Dynamics of A Periodically Pulsed Bio-reactor
Model with A Hydraulic Storage Zone
Sze-Bi Hsu
∗Feng-Bin Wang
†Xiao-Qiang Zhao
‡Abstract
In this paper, we investigate a periodically pulsed bio-reactor model of a flowing water habitat with a hydraulic storage zone in which no flow occurs. The full system can be reduced to a limiting system based on a conservation principle. Then we obtain sufficient conditions in term of principal eigen-values for the persistence of single population and the coexistence of two competing populations for the limiting system by appealing to the theory of monotone dynamical systems. Finally, we use the theory of chain transitive sets to lift the dynamics of the limiting system to the full system.
Keywords. Periodic bioreactor model, hydraulic storage zone, extinction and persistence, periodic coexistence state.
AMS 2010 MSC. 35K55, 37C65, 92D25.
Short Title. A periodic bio-reactor model with a storage zone.
∗Department of Mathematics and The National Center for Theoretical Science, National
Tsing-Hua University, Hsinchu 300, Taiwan. Research partially supported by National Council of Sci-ence, Republic of China.
†Department of Mathematics, National Tsing-Hua University, Hsinchu 300, Taiwan. Research
partially supported by National Council of Science, Republic of China.
‡Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s,
NL A1C 5S7, Canada. Research supported in part by the NSERC of Canada and the MITACS of Canada.
1
Introduction
The chemostat is a basic piece of laboratory apparatus, yet it plays an important role in most theoretical studies of microbial growth and competition [16]. Al-though it provides a simple model for many microbial habitats, the assumption of idealized mixing may be doubtful. Natural environments are usually spatially inho-mogeneous and several models have been introduced where the habitat is not well mixed. For the unstirred chemostat, the authors in [7] removed the “well-mixed” hypothesis and considered a system of reaction-diffusion equations. Basically, the flow enters at one boundary supplying nutrient resource(s), and exits at another, removing nutrients and organisms, while diffusion transports organisms and nutri-ent across the habitat domain. A differnutri-ent environmnutri-ent for microbial growth and competition is the flow reactor model with advection [1, 2, 9, 15]. Let us briefly introduce the flow reactor model of microbial competition for a nutrient in a river-ine reservoir occupying a simple channel of longitudinally invariant cross-section that was formulated by Kung and Baltzis in [9]. The channel is assumed to have constant cross-sectional area A and length L, yielding volume V . A flow of water enters at the upstream end (x = 0), with discharge F (dimensions length3 / time).
An equal flow exits at the downstream end (x = L), which is assumed to be a dam. Based on this flow, a dilution rate D (dimensions time−1) is defined as F/V . The advective flow within the channel is set to maintain water balance, by transporting water with a net velocity ν = DL. The microbial populations Ni, i = 1, 2 compete
for nutrient R. The competition is purely exploitative in the sense that organisms simply consume the nutrient, thereby making it unavailable for its competitor. A flow of medium in the channel with velocity ν in the direction of increasing x brings fresh nutrient at a a time-independent constant concentration into the reactor at x = 0 and carries medium, unutilized nutrient and organisms out of the reactor at x = L. Nutrient and organisms are assumed to diffuse throughout the channel with the same diffusivity δ.
