The global attractor of a competitor–competitor–mutualist reaction–diffusion system with time delays

Nonlinear Analysis 67 (2007) 2623–2631 www.elsevier.com/locate/na

The global attractor of a competitor–competitor–mutualist reaction–diffusion system with time delays C.V. Pao ∗ Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, United States Received 4 May 2006; accepted 11 September 2006

Abstract The aim of this paper is to investigate the asymptotic behavior of time-dependent solutions of a three-species reaction–diffusion system in a bounded domain under a Neumann boundary condition. The system governs the population densities of a competitor, a competitor–mutualist and a mutualist, and time delays may appear in the reaction mechanism. It is shown, under a very simple condition on the reaction rates, that the reaction–diffusion system has a unique constant positive steady-state solution, and for any nontrivial nonnegative initial function the corresponding time-dependent solution converges to the positive steady-state solution. An immediate consequence of this global attraction property is that the trivial solution and all forms of semitrivial solutions are unstable. Moreover, the state–state problem has no nonuniform positive solution despite possible spatial dependence of the reaction and diffusion. All the conclusions for the time-delayed system are directly applicable to the system without time delays and to the corresponding ordinary differential system with or without time delays. c 2006 Elsevier Ltd. All rights reserved.

MSC: 35K57; 35K50; 34K20 Keywords: Reaction–diffusion system; Asymptotic behavior; Positive solutions; Global attraction; Time delays; Upper and lower solutions

1. Introduction Large time behavior of solutions of reaction–diffusion systems is one of the most important concerns in population dynamics. Since many such equations possess multiple steady-state solutions, the determination of the asymptotic behavior of the time-dependent solution in relation to the steady-state solutions becomes rather delicate, especially in determining the exact limit of the time-dependent solution. In this paper we investigate the stability problem for a special reaction–diffusion system, called a competitor–competitor–mutualist system, which governs the population densities of a competitor–mutualist u, a competitor v, and a mutualist w in a bounded domain. The mathematical problem, including possible time delays in the reaction mechanism, is given by u t − L 1 u = β1 (x)u[a1 − b1 u − (c10 v + c100 vτ2 )/(1 + σ10 w + σ100 wτ3 )] vt − L 2 v = β2 (x)v[a2 − (b20 u + b200 u τ1 ) − c2 v] ∗ Tel.: +1 919 515 2382; fax: +1 919 515 3798.

E-mail address: [email protected]. c 2006 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2006.09.027

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wt − L 3 w = β3 (x)w[a3 − b3 w/(1 + σ20 u + σ200 u τ1 )] (t > 0, x ∈ Ω ) ∂u/∂ν = ∂v/∂ν = ∂w/∂ν = 0 (t > 0, x ∈ ∂Ω ) u(t, x) = η1 (t, x) (t ∈ I1 ), v(t, x) = η2 (t, x) (t ∈ I2 ), w(t, x) = η3 (t, x)

(1.1) (t ∈ I3 ), (x ∈ Ω )

where u τ1 ≡ u(t − τ1 , x), vτ2 = v(t − τ2 , x), wτ3 = w(t − τ3 , x) with time delays τi > 0, i = 1, 2, 3, Ω is a smooth bounded domain in Rn with boundary ∂Ω , and ∂/∂ν denotes the outward normal derivative on ∂Ω . For each i = 1, 2, 3, Ii is the interval [−τi , 0], ai , bi , ci and τi , where b2 = b20 + b200 , c1 = c10 + c100 are positive constants, c10 , c100 , b20 , b200 , and σ j0 , σ j00 , j = 1, 2, are all nonnegative constants, and L i is a uniformly elliptic operator in the form Li ≡

n X

(i)

a j,k (x)

j,k=1

n X ∂ ∂2 (i) + . b j (x) ∂ x j ∂ xk ∂ xj j=1

Of special interest is the diffusion–convection operator Li = D

(i)

(x)∇ 2 + c

(i)

(x) · ∇

(i = 1, 2, 3) (i)

(i)

