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

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:

ut−d1Δur1u

1− u K

−fv, x∈Ω, t >0,

vt−d2Δvr2v

1− v δu

, x∈Ω, t >0,

∂νu∂νv0, x∈∂Ω, t >0,

ux,0 u0x>0, vx,0 v0x≥0, x∈Ω,

1.1

where Ω ⊂ ^ÊN 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, u₀x and v₀x are given smooth functions onΩ which satisfy compatibility conditions. The constantsd1,d2, called

diffusion coefficients, are positive, r1 and r2 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 parametersr₁,r₂,K, andδare 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 ofu. 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_uu ≤ 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 functionf in the system1.1takes the form f βu/umvcalled 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 system1.1withfβu/1munv. Using the scaling

r1

Ku−→u, r1

Kδv−→v, r1−→λ, Kδ

r1 β−→β, K

r1m−→m, Kδ

r1 n−→n, 1.2

and takingr21 for simplicity of calculation,1.1becomes

u_t−d₁Δuλu−u²− βuv

1munv g₁u, v, x∈Ω, t >0, vt−d2Δvv

1−v

u

g2u, v, x∈Ω, t >0,

∂_νu∂_νv0, x∈∂Ω, t >0,

ux,0 u₀x>0, vx,0 v₀x≥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−β24λmn

2mn , 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 termv/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:

ut−d1Δuλu−u²− βuv

1munv, x∈Ω, t >0, vt−d2Δ1d3uvv

1−v u

, x∈Ω, t >0,

∂νu∂νv0, x∈∂Ω, t >0,

ux,0 u0x>0, vx,0 v0x≥0. x∈Ω,

1.5

where Δd2d3uv is a cross-diffusion term. If d3 > 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 ifa11 < 1, wherea11 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 r₂. 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_Ωu0}and 0< vx, t≤max{λ,sup_Ωu0,sup_Ωv0}onΩ× 0,∞.

2.1. Global Attractor and Permanence

First, we show that^Ê0 0, λ× 0, λis a global attractor for1.3.

Theorem 2.1. Letux, t, vx, tbe any non-negative solution of 1.3. Then,

tlim→∞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 constantT 0 such thatux, t < λεonΩ× T,∞for an arbitrary constantε >0, and thus,

v_t−d₂Δv≤v

1− v λε

, x, t∈Ω× T,∞. 2.2

Letvtbe 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 asε → 0.

Theorem 2.2. Assume thatβ < nλ1, then the positive solutionux, t, vx, tof1.3satisfies

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−1²4mλ

1nλ−β

. 2.6

Proof. Sinceβ < nλ1, there exists a sufficiently small constantε₁>0 such thatλnλ−βλ ε1>0. In view of Theorem2.1, there exists aT 0 such thatvx, t< λε1inΩ× T,∞.

Thus we have

ut−d1Δu≥ −mu² mλ−nλ−nε₁−1uλ nλ−β

λε₁

1munλε1 u 2.7

forx, t∈Ω× T,∞. Letutbe the unique positive solution of dw

dt −mw² mλ−nλ−nε₁−1wλ nλ−β

λε₁

1mwnλε1 w, t∈ T,∞,

wT min

Ω ux, T>0.

2.8

Then, lim_t→∞inf_Ωux, t≥lim_t→∞ut, where

t→lim∞ut 1

2m m−nλ−1−nε₁

m−nλ−nε₁−1²4m λ

nλ−β

λε₁ . 2.9 By continuity asε₁ → 0, we have lim_t→∞inf_Ωux, t≥K. Similarly, we can prove the second result of2.5.

From Theorems2.1and2.2, we see that the system1.3is permanent ifβ < nλ1.

2.2. Local Stability of Nonnegative Equilibria

Lemma 2.3. The semi-trivial solutionλ,0of1.3is unconditionally unstable.

Proof. The linearization matrix of1.3atλ,0is

J₁

⎛

⎝−λ − βλ 1mλ

0 1

⎞

⎠. 2.10

It is easy to see that 1 is an eigenvalue ofJ1, 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 g₁u, v, dv

dt g₂u, v, 2.11 whereg₁u, vandg₂u, vare given in1.3.

