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,**

^{1}**Ling Guo,**

^{2}**and Shengmao Fu**

^{2}*1**Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China*

*2**Department 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 spaceR* ^{n}*. 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:

*X*_{1}^{} *BX*_{2}−*r*_{1}*X*_{1}−*CX*_{1}−*η*_{1}*X*_{1}^{2}−*η*_{2}*X*_{1}^{3}− *EX*^{2}_{1}*X*3

1*FX*^{2}_{1}*,*

*X*_{2}^{} *CX*1−*r*2*X*2*,*

*X*_{3}^{} −r3*X*_{3}−*η*_{3}*X*^{2}_{3}*AX*_{3} *EX*_{1}^{2}
1*FX*_{1}^{2}*,*

1.1

where*X*1t ,*X*2t denote the density of the immature and mature population of the prey,
respectively, *X*_{3}t is the density of the predator. For the prey, the immature population is
nonlinear density restriction.*X*_{3}is assumed to consume*X*_{1}with Holling type III functional
response*EX*_{1}^{2}*/1FX*^{2}_{1} and contributes to its growth with rate*AEX*^{2}_{1}*/1FX*^{2}_{1} . For more
details on the backgrounds of this model see references11,12.

Using the scaling *u* √

*FX*_{1}*, v* r2

√*F/C X*2*, w* E/r2

√*F X*3*, dτ* *r*_{2}*dt*and
redenoting*τ*by*t, we can reduce the system*1.1 to

*u*^{}*βv*−*au*−*bu*^{2}−*cu*^{3}− *u*^{2}*w*
1*u*^{2} ≡*f*_{1}*,*
*v*^{}*u*−*v*≡*f*2*,*

*w*^{}−kw−*γw*^{2} *αu*^{2}*w*
1*u*^{2} ≡*f*3*,*

1.2

where*β* *BC/r*_{2}^{2}*, a* r1*C /r*2*, b* *η*_{1}*/r*_{2}√

*F, c* *η*_{2}*/r*_{2}*F, k* *r*_{3}*/r*_{2}*, α* *AE/r*_{2}*F, γ*
*η*3

√*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:

*u**t*−*d*1Δu*βv*−*au*−*bu*^{2}−*cu*^{3}− *u*^{2}*w*

1*u*^{2}*,* *x*∈Ω, t >0,
*v** _{t}*−

*d*

_{2}Δv

*u*−

*v,*

*x*∈Ω, t >0,

*w**t*−*d*3Δw−kw−*γw*^{2} *αu*^{2}*w*

1*u*^{2}*,* *x*∈Ω, t >0,

*∂*_{η}*u∂*_{η}*v∂*_{η}*w*0, *x*∈*∂Ω, t >*0,

*ux,*0 *u*_{0}x ≥0, *vx,*0 *v*_{0}x ≥0, *wx,*0 *w*_{0}x ≥0, *x*∈Ω,

1.3

whereΩis a bounded domain inR* ^{n}*with smooth boundary

*∂Ω,η*is the outward unit normal vector on

*∂Ω, and∂*

_{η}*∂/∂η.u*

_{0}x , v0x , w0x are nonnegative smooth functions on Ω.

The diﬀusion coeﬃcients*d**i* 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}* Δ

*d*

_{1}

*α*

_{11}

*uα*

_{13}

*w u βv*−

*au*−

*bu*

^{2}−

*cu*

^{3}−

*u*

^{2}

*w*

1*u*^{2}*,* *x*∈Ω, t >0,
*v**t* Δ*d*2*α*22*v v u*−*v,* *x*∈Ω, t >0,

*w**t* Δd3*α*33*w w*−*kw*−*γw*^{2} *αu*^{2}*w*

1*u*^{2}*,* *x*∈Ω, t >0,

*∂u*

*∂ν* *∂v*

*∂ν* *∂w*

*∂ν* 0, *x*∈*∂Ω, t >*0,

*ux,*0 *u*_{0}x ≥0, *vx,*0 *v*_{0}x ≥0, *wx,*0 *w*_{0}x ≥0, *x*∈Ω,

1.4

where*d*_{1}*, d*_{2}*, d*_{3} are the diﬀusion rates of the three species, respectively. *α** _{ii}* i 1,2,3 are
referred as self-diﬀusion pressures, and

*α*13is 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. For

*α*

*ij*

*/*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 *E*_{0} 0,0,0 . If *β > a, then*1.2 has semitrivial equilibria*E*_{1}m0*, m*_{0}*,*0 , where *m*_{0}

*b*^{2}4cβ−*a *−*b /2c. To discuss the existence of the positive equilibrium point of*1.2 ,
we give the following assumptions:

*α > k,* *β > a,*

*k*

*α*−*k* *< m*_{0}*,* *β*−*a*−*c*
2 *b*^{2}

8c ≤ *b*√*p*_{1}

24c 24

*β*−*a*
*c*^{2}
3b^{2}4c

*β*−*a*−*c*

−*b*√*p*1

*,*
2.1
where*p*19b^{2}24cβ−*a*−*c *≥0. Let one curve*l*1:*g*1u 1*u*^{2} /u β−*a*−*bu*−*cu*^{2} , and
the other curve*l*_{3}:*g*_{3}u *k*γw*αu*^{2}*/1*u^{2} . Obviously,*l*_{1}passes the pointm0*,*0 . Noting

thatβ−*a*−*c u*^{2}−2bu^{3}−3cu^{4}−*βa*attains its maximum at*u* √*p*1−3b /12c, thus when
β−a−c /2b^{2}*/8c*≤*b*√*p*_{1}*/24c24β−a c*^{2}*/3b*^{2}4cβ−a−c −b√*p*_{1} ,*g*_{1}^{}u *<*00*< u < m*_{0} .
*l*3 has the asymptote*w* *α*−*k/γ* and passes the point