(i)

where ∇ and ∇ 2 are the gradient and Laplacian operators in Ω , D (x) ≥ d > 0 in Ω , and c (x) = (i) (i) (c1 (x), . . . , cn (x)). It is assumed that the coefficients of L i and the functions βi and ηi are smooth functions in their respective domains, βi (x) > 0 in Ω , and ηi (t, x) ≥ 0 in Ii × Ω , where Ω = Ω ∪ ∂Ω (see [8,10] for some detailed assumptions). The constants c10 or c100 , b20 or b200 as well as σ j0 or σ j00 , j = 1, 2, are allowed to be zero. In particular, if c100 = b200 = σ100 = σ200 = 0, then problem (1.1) is reduced to the standard competitor–competitor–mutualist system without time delays (see (1.5) below). To investigate the asymptotic behavior of the solution of (1.1) we also consider the corresponding steady-state system −L 1 u = β1 (x)u[a1 − b1 u − c1 v/(1 + σ1 w)] −L 2 v = β2 (x)v[a2 − b2 u − c2 v] −L 3 w = β3 (x)w[a3 − b3 w/(1 + σ2 u)] ∂u/∂ν = ∂v/∂ν = ∂w/∂ν = 0

(1.2)

(x ∈ Ω )

(x ∈ ∂Ω )

where c1 = c10 + c100 ,

b2 = b20 + b200 ,

σ1 = σ10 + σ100 ,

σ2 = σ20 + σ200 .

(1.3)

It is clear that problem (1.2) has the trivial solution (0, 0, 0) and various forms of semitrivial solutions, including (a1 /b1 , 0, 0), (0, a2 /c2 , 0), (0, 0, a3 /b3 ) and some others (see [13,16]). The competitor–competitor–mutualist model was initiated by Rai, Freedman and Addicott [13] for the ordinary differential system u t = u[a1 − b1 u − (c10 v + c100 vτ2 )/(1 + σ10 w + σ100 wτ3 )] vt = v[a2 − (b20 u + b200 u τ1 ) − c2 v] wt = w[a3 − b3 w/(1 + σ20 u + σ200 u τ1 )] u(t) = η1 (t)

(t ∈ I1 ), c100

v(t) = η2 (t) b200

σ100

(1.4)

(t > 0) (t ∈ I2 ),

w(t) = η3 (t)

(t ∈ I3 ),

σ200

without time delays (that is, = = = = 0). They obtain conditions for the boundedness of the global solution and local stability or instability of the various equilibria. Their model was extended by Zheng [16] to the reaction–diffusion system (without time delays) u t − D1 ∇ 2 u = u[a1 − b1 u − c1 v/(1 + σ1 w)] vt − D2 ∇ 2 v = v[a2 − b2 u − c2 v] wt − D3 ∇ 2 w = w[a3 − b3 w/(1 + σ2 u)]

(t > 0, x ∈ Ω )

∂u/∂ν = ∂v/∂ν = ∂w/∂ν = 0, (t > 0, x ∈ ∂Ω ) u(0, x) = η1 (x), v(0, x) = η2 (x), w(0, x) = η3 (x)

(1.5) (x ∈ Ω ).

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Using comparison principles and the monotone method, the author investigated the stability and instability of the various semitrivial solutions of the corresponding steady-state problem under both Dirichlet and Neumann boundary conditions. The stability of a positive steady-state solution is also discussed using spectral analysis of the linearized operator which yields local stability (or instability) of the positive solution. The problem (1.5) has been further extended to periodic systems by several workers in [1,4,11,14,15,17] where the diffusion coefficients Di (or the coefficients of L i ) and the various reaction rates ai , bi , etc. are periodic functions of t. Both existence and asymptotic behavior of time-periodic solutions were discussed in the above papers. There are also many similar three-species reaction–diffusion models, including prey–predator models, competition models, and food chain models (cf. [2,3,5–7, 12]). The purpose of this paper is to investigate the asymptotic behavior of the time-dependent solution of (1.1) in relation to a positive steady-state solution of (1.2) using the method of upper and lower solutions. Specifically, we show, under the simple condition c1 /c2 < a1 /a2 < b1 /b2 , that problem (1.2) has a unique constant positive solution (ρ1 , ρ2 , ρ3 ) and for any nonnegative initial function (η1 , η2 , η3 ) with ηi (0, x)(i = 1, 2, 3) not identically zero the corresponding solution of (1.1) converges to (ρ1 , ρ2 , ρ3 ) as t → ∞ (see Theorem 2). This implies that the constant solution (ρ1 , ρ2 , ρ3 ) is a global attractor of the system (1.1). An immediate consequence of this global attraction property is that the trivial solution and all forms of semitrivial solutions of (1.2) are unstable. Furthermore, despite the dependence of L i and βi on x, problem (1.2) has no nonuniform positive solution (see Corollary 1). All the above conclusions are directly applicable to the system (1.5) without time delays, and to the ordinary differential system (1.4) with or without time delays (see Theorems 3 and 4). The above results are stated in Section 2 and proofs of these results are given in Section 3. 2. The main results In this section we present our main results on the global existence and asymptotic behavior of the timedependent solution in relation to a constant positive steady-state solution. Our first result is the existence of a bounded positive solution of (1.1) without any additional conditions on the various reaction rates. Theorem 1. Given any nonnegative initial function (η1 , η2 , η3 ), there exist positive constants K 1 , K 2 and K 3 such that a unique global solution (u, v, w) to (1.1) exists and satisfies the relation (0, 0, 0) ≤ (u(t, x), v(t, x), w(t, x)) ≤ (K 1 , K 2 , K 3 ),

(t > 0, x ∈ Ω ).

