positive solutions for a three-trophic food chain model with diffusion and beddington–deangelis...

Nonlinear Analysis: Real World Applications 12 (2011) 902–917

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Nonlinear Analysis: Real World Applications

journal homepage: www.elsevier.com/locate/nonrwa

Positive solutions for a three-trophic food chain model with diffusionand Beddington–Deangelis functional response

Jun Zhou a,b,∗, Chunlai Mu b

a School of Mathematics and Statistics, Southwest University, Chongqing, 400715, Chinab College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

a r t i c l e i n f o

Article history:Received 28 September 2009Accepted 6 August 2010

Keywords:Positive solutionsFood chain modelBeddington–Deangelis functional responseFixed point indexExtinctionPermanenceGlobal attractorStability

a b s t r a c t

In this paper, we investigate the existence, stability, permanence, and global attractor ofpositive solutions for the following three species food chain model with diffusion andBeddington–Deangelis functional response

−∆u1 = u1(r − u1) −a1u1u2

e1 + u1 + u2in Ω,

−∆u2 =m1u1u2

e1 + u1 + u2− b1u2 −

a2u2u3

e2 + u2 + u3in Ω,

−∆u3 =m2u2u3

e2 + u2 + u3− b2u3 in Ω,

k1∂u1

∂ν+ u1 = k2

∂u2

∂ν+ u2 = k3

∂u3

∂ν+ u3 = 0 on ∂Ω,

where Ω is a bounded domain of RN , N ≥ 1, with boundary ∂Ω of class C2+α for someα ∈ (0, 1), ν is the outward unit vector on ∂Ω , the parameters r, ai, bi, ei, mi (i = 1, 2)are strictly positive, and ki ≥ 0 (i = 1, 2, 3), ui (i = 1, 2, 3) are the respective densities ofprey, predator, and top predator.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Understanding of spatial and temporal behaviors of interaction species in ecological systems is a center issue inpopulation ecology. One aspect of great interest for a model with multispecies interactions is whether the involved speciescan persist or even stabilize at a coexistence steady state. In the casewhether the species are homogeneously distributed, thiswould be indicated by a constant positive solution of a ordinary differential equation system. In the spatially inhomogeneouscase, the existence of a nonconstant time-independent positive solution, also called stationary pattern,which is an indicationof the richness of the corresponding partial differential equation dynamics. In recent years, stationary pattern induced bydiffusion has been studied extensively, and many important phenomena have been observed.

The role of diffusion in the modeling of many physical, chemical and biological processes has been extensively studied.Starting with Turing’s seminal 1952 paper [1], diffusion and cross-diffusion have been observed as causes of the spon-taneous emergence of ordered structures, called patterns, in a variety of non-equilibrium situations. These include the

J. Zhou is supported by the Fundamental Research Funds for the Central Universities (No. XDJK2009C069) and C.L. Mu is supported by NNSF of China(No. 10771226).∗ Corresponding author at: School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China.

E-mail addresses: [email protected] (J. Zhou), [email protected] (C. Mu).

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J. Zhou, C. Mu / Nonlinear Analysis: Real World Applications 12 (2011) 902–917 903

Gierer–Meinhardt model [2–6], the Sel’kov model [7,8], the Noyes–Field model for Belousov–Zhabotinskii reaction [9],the chemotactic diffusion model [10,11], the Brusselator model [12], the competition model [13–17], the predator–preymodel [18–26], as well as models of semiconductors, plasmas, chemical waves, combustion systems, embryogenesis, etc.,see e.g. [27–29] and references therein. Diffusion-driven instability, also called Turing instability, has also been verifiedempirically [30,31].

This paper studies the stationary solution of the parabolic model

∂u1

∂t− 1u1 = u1(r − u1) −

a1u1u2

e1 + u1 + u2in Ω × R+,

∂u2

∂t− 1u2 =

m1u1u2

e1 + u1 + u2− b1u2 −

a2u2u3

e2 + u2 + u3in Ω × R+,

∂u3

∂t− 1u3 =

m2u2u3

e2 + u2 + u3− b2u3 in Ω × R+,

k1∂u1

∂ν+ u1 = k2

∂u2

∂ν+ u2 = k3

∂u3

∂ν+ u3 = 0 on ∂Ω × R+,

(1.1)

with initial values ui(x, 0) = ui0(x) ≥ 0, ≡ 0, (i = 1, 2, 3) inΩ , whereΩ is a bounded domain ofRN ,N ≥ 1, with boundary∂Ω of class C2+α for some α ∈ (0, 1), ν is the outward unit vector on ∂Ω , the parameters r, ai, bi, ei, mi (i = 1, 2) arestrictly positive, and ki ≥ 0 (i = 1, 2, 3), and ui0 (i = 1, 2, 3) stand for the initial conditions. We denote R+ := (0, ∞).

Problem (1.1) models three species food chain with diffusion and Beddington–Deangelis functional response, whereui (i = 1, 2, 3) are the respective densities of prey, predator, and top predator (cf. [32–34], and the list of references therein).

In our work here, we are mainly characterizing the existence of steady-states of (1.1), which are the solution of

−1u1 = u1(r − u1) −a1u1u2

e1 + u1 + u2in Ω,

−1u2 =m1u1u2

e1 + u1 + u2− b1u2 −

a2u2u3

e2 + u2 + u3in Ω,

−1u3 =m2u2u3

e2 + u2 + u3− b2u3 in Ω,

k1∂u1

∂ν+ u1 = k2

∂u2

∂ν+ u2 = k3

∂u3

∂ν+ u3 = 0 on ∂Ω.

(1.2)

Owing to the classical theory of parabolic equations, the solutions of (1.1) are globally defined in time and satisfy

ui(·, t) ≥ 0, (i = 1, 2, 3), for all t ∈ R+. (1.3)

So, in this paper, we will discuss the nonnegative solutions of (1.2). We have a trivial nonnegative solution (u1, u2, u3) =

(0, 0, 0). As in [35,36], the other nonnegative solutions of (1.2) can be classified by three types:

(i) nonnegative solutions with exactly two components identically zero

(u1, 0, 0), (0, u2, 0), (0, 0, u3);

(ii) nonnegative solutions with exactly one component identically zero

(u⋆1, u

⋆2, 0), (u1, 0, u3), (0, u2, u3);

(iii) nonnegative solutions with no component identically zero.

We call solutions of the first two types semitrivial solutions, while those of the third type we call positive solutions.In the present work, we attempt to further understand the influence of diffusion and functional response on pattern

formation. As a consequence, the existence and nonexistence results for nonconstant positive steady state solution to (1.2)indicate the stationary pattern arises as the diffusion coefficient enter into certain regions. In other words, diffusion doeshelp to create stationary pattern. On the other hand, our results also demonstrate that diffusion and functional response canbecome determining factors in the formation pattern.

The distribution of this paper is the following. In Section 2, we give some fundamental theorems. In Section 3, thenecessary and sufficient conditions for positive solutions of (1.2) are investigated. To achieve this, some degree theoremsare developed, which is playing an important role in the text. Finally, in Section 4, sufficient conditions for the extinctionand permanence to the time-dependent system (1.1) are investigated.