*k/α*−*k,*0 . In this case, *l*1 and
*l*_{3} have unique intersection u^{∗}*, w*^{∗} , as shown in Figure 1. *E*^{∗} u^{∗}*, v*^{∗}*, w*^{∗} is the unique
positive equilibrium point of1.2 , where*v*^{∗} *u*^{∗},*w*^{∗} 1*u*^{∗2} /u^{∗} β−*a*−*bu*^{∗}−*cu*^{∗2} ,
*k*γw^{∗}*αu*^{∗2}*/1*u^{∗2} . In addition, the restriction of the existence of the positive equilibrium
can be removed, if*β < ac.*

The Jacobian matrix of the equilibrium*E*0is

*J*E0

⎛

⎜⎜

⎝

−a β 0 1 −1 0 0 0 −k

⎞

⎟⎟

⎠*.* 2.2

The characteristic equation of*JE*0 isλ*k λ*^{2} 1*a λa*−*β * 0.*E*0 is a saddle for
*β > a. In addition, the dimensions of the local unstable and stable manifold ofE*0are 1 and 2,
respectively.*E*_{0}is locally asymptotically stable for*β < a.*

The Jacobian matrix of the equilibrium*E*1is

*JE*1

⎛

⎜⎜

⎜⎜

⎝

*a*_{11} *β* − *m*^{2}_{0}
1*m*^{2}_{0}

1 −1 0

0 0 *a*33

⎞

⎟⎟

⎟⎟

⎠*,* 2.3

where*a*_{11} −a−2bm_{0}−3cm^{2}_{0}*, a*_{33} −k*αm*^{2}_{0}*/1m*^{2}_{0} . The characteristic equation of*JE*1

is*λ*^{3}*A*1*λ*^{2}*B*1*λC*10, where

*A*_{1}−a11−*a*_{33}1,
*B*_{1}*a*_{11}*a*_{33}−*a*_{33}−

*a*_{11}*β*
*,*
*C*1*a*33

*a*11*β*
*,*
*H*_{1} *A*_{1}*B*_{1}−*C*_{1} *a*_{11}*a*_{33}

*a*_{33}−*a*_{11}*a*_{33}

*a*_{11}*β*

−*a*_{33}
1*β*

−

*a*_{11}*β*
*.*

2.4

According to Routh-Hurwitz criterion,*E*1is locally asymptotically stable for*a*11*β <*0 and
*a*_{33}*<*0, that is,*m*^{2}_{0}α−*k < k*and*m*_{0}*>*

*b*^{2}3cβ−*a *−*b /3c.*

The Jacobian matrix of the equilibrium*E*^{∗}is

*JE*^{∗}

⎛

⎜⎜

⎝

*a*11 *β a*13

1 −1 0
*a*31 0 *a*33

⎞

⎟⎟

⎠*,* 2.5

*k*
*α*−*k*

*m*0

O *u*

*α*−*k*
*γ*

*w*

*l*_{3}
*E*^{∗}

*l*1

**Figure 1**

where

*a*11−a−2bu^{∗}−3cu^{∗2}− 2u^{∗}*w*^{∗}

1*u*^{∗2} ^{2}*,* *a*13 − *u*^{∗2}
1*u*^{∗2}*,*
*a*31 2αu^{∗}*w*^{∗}

1*u*^{∗2} ^{2}*,* *a*33−γw^{∗}*.*

2.6

The characteristic equation of*J*E^{∗} is*λ*^{3}*A*2*λ*^{2}*B*2*λC*20, where
*A*_{2}−a11−*a*_{33}1,

*B*2 *a*11*a*33−*a*13*a*31−*a*33−
*a*11*β*

*,*
*C*_{2}*a*_{33}

*a*_{11}*β*

−*a*_{13}*a*_{31}*,*
*H*2*A*2*B*2−*C*2 *a*11*a*33

*a*13*a*31*a*33−*a*11*a*33

*a*11*β*

−*a*33

1*β*

−

*a*11*β*
*.*
2.7
According to Routh-Hurwitz criterion, *E*^{∗} is locally asymptotically stable for *a*_{11} *β <* 0.

Obviously,*a*11*β <*0 can be checked by2.1 .