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

Theorem 2.4. If a11 < 1, then the unique positive equilibrium u_∗, v_∗ of 2.11 is locally asymptotically stable. Ifa₁₁ >1, thenu∗, v_∗is unstable, wherea₁₁1/β mλ−u_∗²−βu_∗. Proof. The linearization matrix of2.11atu∗, v_∗is

J2

a₁₁ a₁₂ a21 a22

, 2.12

where a₁₁ 1

β

mλ−u_∗²−βu_∗

, a₁₂−λ−u_∗1mu_∗

1 mnu∗ , a₂₁ 1, a₂₂−1. 2.13 A simple calculation shows

detJ₂−a11−a₁₂ mnu²_∗λ

1 mnu∗, traceJ₂a₁₁−1. 2.14 Clearly, detJ₂>0. Ifa₁₁<1, then traceJ₂<0. Hence, all eigenvalues ofJ₂have negative real parts andu∗, v_∗is locally asymptotically stable. Ifa11 >1, then traceJ2 >0, which implies thatJ2has two eigenvalues with positive real parts andu_∗, v_∗is unstable.

Similarly as in 23, 29, let 0 μ1 < μ2 < μ3 < μ4. . . be the eigenvalues of the operator−ΔonΩwith the homogeneous Neumann boundary condition, and letEμibe the eigenspace corresponding toμiinH¹Ω. Let{φij:j1,2, . . . ,dimEμi}be the orthonormal basis ofEμi, X H¹Ω²,Xij{cφij:c∈^Ê2}. Then,

X^∞

i1

Xi, Xi

dimE^μi

j1

Xij. 2.15

Definei₀as the largest positive integer such thatd₁μ_i< a₁₁fori≤i₀. Clearly, if

d1μ2< a11, 2.16

then 2≤i0<∞. In this case, denote d₂min

2≤i≤i0

dⁱ₂ , dⁱ₂ d1μidetJ2

μi

a11−d1μi

. 2.17

The local stability ofu∗, v_∗for1.3can be summarized as follows.

Theorem 2.5. (i) Assume thata11 >1, thenu_∗, v_∗is unstable.

(ii) Assume thata11 <1. Thenu_∗, v_∗is locally asymptotically stable ifa11 ≤d1μ2;u_∗, v_∗ is locally asymptotically stable ifa11 > d1μ2 andd2 < d2;u∗, v_∗is unstable ifa11 > d1μ2 and d2>d2.

Proof. Consider the following linearization operator of1.3atu∗, v_∗:

L

d1Δ a11 a12

a21 d2Δ a22

, 2.18

wherea₁₁,a₁₂,a₂₁, anda₂₂are given in2.13. Supposeφx, ψx^Tis an eigenfunction ofL corresponding to an eigenvalueμ, then

d1Δφ

a11−μ

φa12ψ, d2Δψa21φ

a22−μ ψ_T

0,0^T. 2.19

Setting

φ

1≤i<∞,1≤j≤dimEμi

a_ijφ_ij, ψ

1≤i<∞,1≤j≤dimEμi

b_ijφ_ij, 2.20

we can find that

1≤i<∞,1≤j≤dimE^μi Li

aij

b_ij

φij 0, whereLi

a11−d1μi−μ a12

a₂₁ a₂₂−d₂μ_i−μ

. 2.21

It follows thatμis an eigenvalue ofLif and only if the determinant of the matrixLiis zero for somei≥1, that is,

μ²P_iμQ_i0, 2.22

where

P_i d₁d₂μi−traceJ₂, Q_i−d2μ_i

a₁₁−d₁μ_i

d₁μ_idetJ₂. 2.23 Clearly, Q1 > 0 since μ1 0. If a11 > 1, then traceJ2 > 0 and P1 < 0. Hence,L has two eigenvalues with positive real parts and the steady stateu_∗, v_∗is unstable.

Note thatPi >0 for alli≥1 ifa11 <1, andQi>0 for alli≥1 ifa11 ≤d1μ2. This implies that Reμ < 0 for all eigenvalueμ, and so the steady state u_∗, v_∗ is locally asymptotically stable.

