Volume 2011, Article ID 404696,26pages doi:10.1155/2011/404696

*Research Article*

**Non-Constant Positive Steady States for** **a Predator-Prey Cross-Diffusion Model with** **Beddington-DeAngelis Functional Response**

**Lina Zhang and Shengmao Fu**

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

Correspondence should be addressed to Shengmao Fu,fusm@nwnu.edu.cn Received 13 October 2010; Accepted 30 January 2011

Academic Editor: Dumitru Motreanu

Copyrightq2011 L. Zhang 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.

This paper deals with a predator-prey model with Beddington-DeAngelis functional response under homogeneous Neumann boundary conditions. We mainly discuss the following three problems:1stability of the nonnegative constant steady states for the reaction-diﬀusion system;

2 the existence of Turing patterns; 3the existence of stationary patterns created by cross- diﬀusion.

**1. Introduction**

Consider the following predator-prey system with diﬀusion:

*u**t*−*d*1Δu*r*1*u*

1− *u*
*K*

−*fv,* *x*∈Ω, t >0,

*v**t*−*d*2Δv*r*2*v*

1− *v*
*δu*

*,* *x*∈Ω, t >0,

*∂**ν**u∂**ν**v*0, *x*∈*∂Ω, t >*0,

*ux,*0 *u*0x*>*0, *vx,*0 *v*0x≥0, *x*∈Ω,

1.1

where Ω ⊂ ^{Ê}* ^{N}* is a bounded domain with smooth boundary

*∂Ω*and

*ν*is the outward unit normal vector of the boundary

*∂Ω. In the system*1.1,

*ux, t*and

*vx, t*represent the densities of the species prey and predator, respectively,

*u*

_{0}x and

*v*

_{0}x are given smooth functions onΩ which satisfy compatibility conditions. The constants

*d*1,

*d*2, called

diﬀusion coeﬃcients, are positive, *r*1 and *r*2 are the intrinsic growth rates of the prey
and predator,*K* denotes the carrying capacity of the prey, and *δu* represents the carrying
capacity of the predator, which is in proportion to the prey density. The function *f* is a
functional response function. The parameters*r*_{1},*r*_{2},*K, andδ*are all positive constants. The
homogeneous Neumann boundary conditions indicate that the system is self-contained with
zero population flux across the boundary. For more ecological backgrounds about this model,
one can refer to 1–6.

In recent years there has been considerable interest in investigating the system1.1
with the prey-dependent functional responsei.e.,*f* is only a function of*u. In* 5,6, Du,
Hsu and Wang investigated the global stability of the unique positive constant steady state
and gained some important conclusions about pattern formation for1.1with Leslie-Gower
functional responsei.e.,*fβu. In* 7,8, Peng and Wang studied the long time behavior of
time-dependent solutions and the global stability of the positive constant steady state for1.1
with Holling-Tanner-type functional responsei.e.,*f* *βu/mu. They also established*
some results for the existence and nonexistence of non-constant positive steady states with
respect to diﬀusion and cross-diﬀusion rates. In 9, Ko and Ryu investigated system1.1
when *f* satisfies a general hypothesis: *f0 * 0, and there exists a positive constant *M*
such that 0 *< f** _{u}*u ≤

*M*for all

*u >*0. They studied the global stability of the positive constant steady state and derived various conditions for the existence and non-existence of non-constant positive steady states. When the function

*f*in the system1.1takes the form

*f*

*βu/umv*called ratio-dependent functional response, Peng, and Wang 10studied the global stability of the unique positive constant steady state and gained several results for the non-existence of non-constant positive solutions.

It is known that the prey-dependent functional response means that the predation
behavior of the predator is only determined by the prey, which contrasts with some realistic
observations, such as the paradox of enrichment 11,12. The ratio-dependent functional
response reflects the mutual interference between predator and prey, but it usually raises
controversy because of the low-density problem 13. In 1975, Beddington and DeAngelis
14, 15 proposed a function *f* *βu/1* *munv, commonly known as Beddington-*
DeAngelis functional response. It has an extra term in the denominator which models mutual
interference between predator and prey. In addition, it avoids the low-density problem.

In this paper, we study the system1.1with*fβu/1munv. Using the scaling*

*r*1

*Ku*−→*u,* *r*1

*Kδv*−→*v,* *r*1−→*λ,* *Kδ*

*r*1 *β*−→*β,* *K*

*r*1*m*−→*m,* *Kδ*

*r*1 *n*−→*n,* 1.2

and taking*r*21 for simplicity of calculation,1.1becomes

*u** _{t}*−

*d*

_{1}Δu

*λu*−

*u*

^{2}−

*βuv*

1*munv* ^{}*g*_{1}u, v, *x*∈Ω, t >0,
*v**t*−*d*2Δv*v*

1−*v*

*u*

*g*2u, v, *x*∈Ω, t >0,

*∂*_{ν}*u∂*_{ν}*v*0, *x*∈*∂Ω, t >*0,

*ux,*0 *u*_{0}x*>*0, *vx,*0 *v*_{0}x≥0, *x*∈Ω.

1.3

It is obvious that1.3has two nonnegative constant solutions: the semitrivial solutionλ,0
and the unique positive constant solutionu_{∗}*, v*_{∗}, where

*u*_{∗} *λmn*−1−*β*

*λmn*−1−*β*24λm*n*

2m*n* *,* *v*_{∗}*u*_{∗}*.* 1.4

In the system1.3, the Beddington-DeAngelis functional response is used only in the prey
equation, not the predator, and the predator equation contains a Leslie-Gower term*v/δu*
16. To our knowledge, there are few known results for1.3while there has been relatively
good success for the predator-prey model with the full Beddington-DeAngelis functional
responses. For example, Cantrell and Cosner 17derived criteria for permanence and for
predator extinction, and Chen and Wang 18 proved the nonexistence and existence of
nonconstant positive steady states.

Taking into account the population fluxes of one species due to the presence of the other species, we consider the following cross-diﬀusion system:

*u**t*−*d*1Δu*λu*−*u*^{2}− *βuv*

1*munv,* *x*∈Ω, t >0,
*v**t*−*d*2Δ1*d*3*uvv*

1−*v*
*u*

*,* *x*∈Ω, t >0,

*∂**ν**u∂**ν**v*0, *x*∈*∂Ω, t >*0,

*ux,*0 *u*0x*>*0, *vx,*0 *v*0x≥0. x∈Ω,

1.5

where Δd2*d*3*uv* is a cross-diﬀusion term. If *d*3 *>* 0, the movement of the predator is
directed towards the lower concentration of the prey, which represents that the prey species
congregate and form a huge group to protect themselves from the attack of the predator.