(2.1)

Moreover, (u(t, x), v(t, x), w(t, x)) is positive for t > 0, x ∈ Ω if ηi (0, x) 6≡ 0 in Ω for i = 1, 2, 3. Before stating our main result on the asymptotic behavior of the solution (u, v, w) we observe that if η1 (0, x) ≡ 0 in Ω , then the solution component u(t, x) of (1.1) is identically zero for all t > 0, x ∈ Ω . The same is true for the solution components v(t, x) and w(t, x) if η2 (0, x) or η3 (0, x) is identically zero in Ω . Hence, to ensure the convergence of the time-dependent solution (u, v, w) to a positive steady-state solution, it is necessary that ηi (t, x) is nontrivial in the sense that ηi (0, x) is not identically zero in Ω for every i = 1, 2, 3. In the following theorem we give a simple condition on ai , bi and ci , i = 1, 2, so that the solution (u, v, w) of (1.1) converges uniformly to a unique constant positive solution (ρ1 , ρ2 , ρ3 ) of (1.2). This constant solution is governed by the algebraic equations b1 ρ1 + c1 ρ2 /(1 + σ1 ρ3 ) = a1 ,

b2 ρ1 + c2 ρ2 = a2 ,

a3 (1 + σ2 ρ1 ) − b3 ρ3 = 0.

(2.2)

Theorem 2. Assume that the constants ai , bi , and ci , i = 1, 2, satisfy the relation c1 /c2 < a1 /a2 < b1 /b2 .

(2.3)

Then the steady-state problem (1.2) has a unique constant positive solution (ρ1 , ρ2 , ρ3 ) governed by (2.2). Moreover, for any nontrivial nonnegative initial function (η1 , η2 , η3 ) the corresponding solution (u, v, w) of (1.1) possesses the convergence property lim (u(t, x), v(t, x), w(t, x)) = (ρ1 , ρ2 , ρ3 )

t→∞

uniformly in x ∈ Ω .

(2.4)

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Since the convergence of (u, v, w) to (ρ1 , ρ2 , ρ3 ) holds true for every nontrivial nonnegative (η1 , η2 , η3 ), a direct consequence of the above theorem is the following. Corollary 1. Let condition (2.3) hold. Then (i) the trivial solution and all forms of semitrivial solutions of (1.2) are unstable, and (ii) problem (1.2) has no nonuniform positive solution. The existence and the global attraction property of the constant steady-state solution (ρ1 , ρ2 , ρ3 ) is ensured by condition (2.3). Since this condition is independent of time delays and other rate constants, we see by letting c100 = b200 = σ100 = σ200 = 0 that all the conclusions of Theorems 1 and 2 and Corollary 1 hold true for the system (1.5) without time delays. Specifically, we have the following. Theorem 3. Under the condition (2.3), all the conclusions in Theorem 1, Theorem 2 and Corollary 1 hold true for the system (1.5). On the other hand, by letting L i = 0 for i = 1, 2, 3, we also have the following results for the ordinary differential system (1.4) (with or without delays). Theorem 4. Let condition (2.3) hold. Then for any nonnegative initial function (η1 (t), η2 (t), η3 (t)) with (η1 (0), η2 (0), η3 (0)) > (0, 0, 0), the solution (u(t), v(t), w(t)) of (1.4) possesses the convergence property lim (u(t), v(t), w(t)) = (ρ1 , ρ2 , ρ3 ).

