² is 1/2, andm0<2k/α, we findm0u/1u² −k/α <0.

Therefore,Et ≤0.

iii Let

Et uβv 1

αw, 2.13

then

Et β−a

u−bu²−cu³−k αw−γ

αw². 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,∞

C^2,1Ω×0,∞ ³, and

0≤u≤M1max

⎧⎪

⎨

⎪⎩sup

Ω u0,sup

Ω v0,

b²4c β−a

−b 2c

⎫⎪

⎬

⎪⎭,

0≤v≤M₂M₁,

0≤w≤M₃max

⎧⎪

⎨

⎪⎩sup

Ω w₀, αM1 2

γ 1M1

2− k γ

⎫⎪

⎬

⎪⎭

3.1

onΩ×0,∞ . In particular, ifu₀, v₀, w₀≥/≡ 0, thenu, v, w >0 for 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. In addition,0,0,0 andM₁,M₂,M₃ are a pair of lower- upper solutions of problem1.3 cf. M1,M2,M3 in3.1 . From29, Theorem 5.3.4, we conclude that1.3 exists a unique classical solutionu, v, w satisfying3.1 . According to strong maximum principle, it follows thatux, t , vx, t , wx, t > 0, ∀t > 0, x ∈ Ω. So the proof of the Theorem is completed.

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,sup

Ω v0,β−a b

"

,

M₃max

⎧⎪

⎨

⎪⎩sup

Ω

w₀, αM₁² γ

1M₁²− 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∈$

C¹ Ω%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 F_UE^∗ 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^∗.

3.5

The linearization of1.3 isUt LUat E^∗. For eachi ≥ 1,Xi is invariant under the operator L, andλis an eigenvalue of L onX_i, if and only ifλis an eigenvalue of the matrix

−μiDF_UE^∗ . The characteristic equation isϕ_iλ λ³A_iλ²B_iλC_i0, where Aiμid1d2d3 −a11−a331,

B_iμ²_id1d₂d₁d₃d₂d₃

μid11−a33 −d2a11a33 d31−a11

a₁₁a₃₃−a₁₃a₃₁−a₃₃−

a₁₁β , Ciμ³_id1d2d3μ²_id1d3−a33d1d2−a11d2d3

−μ_i

d₁a₃₃−d₂a11a₃₃−a₁₃a₃₁ d₃

a₁₁β a33

a11β

−a13a31,

H_iA_iB_i−C_iP₃μ³_i P₂μ²_i P₁μ_iP₀,

P3 d1d2 d1d2d1d3d2d3 d²₃d1d2 ,

P₂ d₁d₂d₃ d11−a₃₃ −d₂a11a₃₃ d₃1−a₁₁

−a11d1d2d3 d2d1d3 −a33d3d1d2 , P1d1

a11a33−a13a31−

a11β

−d2

a11β a33

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 byλ_i,1,λ_i,2,λ_i,3 all have negative real parts if and only ifA_i>0, C_i>0, H_i>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

ϕ_iλ μ³_iξ³_i A_iμ²_iξ²_i B_iμ_iξC_i ϕ)_iξ . 3.8 Sinceμi → ∞asi → ∞, it follows that

ilim→ ∞

) ϕ_iξ

μ³_i ξ³ d₁d₂d₃ ξ² d₁d₂d₂d₃d₁d₃ ξd₁d₂d₃ϕξ .) 3.9

Clearly, ϕξ ) has the three roots−d1,−d2,−d3. Letd min{d1, d2, d3}. By continuity, there existsi₀such that the three rootsξ_i1, ξ_i2, ξ_i3ofϕ)_iξ 0 satisfy

Re{ξi1},Re{ξi2},Re{ξi3} ≤ −d

2, i≥i0. 3.10 Let −δ) max_0≤i≤i0{Re{λi1},Re{λi2},Re{λi3}}, then δ >) 0. Let δ min{δ, d/2}, then) 3.7 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 − m²₀ 1m²₀, a₂₁ 1, a₂₂−1, a₂₃0,

a₃₁0, a₃₂0, a₃₃ −k αm²₀ 1m²₀.

3.11

The characteristic equation of−μiDF_UE1 isϕ_iλ λ³A_iλ²B_iλC_i0, where A_iμ_id1d₂d₃ −a₁₁−a₃₃1,

Biμ²_id1d2d1d3d2d3

μ_id11−a₃₃ −d₂a11a₃₃ d₃1−a₁₁ a11a33−a33−

a11β ,

C_iμ³_id₁d₂d₃μ²_id1d₃−a₃₃d₁d₂−a₁₁d₂d₃

−μi

d1a33−d2a11a33d3

a11β a33

a11β , H_iA_iB_i−C_iP₃μ³_i P₂μ²_i P₁μ_iP₀,

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−a11a33

a11β

−a33

1β

−

a11β .

3.12

The three roots ofϕiλ 0 all have negative real parts fora11β <0 anda33<0. Namely,E1

is the locally asymptotically stable, ifm²₀α−k < kandm₀>

b²3cβ−a −b /3c.

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−a ²< 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. Leta, 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 lim_{t→ ∞}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, m²₀α−k < k, and

b²3cβ−a −b /3c < m0 <2k/αhold, then the equilibrium pointE₁of system1.3 is globally asymptotically stable.

iii Assume thatβ < aand k > αhold, then the equilibrium point E0 of 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

21u^{2 2} − uu^{∗ 2} 21u^{∗2 2}1u^{2 2} uuu^∗

1u^∗2 1u^{2 2}

dx−γ α−1

2

*

Ωw−w^{∗ 2}dx.