It is clear that such an environment of prey-predator interaction often occurs in reality. For example, in 19–21, and so forth, with the similar biological interpretation, the authors also introduced the same cross-diﬀusion term as in 1.5 to the prey of various prey-predator models.

The main aim of this paper is to study the eﬀects of the diﬀusion and cross-
diﬀusion pressures on the existence of stationary patterns. We will demonstrate that
the unique positive constant steady state u∗*, v*_{∗} for the reduced ODE system is locally
asymptotically stable if*a*11 *<* 1, where*a*11 1/β{mλ−*u*_{∗}^{2}−*βu*_{∗}}. Butu_{∗}*, v*_{∗}can lose its
stability when it is regarded as a stationary solution of the corresponding reaction-diﬀusion
system see Theorem 2.5 and Turing patterns can be found as a result of diﬀusion see
Theorem 3.5. Moreover, after the cross-diﬀusion pressure is introduced, even though the
unique positive constant steady state is asymptotically stable for the model without cross-
diﬀusion, stationary patterns can also exist due to the emergence of cross-diﬀusion see
Theorem4.4. The main conclusions of this paper continue to hold for any positive constant
*r*_{2}. We also remark here that, there have been some works which are devoted to the studies
of the role of diﬀusion and cross-diﬀusion in helping to create stationary patterns from the
biological processes 22–25.

This paper is organized as follows. In Section2, we study the long time behavior of 1.3. In Section3, we investigate the existence of Turing patterns of1.3by using the Leray- Schauder degree theory. In Section4, we prove the existence of stationary patterns of1.5.

We end with a brief section on conclusions.

**2. The Long Time Behavior of Time-Dependent Solutions**

In this section, we discuss the global behavior of solutions for the system 1.3. By the
standard theory of parabolic equations 26, 27, we can prove that the problem 1.3 has
a unique classical global solution u, v, which satisfies 0 *< ux, t* ≤ max{λ,sup_{Ω}*u*0}and
0*< vx, t*≤max{λ,sup_{Ω}*u*0*,*sup_{Ω}*v*0}onΩ× 0,∞.

**2.1. Global Attractor and Permanence**

**2.1. Global Attractor and Permanence**

First, we show that^{Ê}0^{} 0, λ× 0, λis a global attractor for1.3.

* Theorem 2.1. Let*ux, t, vx, t

*be any non-negative solution of*1.3. Then,

*t*lim→∞sup

Ω

*ux, t*≤*λ,* lim

*t→*∞sup

Ω

*vx, t*≤*λ.* 2.1

*Proof. The first result of* 2.1 follows easily from the comparison argument for parabolic
problems. Then, there exists a constant*T* 0 such that*ux, t* *< λε*onΩ× T,∞for
an arbitrary constant*ε >*0, and thus,

*v** _{t}*−

*d*

_{2}Δv≤

*v*

1− *v*
*λε*

*,* x, t∈Ω× T,∞. 2.2

Let*vt*be the unique positive solution of
*dw*

*dt* *w*

1− *w*
*λε*

*,* *t*∈ T,∞,

*wT* max

Ω *vx, T*≥0.

2.3

The comparison argument yields

*t→*lim∞sup

Ω

*vx, t*≤ lim

*t*→∞*vt λε,* 2.4

which implies the second assertion of2.1by the continuity as*ε* → 0.

* Theorem 2.2. Assume thatβ < nλ1, then the positive solution*ux, t, vx, t

*of*1.3

*satisfies*

*t→*lim∞inf

Ω *ux, t*≥*K,* lim

*t→*∞inf

Ω *vx, t*≥*K,* 2.5

*where*

*K*^{} 1

2m m−*nλ*−1

m−*nλ*−1^{2}4mλ

1*nλ*−*β*

*.* 2.6

*Proof. Sinceβ < nλ1, there exists a suﬃciently small constantε*_{1}*>*0 such that*λ*nλ−*βλ*
*ε*1*>*0. In view of Theorem2.1, there exists a*T* 0 such that*vx, t< λε*1inΩ× T,∞.

Thus we have

*u**t*−*d*1Δu≥ −mu^{2} *mλ*−*nλ*−*nε*_{1}−1u*λ*
*nλ*−*β*

λ*ε*_{1}

1*munλε*1 *u* 2.7

forx, t∈Ω× T,∞. Let*ut*be the unique positive solution of
*dw*

*dt* −mw^{2} *mλ*−*nλ*−*nε*_{1}−1w*λ*
*nλ*−*β*

λ*ε*_{1}

1*mwnλε*1 *w,* *t*∈ T,∞,

*wT* min

Ω *ux, T>*0.

2.8

Then, lim_{t→}_{∞}inf_{Ω}*ux, t*≥lim_{t→}_{∞}*ut, where*

*t→*lim∞*ut * 1

2m m−*nλ*−1−*nε*_{1}

m−*nλ*−*nε*_{1}−1^{2}4m
*λ*

*nλ*−*β*

λ*ε*_{1}
*.*
2.9
By continuity as*ε*_{1} → 0, we have lim_{t→}_{∞}inf_{Ω}*ux, t*≥*K. Similarly, we can prove the second*
result of2.5.

From Theorems2.1and2.2, we see that the system1.3is permanent if*β < nλ*1.

**2.2. Local Stability of Nonnegative Equilibria**

Now, we consider the stability of non-negative equilibria.
**2.2. Local Stability of Nonnegative Equilibria**

* Lemma 2.3. The semi-trivial solution*λ,0

*of*1.3

*is unconditionally unstable.*

*Proof. The linearization matrix of*1.3atλ,0is

*J*_{1}

⎛

⎝−λ − *βλ*
1*mλ*

0 1

⎞

⎠*.* 2.10

It is easy to see that 1 is an eigenvalue of*J*1, thusλ,0is unconditionally unstable.

Now, we discuss the Turing instability ofu∗*, v*_{∗}. Recall that a constant solution is
Turing unstable if it is stable in the absence of diﬀusion, and it becomes unstable when
diﬀusion is present 28. More precisely, this requires the following two conditions.

iIt is stable as an equilibrium of the system of ordinary diﬀerential equations
*du*

*dt* *g*_{1}u, v, *dv*

*dt* *g*_{2}u, v, 2.11
where*g*_{1}u, vand*g*_{2}u, vare given in1.3.

iiIt is unstable as a steady state of the reaction-diﬀusion system1.3.