Now we discuss the global stability of equilibrium points for1.2 .
* Theorem 2.1. (i) Assume that*2.1 ,

*bcu*^{∗}−*u*^{∗}

*β*−*a*−*bu*^{∗}
22√

1*u*^{∗2} *>*

√

*u*^{∗2}1*u*^{∗}_{2}
8u^{∗2}1 ^{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, m*^{2}_{0}α−*k < k, and*

*b*^{2}3cβ−*a *−*b /3c < m*0 *<* 2k/α*hold,*
*then the equilibrium pointE*_{1}*of* 1.2 *is globally asymptotically stable.*

*(iii) Assume thatβ*≤*aholds, then the equilibrium pointE*_{0}*of* 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 of*Et *along the positive solution of1.2 , we have

*E*^{}t −*β*
*u*^{∗}

*v*

*u*u−*u*^{∗} −
*u*

*v*v−*v*^{∗}
2

−u−*u*^{∗} ^{2}

*bcucu*^{∗} *w*^{∗}1−*u*^{∗}*u *
1*u*^{∗2} 1*u*^{2}

− *c*

*α*w−*w*^{∗} ^{2} u−*u*^{∗} w−*w*^{∗}

*u*^{∗}*u*

1*u*^{∗2} 1*u*^{2} − *u*
1*u*^{2}

≤ −u−*u*^{∗} ^{2}

*bcucu*^{∗} *w*^{∗}1−*u*^{∗}*u *

1*u*^{∗2} 1*u*^{2} − *u*^{2}

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

1*u*^{∗2} 1*u*^{2} ^{2}

−
*γ*

*α*−1
2

w−*w*^{∗} ^{2}*.*

2.10
When *u* ∈ 0,∞ , the minimum of 1−*u*^{∗}*u /1u*^{2} and*uuu*^{∗} /1*u*^{2} ^{2} is−u^{∗2}*/2*
2√

1*u*^{∗2} and 0, respectively; the maximum ofu*u*^{∗} /1*u*^{2} is *u/1u*^{2} are u^{∗}

√1*u*^{∗2} /2 and 1/2, respectively. Thus, when 2.8 hold, *E*^{}t ≤ 0. According to the
Lyapunov-LaSalle invariance principle28,*E*^{∗}is globally asymptotically stable if2.1 –2.3
hold.

ii Let

*Et *

*u*−*m*0−*m*0ln *u*
*m*_{0}

*β*

*v*−*m*0−*m*0ln *v*
*m*_{0}

1

*αw.* 2.11
Then

*E*^{}t − *β*
*m*0

*v*

*u*u−*m*_{0} −
*u*

*v*v−*m*_{0}
_{2}

−

b*cucm*_{0} u−*m*_{0} ^{2} *c*
*αw*^{2}−*w*

*m*_{0}*u*
1*u*^{2} −*k*

*α*

*.*

2.12

Noting that the maximum of*u/1u*^{2} is 1/2, and*m*0*<*2k/α, we find*m*0*u/1*u^{2} −*k/α <*0.

Therefore,*E*^{}t ≤0.

iii Let

*Et uβv* 1

*αw,* 2.13

then

*E*^{}t
*β*−*a*

*u*−*bu*^{2}−*cu*^{3}−*k*
*αw*−*γ*

*αw*^{2}*.* 2.14

Thus,*E*^{}t ≤0 for*β*≤*a. This completes the proof of*Theorem 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. Letu*0x , v0x , w0x

*be nonnegative smooth functions on*Ω. Then system1.3

*has a unique nonnegative solution*ux, t , vx, t , wx, t ∈CΩ×0,∞

*C*^{2,1}Ω×0,∞ ^{3}*,*
*and*

0≤*u*≤*M*1max

⎧⎪

⎨

⎪⎩sup

Ω *u*0*,*sup

Ω *v*0*,*

*b*^{2}4c
*β*−*a*

−*b*
2c

⎫⎪

⎬

⎪⎭*,*

0≤*v*≤*M*_{2}*M*_{1}*,*

0≤*w*≤*M*_{3}max

⎧⎪

⎨

⎪⎩sup

Ω *w*_{0}*,* *αM*1
2

*γ*
1*M*1

2− *k*
*γ*

⎫⎪

⎬

⎪⎭

3.1

*on*Ω×0,∞ . In particular, if*u*_{0}*, v*_{0}*, w*_{0}≥*/*≡ 0, then*u, v, w >0 for allt >*0, x∈Ω.

*Proof. It is easily seen that* f1*, f*_{2}*, f*_{3} is suﬃciently smooth in R^{3}_{} and possesses a mixed
quasimonotone property inR^{3}_{}. In addition,0,0,0 and*M*_{1}*,M*_{2}*,M*_{3} are a pair of lower-
upper solutions of problem1.3 cf. *M*1*,M*2*,M*3 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 that*ux, 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 *η*2 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

*M*1*M*2max

! sup

Ω *u*0*,*sup

Ω *v*0*,β*−*a*
*b*

"

*,*

*M*_{3}max

⎧⎪

⎨

⎪⎩sup

Ω

*w*_{0}*,* *αM*_{1}^{2}
*γ*

1*M*_{1}^{2}− *k*
*γ*

⎫⎪

⎬

⎪⎭*.*

3.2

Now we show the local and global stability of constant equilibrium solutions*E*_{0}*, E*_{1}*, 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,m*^{2}_{0}α−*k * *< k, andm*0 *>*

*b*^{2}3cβ−*a *−*b/3chold, then the*
*equilibrium pointE*_{1}*of*1.3 *is locally asymptotically stable.*

iii *Assume thatβ < aholds, then the equilibrium pointE*_{0}*of* 1.3 *is locally asymptotically*
*stable.*

*Proof. Let 0μ*_{1} *< μ*_{2} *< μ*_{3} *<*· · · be the eigenvalues of the operator−ΔonΩwith Neumann
boundary condition, and let*Eμ**i* be the eigenspace corresponding to*μ**i*in*C*^{1}Ω . Let