Assume thata₁₁> d₁μ₂. Ifd₂<d₂, thend₁μ_i < a₁₁andd₂< dⁱ₂ fori∈ 2, i0. It follows thatQi>0 for alli∈ 2, i0. Furthermore, ifi > i0, thend1μi ≥a11 andQi>0. The conclusion leads to the locally asymptotically stability ofu∗, v_∗again. Ifd₂>d₂, then we may assume that the minimum in2.17is attained byk ∈ 2, i0. Thus,d1μk < a11 andd2 > d^k₂ , so we haveQ_k<0. This implies thatu∗, v_∗is unstable.

Remark 2.6. From Theorems2.4and2.5, we can conclude thatu∗, v_∗is Turing unstable if d₁μ₂< a₁₁<1 andd₂>d₂.

2.3. Global Stability of

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 parametersλ,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 thatβ < nλ1 and

β λu_∗

Ku_∗1mu_∗−1mKnK 1mλnλ

<1mu_∗nv_∗1mKnK. 2.24

Thenu∗, v_∗attracts all positive solutions of 1.3.

Proof. Define the Lyapunov function

E1t

Ω

u−2u_∗u_∗² u

dxδ1

Ω

v−v_∗−v_∗ln v v_∗

dx, 2.25

where

δ1 Ku_∗ 1 β

1mu_∗nv_∗1mλnλ

, 2.26

u, v is a positive solution of 1.3. Then E1t ≥ 0 for all t ≥ 0. The straightforward computations give that

dE1

dt

Ω

u²−u_∗²

u² utdxδ1

Ω

v−v_∗ v vtdx

ΩD1dx

Ω

1 u

A1u−u_∗²B1u−u_∗v−v_∗ C1v−v_∗² dx,

2.27

where

D1−

d12u_∗²

u³ |∇u|²δ1d2v_∗ v²|∇v|²

≤0,

A₁ uu_∗ −1 βmv_∗

1mu_∗nv_∗1munv

,

B1δ1− βuu_∗1mu_∗

1mu_∗nv_∗1munv, C1−δ1.

2.28

From Theorems2.1and2.2, there exists at0 0 such thatK−ε < ux, t,vx, t < λεin Ω× t0,∞for an arbitrary and small enough constantε >0. By continuity asε → 0,2.24 implies that

B₁ Ku_∗

1mu_∗nv_∗1mKnK

× 1mu_∗nv_∗1mKnK

−β

uu_∗1mu_∗1mKnK

Ku_∗1munv −1mKnK 1mλnλ

≥0

2.29

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

dE1

dt ≤

ΩD1dx

Ω

1

uA1B1u−u_∗²dx

Ω

1 u

B1

4 C₁

v−v_∗²dx

ΩD₁dx

Ω

1

u δ₁−uu_∗

1 β

1mu_∗nv_∗1munv

u−u_∗²dx

Ω

1 u −3

4δ1− βuu_∗1mu_∗ 41mu_∗nv_∗1munv

v−v_∗²dx

≤0

2.30

inΩ× t0,∞. Similarly as in 24,30, the standard argument concludesux, t, vx, t → u∗, v_∗in L^∞Ω², which thereby shows thatu∗, v_∗attracts all positive solutions of1.3 under our hypotheses. Thus, the proof is complete.

Theorem 2.8. Assume thatβ < nλ1,

β

1mu_∗−1mKnK 1mλnλ

<1mu_∗nv_∗1mKnK, 2.31

β < λmλn2mn

2 . 2.32

Then,u∗, v_∗attracts all positive solutions of 1.3.

Proof. Define the Lyapunov function

E2t

Ω

u_∗−u u ln u

u_∗

dxδ2

Ω v−v_∗−v_∗ln v v_∗

dx, 2.33

whereδ21 β/1mu_∗nv_∗1mλnλ,u, vis a positive solution of1.3. Then dE2

dt

ΩD₂dx

Ω

1 u

A₂u−u_∗²B₂u−u_∗v−v_∗ C₂v−v_∗²

dx, 2.34

where

D2− d1

2u_∗−u

u³ |∇u|²δ2d2

v_∗ v²|∇v|²

,

A2−1 βmv_∗

1mu_∗nv_∗1munv, B2δ2− β1mu_∗

1mu_∗nv_∗1munv, C2−δ2.