However, if we consider the periodic time dependence in the nutrient concen-tration to account for seasonal or daily changes, then the model will become more realistic. The authors in [18] assume that the nutrient concentration in the medium is maintained at the periodically varying concentration R(0)(t + τ ) = R(0)(t) at the
fol-lowing parabolic systems: ∂R ∂t = δ ∂2R ∂x2 − ν ∂R ∂x − q1f1(R)N1− q2f2(R)N2, ∂N1 ∂t = δ ∂2N 1 ∂x2 − ν ∂N1 ∂x + f1(R)N1, (1.1) ∂N2 ∂t = δ ∂2N2 ∂x2 − ν ∂N2 ∂x + f2(R)N2, 0 < x < L, t > 0 with boundary conditions
νR(0, t) − δ∂R ∂x(0, t) = νR (0)(t), νNi(0, t) − δ ∂Ni ∂x (0, t) = 0, (1.2) ∂R ∂x(L, t) = ∂Ni ∂x (L, t) = 0, i = 1, 2, and initial conditions
R(x, 0) = R0(x) ≥ 0,
Ni(x, 0) = Ni0(x) ≥ 0, 0 < x < L, i = 1, 2, (1.3)
where qi is the constant nutrient quota for species i; R(0)(t) satisfies
R(0)(·) ∈ C2(R+, R), R(0)(t) ≥ 0 but R(0)(·) ≡/ 0 on R+ := [0, ∞),
R(0)(t + τ ) = R(0)(t), for some real number τ > 0. (1.4) The nonlinear functions fi(R) describes the nutrient uptake rate and the growth
rate of the organisms Ni at nutrient concentration R. We assume that these
func-tions satisfy
fi(0) = 0, fi0(R) > 0, fi ∈ C2, i = 1, 2.
A usual example is the Monod function fi(R) =
µmax,iR
Kµ,i+ R
.
The aim of this paper is to generalize (1.1)-(1.3) by adding a hydraulic storage zone in which there is no spatial transport. Models with hydraulic storage zones
partition the cross-section of a channel into a flowing zone of area A, and a static zone of area AS. Exchange of nutrient and populations between the flowing and
storage zones occurs by Fickian diffusion with rate α (time−1). Although nutrient concentration and population densities vary with location x in both the flowing channel and the storage zone, advective and diffusive transport occur only in the flowing zone, not the storage zone. Suppose R(x, t), Ni(x, t) denote nutrient
con-centration and population densities in the flowing channel; RS(x, t), NS,i denote
nutrient concentration and population densities in the storage zone. Then the governing equations are
∂R ∂t = δ ∂2R ∂x2 − ν ∂R ∂x − q1f1(R)N1− q2f2(R)N2+ α(RS − R), ∂N1 ∂t = δ ∂2N1 ∂x2 − ν ∂N1 ∂x + α(NS,1− N1) + f1(R)N1, ∂N2 ∂t = δ ∂2N 2 ∂x2 − ν ∂N2 ∂x + α(NS,2− N2) + f2(R)N2, (1.5) ∂RS ∂t = −α A AS (RS− R) − q1f1(RS)NS,1− q2f2(RS)NS,2, ∂NS,1 ∂t = −α A AS (NS,1− N1) + f1(RS)NS,1, ∂NS,2 ∂t = −α A AS (NS,2− N2) + f2(RS)NS,2, 0 < x < L, t > 0
with boundary conditions
νR(0, t) − δ∂R ∂x(0, t) = νR (0)(t), νNi(0, t) − δ ∂Ni ∂x (0, t) = 0, (1.6) ∂R ∂x(L, t) = ∂Ni ∂x (L, t) = 0, i = 1, 2, and initial conditions
R(x, 0) = R0(x) ≥ 0, Ni(x, 0) = Ni0(x) ≥ 0, 0 < x < L,
RS(x, 0) = R0S(x) ≥ 0, NS,i(x, 0) = NS,i0 (x) ≥ 0, i = 1, 2, (1.7)
where R(0)(t) satisfies (1.4). We should point out that the authors in [4] considered
One of the main technical difficulties in our analysis is the lack of compactness of solution maps of the model system. This is because some equations have no diffusion terms. To overcome these problems, we first prove that the solution maps associated with a linearized system around the trivial (or semi-trivial) solution are κ-contractions, where κ is the Kuratowski measure of noncompactness (see, e.g., [3]). By a generalized Krein-Rutman Theorem, we can show that the principal eigenvalue of the associated eigenvalue problems exists, and hence, the stability of those trivial (or semi-trivial) solutions can be determined. Next, we prove the solution maps associated with our systems are asymptotically compact on any bounded set and then conclude that the associated Poincar´e map is κ-contracting and admits a global attractor under some appropriate conditions by using the results in [11].