2. Preliminaries

In this section, we give some fundamental theorems, especially some degree theorems, which play an important role inthe text.

904 J. Zhou, C. Mu / Nonlinear Analysis: Real World Applications 12 (2011) 902–917

For each q ∈ Cα(Ω) (0 < α < 1) and k ≥ 0, denote the principle eigenvalue of−1u + q(x)u = λu in Ω,

k∂u∂ν

+ u = 0 on ∂Ω,(2.1)

by λ1,k(q) and simply denote λ1,k(0) by λ1,k. It is well known that λ1,k(q(x)) is strictly increasing in the sense that q1(x) ≤

q2(x) and q1(x) ≡ q2(x) implies λ1,k(q1(x)) < λ1,k(q2(x)) (see Proposition 1.1 of [37]).

Theorem 2.1 ([37–39]). For q ∈ Cα(Ω) (0 < α < 1) and p be a sufficiently large number such that p > q(x) for all x ∈ Ω ,define an positive compact operator L := (−∆ + p)−1(p − q(x)) : C1

k (Ω) → C1k (Ω) = u ∈ C1(Ω) : k∂νu + u = 0 on ∂Ω

for k ≥ 0 a constant. Denote the spectral radius of L by rk(L). Then we have

(a) λ1,k(q) > 0 ⇔ rk(L) < 1;(b) λ1,k(q) < 0 ⇔ rk(L) > 1;(c) λ1,k(q) = 0 ⇔ rk(L) = 1.

From Theorem 2.1, we see that it is crucial to determine the eigenvalue λ1,k(q). The following theorem is established byTheorem 2.4 of [40] and Theorem 11.10 of [41] (see also [38,42–45]).

Theorem 2.2. Let q(x) ∈ L∞(Ω) and φ ≥ 0, φ ≡ 0 in Ω with k∂νφ + φ = 0 on ∂Ω for k ≥ 0 a constant. Then we have

(a) if 0 ≡ −1φ + q(x)φ ≤ 0, then λ1,k(q) < 0;(b) if 0 ≡ −1φ + q(x)φ ≥ 0, then λ1,k(q) > 0;(c) if − 1φ + q(x)φ ≡ 0, then λ1,k(q) = 0.

Consider the following single equation:−1u = uf (x, u) in Ω,

k∂u∂ν

+ u = 0 on ∂Ω,(2.2)

where Ω is a bounded domain in RN with smooth boundary ∂Ω , k is nonnegative constant, ν is the outward unit vector on∂Ω . Assume that the function f (x, u) : Ω × [0, ∞) → R satisfies the following hypotheses:

(H1) f (x, u) is Cα-function in x, where 0 < α < 1.(H2) f (x, u) is C1-function in uwith fu(x, u) < 0 for all (x, u) ∈ Ω × [0, ∞).(H3) f (x, u) ≤ 0 in Ω × [C, ∞) for some positive constant C .

Theorem 2.3 ([42,45]). (i) The nonnegative solution u(x) of (2.2) satisfies u(x) ≤ C for all x ∈ Ω;(ii) If λ1,k(−f (x, 0)) ≥ 0, then (2.2) has no positive solutions. Moreover, the trivial solution u(x) = 0 is globally asymptoticallystable;(iii) If λ1,k(−f (x, 0)) < 0, then (2.2) has a unique positive solution which is globally asymptotically stable. In this case, the trivialsolution u(x) = 0 is unstable.

Due to above theorem, we denote Θk(ρ(x)) with Θk(ρ(x)) ≤ maxx∈Ω ρ(x) be the unique positive solution of thefollowing equation

−1φ = φ(ρ(x) − φ) in Ω,

k∂φ

∂ν+ φ = 0 on ∂Ω,

(2.3)

if λ1,k(−ρ) < 0, where ρ(x) ∈ Cα(Ω)(0 < α < 1) is a positive function.Now, we state the fixed point index theory, which is a fundamental tool in our proofs.Let E be a Banach space and W ⊂ E is a closed convex set. W is called a total wedge if γW ⊂ W for all γ ≥ 0 and

W − W = E. For y ∈ W , define Wy = x ∈ E : y + γ x ∈ W for some γ > 0 and Sy = x ∈ W y : −x ∈ W y. Then W y is awedge containing W, y, − y, while Sy is a closed subset of E containing y. Let T be a compact linear operator on E whichsatisfies T (W y) ⊂ W y. We say that T has property α onW y if there is a t ∈ (0, 1) and aω ∈ W ySy such that (I− tT )ω ∈ Sy.Let A : W → W is a compact operator with a fixed point y ∈ W and A is Fréchet differentiable at y. Let L = A′(y) be theFréchet derivative of A at y. Then L maps W y into itself. We denote by degW (I − A,D) the degree of I − A in D relative toW , indexW (A, y) the fixed point index of A at y relative to W and

degW (I − A, Φ) =

−y∈Φ

indexW (A, y),

where indexW (A, y) the fixed point index of A at y relative to W and Φ only contain discrete points.


Theorem 2.4 ([39,46,47]). Assume that I − L has no non-trivial kernel in W y. Then, we have(i) If L has property α on W y, then indexW (A, y) = 0;(ii) If L does not have property α on W y, then indexW (A, y) = (−1)σ , where σ is the sum of multiplicities of all eigenvalues ofL which is greater than 1.

Finally, we introduce the degree calculations, which was introduced by Dancer and Du in [35] and we state here forreader’s convenience.

Suppose E1 and E2 are ordered Banach spaces with positive cones W1 and W2, respectively. Let E = E1

E2 andW = W1

W2. Then E is an ordered Banach space with positive cone W .

Let D be an open set in W containing 0 and Ai : D → Wi be completely continuous operators, i = 1, 2. Denote by (u, v)a general element in W with u ∈ W1 and v ∈ W2. Let A : D → W be defined by

A(u, v) = (A1(u, v), A2(u, v)).

Also we define

W2(ϵ) = v ∈ W2 : ‖v‖E2 < ϵ.

Then we have the following results:

Theorem 2.5. Suppose U ⊂ W1 ∩ D is relatively open and bounded, and

A1(u, 0) = u for u ∈ ∂U; A2(u, 0) ≡ 0 for u ∈ U .

Suppose A2 : D → W2 extends to a continuously differentiable mapping of a neighborhood of D into E2, W2 − W2 is dense in E2and Φ = u ∈ U : u = A1(u, 0).

Then the following are true:(i) degW (I − A,U × W2(ϵ), 0) = 0 for ϵ > 0 small, if for any u ∈ Φ , the spectral radius r(A′

2(u, 0)|W2) > 1 and 1 is not aneigenvalue of A′

2(u, 0)|W2 corresponding to a positive eigenvector;(ii) degW (I − A,U × W2(ϵ), 0) = degW1

(I − A1|W1 ,U, 0) for ϵ > 0 small, if for any u ∈ Φ , the spectral radiusrA′

2(u, 0)|W2

< 1.