2.35

From Theorems2.1and2.2, there exists at0 0 such thatK−ε < ux, t,vx, t < λεin Ω× t0,∞for an arbitrary and small enough constantε >0. Thus2.31implies that

B₂ 1

1mu_∗nv_∗1mKnK

× 1mu_∗nv_∗1mKnK

−β

1mu_∗1mKnK

1munv − 1mKnK 1mλnλ

≥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

dE2

dt ≤

ΩD2dx

Ω

1

uA2B2u−u_∗²dx

Ω

1 u

B2

4 C2

v−v_∗²dx

ΩD₂dx

Ω

1 u δ₂−

1 β

1mu_∗nv_∗1munv

u−u_∗²dx

Ω

1 u −3

4δ2− β1mu_∗

41mu_∗nv_∗1munv

v−v_∗²dx

≤0

2.37

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

Remark 2.9. If we choose the common Lyapunov function

E3t

Ω u−u_∗−u_∗ln u u_∗

dxδ3

Ω v−v_∗−v_∗ln v v_∗

dx, 2.38

whereδ₃K{1 β/1mu_∗nv_∗1mλnλ}, 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 choosem1, then2.32holds sinceβ < λn1. It is not hard to verify that the condition2.31in Theorem2.8contains the condition2.24in Theorem2.7. However, if we choosemandnto be sufficiently small, thenu_∗v_∗ → λ/1βandK → 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:

−d1Δuλu−u²− βuv

1munv g1u, v inΩ,

−d2Δvv 1−v

u

g₂u, v inΩ,

∂νu∂νv0 on∂Ω.

3.1

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

In the following, the generic constantsC1,C2,C_∗,C,C, and so forth, will depend on the domainΩand the dimensionN. However, asΩand the dimensionNare 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

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 3.1maximum principle 25. Letg∈CΩ×^Ê1andbj∈CΩ,j1,2, . . . , N.

iIfw∈C²Ω∩C¹Ωsatisfies

Δwx ^N

j1

bjxwxjgx, wx≥0 inΩ,

∂w

∂ν ≤0 on∂Ω,

3.2

andwx0 max_Ωwx, thengx0, wx0≥0.

iiIfw∈C²Ω∩C¹Ωsatisfies

Δwx ^N

j1

bjxwxjgx, wx≤0 inΩ,

∂w

∂ν ≥0 on∂Ω,

3.3

andwx0 min_Ωwx, thengx0, wx0≤0.

Lemma 3.2Harnack, inequality 31. Letw∈C²Ω∩C¹Ωbe a positive solution toΔwx cxwx 0, wherec ∈ CΩ, satisfying the homogeneous Neumann boundary condition. Then there exists a positive constantC_∗which depends only onc_∞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 numberd, there exists a positive constantCΛ, dsuch that every positive solutionu, vof3.1satisfiesC < ux,vx< λifd1≥d.

Proof. Let ux1 max_Ωux, vx2 max_Ωvx, uy1 min_Ωux, vy2 min_Ωvx.

Application of Lemma3.1yields that

λ−ux1− βvx1

1mux1 nvx1 ≥0, λ−u

y1

− βv y₁ 1mu

y₁ nv

y₁ ≤0,

1−vx2

ux2 ≥0, 1−v y₂ u

y₂ ≤0.

3.5

Clearly,ux1< λandvx2≤ux2≤ux1< λ. Moreover, we have v

y1

≤vx2≤ux2≤ux1, 3.6

v y₁

≥v y₂

≥u y₂

≥u y₁

. 3.7

By3.5, we obtain m

u y₁₂

1nv

y₁

−λm u

y₁

β−λn v

y₁

−λ≥0. 3.8 Noting thatuy1≤vy1≤ux1from3.6and3.7,3.8implies that max_Ωux ux1>

C₁for some positive constantC₁C₁Λ.