**Theorem 2.4. If***a*11 *<* *1, then the unique positive equilibrium* u_{∗}*, v*_{∗} *of* 2.11 *is locally*
*asymptotically stable. Ifa*_{11} *>1, then*u∗*, v*_{∗}*is unstable, wherea*_{11}1/β mλ−*u*_{∗}^{2}−*βu*_{∗}.
*Proof. The linearization matrix of*2.11atu∗*, v*_{∗}is

*J*2

*a*_{11} *a*_{12}
*a*21 *a*22

*,* 2.12

where
*a*_{11} 1

*β*

*mλ*−*u*_{∗}^{2}−*βu*_{∗}

*,* *a*_{12}−λ−*u*_{∗}1*mu*_{∗}

1 m*nu*∗ *,* *a*_{21} 1, *a*_{22}−1. 2.13
A simple calculation shows

det*J*_{2}−a11−*a*_{12} m*nu*^{2}_{∗}*λ*

1 *mnu*∗*,* trace*J*_{2}*a*_{11}−1. 2.14
Clearly, det*J*_{2}*>*0. If*a*_{11}*<*1, then trace*J*_{2}*<*0. Hence, all eigenvalues of*J*_{2}have negative real
parts andu∗*, v*_{∗}is locally asymptotically stable. If*a*11 *>*1, then traceJ2 *>*0, which implies
that*J*2has two eigenvalues with positive real parts andu_{∗}*, v*_{∗}is unstable.

Similarly as in 23, 29, let 0 *μ*1 *< μ*2 *< μ*3 *< μ*4*. . .* be the eigenvalues of the
operator−ΔonΩwith the homogeneous Neumann boundary condition, and let*Eμ**i*be the
eigenspace corresponding to*μ**i*in*H*^{1}Ω. Let{φ*ij*:*j*1,2, . . . ,dim*Eμ**i*}be the orthonormal
basis of*Eμ**i*, **X** H^{1}Ω^{2},**X***ij*{cφ*ij*:**c**∈^{Ê}^{2}}. Then,

**X**^{∞}

*i1*

**X***i**,* **X***i*

dim*E*^{μ}^{i}

*j1*

**X***ij**.* 2.15

Define*i*_{0}as the largest positive integer such that*d*_{1}*μ*_{i}*< a*_{11}for*i*≤*i*_{0}. Clearly, if

*d*1*μ*2*< a*11*,* 2.16

then 2≤*i*0*<*∞. In this case, denote
*d*_{2}^{}min

2≤i≤i0

*d*^{i}_{2} *,* *d*^{i}_{2} ^{} *d*1*μ**i*det*J*2

*μ**i*

*a*11−*d*1*μ**i*

*.* 2.17

The local stability ofu∗*, v*_{∗}for1.3can be summarized as follows.

* Theorem 2.5. (i) Assume thata*11

*>1, then*u

_{∗}

*, v*

_{∗}

*is unstable.*

*(ii) Assume thata*11 *<1. Then*u_{∗}*, v*_{∗}*is locally asymptotically stable ifa*11 ≤*d*1*μ*2*;*u_{∗}*, v*_{∗}
*is locally asymptotically stable ifa*11 *> d*1*μ*2 *andd*2 *<* *d*2*;*u∗*, v*_{∗}*is unstable ifa*11 *> d*1*μ*2 *and*
*d*2*>d*2*.*

*Proof. Consider the following linearization operator of*1.3atu∗*, v*_{∗}:

*L*

*d*1Δ *a*11 *a*12

*a*21 *d*2Δ *a*22

*,* 2.18

where*a*_{11},*a*_{12},*a*_{21}, and*a*_{22}are given in2.13. Supposeφx, ψx* ^{T}*is an eigenfunction of

*L*corresponding to an eigenvalue

*μ, then*

*d*1Δφ

*a*11−*μ*

*φa*12*ψ, d*2Δψ*a*21*φ*

*a*22−*μ*
*ψ*_{T}

0,0^{T}*.* 2.19

Setting

*φ*

1≤i<∞,1≤j≤dimEμ*i*

*a*_{ij}*φ*_{ij}*,* *ψ*

1≤i<∞,1≤j≤dim*Eμ**i*

*b*_{ij}*φ*_{ij}*,* 2.20

we can find that

1≤i<∞,1≤j≤dimE^{μ}*i*
*L**i*

*a**ij*

*b*_{ij}

*φ**ij* 0, where*L**i*

*a*11−*d*1*μ**i*−*μ* *a*12

*a*_{21} *a*_{22}−*d*_{2}*μ** _{i}*−

*μ*

*.* 2.21

It follows that*μ*is an eigenvalue of*L*if and only if the determinant of the matrix*L**i*is zero
for some*i*≥1, that is,

*μ*^{2}*P*_{i}*μQ** _{i}*0, 2.22

where

*P*_{i}*d*_{1}*d*_{2}μ*i*−trace*J*_{2}*,* *Q** _{i}*−d2

*μ*

_{i}*a*_{11}−*d*_{1}*μ*_{i}

*d*_{1}*μ** _{i}*det

*J*

_{2}

*.*2.23 Clearly,

*Q*1

*>*0 since

*μ*1 0. If

*a*11

*>*1, then traceJ2

*>*0 and

*P*1

*<*0. Hence,

*L*has two eigenvalues with positive real parts and the steady stateu

_{∗}

*, v*

_{∗}is unstable.

Note that*P**i* *>*0 for all*i*≥1 if*a*11 *<*1, and*Q**i**>*0 for all*i*≥1 if*a*11 ≤*d*1*μ*2. This implies
that Re*μ <* 0 for all eigenvalue*μ, and so the steady state* u_{∗}*, v*_{∗} is locally asymptotically
stable.

Assume that*a*_{11}*> d*_{1}*μ*_{2}. If*d*_{2}*<d*_{2}, then*d*_{1}*μ*_{i}*< a*_{11}and*d*_{2}*< d*^{i}_{2} for*i*∈ 2, i0. It follows
that*Q**i**>*0 for all*i*∈ 2, i0. Furthermore, if*i > i*0, then*d*1*μ**i* ≥*a*11 and*Q**i**>*0. The conclusion
leads to the locally asymptotically stability ofu∗*, v*_{∗}again. If*d*_{2}*>d*_{2}, then we may assume
that the minimum in2.17is attained by*k* ∈ 2, i0. Thus,*d*1*μ**k* *< a*11 and*d*2 *> d*^{k}_{2} , so we
have*Q*_{k}*<*0. This implies thatu∗*, v*_{∗}is unstable.

*Remark 2.6. From Theorems*2.4and2.5, we can conclude thatu∗*, v*_{∗}is Turing unstable if
*d*_{1}*μ*_{2}*< a*_{11}*<*1 and*d*_{2}*>d*_{2}.