The organization of the paper is as follows. In section 2, we study the well-posedness and the conservation principle for system (1.5)-(1.7). Due to this con-servation law, (1.5)-(1.7) can be reduced to a limiting system which generates a monotone dynamical system. In section 3, we consider the single population growth in the limiting system. We show that when the trivial solution is asymptotically stable, then the single population will be washed out; when the trivial solution is unstable, there is a unique periodic positive solution which attracts all solutions with nonzero initial data. Section 4 is devoted to the study of the limiting sys-tem of the two competing species model. We prove the existence of a positive periodic solution (i.e., a periodic coexistence state) if each species can invade the semi-trivial periodic state established by the other species. In section 5, we lift the dynamics of the limiting system to the full system by using the theory of internal chain transitive sets.
2
The conservation principle
This section is devoted to the study of the well-posedness of the initial-boundary-value problem (1.5)-(1.7) and the conservation principle. Let X+ = C([0, L], R6
+) be
the positive cone in the Banach space X = C([0, L], R6) with the usual supremum norm. In order to simplify notations, we set u0 = R, u1 = N1, u2 = N2, u3 = RS,
u4 = NS,1, u5 = NS,2 and u = (u0, u1, u2, u3, u4, u5). We assume that the initial
data in (1.7) satisfying (u0
0, u01, u02, u30, u04, u05) := (R0, N10, N20, R0S, NS,10 , NS,20 ) ∈ X+.
theory developed in [10] where existence and uniqueness and positivity are treated simultaneously (taking delay as zero). The idea is to view the system (1.5)-(1.7) as the abstract ordinary differential equation in X+ and the so-called mild solutions can be obtained for any given initial data. More precisely,
u0(t) = V (t, 0)u00+ Rt 0 T0(t − s)B0(u(s))ds, ui(t) = Ti(t)u0i + Rt 0 Ti(t − s)Bi(u(s))ds, i = 1, 2, ui(t) = u0i + Rt 0 Bi(u(s))ds, i = 3, 4, 5, (2.1)
where Ti(t) is the positive, non-expansive, analytic semigroup on C([0, L], R) (see,
e.g., [15, Chapter 7]) such that u = Ti(t)u0i, i = 0, 1, 2, satisfies the linear initial
value problem ∂u ∂t = δ ∂2u ∂x2 − ν ∂u ∂x, t > 0, 0 < x < L,
νu(0, t) − δ∂u∂x(0, t) = ∂u∂x(L, t) = 0, t > 0, u(x, 0) = u0
i(x), i = 0, 1, 2.
(2.2)
V (t, s), t > s, is the family of affine operators on C([0, L], R) (see, e.g., [13, Chap-ter 5]) such that u = V (t, s)u00 satisfies the linear system with nonhomogeneous, periodic boundary conditions, with start time s, given by
∂u ∂t = δ ∂2u ∂x2 − ν ∂u ∂x, t > s, 0 < x < L,
νu(0, t) − δ∂u∂x(0, t) = νR(0)(t), ∂u
∂x(L, t) = 0, t > s,
u(x, s) = u0 0(x).
(2.3)
Since R(0)(t + τ ) = R(0)(t), it follows that
V (t + τ, s + τ ) = V (t, s), ∀ t > s.
The nonlinear operator Bi : C([0, L], R+) → C([0, L], R) is defined by
B0(u) = −q1f1(u0)u1 − q2f2(u0)u2+ α(u3− u0), B1(u) = α(u4− u1) + f1(u0)u1, B2(u) = α(u5− u2) + f2(u0)u2, B3(u) = −αAA S(u3− u0) − q1f1(u3)u4 − q2f2(u3)u5, B4(u) = −αAA S(u4− u1) + f1(u3)u4, B5(u) = −αAAS(u5− u2) + f2(u3)u5. (2.4)
By standard maximum principle arguments (see, e.g., [15, Chapter 7]), it fol-lows that V (t, s)C([0, L], R+) ⊂ C([0, L], R+), ∀ t > s and Ti(t)C([0, L], R+) ⊂
C([0, L], R+), ∀ t > 0. The operator V and semi-group T0 are related to [10, the
equation (1.9) below ] by setting β(x, t) = νR(0)(t). Since f
i(0) = 0, it follows that
Bi(u) ≥ 0 whenever ui ≡ 0, ∀ 0 ≤ i ≤ 5, and hence, B := (B0, B1, B2, B3, B4, B5)
is quasipositive (see, e.g., [10, Remark 1.1]). By [10, Theorem 1 and Remark 1.1], we have the following results:
Lemma 2.1. The system (1.5)-(1.7) has a unique noncontinuable solution and the solutions to (1.5)-(1.7) remain non-negative on their interval of existence if they are non-negative initially.