*X*

#
*U*∈$

*C*^{1}
Ω%_{3}

*, ∂**η**U*0, x∈*∂Ω*

&

*,* *X**ij*'

*c*·*φ**ij* :*c*∈R^{3}(

*,* 3.3

where{φ*ij*;*j*1, . . . ,dim*Eμ**i* }is an orthonormal basis of*Eμ**i* , then

*X*⊕^{∞}_{i1}*X**i**,* *X**i*⊕^{dimE}^{μ}^{i}

*j1* *X**ij**.* 3.4

i Let*D*diagd1*, d*_{2}*, d*_{3} ,*LDΔ F** _{U}*E

^{∗}

*DΔ*{a

*ij*}, where

*a*_{11} −a−2bu^{∗}−3cu^{∗2}− 2u^{∗}*w*^{∗}

1*u*^{∗2} ^{2}*,* *a*_{12}*β,* *a*_{13} − *u*^{∗2}
1*u*^{∗2}*,*
*a*21 1, *a*22−1, *a*230,

*a*_{31} 2αu^{∗}*w*^{∗}

1*u*^{∗2} ^{2}*,* *a*_{32}0, *a*_{33}−γw^{∗}*.*

3.5

The linearization of1.3 is*U**t* *LU*at *E*^{∗}. For each*i* ≥ 1,*X**i* is invariant under the
operator L, and*λ*is an eigenvalue of L on*X** _{i}*, if and only if

*λ*is an eigenvalue of the matrix

−μ*i**DF** _{U}*E

^{∗}. The characteristic equation is

*ϕ*

*λ*

_{i}*λ*

^{3}

*A*

_{i}*λ*

^{2}

*B*

_{i}*λC*

*0, where*

_{i}*A*

*i*

*μ*

*i*d1

*d*2

*d*3 −

*a*11−

*a*331,

*B*_{i}*μ*^{2}* _{i}*d1

*d*

_{2}

*d*

_{1}

*d*

_{3}

*d*

_{2}

*d*

_{3}

*μ**i*d11−*a*33 −*d*2a11*a*33 *d*31−*a*11

*a*_{11}*a*_{33}−*a*_{13}*a*_{31}−*a*_{33}−

*a*_{11}*β*
*,*
*C**i**μ*^{3}_{i}*d*1*d*2*d*3*μ*^{2}* _{i}*d1

*d*3−

*a*33

*d*1

*d*2−

*a*11

*d*2

*d*3

−*μ*_{i}

*d*_{1}*a*_{33}−*d*_{2}a11*a*_{33}−*a*_{13}*a*_{31} *d*_{3}

*a*_{11}*β*
*a*33

*a*11*β*

−*a*13*a*31*,*

*H*_{i}*A*_{i}*B** _{i}*−

*C*

_{i}*P*

_{3}

*μ*

^{3}

_{i}*P*

_{2}

*μ*

^{2}

_{i}*P*

_{1}

*μ*

_{i}*P*

_{0}

*,*

*P*3 d1*d*2 d1*d*2*d*1*d*3*d*2*d*3 *d*^{2}_{3}d1*d*2 ,

*P*_{2} *d*_{1}*d*_{2}*d*_{3} d11−*a*_{33} −*d*_{2}a11*a*_{33} *d*_{3}1−*a*_{11}

−*a*11*d*1d2*d*3 *d*2d1*d*3 −*a*33*d*3d1*d*2 ,
*P*1*d*1

*a*11*a*33−*a*13*a*31−

*a*11*β*

−*d*2

*a*11*β*
*a*33

*d*_{3}a11*a*_{33}−*a*_{33}−*a*_{13}*a*_{31}

−a11*a*33−1 d11−*a*33 −*d*2a11*a*33 *d*31−*a*11 ,
*P*_{0} *a*_{11}*a*_{33}

*a*_{13}*a*_{31}*a*_{33}−*a*_{11}*a*_{33}

*a*_{11}*β*

−*a*_{33}
1*β*

−

*a*_{11}*β*
*.*

3.6

From Routh-Hurwitz criterion, we can see that three eigenvaluesdenoted by*λ** _{i,1}*,

*λ*

*,*

_{i,2}*λ*

*all have negative real parts if and only if*

_{i,3}*A*

_{i}*>*0, C

_{i}*>*0, H

_{i}*>*0. Noting that

*a*

_{11}

*, a*

_{13}

*, a*

_{33}

*<*0, a

_{31}

*>*

0, we must have*a*11*β <*0. It is easy to check that*a*11*β <*0 if*g*_{1}^{}u1 *<*0seeSection 2 .
We can conclude that there exists a positive constant*δ, such that*

Re{λ*i,1*},Re{λ*i,2*},Re{λ*i,3*} ≤ −δ, *i*≥1. 3.7

In fact, let*λμ**i**ξ, then*

*ϕ** _{i}*λ

*μ*

^{3}

_{i}*ξ*

^{3}

_{i}*A*

_{i}*μ*

^{2}

_{i}*ξ*

^{2}

_{i}*B*

_{i}*μ*

_{i}*ξC*

_{i}*ϕ*)