**2.3. Global Stability of**

u∗**2.3. Global Stability of**

*, v*∗

The following three theorems are the global stability results of the positive constant solution
u∗*, v*_{∗}. In the sense of biology, our conclusion of the global stability ofu∗*, v*_{∗}implies that, in
some ranges of the parameters*λ,β,m, andn, both the prey and the predator will be spatially*
homogeneously distributed as time converges to infinity, no matter how quickly or slowly
they diﬀuse.

**Theorem 2.7. Assume that**β < nλ1 and

*β* *λu*_{∗}

*Ku*_{∗}1*mu*_{∗}−1*mKnK*
1*mλnλ*

*<*1*mu*_{∗}*nv*_{∗}1*mKnK.* 2.24

*Then*u∗*, v*_{∗}*attracts all positive solutions of* 1.3.

*Proof. Define the Lyapunov function*

*E*1t

Ω

*u*−2u_{∗}*u*_{∗}^{2}
*u*

*dxδ*1

Ω

*v*−*v*_{∗}−*v*_{∗}ln *v*
*v*_{∗}

*dx,* 2.25

where

*δ*1 *Ku*_{∗} 1 *β*

1*mu*_{∗}*nv*_{∗}1*mλnλ*

*,* 2.26

u, v is a positive solution of 1.3. Then *E*1t ≥ 0 for all *t* ≥ 0. The straightforward
computations give that

*dE*1

*dt*

Ω

*u*^{2}−*u*_{∗}^{2}

*u*^{2} *u**t**dxδ*1

Ω

*v*−*v*_{∗}
*v* *v**t**dx*

Ω*D*1*dx*

Ω

1
*u*

*A*1u−*u*_{∗}^{2}*B*1u−*u*_{∗}v−*v*_{∗} *C*1v−*v*_{∗}^{2}
*dx,*

2.27

where

*D*1−

*d*12u_{∗}^{2}

*u*^{3} |∇u|^{2}*δ*1*d*2*v*_{∗}
*v*^{2}|∇v|^{2}

≤0,

*A*_{1} *uu*_{∗} −1 *βmv*_{∗}

1*mu*_{∗}*nv*_{∗}1*munv*

*,*

*B*1*δ*1− *βuu*_{∗}1*mu*_{∗}

1*mu*_{∗}*nv*_{∗}1*munv,* *C*1−δ1*.*

2.28

From Theorems2.1and2.2, there exists a*t*0 0 such that*K*−*ε < ux, t,vx, t* *< λε*in
Ω× t0*,*∞for an arbitrary and small enough constant*ε >*0. By continuity as*ε* → 0,2.24
implies that

*B*_{1} *Ku*_{∗}

1*mu*_{∗}*nv*_{∗}1*mKnK*

× 1*mu*_{∗}*nv*_{∗}1*mKnK*

−β

u*u*_{∗}1*mu*_{∗}1*mKnK*

K*u*_{∗}1*munv* −1*mKnK*
1*mλnλ*

≥0

2.29

inΩ× t0*,*∞. Applying the Young inequality to2.27, we have

*dE*1

*dt* ≤

Ω*D*1*dx*

Ω

1

*u*A1*B*1u−*u*_{∗}^{2}*dx*

Ω

1
*u*

*B*1

4 *C*_{1}

v−*v*_{∗}^{2}*dx*

Ω*D*_{1}*dx*

Ω

1

*u* *δ*_{1}−u*u*_{∗}

1 *β*

1*mu*_{∗}*nv*_{∗}1*munv*

u−*u*_{∗}^{2}*dx*

Ω

1
*u* −3

4*δ*1− *βuu*_{∗}1*mu*_{∗}
41*mu*_{∗}*nv*_{∗}1*munv*

v−*v*_{∗}^{2}*dx*

≤0

2.30

inΩ× t0*,*∞. Similarly as in 24,30, the standard argument concludesux, t, vx, t →
u∗*, v*_{∗}in L^{∞}Ω^{2}, which thereby shows thatu∗*, v*_{∗}attracts all positive solutions of1.3
under our hypotheses. Thus, the proof is complete.

**Theorem 2.8. Assume that**β < nλ1,

*β*

1*mu*_{∗}−1*mKnK*
1*mλnλ*

*<*1*mu*_{∗}*nv*_{∗}1*mKnK,* 2.31

*β <* λm*λn*2m*n*

2 *.* 2.32

*Then,*u∗*, v*_{∗}*attracts all positive solutions of* 1.3.

*Proof. Define the Lyapunov function*

*E*2t

Ω

*u*_{∗}−*u*
*u* ln *u*

*u*_{∗}

*dxδ*2

Ω *v*−*v*_{∗}−*v*_{∗}ln *v*
*v*_{∗}

*dx,* 2.33

where*δ*21 β/1*mu*_{∗}*nv*_{∗}1*mλnλ,*u, vis a positive solution of1.3. Then
*dE*2

*dt*

Ω*D*_{2}*dx*

Ω

1
*u*

*A*_{2}u−*u*_{∗}^{2}*B*_{2}u−*u*_{∗}v−*v*_{∗} *C*_{2}v−*v*_{∗}^{2}

*dx,* 2.34

where

*D*2− *d*1

2u_{∗}−*u*

*u*^{3} |∇u|^{2}*δ*2*d*2

*v*_{∗}
*v*^{2}|∇v|^{2}

*,*

*A*2−1 *βmv*_{∗}

1*mu*_{∗}*nv*_{∗}1*munv,*
*B*2*δ*2− *β1mu*_{∗}

1*mu*_{∗}*nv*_{∗}1*munv,* *C*2−δ2*.*

2.35

From Theorems2.1and2.2, there exists a*t*0 0 such that*K*−*ε < ux, t,vx, t* *< λε*in
Ω× t0*,*∞for an arbitrary and small enough constant*ε >*0. Thus2.31implies that

*B*_{2} 1

1*mu*_{∗}*nv*_{∗}1*mKnK*

× 1*mu*_{∗}*nv*_{∗}1*mKnK*

−β

1*mu*_{∗}1*mKnK*

1*munv* − 1*mKnK*
1*mλnλ*

≥0

2.36

inΩ× t0*,*∞. On the other hand,2.32guarantees that 2u_{∗}−u >0 inΩ× t0*,*∞. Applying
the Young inequality to2.34, we have

*dE*2

*dt* ≤

Ω*D*2*dx*

Ω

1

*u*A2*B*2u−*u*_{∗}^{2}*dx*

Ω

1
*u*

*B*2

4 *C*2

v−*v*_{∗}^{2}*dx*

Ω*D*_{2}*dx*

Ω

1
*u* *δ*_{2}−

1 *β*

1*mu*_{∗}*nv*_{∗}1*munv*

u−*u*_{∗}^{2}*dx*

Ω

1
*u* −3

4*δ*2− *β1mu*_{∗}

41*mu*_{∗}*nv*_{∗}1*munv*

v−*v*_{∗}^{2}*dx*

≤0

2.37

inΩ× t0*,*∞. Consequently, our analysis confirms that Theorem2.8holds.