Letcx d₁⁻¹λ−u−βv/1munv. Then,cx_∞ ≤ 2βλ/d. The Harnack inequality shows that there exists a positive constantC_∗C_∗λ, β, dsuch that

maxΩ ux≤C_∗min

Ω ux. 3.9

Combining 3.9 with max_Ωux > C1, we find that min_Ωux > C1 for some positive constantC CΛ, d. It follows from3.7that min_Ωvx vy2 ≥ uy1 > C. The proof is completed.

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 coefficientd1varies while the other parametersd2,λ,β,m, andnare fixed.

Theorem 3.4. For any positive numberd, there exists a positive constantD DΛ, d > dsuch that3.1has no non-constant positive solution ifd₁> D.

Proof. For anyϕ∈L¹Ω, let

ϕ 1

|Ω|

Ωϕ dx. 3.10

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

Ω

d1u

u² |∇u|² d2v v² |∇v|²

dx

Ωg1u, vu−u u dx

Ωg2u, vv−v v dx

Ω −1 βmv

1munv1munv

u−u²dx

Ω − β1mu

1munv1munv v uu

u−uv−vdx

Ω

−1 u

v−v²dx.

3.11

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

Ω

d₁|∇u|²d₂|∇v|² dx≤C₂

−1βm n C₃

Ωu−u²dxC₂

Ω

ε− 1

u

v−v²dx 3.12 for some positive constantsC₂ C₂Λ, d,C₃ C₃Λ, d, ε, whereεis the arbitrary small positive constant arising from Young’s inequality. By Theorem3.3, we can chooseε∈0,1/λ.

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

μ₂

Ω

d₁u−u²d₂v−v²

dx≤C₄

Ωu−u²dxC₂

Ω

ε− 1

u

v−v²dx, 3.13

which implies thatuuconstant andvvconstant ifd1> Dmax{C4/μ2, d}.

3.3. Existence of Non-Constant Positive Steady States

Throughout this subsection, we always assume thata11 >0. First, we study the linearization of3.1atu_∗, v_∗. Let

Y u, v:u, v∈ C¹

Ω₂

, ∂_νu∂_νv0 on∂Ω

. 3.14

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

Fe

a11−d1Δ⁻¹

g1u, v a11u

−a22−d₂Δ⁻¹

g₂u, v−a₂₂v

, 3.15

wheree ux, vx^T,a11−d1Δ⁻¹, and−a22−d2Δ⁻¹are the inverses of the operators a11 −d1Δand−a22 −d2ΔinY 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− Fee∗Ψ μΨ, Ψ/0, 3.16

whereΨ ψ1, ψ2^T ande_∗ u_∗, v_∗^T. If 0 is not an eigenvalue of3.16, then the Leray- Schauder Theorem 27implies that

indexI− F,e_∗ −1^γ, 3.17

whereγis the sum of the algebraic multiplicities of the positive eigenvalues of−I− F_ee_∗, 3.16can be rewritten as

− μ1

d1Δψ1

−μ1

a11ψ1a12ψ2,

− μ1

d2Δψ2a21ψ1 μ1

a22ψ2.

3.18

As in the proof of Theorem2.5, we can conclude thatμis an eigenvalue of−I− Fee∗on Xijif and only if it is a root of the characteristic equation detBi0, where

B_i

−μ1 a₁₁−

μ1

d₁μ_i a₁₂

a₂₁

μ1

a₂₂− μ1

d₂μ_i

. 3.19

The characteristic equation detB_i0 can be written as

μ² 2d1μi

a11d1μiμ −d2μ_i

a₁₁−d₁μ_i

d₁μ_idetJ₂ a11d1μi

−a22d2μi

0. 3.20

Note that−d2μ_ia11−d₁μ_i d₁μ_idetJ₂ Q_i, whereQ_iis given in2.23. Therefore, if 0 is not a root of3.20for alli≥1, we have

indexI− F,e∗ −1^γ, 3.21

whereγis the sum of the algebraic multiplicities of the positive roots of3.20.