(2.5)

t→∞

Remark 2.1. (a) Theorem 2 implies that under condition (2.3), the constant steady-state (ρ1 , ρ2 , ρ3 ) is a global attractor (relative to all positive time-dependent solutions), and the competition model (1.1) is persistent. This conclusion holds true for any time delays and any other reaction rate constants in the system. In particular, if σ10 = σ100 = 0, then the conclusion of Theorem 2 is reduced to that in [10] for the two-species competition model between u and v. (b) Despite the dependence of L i and βi on the spatial variable x, the steady-state problem (1.2) cannot sustain a nonuniform positive solution if condition (2.3) holds. It is important to point out that the existence of a unique positive solution to the algebraic equations in (2.2) does not rule out the possibility of existing nonuniform solutions of the steady-state problem (1.2). Our conclusion on the nonexistence of nonuniform steady-state solution is based on the global attraction property of (ρ1 , ρ2 , ρ3 ), which is obtained from the fact that the maximal and minimal solutions of (1.2) are constants and both of them coincide with (ρ1 , ρ2 , ρ3 ) (cf. [10]). 3. Proof of main theorems Before proving the main theorems, we review some known results from [8–10] for the present problems (1.1) and (m) (m) (m) (1.2). Let Q T = (0, T ] × Ω , Q T = [0, T ] × Ω and ST = (0, T ] × ∂Ω , and let C (Q) = C (Q) × C (Q) × (m) (m) C (Q) and R M = R+ ×[0, M]×R+ , where T and M are arbitrarily large positive constants and C (Q) is the set of m-times-continuously differentiable functions in Q. By letting z = M −v, we transform problem (1.1) into the form u t − L 1 u = β1 (x)u[a1 − b1 u − (c10 (M − z) + c100 (M − z τ2 ))/(1 + σ10 w + σ100 wτ3 )] ≡ f 1 (x, u, z, w, z τ2 , wτ3 ) z t − L 2 z = −β2 (x)(M − z)[a2 − (b20 u + b200 u τ1 ) − c2 (M − z)] ≡ f 2 (x, u, z, u τ1 ) wt − L 3 w = β3 (x)w[a3 − b3 w/(1 + σ20 u + σ200 u τ1 )] ≡ f 3 (x, u, w, u τ1 ), ∂u/∂ν = ∂z/∂ν = ∂w/∂ν = 0, ((t, x) ∈ ST ) u(t, x) = η1 (t, x) (t ∈ I1 ), z(t, x) = η2∗ (t, x)

(t ∈ I2 ),

((t, x) ∈ Q T )

w(t, x) = η3 (t, x)

(3.1)

(t ∈ I3 ), (x ∈ Ω ),

where η2∗ = M − η2 (t, x). Similarly, the steady-state problem (1.2) is transformed into the form −L 1 u = β1 (x)u[a1 − b1 u − c1 (M − z)/(1 + σ1 w)] ≡ f 1∗ (x, u, z, w) −L 2 z = −β2 (x)(M − z)[a2 − b2 u − c2 (M − z)] ≡ f 2∗ (x, u, z) −L 3 w = β3 (x)w[a3 − b3 w/(1 + σ2 u)] ≡ f 3∗ (x, u, w) ∂u/∂ν = ∂z/∂ν = ∂w/∂ν = 0,

(x ∈ ∂Ω ).

(x ∈ Ω )

(3.2)

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Hence, to show the existence and asymptotic behavior of solutions of (1.1) in relation to solutions of (1.2), it suffices to show the same for the solutions of the transformed system (3.1) in relation to solutions of (3.2). It is easy to see that the reaction function f(x, U, Uτ ) ≡ ( f 1 (x, u, z, w, z τ2 , wτ3 ), f 2 (x, u, z, u τ1 ), f 3 (x, u, w, u τ1 )),

(3.3)

in the transformed system (3.1) is quasimonotone nondecreasing in R M , where U ≡ (u, z, w), Uτ ≡ (u τ1 , z τ2 , wτ3 ). Specifically, the components f i (x, U, Uτ ), i = 1, 2, 3, of f(x, U, Uτ ) possess the property ∂ fi (x, U, Uτ ) ≥ 0 ∂u j

for j 6= i

and

∂ fi ≥0 ∂u τ j

for all j (i = 1, 2, 3),

(3.4)

where (u 1 , u 2 , u 3 ) = (u, z, w) and (u τ1 , u τ2 , u τ3 ) = (u τ1 , z τ2 , wτ3 ). In view of the above quasimonotone nondecreasing property we call a function U˜ ≡ (u, ˜ z˜ , w) ˜ ∈ C 2 (Q T ) ∩ C(Q T ) an upper solution of (3.1) if it satisfies the inequalities u˜ t − L 1 u˜ ≥ β1 (x)u[a ˜ 1 − b1 u˜ − (c10 (M − z˜ ) + c100 (M − z˜ τ2 ))/(1 + σ10 w˜ + σ100 w˜ τ3 )] z˜ t − L 2 z˜ ≥ −β2 (x)(M − z˜ )[a2 − (b20 u˜ + b200 u˜ τ1 ) − c2 (M − z˜ )] w˜ t − L 3 w˜ ≥ β3 (x)w[a ˜ 3 − b3 w/(1 ˜ + σ20 u˜ + σ200 u˜ τ1 )]