*Remark 2.9. If we choose the common Lyapunov function*

*E*3t

Ω *u*−*u*_{∗}−*u*_{∗}ln *u*
*u*_{∗}

*dxδ*3

Ω *v*−*v*_{∗}−*v*_{∗}ln *v*
*v*_{∗}

*dx,* 2.38

where*δ*_{3}*K{1* β/1*mu*_{∗}*nv*_{∗}1*mλnλ}, we can also derive the global stability of*
u_{∗}*, v*_{∗}for1.3under a stronger condition than2.24. Thus, the Lyapunov function defined
by2.25is better than2.38in discussing the global stability ofu_{∗}*, v*_{∗}for1.3.

*Remark 2.10. If we choosem*1, then2.32holds since*β < λn*1. It is not hard to verify that
the condition2.31in Theorem2.8contains the condition2.24in Theorem2.7. However, if
we choose*m*and*n*to be suﬃciently small, then*u*_{∗}*v*_{∗} → *λ/1β*and*K* → *λ1*−*β. We*
can see that the range of parameters satisfying2.24is wider than that satisfying2.32. This
means that we can derive various conditions for the global stability ofu∗*, v*_{∗}by choosing
diﬀerent Lyapunov functions.

**3. Stationary Patterns for the PDE System without Cross-Diffusion**

In this section, we discuss the corresponding steady-state problem of1.3:

−d1Δu*λu*−*u*^{2}− *βuv*

1*munv* *g*1u, v inΩ,

−d2Δv*v*
1−*v*

*u*

*g*_{2}u, v inΩ,

*∂**ν**u∂**ν**v*0 on*∂Ω.*

3.1

The existence and non-existence of the non-constant positive solutions of3.1will be given.

In the following, the generic constants*C*1,*C*2,*C*_{∗},*C,C, and so forth, will depend on*
the domainΩand the dimension*N. However, as*Ωand the dimension*N*are fixed, we will

not mention the dependence explicitly. Also, for convenience, we shall writeΛinstead of the collective constantsλ, β, m, n.

**3.1. A Priori Upper and Lower Bounds**

**3.1. A Priori Upper and Lower Bounds**

The main purpose of this subsection is to give a priori upper and lower bounds for the positive solutions to3.1. To this aim, we first cite two known results.

**Lemma 3.1**maximum principle 25. Let*g*∈*CΩ*×^{Ê}^{1}*andb**j*∈*CΩ,j*1,2, . . . , N.

i*Ifw*∈*C*^{2}Ω∩*C*^{1}Ω*satisfies*

Δwx ^{N}

*j1*

*b**j*xw*x**j**gx, wx*≥0 inΩ,

*∂w*

*∂ν* ≤0 on*∂Ω,*

3.2

*andwx*0 max_{Ω}*wx, thengx*0*, wx*0≥*0.*

ii*Ifw*∈*C*^{2}Ω∩*C*^{1}Ω*satisfies*

Δwx ^{N}

*j1*

*b**j*xw*x**j**gx, wx*≤0 inΩ,

*∂w*

*∂ν* ≥0 on*∂Ω,*

3.3

*andwx*0 min_{Ω}*wx, thengx*0*, wx*0≤*0.*

**Lemma 3.2**Harnack, inequality 31. Let*w*∈*C*^{2}Ω∩*C*^{1}Ω*be a positive solution to*Δwx
*cxwx * *0, wherec* ∈ *CΩ, satisfying the homogeneous Neumann boundary condition. Then*
*there exists a positive constantC*_{∗}*which depends only on*c_{∞}*such that*

maxΩ *w*≤*C*_{∗}min

Ω *w.* 3.4

*The results of upper and lower bounds can be stated as follows.*

**Theorem 3.3. For any positive number**d, there exists a positive constantCΛ, dsuch that every*positive solutionu, vof*3.1*satisfiesC < ux,vx< λifd*1≥*d.*

*Proof. Let* *ux*1 max_{Ω}*ux,* *vx*2 max_{Ω}*vx, uy*1 min_{Ω}*ux, vy*2 min_{Ω}*vx.*

Application of Lemma3.1yields that

*λ*−*ux*1− *βvx*1

1*mux*1 *nvx*1 ≥0,
*λ*−*u*

*y*1

− *βv*
*y*_{1}
1*mu*

*y*_{1}
*nv*

*y*_{1} ≤0,

1−*vx*2

*ux*2 ≥0, 1−*v*
*y*_{2}
*u*

*y*_{2} ≤0.

3.5

Clearly,*ux*1*< λ*and*vx*2≤*ux*2≤*ux*1*< λ. Moreover, we have*
*v*

*y*1

≤*vx*2≤*ux*2≤*ux*1, 3.6

*v*
*y*_{1}

≥*v*
*y*_{2}

≥*u*
*y*_{2}

≥*u*
*y*_{1}

*.* 3.7

By3.5, we obtain
*m*

*u*
*y*_{1}_{2}

1*nv*

*y*_{1}

−*λm*
*u*

*y*_{1}

*β*−*λn*
*v*

*y*_{1}

−*λ*≥0. 3.8
Noting that*uy*1≤*vy*1≤*ux*1from3.6and3.7,3.8implies that max_{Ω}*ux ux*1*>*

*C*_{1}for some positive constant*C*_{1}*C*_{1}Λ.

Let*cx* ^{}*d*_{1}^{−1}λ−*u*−βv/1*munv. Then,*cx_{∞} ≤ 2*βλ/d. The Harnack*
inequality shows that there exists a positive constant*C*_{∗}*C*_{∗}λ, β, dsuch that

maxΩ *ux*≤*C*_{∗}min

Ω *ux.* 3.9

Combining 3.9 with max_{Ω}*ux* *> C*1, we find that min_{Ω}*ux* *> C*1 for some positive
constant*C* *CΛ, d. It follows from*3.7that min_{Ω}*vx * *vy*2 ≥ *uy*1 *> C. The proof*
is completed.

**3.2. Non-Existence of Non-Constant Positive Steady States**

**3.2. Non-Existence of Non-Constant Positive Steady States**

In the following theorem we will discuss the non-constant positive solutions to3.1when
the diﬀusion coeﬃcient*d*1varies while the other parameters*d*2,*λ,β,m, andn*are fixed.