Theorem 3.5. Assume that the parametersλ,β,m,n, andd1are fixed and 0< a11 <1. Ifa11/d1 ∈ μn, μ_n1for somen≥ 2 and

2≤i≤n, Qi<0dimEμiis odd, then the problem3.1has at least one non-constant positive solution for anyd2 > d2, whereQi and d2 are given in2.23 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 d2 d˘₂>d₂. In the follow we fixd₂d˘₂. Takingda₁₁/μ₂in Theorems3.3and3.4, we obtain a positive constantD. Fixedd1D1 andd21. Forθ∈ 0,1, define a homotopy

Fθ;e

⎛

⎜⎝

a₁₁−

θd₁ 1−θd₁ Δ₋₁

g₁u, v a₁₁u −a22−

θd₂ 1−θd₂ Δ₋₁

g₂u, v−a₂₂v

⎞

⎟⎠. 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 Csuch 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∈∂BCandFθ;·: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 a₁₁/d₁ ∈ μn, μ_n1 for some n ≥ 2, then i₀ n and d₂ min_2≤i≤ndⁱ₂ in 2.17. Since d2 d˘2 >d2, thenQk<0 for somek, 2≤k ≤n. Letik. Then,3.20has one positive root and a negative root. Furthermore, we haveQ_i>0 fori1 and alli≥n1. Therefore, when i1 andi≥n1, the characteristic equation3.20has no roots with non-negative real parts.

In addition, if

2≤i≤n, Qi<0dimEμiis odd, we have

indexI− F1;·,e_∗ −1^{2≤i≤n, Qi<0dimEμi}−1. 3.24

By our supposition, the equationF1;e e has only the positive solution e_∗inBC, and hence

degI− F1;·, BC,0 indexI− F1;·,e_∗ −1. 3.25

Similar argument showsμis an eigenvalue of−I− F_e0;e_∗if and only if it is a root of the characteristic equation

μ² 2d₁μ_i

a₁₁d₁μ_iμ−d₂μ_i

a₁₁−d₁μ_i

d₁μ_idetJ₂

a11d1μi

−a22d2μi

0. 3.26

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

indexI− F0;·,e∗ −1⁰1. 3.27

In view of Theorem3.4, it follows that the equationF0;e e has only the positive solution e_∗inBC, and therefore,

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

Example 3.6. LetΩ 0,1. Then, the parametersλ 2,β 6,m 3,n 0.1,d₁ 0.0152, andd24.1309 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−u²− 6uv 13u0.1v, dv

dt v 1−v

u

,

3.29

but it is an unstable steady state for the system ut−0.0152uxx2u−u²− 6uv

13u0.1v, x∈0,1, t >0, vt−4.1309vxxv

1−v u

, x∈0,1, t >0, uxvx0, x0,1, t >0,

ux,0 u₀x>0, vx,0 v₀x≥0, x∈0,1.

3.30

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

3.4. Bifurcation

In this subsection, we discuss the bifurcation of non-constant positive solutions of3.1with respect to the diffusion coefficientd₂. In the consideration of bifurcation with respect tod₂, we recall that, for a constant solutione∗,d2;e∗ ∈0,∞×Y is a bifurcation point of3.1 if, for anyδ ∈ 0, d2, there exists ad2 ∈ d2−δ, d2δsuch that3.1has a non-constant positive solution close toe∗. 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 d₂;μ

d₁d₂μ²−d2a₁₁−d₁μdetJ₂. 3.31 It is clear thatQd2;μ 0 has at most two roots for any fixedd2 >0. Noting that detJ2 >0 in the proof of Theorem2.4, if

Rd2d2a11d1²4d1d2a12>0, 3.32

thenQd2, μ 0 has two different real roots with same symbols. Let S_p!

μ₁, μ₂, μ₃, . . ."

, Σd₂ !

μ_i >0|Q d₂;μ_i

0, d₁μ_i< a₁₁"

, Γ

d₂|d₂dⁱ₂ d1μi−detJ2

μi

a11−d1μi

, μ_i>0, d₁μ_i< a₁₁

. 3.33

We note that for eachd₂ > 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 3.7bifurcation with respect tod₂.