3.17

It is not hard to verify that

Et ≤ −l1

*

Ωu−u^{∗ 2}dx−l₃

*

Ωw−w^{∗ 2}dx, 3.18 if3.14 hold. ApplyingLemma 3.5, we can obtain

t→ ∞lim

*

Ωu−u^{∗ 2}dx0, lim

t→ ∞

*

Ωw−w^{∗ 2}dx0. 3.19

RecomputingEt , we find Et ≤ −

*

Ω

d1u^∗

u² |∇u|²βd2v^∗

v² |∇v|²d3w^∗ αw² |∇w|²

dx

≤ −C

*

Ω

|∇u|²|∇v|²|∇w|²

dx−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,

t→ ∞lim

*

Ω

|∇u|²|∇v|²|∇w|²

dx0. 3.21

Using the Pioncar´e inequality, we have

t→ ∞lim

*

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^{∗ 2}dx≤2

*

Ωu−u ²dx2

*

Ωu−u^{∗ 2}dx,

|Ω||wt −w^∗|²

*

Ωw−w^{∗ 2}dx≤2

*

Ωw−w ²dx2

*

Ωw−w^{∗ 2}dx,

3.23

according to3.19 and3.22 , we can see

ut → u^∗, wt → w^∗ t → ∞ . 3.24 Thus, there exists{tm}, utm → 0 ast_m → ∞. 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^∗ c

u²uu^∗u^∗2

u−u^∗ −duw−w^∗ % dx--

--tm

. 3.26

According to3.19 to compute the limit of the previous equation and using the uniqueness of the limit, we havev,v^∗, and

tmlim→ ∞vtm v^∗. 3.27

It follows from 3.15 that there exists a subsequence of {tm}, denoted still by {tm}, and nonnegative functionsg_i ∈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^∗ inC²

Ω

. 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 αw²−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

u²βv²w²

dx. 3.32

Then

Et −

*

Ω

d1|∇u|²βd2|∇v|²d3|∇w|² dx

*

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

2X²₁

12X₁², x∈Ω, t >0, X_2t−D₂ΔX21.4X₁−X₂, x∈Ω, t >0,

X_3t−D₃ΔX3−X3−√

2X²₃X₃ 2X²₁

12X₁², x∈Ω, t >0,

∂ηXi0, i1,2,3, x∈∂Ω, t >0, X_ix,0 X_i0x ≥0, i1,2,3, x∈Ω.

3.36

Using the software Matlab, one can obtainu^∗v^∗1.1274,w^∗0.1199. It is easy to see that the previous system satisfies the all conditions ofTheorem 3.6i . So the positive equilibrium point0.5637,0.5637,0.1199 of the previous system is globally asymptotically stable.

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 ∈W_p¹Ω , p > n, then1.4 has a unique nonnegative solutionu, v, w∈ C0, T , W_p¹Ω

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 W_p¹Ω ,v·, t W_p¹Ω ,w·, t W_p¹Ω : 0< t < T(

<∞, 4.1

thenT ∞. If, in addition,u₀, v₀, w₀∈W_p²Ω , thenu, v, w∈C0,∞ , W_p²Ω .

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 ∈ C^2λΩ 0 < λ < 1 are nonnegative functions and satisfy zero Neumann boundary conditions.

Then1.4 has a unique nonnegative solutionu, v, w∈C^2λ,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 Q_T≡Ω×0, T , sup

0<t<T

u·, t _L¹_Ω ,sup

0<t<T

v·, t _L¹_Ω ≤C₁T , u_L2QT ,v_L2QT ≤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

u aβ

v

dx≤ −a

*

Ωv dx

*

Ω

βu−bu²

dx. 4.3

Using Young inequality and H ¨older inequality, we have

*

Ω

βu−bu²

dx≤C_2,1− a aβ

*

Ωu dx, 4.4

whereC2,1 1/4b βa/aβ ²|Ω|.It follows from4.3 and4.4 that d

dt

*

Ω

u aβ

v

dx≤C_2,1− a aβ

*

Ω

u aβ

v

dx. 4.5

Thus,

u·, t _L1Ω ,v·, t _L1Ω ≤C_2,2, 4.6 whereC2,2depends onv0_L1Ω ,u0_L1Ω and coefficients of1.4 . In addition, there exists a positive constantC₁T , such that

sup

0<t<T

u·, t _L¹_Ω ,sup

0<t<T

v·, t _L¹_Ω ≤C₁T . 4.7

Integrating the first equation of1.4 overΩ, we have d

dt

*

Ωu dx≤β

*

Ωv dx−b

*

Ωu²dx. 4.8

Integrating4.8 from 0 toT, we have

*

Ωux, T dx−

*

Ωux,0 dx≤β

*_T

0

*

Ωv dx dt−b

*_T

0

*

Ωu²dx dt. 4.9 According to4.7 , there exists a positive constantC₂T , such that

u_L2QT ≤C₂T . 4.10 Multiplying the second equation of1.4 byvand integrating it overΩ, we obtain

1 2

d dt

*

Ωv²dx−

*

Ωd22α₂₂v |∇v|²dx

*

Ω

uv−v² dx

≤ 1 2

*

Ωu²dx− 1 2

*

Ωv²dx.

4.11

Integrating the previous inequation from 0 toT, we have

v_L2QT ≤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 onw0_W¹

2Ω andw0_L∞Ω , such that w1_W^2,1

2 Qτ ≤C₃τ . 4.13 Furthermore∇w1∈V₂Qτ and∇w1∈L^{2n2 /n}Qτ .