*ξ . 3.8 Since*

_{i}*μ*

*i*→ ∞as

*i*→ ∞, it follows that

*i*lim→ ∞

)
*ϕ** _{i}*ξ

*μ*^{3}_{i}*ξ*^{3} *d*_{1}*d*_{2}*d*_{3} ξ^{2} *d*_{1}*d*_{2}*d*_{2}*d*_{3}*d*_{1}*d*_{3} ξ*d*_{1}*d*_{2}*d*_{3}*ϕξ .*) 3.9

Clearly, *ϕξ *) has the three roots−d1*,*−d2*,*−d3. Let*d* min{d1*, d*2*, d*3}. By continuity, there
exists*i*_{0}such that the three roots*ξ*_{i1}*, ξ*_{i2}*, ξ** _{i3}*of

*ϕ*)

*ξ 0 satisfy*

_{i}Re{ξ*i1*},Re{ξ*i2*},Re{ξ*i3*} ≤ −*d*

2*,* *i*≥*i*0*.* 3.10
Let −*δ*) max_{0≤i≤i}_{0}{Re{λ*i1*},Re{λ*i2*},Re{λ*i3*}}, 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 of*E*^{∗}.

ii The linearization of1.4 is*U**t**LU*at*E*1, where*LDΔ F**U*E1 *DΔ *{a*ij*},
and

*a*11 −a−2bm0−3cm^{2}_{0}*,* *a*12*β,* *a*13 − *m*^{2}_{0}
1*m*^{2}_{0}*,*
*a*_{21} 1, *a*_{22}−1, *a*_{23}0,

*a*_{31}0, *a*_{32}0, *a*_{33} −k *αm*^{2}_{0}
1*m*^{2}_{0}*.*

3.11

The characteristic equation of−μ*i**DF** _{U}*E1 is

*ϕ*

*λ*

_{i}*λ*

^{3}

*A*

_{i}*λ*

^{2}

*B*

_{i}*λC*

*0, where*

_{i}*A*

_{i}*μ*

*d1*

_{i}*d*

_{2}

*d*

_{3}−

*a*

_{11}−

*a*

_{33}1,

*B**i**μ*^{2}* _{i}*d1

*d*2

*d*1

*d*3

*d*2

*d*3

*μ** _{i}*d11−

*a*

_{33}−

*d*

_{2}a11

*a*

_{33}

*d*

_{3}1−

*a*

_{11}

*a*11

*a*33−

*a*33−

*a*11*β*
*,*

*C*_{i}*μ*^{3}_{i}*d*_{1}*d*_{2}*d*_{3}*μ*^{2}* _{i}*d1

*d*

_{3}−

*a*

_{33}

*d*

_{1}

*d*

_{2}−

*a*

_{11}

*d*

_{2}

*d*

_{3}

−*μ**i*

*d*1*a*33−*d*2*a*11*a*33*d*3

*a*11*β*
*a*33

*a*11*β*
*,*
*H*_{i}*A*_{i}*B** _{i}*−

*C*

_{i}*P*

_{3}

*μ*

^{3}

_{i}*P*

_{2}

*μ*

^{2}

_{i}*P*

_{1}

*μ*

_{i}*P*

_{0}

*,*

*P*3 d1*d*2 d1*d*2*d*1*d*3*d*2*d*3 *d*^{2}_{3}d1*d*2 ,

*P*_{2} *d*_{1}*d*_{2}*d*_{3} d11−*a*_{33} −*d*_{2}a11*a*_{33} *d*_{3}1−*a*_{11}

−*a*11*d*1d2*d*3 *d*2d1*d*3 −*a*33*d*3d1*d*2 ,
*P*_{1} *d*_{1}

*a*_{11}*a*_{33}−

*a*_{11}*β*

−*d*_{2}
*a*_{11}*β*

*a*_{33}

*d*_{3}a11*a*_{33}−*a*_{33}

−a11*a*_{33}−1 d11−*a*_{33} −*d*_{2}a11*a*_{33} *d*_{3}1−*a*_{11} ,
*P*0 *a*11*a*33

*a*33−*a*11*a*33

*a*11*β*

−*a*33

1*β*

−

*a*11*β*
*.*

3.12

The three roots of*ϕ**i*λ 0 all have negative real parts for*a*11*β <*0 and*a*33*<*0. Namely,*E*1

is the locally asymptotically stable, if*m*^{2}_{0}α−*k < k*and*m*_{0}*>*

*b*^{2}3cβ−*a *−*b /3c.*

iii The linearization of1.3 is*U**t**LU*at*E*0, where*LDΔ F**U*E0 *DΔ *{a*ij*},
and

*a*11 −a, *a*12 *β,* *a*130,
*a*_{21} 1, *a*_{22}−1, *a*_{23}0,
*a*_{31}0, *a*_{32} 0, *a*_{33}−k.