3. Existence and non-existence of positive solutions

3.1. Existence of positive solutions

To give some sufficient conditions for the existence of positive solutions of (1.2) by using fixed point index theory, weneed an a priori estimate for nonnegative solutions of (1.2). So, we first give the following theorem:

Theorem 3.1. If m1r > b1(e1 + r) and m1m2r + b1b2(e1 + r − e2) > m2b1(e1 + r) + m1b2r, any nonnegative solution(u1, u2, u3) of (1.2) has an a priori bounds:

u1(x) ≤ Q1, u2(x) ≤ Q2, and u3(x) ≤ Q3, (3.1)

where

Q1 = r, Q2 =m1r − b1(e1 + r)

b1,

and

Q3 =m1m2r + b1b2(e1 + r − e2) − m2b1(e1 + r) − m1b2r

b1b2.

Proof. From the first equation of (1.2), we have−1u1 ≤ u1(r − u1) in Ω,

k1∂u1

∂ν+ u1 = 0 on ∂Ω,

(3.2)

we get u1(x) ≤ r by maximum principle. Taking account of u(x) ≤ r , we get from the second equation of (1.2) that−1u2 ≤ u2

m1r

e1 + r + u2− b1

in Ω,

k2∂u2

∂ν+ u2 = 0 on ∂Ω,

(3.3)


so, by maximum principle, we have

u2(x) ≤m1r − b1(e1 + r)

b1.

Similarly, we get from the third equation of (1.2) that−1u3 ≤ u3

m1m2r − m2b1(e1 + r)

b1e2 + m1r − b1(e1 + r) + b1u3− b2

in Ω,

k3∂u3

∂ν+ u3 = 0 on ∂Ω,

(3.4)

so, by maximum principle, we have

u3(x) ≤m1m2r + b1b2(e1 + r − e2) − m2b1(e1 + r) − m1b2r

b1b2.

The proof is complete.

Remark 3.2. From the proof of Theorem 3.1, we see that m1r > b1(e1 + r) and m1m2r + b1b2(e1 + r − e2) > m2b1(e1 +

r)+m1b2r are the necessary conditions for problem (1.2) have positive solutions. So, throughout this subsection, we assumethat

(H) m1r > b1(e1 + r), and m1m2r + b1b2(e1 + r − e2) > m2b1(e1 + r) + m1b2r.

We introduce the following notations:

E = C1k1(Ω)

C1k2(Ω)

C1k3(Ω),

where

C1ki(Ω) =

φ ∈ C1(Ω) : ki

∂φ

∂ν+ φ = 0, on ∂Ω

, i = 1, 2, 3,

W = K1

K2

K3, where Ki = φ ∈ C1

ki(Ω) : φ ≥ 0 on Ω, i = 1, 2, 3,

D = (u1, u2, u3) ∈ W : u1 < Q1 + 1, u2 < Q2 + 1, u3 < Q3 + 1 ,

where Q1, Q2, Q3 are defined in Theorem 3.1.From Theorem 3.1, we see that the nonnegative solutions of (1.2) must be in D. Take p sufficiently large positive with p >

maxr+a1Q2, b1+a2Q3, b2 such that u1(r−u1)−a1u1u2

e1+u1+u2+pu1,

m1u1u2e1+u1+u2

−b1u2−a2u2u3

e2+u2+u3+pu2 and

m2u2u3e2+u2+u3

−b2u3+pu3

are respective monotone increasing with respect to u1, u2 and u3 for all (u1, u2, u3) ∈ [0,Q1] × [0,Q2] × [0,Q3].Define a positive and compact operator A : E → E by

A(u1, u2, u3) = (−∆ + p)−1

u1

r − u1 −

a1u2

e1 + u1 + u2

+ pu1

u2

m1u1

e1 + u1 + u2− b1 −

a2u3

e2 + u2 + u3

+ pu2

u3

m2u2

e2 + u2 + u3− b2

+ pu3

.

Remark 3.3. Note that (1.2) is equivalent to (u1, u2, u3) = A(u1, u2, u3) by elliptic regularity, and therefore it suffices toprove A has a positive fixed point in interior of D to show that (1.2) has a positive solution.

The following lemma give the degree of I − A in D relative to W and the fixed point index of A at the trivial solution(0, 0, 0) of (1.2) relative to W .

Lemma 3.4. Let (H) hold and assume that r > λ1,k1 , then we have(i) degW (I − A,D) = 1; (ii) indexW (A, (0, 0, 0)) = 0.

Proof. (i) It is easy to see that A has no fixed point on ∂D, so the degW (I − A,D) is well defined. For θ ∈ [0, 1], we define apositive and compact operator Aθ : E → E by

Aθ (u1, u2, u3) = (−∆ + p)−1

θu1

r − u1 −

a1u2

e1 + u1 + u2

+ pu1

θu2

m1u1

e1 + u1 + u2− b1 −

a2u3

e2 + u2 + u3

+ pu2

θu3

m2u2

e2 + u2 + u3− b2

+ pu3

,


then A1 = A. For each θ , a fixed point of Aθ is a solution of the following problem

−1u1 = θu1

r − u1 −

a1u2

e1 + u1 + u2

in Ω,

−1u2 = θu2

m1u1

e1 + u1 + u2− b1 −

a2u3

e2 + u2 + u3

in Ω,

−1u3 = θu3

m2u2

e2 + u2 + u3− b2

in Ω,

k1∂u1

∂ν+ u1 = k2

∂u2

∂ν+ u2 = k3

∂u3

∂ν+ u3 = 0 on ∂Ω.

(3.5)

As in Theorem 3.1, we see that the fixed point of Aθ satisfies u(x) ≤ Q1, v(x) ≤ Q2, w(x) ≤ Q3 for each θ ∈ [0, 1]. So Aθ

has no fixed point on ∂D, the degW (I − Aθ ,D) is well defined and degW (I − Aθ ,D) is independent of θ . Therefore

degW (I − A,D) = degW (I − A1,D) = degW (I − A0,D).

Note that (3.5) has only the trivial solution (0, 0, 0) when θ = 0. Set

L = A′

0(0, 0, 0) = (−∆ + p)−1

p 0 00 p 00 0 p

.

Assume thatL(ξ1, ξ2, ξ3) = (ξ1, ξ2, ξ3) for some (ξ1, ξ2, ξ3) ∈ W (0,0,0) = K1×K2×K3. It is easy to see (ξ1, ξ2, ξ3) = (0, 0, 0).Thus I − L has no non-trivial kernel in W (0,0,0). Since λ1,ki > 0, we see that rki((−∆ + p)−1(p)) < 1 for i = 1, 2, 3 byTheorem 2.1. This implies that L does not have property α. So, by Theorem 2.4, we get

degW (I − A,D) = degW (I − A0,D) = indexW (A0, (0, 0, 0)) = 1.

(ii) Observe that A(0, 0, 0) = (0, 0, 0). Let L = A′(0, 0, 0), then

L = A′(0, 0, 0) = (−∆ + p)−1

r + p 0 00 −b1 + p 00 0 −b2 + p

.

Assume that L(ξ1, ξ2, ξ3) = (ξ1, ξ2, ξ3) for some (ξ1, ξ2, ξ3) ∈ W (0,0,0), then−1ξ1 = rξ1 in Ω,

k1∂ξ1

∂ν+ ξ1 = 0 on ∂Ω,

(3.6)

−1ξ2 = −b1ξ2 in Ω,

k2∂ξ2

∂ν+ ξ2 = 0 on ∂Ω,

(3.7)

−1ξ3 = −b2ξ3 in Ω,

k3∂ξ3

∂ν+ ξ3 = 0 on ∂Ω.