((t, x) ∈ Q T )

∂ u/∂ν ˜ ≥ 0, ∂ z˜ /∂ν ≥ 0, ∂ w/∂ν ˜ ≥ 0 ((t, x) ∈ ST ) u(t, ˜ x) ≥ η1 (t, x) (t ∈ I1 ), z˜ (t, x) ≥ η2∗ (t, x) (t ∈ I2 ),

w(t, ˜ x) ≥ η3 (t, x)

(3.5) (t ∈ I3 ),

(x ∈ Ω ).

Similarly, Uˆ ≡ (u, ˆ zˆ , w) ˆ is called a lower solution of (3.1) if it satisfies the inequalities in (3.5) in reverse order. The pair U˜ and Uˆ are said to be ordered if (u, ˜ z˜ , w) ˜ ≥ (u, ˆ zˆ , w). ˆ In terms of ordered upper and lower solutions, we have the following global existence theorem from [9] (see also [10]). Theorem A. Let (u, ˜ z˜ , w), ˜ (u, ˆ zˆ , w) ˆ be a pair of ordered upper and lower solutions of (3.1). Then, problem (3.1) has a unique solution (u, z, w) such that (u, ˆ zˆ , w) ˆ ≤ (u, z, w) ≤ (u, ˜ z˜ , w), ˜

((t, x) ∈ Q T ).

(3.6)

For the steady-state problem (3.2), the definitions of upper and lower solutions, denoted by U˜ s ≡ (u˜ s , z˜ s , w˜ s ) and ˆ Us ≡ (uˆ s , zˆ s , wˆ s ), are the same as those in (3.5) except without the time derivative terms and the initial conditions (that is, with all the equality signs = in (3.2) replaced, respectively, by the inequality signs ≥ and ≤). It is clear from this definition that every pair of upper and lower solutions of (3.2) is also a pair of upper and lower solutions of (3.1) whenever (uˆ s , zˆ s , wˆ s ) ≤ (η1 , η2∗ , η3 ) ≤ (u˜ s , z˜ s , w˜ s ). For a given pair of ordered upper and lower solutions (u˜ s , z˜ s , w˜ s ), (uˆ s , zˆ s , wˆ s ) we set S ≡ {(u s , z s , ws ) ∈ C(Ω ); uˆ s ≤ u s ≤ u˜ s , zˆ s ≤ z s ≤ z˜ s , wˆ s ≤ ws ≤ w˜ s } S0 ≡ {(u s , vs , ws ) ∈ C(Ω ); uˆ s ≤ u s ≤ u˜ s , vˆs ≤ vs ≤ v˜s , wˆ s ≤ ws ≤ w˜ s }

(3.7)

where vˆs = M − z˜ s and v˜s = M − zˆ s . It is known that if problem (3.2) has a pair of ordered upper and lower solutions U˜ s ≡ (u˜ s , z˜ s , w˜ s ), Uˆ s ≡ (uˆ s , zˆ s , wˆ s ) then it has a maximal solution U s ≡ (u s , z s , w s ) and a minimal solution U s ≡ (u s , z s , ws ) such that Uˆ s ≤ U s ≤ U s ≤ U˜ s , where inequality between vectors is in the componentwise sense (cf. [8]). For the present Neumann boundary problem, both U s and U s are constant solutions of (3.2) if U˜ s and Uˆ s are constant upper and lower solutions (cf. [10]). Moreover, the time-dependent solution U (t, x) of (3.1) with (η1 , η2∗ , η3 ) = U˜ s converges monotonically from above to U s as t → ∞, while the solution U (t, x) with (η1 , η2∗ , η3 ) = Uˆ s converges monotonically from below to U s . For arbitrary initial function (η1 , η2∗ , η3 ) in S, the corresponding solution U (t, x) of (3.1) satisfies the relation U (t, x) ≤ U (t, x) ≤ U (t, x) (cf. [9]). It is clear from the above monotone property and the continuity of U s and U s that the convergence of U (t, x) to U s (or U (t, x) to U s ) is uniform in x. In particular, if U s = U s (≡ Us∗ ), then Us∗ ≡ (u ∗ , z ∗ , w∗ ) is the unique solution of (3.2) in S and the solution U (t, x) of (3.1) converges uniformly to Us∗ as t → ∞. In terms of the original system (1.1), it implies that (u ∗s , vs∗ , w∗ ), where

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v ∗ = M − z ∗ , is the unique solution of (1.2) in S0 , and for any initial function (η1 , η2 , η3 ) in S0 the corresponding solution (u, v, w) of (1.1) possesses the convergence property lim (u(t, x), v(t, x), w(t, x)) = (u ∗s (x), vs∗ (x), ws∗ (x))

t→∞

uniformly in x.