1Suppose thatd2∈/Γ. Then,d2;e∗is a regular point of3.1.

2Suppose that d2 ∈ Γ and Rd2 > 0. If

μi∈Σd2dimEμi is odd, thend2;e_∗ is a bifurcation point of 3.1with respect to the curved2;e∗, d2>0. In this case, there exists an intervalσ1, σ2⊂R, where

id2σ1< σ2<∞andσ2∈Γor ii0< σ₁< σ₂ d₂andσ₁∈Γor iii σ1, σ₂ d2,∞, or iv σ1, σ2 0, d2,

such that for everyd2 ∈σ1, σ2,3.1admits 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:

−d1Δuλu−u²− βuv

1munv inΩ,

−d2Δ1d3uvv 1−v

u

inΩ,

∂νu∂νv0 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

Theorem 4.1. Ifd1, d2 ≥ d and d3/d2 ≤ D, whered and D are fixed positive numbers. Then, there exist positive constantsCΛ, d, D,CΛ, d, Dsuch that every positive solutionu, vof4.1 satisfies

C < ux, vx< CΛ, d, D, ∀x∈Ω. 4.2

Proof. We first prove that there exists a positive constantCCΛ, d, Dsuch 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_Ωu ≤ Cmin_Ωu for some positive constant CΛ, d, D. Define ϕx d21d3uvandϕx0 max_Ωϕ. Applying Lemma3.1again to the second equation of4.1, we havevx0≤ux0< λ, which implies

maxΩ v≤d⁻¹₂ max

Ω ϕ <1d₃λλ. 4.4

On the other hand,ϕsatisfies

−Δϕ u−v

d21d3uuϕ inΩ,

∂ϕ

∂ν 0 on∂Ω.

4.5

Denotecx u−v/d21d3uu. we have

cx_∞≤ 1

d2 max_Ωv d2min_Ωu ≤ 1

d2 1d3ux0vx0 d2min_Ωu

< 1

d2 1d3λux0 d2min_Ωu ≤ 1

d2 1d3λ

d2 ·max_Ωu

min_Ωu ≤CΛ, d, D.

4.6

Hence, Lemma3.2implies that there exists a positive constantCΛ, d, Dsuch that max_Ωϕ≤ Cmin_Ωϕ. Moreover, we have

max_Ωv

min_Ωv ≤ max_Ωϕ

min_Ωϕ ·max_Ω1d3u

min_Ω1d3u ≤C·max_Ωu

min_Ωu ≤C. 4.7

Thus,4.3is proved.

Note that min_Ωv < vx0 ≤ ux0 ≤ max_Ωu < λ,4.3implies that there exists a positive constantCΛ, d, Dsuch thatux, 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}^∞_i1withd_1,i, d_2,i∈ d,∞× d,∞, d3,i∈0,∞such that the corresponding positive solutionsui, viof4.1satisfy

minΩ u_i−→0 or min

Ω v_i −→0, asi−→ ∞, 4.8

andui, visatisfies

−d1,iΔuiλu_i−u²_i − βu_iv_i 1muinvi

inΩ,

−d2,iΔ1d_3,iu_iviv_i

1−v_i u_i

inΩ,

∂νui∂νvi0 on∂Ω.

4.9

Integrating by parts, we obtain that

Ωui

λ−ui− βvi

1muinvi

dx0,

Ωv_i

1− v_i ui

dx0.

4.10

By the second equation of4.10, there existsxi ∈Ωsuch thatvixi uixi, for alli≥1. By 4.8, this implies that

minΩ u_i−→0, min

Ω v_i−→0 asi−→ ∞. 4.11 Combining4.3yields

maxΩ u_i−→0, max

Ω v_i−→0 asi−→ ∞. 4.12 So we have

λ−ui− βv_i

1mu_inv_i >0 onΩ, ∀i 1. 4.13

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

Ωui

λ−ui− βvi

1muinvi

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

Theorem 4.2. Ifd2 > 1/μ2 andd3/d2 ≤ D, whereDis a fixed positive number, then the problem 4.1has no non-constant positive solution ifd1is sufficiently large.