(3.8)

Since bi > 0 (i = 1, 2), we see ξ2 = ξ3 ≡ 0 from (3.7) and (3.8). If ξ1 ∈ K1, ξ1 ≡ 0, then λ1,k1 = r from (3.6), whichcontradicts the assumption λ1,k1 < r . So (ξ1, ξ2, ξ3) = (0, 0, 0). Thus I − L has no non-trivial kernel in W (0,0,0).

Since r > λ1,k1 , by Theorem 2.1, we see that r0 = rk1((−∆ + p)−1(r + p)) > 1 and r0 is the principle eigenvalueof (−∆ + p)−1(r + p) with a corresponding eigenfunction φ0 > 0. Since S(0,0,0) = (0, 0, 0), we see that (φ0, 0, 0) ∈

W (0,0,0)S(0,0,0). Set t = 1/r0 ∈ (0, 1), then (I − tL)(φ0, 0, 0) = (0, 0, 0) ∈ S(0,0,0). This shows that L has property α. ThusindexW (A, (0, 0, 0)) = 0 by Theorem 2.4. The proof is complete.

The next lemma gives the index of the semitrivial solution (Θk1(r), 0, 0) of (1.2).

Lemma 3.5. Let (H) hold and assume λ1,k1 < r and −b1 = λ1,k2(−m1Θk1 (r)e1+Θk1 (r) ), then we have

(i) if −b1 > λ1,k2(−m1Θk1 (r)e1+Θk1 (r) ), then, indexW (A, (Θk1(r), 0, 0)) = 0;

(ii) if −b1 < λ1,k2(−m1Θk1 (r)e1+Θk1 (r) ), then, indexW (A, (Θk1(r), 0, 0)) = 1.

Proof. Observe A(Θk1(r), 0, 0) = (Θk1(r), 0, 0). Let L = A′(Θk1(r), 0, 0), then


L = A′(Θk1(r), 0, 0)

= (−∆ + p)−1

r − 2Θk1(r) + p −

a1Θk1(r)e1 + Θk1(r)

0

0m1Θk1(r)

e1 + Θk1(r)− b1 + p 0

0 0 −b2 + p

.

Let L(ξ1, ξ2, ξ3) = (ξ1, ξ2, ξ3) for some (ξ1, ξ2, ξ3) ∈ W (Θk1 (r),0,0) = C1k1

(Ω) × K2 × K3. Then

−1ξ1 + (2Θk1(r) − r)ξ1 = −a1Θk1(r)

e1 + Θk1(r)ξ2 in Ω,

−1ξ2 +

b1 −

m1Θk1(r)e1 + Θk1(r)

ξ2 = 0 in Ω,

−1ξ3 + b2ξ3 = 0 in Ω,

k1∂ξ1

∂ν+ ξ1 = k2

∂ξ2

∂ν+ ξ2 = k3

∂ξ3

∂ν+ ξ3 = 0 on ∂Ω.

(3.9)

Since ξ2 ∈ K2, ξ3 ∈ K3, −b1 = λ1,k2(−m1Θk1 (r)e1+Θk1 (r) ) and b2 > 0, we see that ξ2 = ξ3 ≡ 0. So, we get from the first equation of

(3.9) that−1ξ1 + (2Θk1(r) − r)ξ1 = 0 in Ω,

k1∂ξ1

∂ν+ ξ1 = 0 on ∂Ω.

(3.10)

If ξ1 ≡ 0, then λ1,k1(2Θk1(r) − r) = 0 by Theorem 2.2. On the other hand, λ1,k1(2Θk1(r) − r) > λ1,k1(Θk1(r) − r) = 0, weget a contradiction. Therefore, (ξ1, ξ2, ξ3) = (0, 0, 0), i.e., I − L has no non-trivial kernel in W (Θk1 (r),0,0).

(i) Since −b1 > λ1,k2(−m1Θk1 (r)e1+Θk1 (r) ), we have r0 = rk2((−∆ + p)−1(

m1Θk1 (r)e1+Θk1 (r) − b1 + p)) > 1 is an eigenvalue of (−∆ +

p)−1(m1Θk1 (r)e1+Θk1 (r) −b1 +p)with a corresponding eigenfunction φ0 > 0 by Theorem 2.1. Since S(Θk1 (r),0,0) = C1

k1(Ω)×0×0,

we see (0, φ0, 0) ∈ W (Θk1 (r),0,0)S(Θk1 (r),0,0). Set t = 1/r0 ∈ (0, 1), then

(I − tL)

0φ00

=

0φ00

− t(−∆ + p)−1

−


φ0m1Θk1(r)

e1 + Θk1(r)− b1 + p

φ0

0

=

t(−∆ + p)−1 a1Θk1(r)e1 + Θk1(r)

φ0

00

∈ S(Θk1 (r),0,0).

So, L has property α on W (Θk1 (r),0,0). Therefore, indexW (A, (Θk1(r), 0, 0)) = 0 according to Theorem 2.4.(ii) First, we prove that L does not have property α on W (Θk1 (r),0,0).

Since −b1 < λ1,k2(−m1Θk1 (r)e1+Θk1 (r) ), we have rk2((−∆ + p)−1(

m1Θk1 (r)e1+Θk1 (r) − b1 + p)) < 1. On the contrary, we suppose

that L has property α on W (Θk1 (r),0,0). Then there exists t ∈ (0, 1) and (φ1, φ2, φ3) ∈ W (Θk1 (r),0,0)S(Θk1 (r),0,0) such that(I − tL)(φ1, φ2, φ3) ∈ S(Θk1 (r),0,0). So,

(−∆ + p)−1


− b1 + p

φ2 =1tφ2. (3.11)

It follows from (3.11) that 1/t > 1 is an eigenvalue of the operator (−∆ + p)−1(m1Θk1 (r)e1+Θk1 (r) − b1 + p), which contradicts

rk2((−∆ + p)−1(m1Θk1 (r)e1+Θk1 (r) − b1 + p)) < 1. So, L does not have property α on W (Θk1 (r),0,0). By Theorem 2.4, we have

indexW (A, (Θk1(r), 0, 0)) = (−1)σ , (3.12)

where σ is the sum of the multiplicities of all eigenvalues of L which are greater than 1. Next, we will show σ = 0.


Suppose 1/ρ > 1 is an eigenvalue of L with corresponding eigenfunction (ξ1, ξ2, ξ3), then L(ξ1, ξ2, ξ3) = 1/ρ(ξ1, ξ2, ξ3). Equivalently,

−1ξ1 + pξ1 = ρ

(r − 2Θk1(r) + p)ξ1 −


ξ2

in Ω,

−1ξ2 + pξ2 = ρ

m1Θk1(r)

e1 + Θk1(r)− b1 + p

ξ2 in Ω,

−1ξ3 + pξ3 = ρ(−b2 + p)ξ3 in Ω,

k1∂ξ1

∂ν+ ξ1 = k2

∂ξ2

∂ν+ ξ2 = k3

∂ξ3

∂ν+ ξ3 = 0 on ∂Ω.