3.13

Similar toi ,*E*_{1}is locally asymptotically stable, when*β < a.*

*Remark 3.4. When* *c* 0, denote *E*0 0,0,0 . If *β > a, then* 1.3 has the semitrivial
equilibrium point*E*_{1} m0*, m*_{0}*,*0 , where*m*_{0} β−a /b. If*α > k, β > a, kb*^{2}*<*α−k β−*a *^{2}*<*

27b^{2}α−*k , then*1.3 has a unique positive equilibrium point*E*^{∗} u^{∗}*, v*^{∗}*, w*^{∗} . Similar as
Theorem 3.3, we have the following.

i If*β > a,α > k, and* *kb*^{2} *<* α−*k β*−*a *^{2} *<* 27b^{2}α−*k namely,α > k,β > a,*
*k/α*−*k <*β−*a /b <*3√

3 , then*E*^{∗}is locally asymptotically stable.

ii If*β > a*andα−*k β*−*a *^{2}*< kb*^{2}, then*E*1is locally asymptotically stable.

iii If*β < a, thenE*_{0}is 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* ∈ *C*^{1}a,∞ ,*ψt * ≥ *0, and* *φ* *is*
*bounded from below. Ifφ*^{}t ≤ −bψt *andψ*^{}t ≤ *K*∀t≥ *a for some positive constantK, then*
lim_{t→ ∞}*ψt *0.

**Theorem 3.6.** i *Assume that*2.1 ,

*bcu*^{∗}−*u*^{∗}

*β*−*a*−*bu*^{∗}
22√

1*u*^{∗2} *>*

√

*u*^{∗2}1*u*^{∗}_{2}
8u^{∗2}1 ^{2} 1

8*,*
*γ*

*α>* 1
2*,*

3.14

*hold, then the equilibrium pointE*^{∗}*of system*1.3 *is globally asymptotically stable.*

ii *Assume thatβ > a, m*^{2}_{0}α−*k < k, and*

*b*^{2}3cβ−*a *−*b /3c < m*0 *<*2k/α*hold,*
*then the equilibrium pointE*_{1}*of system*1.3 *is globally asymptotically stable.*

iii *Assume thatβ < aand* *k > αhold, then the equilibrium point* *E*0 *of system*1.3 *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 of*x*∈ Ωand*t* ≥ 0 such thatu·, t _{∞}*,*v·, t _{∞}*,*
w·, t ∞≤*C, fort*≥0. By33, Theorem*A*_{2},

u·, t * _{C}*2αΩ

*,*v·, t

*2αΩ*

_{C}*,*w·, t

*2αΩ ≤*

_{C}*C,*∀t≥

*t*

_{0}

*,*∀t0

*>*0. 3.15

i Define the Lyapunov function

*Et *

*

Ω

*u*−*u*^{∗}−*u*^{∗}ln *u*
*u*^{∗}

dx*β*

*

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
*u*_{0}*, v*_{0}*, w*_{0}≥*/*≡ 0. It is easily see that*Et *≥0 and*Et *0 if and only if*uu*^{∗}.

Calculating the derivative of*Et * along positive solution of1.3 by integration by
parts and the Cauchy inequality, we have

*E*^{}t −

*

Ω

*d*_{1}*u*^{∗}

*u*^{2} |∇u|^{2}*βd*_{2}*v*^{∗}

*v*^{2} |∇v|^{2}*d*_{3}*w*^{∗}
*αw*^{2}|∇w|^{2}

dx

*

Ω

u−*u*^{∗} *f*_{1}u, v, w

*u* *βv*−*v*^{∗} *f*_{2}u, v, w

*v* 1

*α*w−*w*^{∗} *f*_{3}u, v, w
*w*

dx

≤ −

*

Ωu−*u*^{∗} ^{2}

*bcucu*^{∗} *w*^{∗}1−*u*^{∗}*u *

1*u*^{∗2} 1*u*^{2} − *u*^{2}

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

1*u*^{∗2} 1*u*^{2} ^{2}

dx−*γ*
*α*−1

2

*

Ωw−*w*^{∗} ^{2}dx.

3.17

It is not hard to verify that

*E*^{}t ≤ −l1

*

Ωu−*u*^{∗} ^{2}dx−*l*_{3}

*

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

Recomputing*E*^{}t , we find
*E*^{}t ≤ −

*

Ω

*d*1*u*^{∗}

*u*^{2} |∇u|^{2}*βd*2*v*^{∗}

*v*^{2} |∇v|^{2}*d*3*w*^{∗}
*αw*^{2} |∇w|^{2}

dx

≤ −C

*

Ω

|∇u|^{2}|∇v|^{2}|∇w|^{2}

dx−gt .

3.20

From3.15 , we can see that*g*^{}t is bounded int0*,*∞ ,*t*_{0}*>*0. It follows fromLemma 3.5and
3.15 that*g*t → 0 as*t* → ∞. Namely,

*t→ ∞*lim

*

Ω

|∇u|^{2}|∇v|^{2}|∇w|^{2}

dx0. 3.21

Using the Pioncar´e inequality, we have

*t→ ∞*lim

*

Ωu−*u *^{2}dx lim

*t→ ∞*

*

Ωv−*v *^{2}dx lim

*t→ ∞*

*

Ωw−*w *^{2}dx0, 3.22
where*ut 1/|Ω| *+

Ω*u*dx, vt 1/|Ω| +

Ω*v*dx, wt 1/|Ω| +

Ω*w* dx.Noting that

|Ω||ut −*u*^{∗}|^{2}

*

Ωu−*u*^{∗} ^{2}dx≤2

*

Ωu−*u *^{2}dx2

*

Ωu−*u*^{∗} ^{2}dx,

|Ω||wt −*w*^{∗}|^{2}

*

Ωw−*w*^{∗} ^{2}dx≤2

*

Ωw−*w *^{2}dx2

*

Ωw−*w*^{∗} ^{2}dx,

3.23

according to3.19 and3.22 , we can see

*ut * → *u*^{∗}*,* *wt * → *w*^{∗} t → ∞ . 3.24
Thus, there exists{t*m*}, u^{}t*m* → 0 as*t** _{m}* → ∞. Applying the boundness of{vt