**Theorem 3.4. For any positive number**d, there exists a positive constantD*DΛ, d* *> dsuch*
*that*3.1*has no non-constant positive solution ifd*_{1}*> D.*

*Proof. For anyϕ*∈*L*^{1}Ω, let

*ϕ* 1

|Ω|

Ω*ϕ dx.* 3.10

Assume thatu, vis a positive solution of3.1, multiplying the two equations of 3.1by
u−*u/u*andv−*v/v, respectively, and then integrating over*Ωby parts, we have

Ω

*d*1*u*

*u*^{2} |∇u|^{2} *d*2*v*
*v*^{2} |∇v|^{2}

*dx*

Ω*g*1u, v*u*−*u*
*u* *dx*

Ω*g*2u, v*v*−*v*
*v* *dx*

Ω −1 *βmv*

1*munv1munv*

u−*u*^{2}*dx*

Ω − *β1mu*

1*munv1munv* *v*
*uu*

u−*uv*−*vdx*

Ω

−1
*u*

v−*v*^{2}*dx.*

3.11

From Theorem3.3and Young’s inequality, we obtain

Ω

*d*_{1}|∇u|^{2}*d*_{2}|∇v|^{2}
*dx*≤*C*_{2}

−1*βm*
*n* *C*_{3}

Ωu−*u*^{2}*dxC*_{2}

Ω

*ε*− 1

*u*

v−*v*^{2}*dx*
3.12
for some positive constants*C*_{2} *C*_{2}Λ, d,*C*_{3} *C*_{3}Λ, d, ε, where*ε*is the arbitrary small
positive constant arising from Young’s inequality. By Theorem3.3, we can choose*ε*∈0,1/λ.

Then applying the Poincar´e inequality to3.12we obtain

*μ*_{2}

Ω

*d*_{1}u−*u*^{2}*d*_{2}v−*v*^{2}

*dx*≤*C*_{4}

Ωu−*u*^{2}*dxC*_{2}

Ω

*ε*− 1

*u*

v−*v*^{2}*dx,* 3.13

which implies that*uu*constant and*vv*constant if*d*1*> D*max{C4*/μ*2*, d}.*

**3.3. Existence of Non-Constant Positive Steady States**

**3.3. Existence of Non-Constant Positive Steady States**

Throughout this subsection, we always assume that*a*11 *>*0. First, we study the linearization
of3.1atu_{∗}*, v*_{∗}. Let

**Y** u, v:u, v∈
*C*^{1}

Ω_{2}

*, ∂*_{ν}*u∂*_{ν}*v*0 on*∂Ω*

*.* 3.14

For the sake of convenience, we define a compact operatorF:**Y** → **Y by**

Fe^{}

a11−*d*1Δ^{−1}

*g*1u, v *a*11*u*

−a22−*d*_{2}Δ^{−1}

*g*_{2}u, v−*a*_{22}*v*

*,* 3.15

where**e** ux, vx* ^{T}*,a11−

*d*1Δ

^{−1}, and−a22−

*d*2Δ

^{−1}are the inverses of the operators a11 −

*d*1Δand−a22 −

*d*2Δin

**Y with the homogeneous Neumann boundary conditions.**

Then the system3.1is equivalent to the equationI− Fe0. To apply the index theory, we investigate the eigenvalue of the problem

−I− F**e**e∗**Ψ ***μΨ,* **Ψ***/***0,** 3.16

where**Ψ ψ**1*, ψ*2* ^{T}* and

**e**

_{∗}u

_{∗}

*, v*

_{∗}

*. If 0 is not an eigenvalue of3.16, then the Leray- Schauder Theorem 27implies that*

^{T}indexI− F,**e**_{∗} −1^{γ}*,* 3.17

where*γ*is the sum of the algebraic multiplicities of the positive eigenvalues of−I− F** _{e}**e

_{∗}, 3.16can be rewritten as

−
*μ*1

*d*1Δψ1

−*μ*1

*a*11*ψ*1*a*12*ψ*2*,*

−
*μ*1

*d*2Δψ2*a*21*ψ*1
*μ*1

*a*22*ψ*2*.*

3.18

As in the proof of Theorem2.5, we can conclude that*μ*is an eigenvalue of−I− F**e**e∗on
**X***ij*if and only if it is a root of the characteristic equation det*B**i*0, where

*B*_{i}

−*μ*1
*a*_{11}−

*μ*1

*d*_{1}*μ*_{i}*a*_{12}

*a*_{21}

*μ*1

*a*_{22}−
*μ*1

*d*_{2}*μ*_{i}

*.* 3.19

The characteristic equation det*B** _{i}*0 can be written as

*μ*^{2} 2d1*μ**i*

*a*11*d*1*μ**i**μ* −d2*μ*_{i}

*a*_{11}−*d*_{1}*μ*_{i}

*d*_{1}*μ** _{i}*det

*J*

_{2}

*a*11

*d*1

*μ*

*i*

−a22*d*2*μ**i*

0. 3.20

Note that−d2*μ** _{i}*a11−

*d*

_{1}

*μ*

_{i}*d*

_{1}

*μ*

*det*

_{i}*J*

_{2}

*Q*

*, where*

_{i}*Q*

*is given in2.23. Therefore, if 0 is not a root of3.20for all*

_{i}*i*≥1, we have

indexI− F,**e**∗ −1^{γ}*,* 3.21

where*γ*is the sum of the algebraic multiplicities of the positive roots of3.20.

* Theorem 3.5. Assume that the parametersλ,β,m,n, andd*1

*are fixed and 0< a*11

*<1. Ifa*11

*/d*1 ∈ μ

*n*

*, μ*

_{n1}*for somen*≥

*2 and*

2≤i≤n, Q*i**<0*dim*Eμ**i**is odd, then the problem*3.1*has at least one*
*non-constant positive solution for anyd*2 *>* *d*2*, whereQ**i* *and* *d*2 *are given in*2.23 *and* 2.17,
*respectively.*

*Proof. The proof, which is by contradiction, is based on the homotopy invariance of the*
topological degree. Suppose, on the contrary, that the assertion is not true for some *d*2
*d*˘_{2}*>d*_{2}. In the follow we fix*d*_{2}*d*˘_{2}. Taking*da*_{11}*/μ*_{2}in Theorems3.3and3.4, we obtain a
positive constant*D. Fixedd*1*D*1 and*d*21. For*θ*∈ 0,1, define a homotopy

Fθ;**e**^{}

⎛

⎜⎝

*a*_{11}−

*θd*_{1} 1−*θd*_{1}
Δ_{−1}

*g*_{1}u, v *a*_{11}*u*
−a22−

*θd*_{2} 1−*θd*_{2}
Δ_{−1}

*g*_{2}u, v−*a*_{22}*v*

⎞

⎟⎠. 3.22

Then, **e is a positive solution of** 3.1 if and only if it is a positive solution ofF1;**e ** **e.**

It is obvious that **e**_{∗} is the unique constant positive solution of 3.22for any 0 ≤ *θ* ≤ 1.