(3.13)

Since 0 < ρ < 1 and b2 > 0, it follows from the third equation of (3.13) that ξ3 ≡ 0.If ξ2 ≡ 0, it follows from the second equation of (3.13) and Theorem 2.2 that

0 = λ1,k2

p(1 − ρ) − ρ

m1Θk1(r)

e1 + Θk1(r)− b1

> λ1,k2

−


+ b1

= λ1,k2

−


+ b1,

which contradicts −b1 < λ1,k2(−m1Θk1 (r)e1+Θk1 (r) ), and so ξ3 ≡ 0.

If ξ1 ≡ 0, it follows from the first equation of (3.13) that

0 = λ1,k1

p(1 − ρ) − ρ(r − 2Θk1(r))

≥ λ1,k1

2Θk1(r) − r

> λ1,k1

Θk1(r) − r

= 0.

This contradiction shows that ξ1 ≡ 0. So, (ξ1, ξ2, ξ3) ≡ (0, 0, 0), which implies that L has no eigenvalue being greater than1. Consequently, σ = 0, hence indexW (A, (Θk1(r), 0, 0)) = 1. We complete the proof.

In order to study the other semitrivial solution of (1.2). Let us consider the following three sub-systems−1u1 = u1(r − u1) −

a1u1u2

e1 + u1 + u2in Ω,

−1u2 =m1u1u2

e1 + u1 + u2− b1u2 in Ω,

k1∂u1

∂ν+ u1 = k2

∂u2

∂ν+ u2 = 0 on ∂Ω,

(3.14)

−1u1 = u1(r − u1) in Ω,−1u3 = −b2u3 in Ω,

k1∂u1

∂ν+ u1 = k3

∂u3

∂ν+ u3 = 0 on ∂Ω,

(3.15)

−1u2 = −b1u2 −

a2u2u3

e2 + u2 + u3in Ω,

−1u3 =m2u2u3

e2 + u2 + u3− b2u3 in Ω,

k2∂u2

∂ν+ u2 = k3

∂u3

∂ν+ u3 = 0 on ∂Ω.

(3.16)

It is easy to see (3.16) only have trivial solution (u2, u3) = (0, 0), and (3.15) only have a trivial solution (u1, u3) = (0, 0) anda semitrivial solution (u1, u3) = (Θk1(r), 0) if and only if r > λ1,k1 . Next, we give some results about the positive solutionsof (3.14), which are followed from the results in [38,44,48–51] and simple comparison arguments for elliptic problems. Wepoint out that the results in the above references were obtained under Dirichlet boundary conditions, but the correspondingmain results are still valid, even if the boundary condition is replaced by the Robin boundary condition. The next theoremgives some propositions about (3.14).

Theorem 3.6. (3.14) has a positive solution (u⋆1, u

⋆2) with u⋆

1 ≤ Θk1(r) if and only if r > λ1,k1 and −b1 > λ1,k2(−m1Θk1 (r)e1+Θk1 (r) ). In

addition, if r − a1 > λ1,k1 and −b1 > λ1,k2(−m1Θk1 (r−a1)e1+Θk1 (r−a1)

), then the positive solution (u⋆1, u

⋆2) satisfies Θk1(r − a1) ≤ u⋆

1 and

u2 ≤ u⋆2, where u2 is the unique positive solution of the following problem

910 J. Zhou, C. Mu / Nonlinear Analysis: Real World Applications 12 (2011) 902–917−1u2 = u2

m1Θk1(r − a1)

e1 + Θk1(r − a1) + u2− b1

in Ω,

k2∂u2

∂ν+ u2 = 0 on ∂Ω.

(3.17)

Remark 3.7. From Theorem 2.3, we see that if r − a1 > λ1,k1 and −b1 > λ1,k2(−m1Θk1 (r−a1)e1+Θk1 (r−a1)

), (3.17) has a unique positive

solution u2.

Denote Φ = (u⋆1, u

⋆2) : (u⋆

1, u⋆2) is the positive solution of (3.14) and Ψ = (u⋆

1, u⋆2, 0) : where (u⋆

1, u⋆2) ∈ Φ, it is easy

to see that (1.2) has semitrivial solution (u⋆1, u

⋆2, 0) ∈ Ψ from Theorem 3.6. Next, we give the degree of I − A in Ψ relative

to W .

Lemma 3.8. Let (H) hold and assume Ψ = ∅, then we have(i) if −b2 > λ1,k3(−

m2u⋆2

e2+u⋆2) for any (u⋆

1, u⋆2, 0) ∈ Ψ , then, degW (I − A, Ψ ) = 0;

(ii) if −b2 < λ1,k3(−m2u⋆

2e2+u⋆

2) for any (u⋆

1, u⋆2, 0) ∈ Ψ , then, degW (I − A, Ψ ) = 1.

Remark 3.9. (i) If Ψ = ∅, i.e., Φ = ∅, it is easy to see degW (I − A, Ψ ) = 0 from the Leray–Schauder degree theory [52];(ii) From Theorem 3.6, we know that Ψ = ∅ ⇔ Φ = ∅ ⇔ r > λ1,k1 and −b1 > λ1,k2(−

m1Θk1 (r)e1+Θk1 (r) ).

Proof. By using Theorem 2.5, the proof is similar to the Section 2 of [35]. We give the details for reader’s convenience. Recallthe definition of E, W and D is Section 3. Let

E1 = C1k1(Ω)

C1k2(Ω), E2 = C1

k3(Ω), W1 = K1

K2 and W2 = K3.

Define

A1(u1, u2, u3) = (−∆ + p)−1

u1

r − u1 −

a1u2

e1 + u1 + u2

+ pu1

u2

m1u1

e1 + u1 + u2− b1 −

a2u3

e2 + u2 + u3

+ pu2

.

A2(u1, u2, u3) = (−∆ + p)−1u3

m2u2

e2 + u2 + u3− b2

+ pu3

.

Then A = (A1, A2) and we are in the framework to use Theorem 2.5. We choose a neighborhood U ⊂ W1 ∩ D of Ψ ∩ W1such that (Θk1(r), 0) ∈ U .

Now, A1(u1, u2, 0) = (u1, u2) with (u1, u2) ∈ U if and only if (u1, u2, 0) ∈ Ψ . By the results in [38,44], we have

degW1(I − A1|W1 ,U, 0) =

1 if r > λ1,k1 and − b1 > λ1,k2

−


,

−1 if r < λ1,k1 and − b1 < λ1,k2

−


,

0 if (r − λ1,k1) ·

[−b1 − λ1,k2

−


]< 0.

For any (u⋆1, u

⋆2, 0) ∈ Ψ , a direct calculation shows

A′

2(u⋆1, u

⋆2, 0)|W2u3 = (−∆ + p)−1

p − b2 +

m2u⋆2

e2 + u⋆2

u3.