(3.8)

For arbitrary initial function (η1 , η2 , η3 ) the corresponding solution U (t, x) = (u(t, x), v(t, x), w(t, x)) of (1.1) possesses the convergence property (3.8) if there exists t ∗ > 0 such that (u(t ∗ , x), v(t ∗ , x), w(t ∗ , x)) ∈ S0

(3.9)

(cf. [9]). To summarize the above conclusions for the application to the systems (1.1) and (1.2), we state the following. Theorem B. Let U˜ s ≡ (u˜ s , z˜ s , w˜ s ), Uˆ s ≡ (uˆ s , zˆ s , wˆ s ) be a pair of constant ordered upper and lower solutions of (3.2), and let U s ≡ (u s , z s , w s ), U s ≡ (u s , z s , ws ) be the constant maximal and minimal solutions of (3.2). Assume that U s = U s ≡ (u ∗s , z s∗ , ws∗ ). Then (i) the function Us∗ ≡ (u ∗s , vs∗ , ws∗ ), where vs∗ = M − z s∗ , is the unique constant solution of (1.2), (ii) for any initial function (η1 , η2 , η3 ) in S0 , the corresponding solution U ≡ (u, v, w) of (1.1) possesses the convergence property (3.8), and (iii) the convergence property (3.8) holds true for the solution (u, v, w) of (1.1) with arbitrary initial function (η1 , η2 , η3 ) if it has the property (3.9) for some t ∗ > 0. We now prove the main theorems stated in Section 2 using the conclusions in Theorems A and B. Proof of Theorem 1. Choose any constant M > a2 /c2 in the transformed system (3.1). It is easy to verify that for any large constants K i , i = 1, 2, 3, satisfying K 1 ≥ max{a1 /b1 , a2 /c2 },

K 2 = M,

K 3 ≥ (a3 b3 )(1 + σ2 K 1 ),

the pair (u, ˜ z˜ , w) ˜ = (K 1 , K 2 , K 3 ) and (u, ˆ zˆ , w) ˆ = (0, 0, 0) are ordered upper and lower solutions of (3.1) whenever (0, 0, 0) ≤ (η1 , η2∗ , η3 ) ≤ (K 1 , K 2 , K 3 ). Since M, K 1 and K 3 can be chosen arbitrarily large, we conclude from Theorem A that for any nonnegative initial function (η1 , η2∗ , η3 ) there exist positive constants (K 1 , K 2 , K 3 ) such that problem (3.1) has a unique nonnegative global solution (u, z, w) which is bounded by (K 1 , K 2 , K 3 ). The equivalence between (3.1) and (1.1) shows that (u, v, w), where v = M − z, is the unique solution of (1.1) satisfying the relation (2.1). When ηi (0, x) 6≡ 0 in Ω , the positivity of (u, v, w) follows from the maximum principle.  Proof of Theorem 2. In view of Theorem B, it suffices (a) to find a pair of ordered upper and lower solutions of (3.2) which are constant and positive, (b) to show that the maximal and minimal solutions coincide, and (c) to verify property (3.9) for some t ∗ > 0. (a) Construction of upper and lower solutions. We seek a pair of constant upper and lower solutions of (3.2) in the form U˜ s = (M1 , M − δ2 , M3 ),

Uˆ s = (δ1 , M − M2 , δ3 )

(3.10)

where Mi > δi > 0 and δi is sufficiently small, i = 1, 2, 3. It is easy to verify that the components of U˜ s and Uˆ s in (3.10) satisfy the inequalities and the revised inequalities in (3.5), respectively, (without the time derivative terms and the initial conditions) if 0 ≥ M1 β1 (x)[a1 − b1 M1 − c1 δ2 /(1 + σ1 M3 )] 0 ≥ −δ2 β2 (x)[a2 − b2 M1 − c2 δ2 ] 0 ≥ M3 β3 (x)[a3 − b3 M3 /(1 + σ2 M1 )] 0 ≤ δ1 β1 (x)[a1 − b1 δ1 − c1 M2 /(1 + σ1 δ3 )]

(3.11)