By Theorem3.3, there exists a positive constant *C*such that, for all 0 ≤ *θ* ≤ 1, the positive
solutions of the problem Fθ;**e ** **e are contained in** *BC* ^{} {e ∈ **Y** | *C*^{−1} *< u, v < C}.*

SinceFθ;**e***/***e for all e**∈*∂BC*andFθ;·:*BC*× 0,1 → **Y is compact, we can see that**
the degree degI− Fθ;·, BC,0is well defined. Moreover, by the homotopy invariance
property of the topological degree, we have

degI− F0;·, BC,0 degI− F1;·, BC,0. 3.23

If *a*_{11}*/d*_{1} ∈ μ*n**, μ** _{n1}* for some

*n*≥ 2, then

*i*

_{0}

*n*and

*d*

_{2}min

_{2≤i≤n}

*d*

^{i}

_{2}in 2.17. Since

*d*2

*d*˘2

*>d*2, then

*Q*

*k*

*<*0 for some

*k, 2*≤

*k*≤

*n. Letik. Then,*3.20has one positive root and a negative root. Furthermore, we have

*Q*

_{i}*>*0 for

*i*1 and all

*i*≥

*n*1. Therefore, when

*i*1 and

*i*≥

*n*1, the characteristic equation3.20has no roots with non-negative real parts.

In addition, if

2≤i≤n, Q*i**<0*dim*Eμ**i*is odd, we have

indexI− F1;·,**e**_{∗} −1^{}^{2≤i≤}^{n, Qi<}^{0}^{dim}^{Eμ}^{i}^{}−1. 3.24

By our supposition, the equationF1;**e ** **e has only the positive solution e**_{∗}in*BC, and*
hence

degI− F1;·, BC,0 indexI− F1;·,**e**_{∗} −1. 3.25

Similar argument shows*μ*is an eigenvalue of−I− F** _{e}**0;

**e**

_{∗}if and only if it is a root of the characteristic equation

*μ*^{2} 2*d*_{1}*μ*_{i}

*a*_{11}*d*_{1}*μ*_{i}*μ*−*d*_{2}*μ*_{i}

*a*_{11}−*d*_{1}*μ*_{i}

*d*_{1}*μ** _{i}*det

*J*

_{2}

*a*11*d*1*μ**i*

−a22*d*2*μ**i*

0. 3.26

It is easy to check that all eigenvalues of3.26have negative real parts for all*i*≥ 1, which
implies

indexI− F0;·,**e**∗ −1^{0}1. 3.27

In view of Theorem3.4, it follows that the equationF0;**e e has only the positive solution**
**e**_{∗}in*BC, and therefore,*

degI− F0;·, BC,0 indexI− F0;·,**e**∗ 1. 3.28
This contradicts3.23, and the proof is complete.

*Example 3.6. Let*Ω 0,1. Then, the parameters*λ* 2,*β* 6,*m* 3,*n* 0.1,*d*_{1} 0.0152,
and*d*24.1309 satisfy all the conditions of Theorem3.5. This means thatu_{∗}*, v*_{∗} 2√

159− 4/31,2√

159−4/31is a locally asymptotically stable equilibrium point for the system
*du*

*dt* 2u−*u*^{2}− 6uv
13u0.1v*,*
*dv*

*dt* *v*
1−*v*

*u*

*,*

3.29

but it is an unstable steady state for the system
*u**t*−0.0152u*xx*2u−*u*^{2}− 6uv

13u0.1v*,* *x*∈0,1, t >0,
*v**t*−4.1309v*xx**v*

1−*v*
*u*

*,* *x*∈0,1, t >0,
*u**x**v**x*0, *x*0,1, t >0,

*ux,*0 *u*_{0}x*>*0, *vx,*0 *v*_{0}x≥0, *x*∈0,1.

3.30

Moreover, the above reaction-diﬀusion system has at least one non-constant positive steady state.

**3.4. Bifurcation**

**3.4. Bifurcation**

In this subsection, we discuss the bifurcation of non-constant positive solutions of3.1with
respect to the diﬀusion coeﬃcient*d*_{2}. In the consideration of bifurcation with respect to*d*_{2},
we recall that, for a constant solution**e**∗,d2;**e**∗ ∈0,∞×**Y is a bifurcation point of**3.1
if, for any*δ* ∈ 0, d2, there exists a*d*2 ∈ d2−*δ, d*2*δ*such that3.1has a non-constant
positive solution close to**e**∗. Otherwise, we say thatd2;**e**∗is a regular point 27.

We will consider the bifurcation of 3.1 at the equilibrium pointsd2;**e**∗, while all
other parameters are fixed. From2.23, we define

*Q*
*d*_{2};*μ*

*d*_{1}*d*_{2}*μ*^{2}−d2*a*_{11}−*d*_{1}μdet*J*_{2}*.* 3.31
It is clear that*Qd*2;*μ *0 has at most two roots for any fixed*d*2 *>*0. Noting that det*J*2 *>*0
in the proof of Theorem2.4, if

*Rd*2^{}d2*a*11*d*1^{2}4d1*d*2*a*12*>*0, 3.32

then*Qd*2*, μ *0 has two diﬀerent real roots with same symbols. Let
*S** _{p}*!

*μ*_{1}*, μ*_{2}*, μ*_{3}*, . . .*"

*,* Σ*d*_{2} !

*μ*_{i}*>*0|*Q*
*d*_{2};*μ*_{i}

0, d_{1}*μ*_{i}*< a*_{11}"

*,*
Γ

*d*_{2}|*d*_{2}*d*^{i}_{2} *d*1*μ**i*−det*J*2

*μ**i*

*a*11−*d*1*μ**i*

*, μ*_{i}*>*0, d_{1}*μ*_{i}*< a*_{11}

*.* 3.33

We note that for each*d*_{2} *>* 0,Σd2may have 0 or 2 elements. The result is contained in the
following theorem. Its proof is based on the topological degree arguments used earlier in this
paper. We shall omit it but refer the reader to similar treatments in 24,32,33.

**Theorem 3.7**bifurcation with respect to*d*_{2}.

1*Suppose thatd*2∈*/*Γ. Then,d2;**e**∗*is a regular point of*3.1.