Notice that by our choice of p, p − b2 +m2u⋆

2e2+u⋆

2> 0 in Ω for any (u⋆

1, u⋆2, 0) ∈ Ψ . Therefore, it follows from the maximum

principle that A′

2(u⋆1, u

⋆2, 0)|W2 is u0-positive in the sense of [53] with u0 = (−∆)−11. Hence r(A′

2(u⋆1, u

⋆2, 0)|W2) > 0 and is

the only eigenvalue corresponding to a positive eigenvector. By a simple calculation (cf. [38] or [44]) we can easily show thatr(A′

2(u⋆1, u

⋆2, 0)|W2) > 1 if and only if −b2 > λ1,k3(−

m2u⋆2

e2+u⋆2) and r(A′

2(u⋆1, u

⋆2, 0)|W2) < 1 if and only if −b2 < λ1,k3(−

m2u⋆2

e2+u⋆2).

Therefore, by Theorem 2.5 and the above discussion, we have

degW (I − A,U × W2(ϵ), 0) =

0 if − b2 > λ1,k3

−

m2u⋆2

e2 + u⋆2

,

degW1(I − A1|W1 ,U, 0) if − b2 < λ1,k3

−

m2u⋆2

e2 + u⋆2

.


Again, since the above degree does not depend on the particular choice of U and ϵ, and Ψ = ∅ means r > λ1,k1 and

−b1 > λ1,k2(−m1Θk1 (r)e1+Θk1 (r) ) by (ii) of Remark 3.9. By the above discussion we get the results. The proof is complete.

Based on above analysis, we can give the result about the existence of positive solutions of (1.2).

Theorem 3.10. Let (H) hold and assume r > λ1,k1 . If

r − a1 > λ1,k1 , −b1 > λ1,k2

−

m1Θk1(r − a1)e1 + Θk1(r − a1)

, and − b2 > λ1,k3

−

m2u2

e2 + u2

,

then (1.2) has least one positive solution, where u2 is defined in Theorem 3.6.

Proof. Since (H) hold and r > λ1,k1 , we obtain degW (I − A,D) = 1 and indexW (A, (0, 0, 0)) = 0 from Lemma 3.4. Thus, itis sufficient to prove that

indexW (A, (Θk1(r), 0, 0)) + degW (I − A, Ψ ) = 1. (3.18)

Since −b1 > λ1,k2(−m1Θk1 (r−a1)e1+Θk1 (r−a1)

) > λ1,k2(−m1Θk1 (r)e1+Θk1 (r) ), we have indexW (A, (Θk1(r), 0, 0)) = 0 from Lemma 3.5. Moreover,

Theorem 3.6 implies Ψ = ∅. Let (u⋆1, u

⋆2, 0) ∈ Ψ . Since u⋆

1 ≥ Θk1(r − a1) and u⋆2 ≥ u2 by Theorem 3.6, we have

−b2 > λ1,k3(−m2u2e2+u2

) ≥ λ1,k3(−m2u⋆

2e2+u⋆

2) by comparison principle of principle eigenvalue, and thus degW (I − A, Ψ ) = 0

from Lemma 3.8. Therefore, indexW (A, (Θk1(r), 0, 0)) + degW (I − A, Ψ ) = 0. So, (3.18) holds. The proof is complete.

3.2. Non-existence of positive solutions

Before closing this section, we consider the nonexistence of positive solutions of (1.2).

Theorem 3.11. If any one of the following conditions holds, then (1.2) has no positive solution

(i) r ≤ λ1,k1 or m1 − b1 ≤ λ1,k2;

(ii) r > λ1,k1 and − b1 ≤ λ1,k2

−

m1Θk1 (r)e1+Θk1 (r)

;

(iii) r − a1 > λ1,k1 and −b1 ≤ λ1,k2

−

m1Θk1 (r−a1)e1+Θk1 (r−a1)

;

(iv) r > λ1,k1 , m1r > b1(e1 + r), and m2Q2e2+Q2

+ b2 ≤ λ1,k2 , where Q2 =m1r−b1(e1+r)

b1.

Proof. Assume (1.2) has positive solutions (u1, u2, u3); then u1, u2, u3 satisfy the following three equations respectively,−1u1 = u1(r − u1) −

a1u1u2

e1 + u1 + u2in Ω,

k1∂u1

∂ν+ u1 = 0 on ∂Ω,

(3.19)

−1u2 =

m1u1u2

e1 + u1 + u2− b1u2 −

a2u2u3

e2 + u2 + u3in Ω,

k2∂u2

∂ν+ u2 = 0 on ∂Ω,

(3.20)

−1u3 =

m2u2u3

e2 + u2 + u3− b2u3 in Ω,

k3∂u3

∂ν+ u3 = 0 on ∂Ω.

(3.21)

(i) Since u1 > 0 in Ω , we get from (3.19) that

r = λ1,k1

u1 +

a1u2

e1 + u1 + u2

> λ1,k1 ,

by Theorem 2.2 and comparison property of principle eigenvalue, this contradicts r ≤ λ1,k1 .Similarly, we get from (3.20) that

−b1 = λ1,k2

a2u3

e2 + u2 + u3−

m1u1

e1 + u1 + u2

> λ1,k2(−m1) = −m1 + λ1,k2 ,

by Theorem 2.2 and comparison property of principle eigenvalue, this contradictsm1 − b1 ≤ λ1,k2 .


(ii) Since u2 > 0 in Ω , we get from (3.20) that

−b1 = λ1,k2

a2u3

e2 + u2 + u3−

m1u1

e1 + u1 + u2

> λ1,k2

−


,

by Theorem 2.2 and comparison property of principle eigenvalue, this contradicts −b1 ≤ λ1,k2(−m1Θk1(r)e1+Θk1 (r) ).

(iii) Since a1u2e1+u1+u2

< a1, we get form (3.19) that−1u1 ≥ u1(r − u1 − a1) in Ω,

k1∂u1

∂ν+ u1 = 0 on ∂Ω,

(3.22)

then, we have u1 ≥ Θk1(r − a1) by comparison principle. Since u2 > 0 in Ω , we get from (3.20) that

−b1 = λ1,k2

a2u3

e2 + u2 + u3−

m1u1

e1 + u1 + u2

> λ1,k2

−

m1Θk1(r − a1)e1 + Θk1(r − a1)

,

by Theorem 2.2 and comparison property of principle eigenvalue, this contradicts −b1 ≤ λ1,k2(−m1Θk1 (r−a1)e1+Θk1 (r−a1)

).(iv) From Theorem 3.1, we know u2(x) ≤ Q2, so we get from (3.21) that

−b2 = λ1,k3

−

m2u2

e2 + u2 + u3

> λ1,k3

−

m2Q2

e2 + Q2

= −

m2Q2

e2 + Q2+ λ1,k3 ,

by Theorem 2.2 and comparison property of principle eigenvalue, this contradicts m2Q2e2+Q2

+ b2 ≤ λ1,k2 . The proof iscomplete.

4. Asymptotic behavior: extinction and global attractor

In this section, the asymptotic behavior of the time-dependent solutions of (1.1) is considered, that is, sufficientconditions for the extinction and permanence to system (1.1) are investigated.