0 ≤ −M2 β2 (x)[a2 − b2 δ1 − c2 M2 ] 0 ≤ δ3 β3 (x)[a3 − b3 δ3 /(1 + σ2 δ1 )] where c1 , b2 , σ1 and σ2 are given by (1.3). By choosing M1 = a1 /b1 ,

M2 = a2 /c2 ,

M3 = (a3 /b3 )[1 + σ2 (a1 /b1 )]

(3.12)

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and δi , i = 1, 2, 3, sufficiently small, we see that the first, third, and the last two inequalities in (3.11) are satisfied, while the second and the fourth inequalities are reduced to 0 ≤ a2 − (b2 a1 /b1 ) − c2 δ2 0 ≤ a1 − b1 δ1 − (c1 a2 /c2 )/(1 + σ1 δ3 ). In view of condition (2.3), the above two inequalities are satisfied by some sufficiently small δi > 0, i = 1, 2, 3. This shows that the pair U˜ s , Uˆ s in (3.10) are constant ordered upper and lower solutions of (3.2), and therefore problem (3.2) has a constant maximal solution U s ≡ (u s , z s , w s ) and a constant minimal solution U s ≡ (u s , z s , ws ) such that (δ1 , M − M2 , δ3 ) ≤ (u s , z s , ws ) ≤ (u s , z s , w s ) ≤ (M1 , M − δ2 , M3 )

(3.13)

(cf. [10]). It follows that (u s , v s , w s ) and (u s , v s , ws ), where v s = M − z s and v s = M − z s , are constant positive solutions of (1.2) and satisfy the relation δ1 ≤ u s ≤ u s ≤ M 1 ,

δ2 ≤ v s ≤ v s ≤ M2 ,

δ3 ≤ w s ≤ w s ≤ M3 .

(3.14)

(b) Uniqueness of positive solutions. We next show that (u s , v s , w s ) = (u s , v s , ws ), which is equivalent to (u s , z s , w s ) = (u s , z s , ws ). Since both (u s , v s , w s ) and (u s , v s , ws ) are constant positive solutions of (1.2) and βi (x) > 0 in Ω , they satisfy the algebraic equations a1 − b1 ρ1 − c1 ρ2 /(1 + σ1 ρ3 ) = 0 a2 − b2 ρ1 − c2 ρ2 = 0

(3.15)

a3 − b3 ρ3 /(1 + σ2 ρ1 ) = 0 which is equivalent to (2.2). By the last two equations in (3.15) we have ρ2 = (a2 − b2 ρ1 )/c2 ,

ρ3 = (a3 /b3 )(1 + σ2 ρ1 ).

(3.16)

Substitution of the above relation into the first equation leads to the quadratic equation αρ12 + βρ1 + γ = 0

(3.17)

where α = b1 σ1 σ2 (a3 /b3 ), β = b1 (1 + σ1 a3 /b3 ) − b2 (c1 /c2 ) − a1 σ1 σ2 (a3 /b3 )

(3.18)

γ = a2 [c1 /c2 − (a1 /a2 )(1 + σ1 a3 /b3 )]. Since α > 0 and, by the hypothesis c1 /c2 < a1 /a2 , γ < 0, we see that Eq. (3.17) has exactly one positive solution ρ1 . This implies that (ρ1 , ρ2 , ρ3 ), where ρ2 and ρ3 are given by (3.16), is the unique constant solution of (1.2) and ρ3 > 0. To show that ρ2 is also positive, we observe from (3.15) that ρ1 = (1/b1 )[a1 − c1 ρ2 /(1 + σ1 ρ3 )]

and

ρ1 = (1/b2 )(a2 − c2 ρ2 ).