In the followings, we demonstrate that (1.5)-(1.7) have mass conservation in the flow and storage zones. Let
W (x, t) = R(x, t) + q1N1(x, t) + q2N2(x, t) and
WS(x, t) = RS(x, t) + q1NS,1(x, t) + q2NS,2(x, t). (2.5)
Then W (x, t) and WS(x, t) satisfy the following coupled differential equations
∂W ∂t = δ ∂2W ∂x2 − ν ∂W ∂x + αWS− αW, ∂WS ∂t = −α A AS WS+ α A AS W, 0 < x < L, t > 0 (2.6) with boundary conditions
νW (0, t) − δ∂W
∂x (0, t) = νR
(0)(t), ∂W
∂x (L, t) = 0, (2.7)
and initial conditions
W (x, 0) = W0(x), WS(x, 0) = WS0(x). (2.8)
Note that (1.5)-(1.7) reduces to (2.6)-(2.8) for W = R and WS = RS when Ni =
NS,i = 0, i = 1, 2.
We first consider the following auxiliary system which will be used in our dis-cussions later: ∂U ∂t = δ ∂2U ∂x2 − ν ∂U ∂x + αUS− αU, ∂US ∂t = −α A AS US+ α A AS U, 0 < x < L, t > 0 (2.9)
with boundary conditions
νU (0, t) − δ∂U
∂x(0, t) = 0, ∂U
∂x(L, t) = 0, (2.10)
and initial conditions
U (x, 0) = U0(x), US(x, 0) = US0(x). (2.11)
It is easy to see that the system (2.9)-(2.11) is a linear and cooperative system. According to [15, pp. 147-148], the eigenvalue problem
(
λφ(x) = δφ00(x) − νφ0(x), 0 < x < L,
νφ(0) − δφ0(0) = φ0(L) = 0, (2.12)
has a principal eigenvalue, denoted by λ0, with an associated eigenvector φ0 0, and λ0 < 0. This implies that U = 0 is globally asymptotically stable for ∂U∂t = δ∂∂x2U2 − ν
∂U
∂x subject to (2.10).
Substituting U (x, t) = eλtφ(x) and U
S(x, t) = eλtϕ(x), we obtain the associated
eigenvalue problem λφ(x) = δφ00(x) − νφ0(x) − αφ(x) + αϕ(x), λϕ(x) = −αAA Sϕ(x) + α A ASφ(x), 0 < x < L, νφ(0) − δφ0(0) = φ0(L) = 0. (2.13)
Let T (t) be the solution semigroup generated by (2.9)-(2.11) on C([0, L], R2).
It is easy to see that T (t) is a strongly positive operator for each t > 0. By the similar arguments as in the proof of Lemma 3.3 (see also [8]), we can prove that for each t > 0, T (t) is an κ-contraction on C([0, L], R2) in the sense that
κ(T (t)B) ≤ e−αAAStκ(B),
for any bounded subset B of C([0, L], R2), where κ is the Kuratowski measure of noncompactness in the Banach space C([0, L], R2). By the proof of [15, Theorem
7.6.1] and a generalized Krein-Rutman Theorem (see [12] or [8, Lemma 2.2]), it follows that the problem (2.13) has a principal eigenvalue, denoted by λ∗, with an associated eigenvector (φ∗, ϕ∗) 0. Further, for each t > 0, the spectrum radius of T (t), r(T (t)), is the principal eigenvalue of T (t), and hence,
Lemma 2.2. Let λ∗ be the principal eigenvalue of (2.13). Then λ∗ = 1 2 " (λ0− α − αA AS ) + r (λ0− α − αA AS )2+ 4λ0αA AS # , and hence λ∗ < 0.