Theorem 4.1. Let (u1(x, t), u2(x, t), u3(x, t)) be a positive solution of (1.1), then we have(i) If r ≤ λ1,k1 , then (u1, u2, u3) → (0, 0, 0) as t → ∞;(ii) If r > λ1,k1 and −b1 ≤ λ1,k2(−

m1Θk1 (r)e1+Θk1 (r) ), then (u1, u2, u3) → (Θk1(r), 0, 0) as t → ∞.

Proof. (i) First, we observe that any time-dependent solution (u1, u1, u2) of (1.1) satisfies∂u1

∂t− 1u1 ≤ u1(r − u1) in Ω × R+,

k1∂u1

∂ν+ u1 = 0 on ∂Ω × R+.

(4.1)

Let ε be a sufficiently small positive constant such that

ε < minb1e1m1

,b2e2m2

.

Since r ≤ λ1,k1 , from Theorem 2.3, we see that u1(x, t) → 0 uniformly as t → ∞ by using comparison principle for ellipticproblems.

Then, there exists a Tε ≥ 0 such that u1(x, t) ≤ ε for all t > Tε . Therefore, we have∂u2

∂t− 1u2 ≤ u2

m1ε

e1− b1

< 0 in Ω × (Tε, ∞),

k2∂u2

∂ν+ u2 = 0 on ∂Ω × (Tε, ∞),

(4.2)

that concludes u2(x, t) → 0 uniformly as t → ∞ by Theorem 2.3.Then, there exists a T ′

ε ≥ 0 such that u2(x, t) ≤ ε for all t > T ′ε . Therefore, we have

∂u3

∂t− 1u3 ≤ u3

m2ε

e2− b2

< 0 in Ω × (T ′

ε, ∞),

k3∂u3

∂ν+ u3 = 0 on ∂Ω × (T ′

ε, ∞),

(4.3)

that concludes u3(x, t) → 0 uniformly as t → ∞ by Theorem 2.3.


(ii) Let ε be a sufficiently small positive constant such that

(a) ε <e1λ1,k2

−

m1Θk1 (r)e1+Θk1 (r)

+ b1

m1

,

(b) ε <r − λ1,k1

a1.

Since r > λ1,k1 , we get from (4.1) and Theorem 2.3 that,

lim supt→∞

u1(x, t) ≤ Θk1(r). (4.4)

Then, there exist a Tε ≥ 0 such that u1(x, t) ≤ Θk1(r) + ε for all t > Tε . Therefore, we have

∂u2

∂t− 1u2 ≤ u2

m1(Θk1(r) + ε)

e1 + Θk1(r) + ε + u2− b1

≤ u2

m1Θk1(r)

e1 + Θk1(r) + u2+

m1ε

e1− b1

in Ω × (Tε, ∞),

k2∂u2

∂ν+ u2 = 0 on ∂Ω × (Tε, ∞).

(4.5)

Since ε satisfies (a) and−b1 ≤ λ1,k2(−m1Θk1 (r)e1+Θk1 (r) ), we conclude u2(x, t) → 0 uniformly as t → ∞ by Theorem 2.3. Similarly,

we see u3(x, t) → 0 uniformly as t → ∞.Then, there exists a T ′

ε ≥ 0 such that u2(x, t) u3(x, t) < ε for all t > T ′ε . Therefore, we have

∂u1

∂t− 1u1 ≥ u1(r − u1 − a1ε) in Ω × (T ′

ε, ∞),

k1∂u1

∂ν+ u1 = 0 on ∂Ω × (T ′

ε, ∞).

(4.6)

Since ε satisfies (b), we get from Theorem 2.3 that

lim inft→∞

u1(x, t) ≥ Θk1(r − a1ε). (4.7)

So, by (4.4) and (4.7), we have

Θk1(r − a1ε) ≤ lim inft→∞

u1(x, t) ≤ lim supt→∞

u1(x, t) ≤ Θk1(r). (4.8)

Letting ε → 0 in (4.8), we conclude u1(x, t) → Θk1(r) uniformly as t → ∞. The proof is complete.

Definition 4.2. A pair of functions (u1, u2, u3) and (u1, u2, u3) ∈ C2(Ω) ∩ C1(Ω) are called ordered upper and lowersolutions of (1.2) if they satisfy the relations u1 ≥ u1, u2 ≥ u2, u3 ≥ u3 and the following inequalities

−1u1 ≥ u1(r − u1) −a1u1u2

e1 + u1 + u2in Ω,

−∆u1 ≤ u1(r − u1) −a1u1u2

e1 + u1 + u2in Ω,

−1u2 ≥m1u1u2

e1 + u1 + u2− b1u2 −

a2u2u3

e2 + u2 + u3in Ω,

−1u2 ≤m1u1u2

e1 + u1 + u2− b1u2 −

a2u2u3

e2 + u2 + u3in Ω,

−1u3 ≥m2u2u3

e2 + u2 + u3− b2u3 in Ω,

−1u3 ≤m2u2u3

e2 + u2 + u3− b2u3 in Ω,

k1∂u1

∂ν+ u1 ≥ 0 ≥ k1

∂u1

∂ν+ u1 on ∂Ω,

k2∂u2

∂ν+ u2 ≥ 0 ≥ k2

∂u2

∂ν+ u2 on ∂Ω,

k3∂u3

∂ν+ u3 ≥ 0 ≥ k3

∂u3

∂ν+ u3 on ∂Ω.

(4.9)


The following theorem provides sufficient conditions for permanence of the time-dependent system (1.1).

Theorem 4.3. If

r > a1 + λ1,k1 ,

−b1 > a2 + λ1,k2

−

m1Θk1(r − a1)e1 + Θk1(r − a1)

,

−b2 > λ1,k3

−

m2uH2

e2 + uH2

,

(4.10)

where uH2 is the unique positive solution of the following problem−1v = v

m1Θk1(r − a1)

e1 + Θk1(r − a1) + v− b1 − a2

in Ω,

k2∂v

∂ν+ v = 0 on ∂Ω.

(4.11)

Then, there exists a pair of functions (u1, u2, u3) and (u1, u2, u3) ∈ C2(Ω) ∩ C1(Ω) such that

−1u1 = u1(r − u1) −a1u1u2

e1 + u1 + u2in Ω,

−1u1 = u1(r − u1) −a1u1u2

e1 + u1 + u2in Ω,

−1u2 =m1u1u2

e1 + u1 + u2− b1u2 −

a2u2u3

e2 + u2 + u3in Ω,

−1u2 =m1u1u2

e1 + u1 + u2− b1u2 −

a2u2u3

e2 + u2 + u3in Ω,

−1u3 =m2u2u3

e2 + u2 + u3− b2u3 in Ω,

−1u3 =m2u2u3

e2 + u2 + u3− b2u3 in Ω,

k1∂ u1

∂ν+ u1 = 0 = k1

∂ u1

∂ν+ u1 on ∂Ω,

k2∂ u2

∂ν+ u2 = 0 = k2

∂ u2

∂ν+ u2 on ∂Ω,

k3∂ u3

∂ν+ u3 = 0 = k3

∂ u3

∂ν+ u3 on ∂Ω,

(4.12)

and satisfy the following estimates,

Θk1(r − a1) ≤ u1 ≤ u1 ≤ Θk1(r),

uH2 ≤ u2 ≤ u2 ≤ uN

2 ,

uH3 ≤ u3 ≤ u3 ≤ uN

3 ,

(4.13)

where uN2 is the unique positive solution of the following problem−1w = w

m1Θk1(r)

e1 + Θk1(r) + w− b1

in Ω,

k2∂w

∂ν+ w = 0 on ∂Ω,

(4.14)

uH3 is the unique positive solution of the following problem

−1h = h

m2uH2

e2 + uH2 + h

− b2

in Ω,

k3∂h∂ν

+ h = 0 on ∂Ω,

(4.15)

uN3 is the unique positive solution of the following problem

J. Zhou, C. Mu / Nonlinear Analysis: Real World Applications 12 (2011) 902–917 915−1z = z

m2uN

2

e2 + uN2 + z

− b2

in Ω,

k3∂z∂ν

+ z = 0 on ∂Ω.