*m*}, there exists a subsequence of{vt

*m*}, denoted still by{vt

*m*}, such that

*vt*

*m*→

*v.*, On the one hand

*

Ω*u**t*dx--
--*t**m*

|Ω|u^{}t*m* −→0, *t**m*−→ ∞. 3.25

On the other hand

*

Ω*u**t*dx--
--*t**m*

*

Ω

*d*1Δu*f*1u, v, w
dx--

--*t**m*

*

Ω*f*1u, v, w dx--
--*t**m*

*

Ω

$*βv*−*v*^{∗} −

*abuu*^{∗} c

*u*^{2}*uu*^{∗}*u*^{∗2}

u−*u*^{∗} −duw−*w*^{∗} %
dx--

--*t**m*

*.*
3.26

According to3.19 to compute the limit of the previous equation and using the uniqueness
of the limit, we have*v*,*v*^{∗}, and

*t**m*lim→ ∞*vt**m* *v*^{∗}*.* 3.27

It follows from 3.15 that there exists a subsequence of {t*m*}, denoted still by {t*m*}, and
nonnegative functions*g** _{i}* ∈

*C*

^{2}Ω , i1,2,3, such that

*u·, t**m* −→*g*_{1}· , *v·, t**m* −→*g*_{2}· , *w·, t**m* −→*g*_{3}· in*C*^{2}
Ω

*.* 3.28

Applying3.19 –3.27 , we obtain that*g*1*u*^{∗}*, g*2 *v*^{∗}*, g*3*w*^{∗}, and
*u·, t**m* −→*u*^{∗}*,* *v·, t**m* −→*v*^{∗}*,* *w·, t**m* −→*w*^{∗} in*C*^{2}

Ω

*.* 3.29

In view ofTheorem 3.3, we can conclude that*E*^{∗}is globally asymptotically stable.

ii Let

*Et *

*

Ω

*u*−*m*0−*m*0ln *u*
*m*0

dx*β*

*

Ω

*v*−*m*0−*m*0ln *v*
*m*0

dx 1

*α*

*

Ω*w*dx. 3.30
Then

*E*^{}t −m0

*

Ω

*d*_{1}

*u*^{2}|∇u|^{2}*βd*_{2}
*v*^{2}|∇v|^{2}

dx

*

Ω

u−*u*^{∗} *f*_{1}u, v, w

*u* *βv*−*v*^{∗} *f*_{2}u, v, w

*v* 1

*αf*_{3}u, v, w

dx

≤ −

*

Ω

*β*
*m*_{0}

*v*

*u*u−*m*0 −
*u*

*v*v−*m*0

_{2}

−

*

Ω

b*cucm*0 u−*m*0 2 *γ*
*αw*^{2}−*w*

*m*0*u*
1*u*^{2} −*k*

*α*

dx.

3.31

Therefore,*E*^{}t ≤ −b*cm*_{0} +

Ωu−*m*_{0} ^{2}dx−*γ*
*α*
+

Ω*w*^{2}dx.It follows that the equilibrium point
*E*1of1.3 is globally asymptotically stable.

iii Define

*Et * 1
2

*

Ω

*u*^{2}*βv*^{2}*w*^{2}

dx. 3.32

Then

*E*^{}t −

*

Ω

*d*1|∇u|^{2}*βd*2|∇v|^{2}*d*3|∇w|^{2}
dx

*

Ω

*uf*_{1}u, v, w *βvf*_{2}u, v, w *wf*_{3}u, v, w
dx.

3.33

When*a > β, k > α,*

*E*^{}t ≤ −

*

Ω

$

*au*^{2}*βv*^{2} *k*−*α w*^{2}%

dx. 3.34

The following proof is similar toi .

*Remark 3.7. Whenc*0,Theorem 3.6shows the following.

i Assume that*β > a, α > k,*

*k/α*−*k <*β−*a /b <*3√
3,

*b*−*u*^{∗}

*β*−*a*−*bu*^{∗}
22√

1*u*^{∗2} *>*

√

*u*^{∗2}1*u*^{∗2}2

8u^{∗2}1 ^{2} 1
8*,* *γ*

*α* *>* 1

2*,* 3.35

hold, then the equilibrium point*E*^{∗}of1.3 is globally asymptotically stable.

ii Assume that*β > a*and *b*^{2}*k/β*−*a * *>* max{α−*k β*−*a , bα/2}*hold, then the
equilibrium point*E*_{1}of1.3 is globally asymptotically stable.

iii Assume that*β < a*and*k > α*hold, then the equilibrium point*E*0of1.3 is globally
asymptotically stable.

*Example 3.8. Consider the following system:*

*X*1t−*D*1ΔX15X2−0.6X1−1.4X1−2X_{1}^{2}−6X^{3}_{1}−*X*3

2X^{2}_{1}

12X_{1}^{2}*,* *x*∈Ω, t >0,
*X*_{2t}−*D*_{2}ΔX21.4X_{1}−*X*_{2}*,* *x*∈Ω, t >0,

*X*_{3t}−*D*_{3}ΔX3−X3−√

2X^{2}_{3}*X*_{3} 2X^{2}_{1}

12X_{1}^{2}*,* *x*∈Ω, t >0,

*∂**η**X**i*0, *i*1,2,3, x∈*∂Ω, t >*0,
*X** _{i}*x,0

*X*

*x ≥0,*

_{i0}*i*1,2,3, x∈Ω.