This leads to the relation [(c2 /b2 ) − (c1 /b1 )/(1 + σ1 ρ3 )]ρ2 = (a2 /b2 ) − (a1 /b1 ). By condition (2.3) and ρ3 > 0, we must have ρ2 > 0. This shows that (ρ1 , ρ2 , ρ3 ) is the unique constant positive solution of (1.2), and therefore (u s , v s , w s ) = (u s , v s , ws ) = (ρ1 , ρ2 , ρ3 ). By Theorem B (with (u ∗s , vs∗ , ws∗ ) = (ρ1 , ρ2 , ρ3 )) the time-dependent solution (u, v, w) of (1.1) converges to (ρ1 , ρ2 , ρ3 ) as t → ∞ whenever (δ1 , δ2 , δ3 ) ≤ (η1 , η2 , η3 ) ≤ (M1 , M2 , M3 ). This proves the theorem for the class of initial functions in S0 . (c) Global attraction of (ρ1 , ρ2 , ρ3 ). To show the convergence property (2.4) for an arbitrary nontrivial nonnegative initial function (η1 , η2 , η3 ) it suffices to show the property (3.9) for some t ∗ > 0. It is easy to see from the positive property of (u, v, w) in (0, ∞) × Ω that given any t0 > 0 there exists 0 > 0 such that (u, v, w) ≥ (0 , 0 , 0 ) at t = t0 , x ∈ Ω . By considering problem (1.1) (and (3.1)) in the domain (t0 , ∞) × Ω , if necessary, we may assume that ηi (0, x) ≥ 0 in Ω , i = 1, 2, 3. Let S be the sector given by (3.7) with (u˜ s , z˜ s , w˜ s ) = (M1 , M − 0 , M3 ) and (uˆ s , zˆ s , wˆ s ) = (0 , M − M2 , 0 ), and choose any (η1 , η∗2 , η3 ) ∈ S such that (η1 , η∗2 , η3 ) ≤ (η1 , η2∗ , η3 ), where M is the

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positive constant in the transformed system (3.1) and M1 , M2 and M3 are given by (3.12). We choose M sufficiently large such that M > max{M2 , η2 }, where η2 is the maximum value of η2 (t, x) on I2 × Ω . By the conclusion obtained in (b) above, the solution (u, z, w) of the transformed system (3.1) with the initial function (η1 , η∗2 , η3 ) converges to the positive limit (ρ1 , M − ρ2 , ρ3 ) as t → ∞. Since (η1 , η2∗ , η3 ) ≥ (η1 , η∗2 , η3 ) and the function f(x, U, Uτ ) in (3.3) is quasimonotone nondecreasing, the positivity Lemma 3.1 of [9] ensures that (u, z, w) ≥ (u, z, w) in (0, ∞) × Ω . This implies that lim inf(u, z, w) ≥ lim (u, z, w) = (ρ1 , M − ρ2 , ρ3 )

t→∞

t→∞

and, in particular, the solution components u, w of (u, v, w) are strictly positive in (0, ∞) × Ω . Consider the scalar boundary problem Vt − L 2 V = β2 V (a2 −  − c2 V ),

∂ V /∂ν = 0,

V (0, x) = η2 (0, x),

(3.19)

where  > 0 is a given small constant. It is known that the solution V of (3.19) is positive and converges to (a2 −)/c2 as t → ∞ (cf. [8] p. 201). By choosing  ≤ b10 u, a comparison between the second equation in (1.1) and (3.19) shows that v(t, x) ≤ V (t, x) in (0, ∞) × Ω . This implies that there exists T > 0 such that v(t, x) ≤ a2 /c2 for all t ≥ T . It follows from the positive property of (u, v, w) that there exists δ2 > 0 such that δ2 ≤ v(t, x) ≤ a2 /c2

for all t ≥ T .

(3.20)

Using the strictly positive property of v(t, x) in the first equation and then the last equation in (1.1), a similar argument gives δ1 ≤ u(t, x) ≤ a1 /b1 δ3 ≤ w(t, x) ≤ (a3 /b3 )[1 + (a1 /b1 )]

for t ≥ T , x ∈ Ω

(3.21)

for some positive constants δ2 , δ3 with possibly a different T . In view of the choice of (M1 , M2 , M3 ) in (3.12) the relations in (3.20) and (3.21) imply that for some t ∗ > 0, (δ1 , δ2 , δ3 ) ≤ (u(t ∗ , x), v(t ∗ , x), w(t ∗ , x)) ≤ (M1 , M2 , M3 ).

(3.22)

This proves the property (3.9), where S0 is given by (3.7) with (u˜ s , v˜s , w˜ s ) = (M1 , M2 , M3 ), (uˆ s , vˆs , wˆ s ) = (δ1 , δ2 , δ3 ). It follows from Theorem B (with (u ∗s , vs∗ , ws∗ ) = (ρ1 , ρ2 , ρ3 )) that the solution (u, v, w) of (1.1) possesses the convergence property (2.4). This completes the proof of the theorem.  Proof of Theorem 3. The conclusion of the theorem is a direct consequence of Theorem 2 either by letting c100 = b200 = σ100 = σ200 = 0 and c1 = c10 , b2 = b20 , σ1 = σ10 , σ2 = σ20 , or by letting τ1 = τ2 = τ3 = 0 and c1 , b2 , σ1 and σ2 given by (1.3).  Proof of Theorem 4. This follows from the proof of Theorem 2 by letting L i = 0, i = 1, 2, 3. Details are omitted.  References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

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