(4.16)

Furthermore, [u1, u1] × [u2, u2] × [u3, u3] is a positive global attractor of (1.1).

Remark 4.4. (i) We point out such functions (u1, u2, u3) and (u1, u2, u3), called quasisolutions of (1.2) in this paper;(ii) The existence and uniqueness of uH

2 , uN2 , uH

3 , uN3 follows from Theorem 2.3 under condition (4.10).

Proof. It is easy to see that (Θk1(r), uN2 , u

N3 ) and (Θk1(r−a1), uH

2 , uH3 ) are a pair of ordered upper and lower solutions of (1.2)

under assumption (4.10). Using the iteration scheme in [45], the existence of a pair of functions (u1, u2, u3) and (u1, u2, u3)can be shown easily.

Next, we show that [u1, u1] × [u2, u2] × [u3, u3] is a positive global attractor of (1.1). Since u1, u2, u3 > 0 in Ω , thepositivity follows easily. So, it is sufficient to prove that [u1, u1] × [u2, u2] × [u3, u3] is a global attractor.

Let ε be sufficiently small such that

(a) ε <e2−λ1,k3

−

m2uH2

e2+uH2

− b2

m2

≤

e2−λ1,k3

−

m2uN2

e2+uN2

− b2

m2

,

(b) ε <e1−λ1,k2

−

m1Θk1 (r−a1)e1+Θk1 (r−a1)

− b1 − a2

m1

.

Since

∂u1

∂t− 1u1 ≤ u1(r − u1) in Ω × R+,

∂u2

∂t− 1u2 ≤ u2

m1Θk1(r)

e1 + Θk1(r) + u2− b1

in Ω × R+,

k1∂u1

∂ν+ u1 = k2

∂u2

∂ν+ u2 = 0 on ∂Ω × R+.

(4.17)

Then, by Theorem 2.3, assumption (4.10), and comparison principle, it is obvious that

lim supt→∞

u1(x, t) ≤ Θk1(r) and lim supt→∞

u2(x, t) ≤ uN2 . (4.18)

This implies that there exists a Tε ≥ 0 such that

u1(x, t) ≤ Θk1(r) + ε and u2(x, t) ≤ uN2 + ε for all t > Tε. (4.19)

Thus,

∂u3

∂t− 1u3 ≤ u3

m2(uN

2 + ε)

e2 + uN2 + ε + u3

− b2

≤ u3

m2uN

2

e2 + uN2 + u3

+m2ε

e2− b2

in Ω × (Tε, ∞),

k3∂u3

∂ν+ u3 = 0 on ∂Ω × (Tε, ∞).

(4.20)

Since ε satisfies (a), we get from Theorem 2.3 that

lim supt→∞

u3(x, t) ≤ uN3 . (4.21)

This implies that there exists a T ′ε such that

u3(x, t) ≤ uN3 + ε for all t > T ′

ε. (4.22)

On the other hand,∂u1

∂t− 1u1 ≥ u1(r − u1 − a1) in Ω × R+,

k1∂u1

∂ν+ u1 = 0 on ∂Ω × R+.

(4.23)


Then, by Theorem 2.3, assumption (4.10), and comparison principle, it is obvious that

lim inft→∞

u1(x, t) ≥ Θk1(r − a1). (4.24)

This implies that there exists a T ′′ε such that

u1(x, t) ≥ Θk1(r − a1) − ε for all t > T ′′

ε . (4.25)

Thus,

∂u2

∂t− 1u2 ≥ u2

m1(Θk1(r − a1) − ε)

e1 + Θk1(r − a1) − ε + u2− b1 − a2

≥ u2

m1Θk1(r − a1)

e1 + Θk1(r − a1) + u2−

m1ε

e1− b1 − a2

in Ω × (T ′′

ε , ∞),

k2∂u2

∂ν+ u2 = 0 on ∂Ω × (T ′′

ε , ∞).

(4.26)

Since ε satisfies (b), we get from Theorem 2.3 that

lim inft→∞

u2(x, t) ≥ uH2 . (4.27)

This implies that there exists a T ′′′ε ≥ 0 such that

u2(x, t) ≥ uH2 − ε for all t > T ′′′

ε . (4.28)

Thus,

∂u3

∂t− 1u3 ≥ u3

m2(uH

2 − ε)

e2 + uH2 − ε + u3

− b2

≥ u3

m2uH

2

e2 + uH2 + u3

−m2ε

e2− b2

in Ω × (T ′′′

ε , ∞),

k3∂u3

∂ν+ u3 = 0 on ∂Ω × (T ′′′

ε , ∞).

(4.29)

Since ε satisfies (a), we get from Theorem 2.3 that

lim inft→∞

u3(x, t) ≥ uH3 . (4.30)

This implies that there exists a T ′′′′ε ≥ 0 such that

u3(x, t) ≥ uH3 − ε for all t > T ′′′′

ε . (4.31)

Finally, using (4.19), (4.22), (4.25), (4.28), and (4.31), it is concluded that there exist T = maxTε, T ′ε, T ′′

ε , T ′′′ε , T ′′′′

ε suchthat for any non-trivial initial condition ui0(x) (i = 1, 2, 3), the time-dependent solution (u1, u2, u3) of (1.1) satisfies

(u1, u2, u3) ∈ [Θk1(r − a1) − ε, Θk1(r) + ε] × [uH2 − ε, uN

2 + ε] × [uH3 − ε, uN

3 + ε] for all t > T . (4.32)

Then, our result follows by Corollary 2.1 and Theorem 2.1 in [54]. We complete the proof.

The next final theorem gives sufficient conditions for a global attractor in the case that exactly one species is dying out.This can be proved similarly as in the above theorem, so that the results are only stated.

Theorem 4.5. If

r > a1 + λ1,k1 ,

−b1 > a2 + λ1,k2

−

m1Θk1(r − a1)e1 + Θk1(r − a1)

,

−b2 < λ1,k3

−

m2uN2

e2 + uN2

,

(4.33)

where uN2 is defined in Theorem 4.3. Then, there exists a pair of quasisolutions (u1, u2) and (u1, u2) of (3.14) with u1 > u1 and

u2 > u2. Moreover, [u1, u1] × [u2, u2] × 0 is a global attractor of (1.1).

Acknowledgement

The authors would like to thank the referee for the careful reading of this paper and for the valuable suggestions thatimproved the presentation and style of the paper.


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