3.36

Using the software Matlab, one can obtain*u*^{∗}*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. If**u_{0}*, v*_{0}*, w*_{0} ∈*W*_{p}^{1}Ω , p > n, then1.4 *has a unique nonnegative solutionu, v, w*∈
*C0, T* , W_{p}^{1}Ω

*C*^{∞}0, T , C^{∞}Ω , where*T* ≤∞*is the maximal existence time of the solution.*

*If the solution*u, v, w *satisfies the estimate*

sup'

u·, t W_{p}^{1}Ω ,v·, t W_{p}^{1}Ω ,w·, t W_{p}^{1}Ω : 0*< t < T*(

*<*∞, 4.1

*thenT* ∞. If, in addition,*u*_{0}*, v*_{0}*, w*_{0}∈*W*_{p}^{2}Ω , then*u, v, w*∈*C0,*∞ , W_{p}^{2}Ω .

In this section, we consider the existence and the convergence of global solutions to the system1.4 .

**Theorem 4.2. Let***α*11*, α*22 *>* *0 and the space dimension* *n <* *6. Suppose that* *u*0*, v*0*, w*0 ∈
*C*^{2λ}Ω 0 *< λ <* 1 *are nonnegative functions and satisfy zero Neumann boundary conditions.*

*Then*1.4 *has a unique nonnegative solutionu, v, w*∈*C*^{2λ,1λ/2}Ω×0,∞ .
In order to proveTheorem 4.2, some preparations are collected firstly.

* Lemma 4.3. Let*u, v, w

*be a solution of*1.4 . Then

*u, v*≥0, 0≤*w*≤*M*_{1}*,* *in Q** _{T}*≡Ω×0, T ,
sup

0<t<T

u·, t _{L}^{1}_{Ω }*,*sup

0<t<T

v·, t _{L}^{1}_{Ω }≤*C*_{1}T ,
u* _{L}*2Q

*T*

*,*v

*2Q*

_{L}*T*≤

*C*

_{2}T ,

4.2

*whereM*_{1}max{α/γ,w0*L*^{∞}Ω }.

*Proof. From the maximum principle for parabolic equations, it is not hard to verify that*
*u, v, w*≥0 and*w*is bounded.

Multiplying the second equation of1.4 bya*β , adding up the first equation of*
1.4 , and integrating the result overΩ, we obtain

*d*
*dt*

*

Ω

*u*
*aβ*

*v*

*dx*≤ −a

*

Ω*v dx*

*

Ω

*βu*−*bu*^{2}

*dx.* 4.3

Using Young inequality and H ¨*older inequality, we have*

*

Ω

*βu*−*bu*^{2}

*dx*≤*C*_{2,1}− *a*
*aβ*

*

Ω*u dx,* 4.4

where*C*2,1 1/4b β*a/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 * _{L}*1Ω

*,*v·, t

*1Ω ≤*

_{L}*C*

_{2,2}

*,*4.6 where

*C*2,2depends onv0

*1Ω*

_{L}*,*u0

*1Ω and coeﬃcients of1.4 . In addition, there exists a positive constant*

_{L}*C*

_{1}T , such that

sup

0<t<T

u·, t _{L}^{1}_{Ω }*,*sup

0<t<T

v·, t _{L}^{1}_{Ω }≤*C*_{1}T . 4.7

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

*dt*

*

Ω*u dx*≤*β*

*

Ω*v dx*−*b*

*

Ω*u*^{2}*dx.* 4.8

Integrating4.8 from 0 to*T*, we have

*

Ω*ux, T* dx−

*

Ω*ux,*0 dx≤*β*

*_{T}

0

*

Ω*v dx dt*−*b*

*_{T}

0

*

Ω*u*^{2}*dx dt.* 4.9
According to4.7 , there exists a positive constant*C*_{2}T , such that

u* _{L}*2Q

*T*≤

*C*

_{2}T . 4.10 Multiplying the second equation of1.4 by

*v*and integrating it overΩ, we obtain

1 2

*d*
*dt*

*

Ω*v*^{2}*dx*−

*

Ωd22α_{22}*v |∇v|*^{2}*dx*

*

Ω

*uv*−*v*^{2}
*dx*

≤ 1 2

*

Ω*u*^{2}*dx*− 1
2

*

Ω*v*^{2}*dx.*

4.11

Integrating the previous inequation from 0 to*T*, we have

v* _{L}*2Q

*T*≤

*C*

_{2}T . 4.12

* Lemma 4.4. Let*u, v, w

*be a solution of*1.4 ,

*w*1 d3

*α*33

*w w, andτ < T. Then there exists a*

*positive constantC*

_{3}τ

*depending on*w0

_{W}^{1}

2Ω *and*w0* _{L}*∞Ω

*, such that*w1

_{W}^{2,1}

2 Q*τ* ≤*C*_{3}τ . 4.13
*Furthermore*∇w1∈*V*_{2}Q*τ* *and*∇w1∈*L*^{2n2 /n}Q*τ* .