2*Suppose that* *d*2 ∈ Γ *and* *Rd*2 *>* *0. If*

*μ**i*∈Σd2dim*Eμ**i* *is odd, then*d2;**e**_{∗} *is a*
*bifurcation point of* 3.1*with respect to the curve*d2;**e**∗, d2*>0. In this case, there exists*
*an interval*σ1*, σ*2⊂**R**^{}*, where*

i*d*2*σ*1*< σ*2*<*∞*andσ*2∈Γ*or*
ii0*< σ*_{1}*< σ*_{2} *d*_{2}*andσ*_{1}∈Γ*or*
iii σ1*, σ*_{2} d2*,*∞, or
iv σ1*, σ*2 0, d2,

*such that for everyd*2 ∈σ1*, σ*2,3.1*admits a non-constant positive solution.*

**4. Stationary Patterns for the PDE System with Cross-Diffusion**

In this section, we discuss the corresponding steady-state problem of the system1.5:

−d1Δu*λu*−*u*^{2}− *βuv*

1*munv* inΩ,

−d2Δ1*d*3*uvv*
1−*v*

*u*

inΩ,

*∂**ν**u∂**ν**v*0 on*∂Ω.*

4.1

The existence and non-existence of the non-constant positive solutions of4.1will be given.

**4.1. A Priori Upper and Lower Bounds**

**4.1. A Priori Upper and Lower Bounds**

* Theorem 4.1. Ifd*1

*, d*2 ≥

*d*

*and*

*d*3

*/d*2 ≤

*D, whered*

*and*

*D*

*are fixed positive numbers. Then,*

*there exist positive constantsCΛ, d, D,CΛ, d, Dsuch that every positive solution*u, v

*of*4.1

*satisfies*

*C < ux, vx< CΛ, d, D,* ∀x∈Ω. 4.2

*Proof. We first prove that there exists a positive constantCCΛ, d, D*such that
maxΩ *u*≤*Cmin*

Ω *u,* max

Ω *v*≤*Cmin*

Ω *v.* 4.3

A direct application of Lemma 3.1 to the first equation of 4.1 gives *u < λ* on Ω. From
Lemma 3.2, we have max_{Ω}*u* ≤ *Cmin*_{Ω}*u* for some positive constant *CΛ, d, D. Define*
*ϕx d*21*d*3*uv*and*ϕx*0 max_{Ω}*ϕ. Applying Lemma*3.1again to the second equation
of4.1, we have*vx*0≤*ux*0*< λ, which implies*

maxΩ *v*≤*d*^{−1}_{2} max

Ω *ϕ <*1*d*_{3}*λλ.* 4.4

On the other hand,*ϕ*satisfies

−Δϕ *u*−*v*

*d*21*d*3*uuϕ* inΩ,

*∂ϕ*

*∂ν* 0 on*∂Ω.*

4.5

Denote*cx u*−*v/d*21*d*3*uu. we have*

cx_{∞}≤ 1

*d*2 max_{Ω}*v*
*d*2min_{Ω}*u* ≤ 1

*d*2 1*d*3*ux*0vx0
*d*2min_{Ω}*u*

*<* 1

*d*2 1*d*3*λux*0
*d*2min_{Ω}*u* ≤ 1

*d*2 1*d*3*λ*

*d*2 ·max_{Ω}*u*

min_{Ω}*u* ≤*CΛ, d, D.*

4.6

Hence, Lemma3.2implies that there exists a positive constant*C*^{}Λ, d, Dsuch that max_{Ω}*ϕ*≤
*C*^{}min_{Ω}*ϕ. Moreover, we have*

max_{Ω}*v*

min_{Ω}*v* ≤ max_{Ω}*ϕ*

min_{Ω}*ϕ* ·max_{Ω}1*d*3*u*

min_{Ω}1*d*3*u* ≤*C*^{}·max_{Ω}*u*

min_{Ω}*u* ≤*C.* 4.7

Thus,4.3is proved.

Note that min_{Ω}*v < vx*0 ≤ *ux*0 ≤ max_{Ω}*u < λ,*4.3implies that there exists a
positive constant*CΛ, d, D*such that*ux, vx< C, for allx*∈Ω.

Turn now to the lower bound. Suppose, on the contrary, that the first result of4.1
does not hold. Then, there exists a sequence{d1,i*, d*_{2,i}*, d*_{3,i}}^{∞}* _{i1}*with

*d*

_{1,i}

*, d*

_{2,i}∈ d,∞× d,∞,

*d*3,i∈0,∞such that the corresponding positive solutionsu

*i*

*, v*

*i*of4.1satisfy

minΩ *u** _{i}*−→0 or min

Ω *v** _{i}* −→0, as

*i*−→ ∞, 4.8

andu*i**, v**i*satisfies

−d1,iΔu*i**λu** _{i}*−

*u*

^{2}

*−*

_{i}*βu*

_{i}*v*

*1*

_{i}*mu*

*i*

*nv*

*i*

inΩ,

−d2,iΔ1*d*_{3,i}*u** _{i}*v

*i*

*v*

_{i}1−*v*_{i}*u*_{i}

inΩ,

*∂**ν**u**i**∂**ν**v**i*0 on*∂Ω.*

4.9

Integrating by parts, we obtain that

Ω*u**i*

*λ*−*u**i*− *βv**i*

1*mu**i**nv**i*

*dx*0,

Ω*v*_{i}

1− *v*_{i}*u**i*

*dx*0.

4.10

By the second equation of4.10, there exists*x**i* ∈Ωsuch that*v**i*x*i* *u**i*x*i*, for all*i*≥1. By
4.8, this implies that

minΩ *u** _{i}*−→0, min

Ω *v** _{i}*−→0 as

*i*−→ ∞. 4.11 Combining4.3yields

maxΩ *u** _{i}*−→0, max

Ω *v** _{i}*−→0 as

*i*−→ ∞. 4.12 So we have

*λ*−*u**i*− *βv*_{i}

1*mu*_{i}*nv*_{i}*>*0 onΩ, ∀i 1. 4.13

Integrating the first equation of4.9overΩby parts, we have

Ω*u**i*

*λ*−*u**i*− *βv**i*

1*mu**i**nv**i*

*dx >*0, ∀i 1, 4.14

which is a contradiction to the first equation of4.10. The proof is completed.

**4.2. Non-Existence of Non-Constant Positive Steady States**

**4.2. Non-Existence of Non-Constant Positive Steady States**

* Theorem 4.2. Ifd*2

*>*1/μ2

*andd*3

*/d*2 ≤

*D, whereDis a fixed positive number, then the problem*4.1

*has no non-constant positive solution ifd*1

*is suﬃciently large.*