Proof. Our proof is essentially the same as that of [19, Lemma 3.1]. For the sake of completeness, we provide its details below. Suppose that λ∗ and (φ∗, ϕ∗) 0 are the eigenvalue-eigenvector pair corresponding to (2.13). From the second equation of (2.13), it follows that −αφ∗(x) + αϕ∗(x) = − αλ ∗ λ∗+ (αA/A S) φ∗(x). From the first equation of (2.13), it follows that
(
(λ∗+ λ∗+(αA/Aαλ∗ S))φ
∗(x) = δφ∗00(x) − νφ∗0(x), 0 < x < L,
νφ∗(0) − δφ∗0(0) = φ∗0(L) = 0. (2.15)
Since φ∗(x) > 0 in (0, L), it deduces that λ0 = λ∗+ αλ∗
λ∗+(αA/A
S), and hence, λ
∗ is a
real zero of the quadratic equation P (λ) := λ2+ (αA
AS
+ α − λ0)λ − λ0αA AS
= 0. (2.16)
It remains to show that λ∗is the maximum root of equation (2.15). Suppose that λ is a given zero of (2.15). Obviously, P (−αAA
S) = −
α2A
AS < 0 and then λ +
αA AS 6= 0. By (2.16), it follows that λ0(λ +αA
AS) = λ[(λ + αA AS) + α], that is, λ 0 = λ + αλ λ+αA AS . Note that (λ0, φ0(x)) satisfies (2.12). Let ϕ0(x) =
αA AS
λ+αA
AS
φ0(x). Hence, λ is an eigenvalue
of (2.13) with eigenfunction (φ0(x), ϕ0(x)). Thus, λ ≤ λ∗ since λ∗ is the principal
eigenvalue of (2.13). Then λ∗ is the maximum root of equation (2.15), and hence
λ∗ = 1 2 " (λ0− α − αA AS ) + r (λ0− α − αA AS )2+ 4λ0αA AS # .
By Lemma 2.2 and (2.14), it follows that r(T (t)) < 1 for each t > 0. This implies that (0, 0) is globally asymptotically stable for (2.9)-(2.11), and hence, (0, 0) is the unique steady-state solution for (2.9)-(2.11).
Lemma 2.3. The system (2.6)-(2.8) admit a unique positive τ -periodic solution (W∗(x, t), WS∗(x, t)) 0 and for any (W0(x), W0S(x)) ∈ C([0, L], R2), the unique
mild solution (W (x, t), WS(x, t)) of (2.6)-(2.8) with
(W (x, 0), WS(x, 0)) = (W0(x), W0S(x)) satisfies lim t→∞((W (x, t), WS(x, t)) − (W ∗ (x, t), WS∗(x, t))) = (0, 0) uniformly for x ∈ [0, L]. (2.17) Proof. Let U (x, t) = W (x, t) − R(0)(t), US(x, t) = WS(x, t).
Then U (x, t) and US(x, t) satisfy the following coupled differential equations
∂U ∂t = δ ∂2U ∂x2 − ν ∂U ∂x + αUS− αU + R1(t), ∂US ∂t = −α A AS US+ α A AS U + R2(t), 0 < x < L, t > 0 (2.18)
with boundary conditions
νU (0, t) − δ∂U
∂x(0, t) = 0, ∂U
∂x(L, t) = 0, (2.19)
and initial conditions
U (x, 0) = U0(x), US(x, 0) = US0(x), (2.20) where R1(t) = −dR 0(t) dt − αR (0)(t), R 2(t) = αAA SR (0)(t).
The boundary condition (2.19) is homogeneous and we rewrite (2.18)-(2.20) as an abstract system of ordinary differential equations in C([0, L], R2) given by
d dt U US = A U US + R1(t) R2(t) , t > 0, (2.21)
with initial condition U (0) US(0) = U 0(x) U0 S(x) , (2.22)
where A is the closure in C([0, L]) × C([0, L]) of
A0 = δ ∂2 ∂x2 − ν ∂ ∂x− α α αAA S −α A AS ! , with domain D(A0) = {(U0, US0) ∈ C2((0, L), R2) ∩ C1([0, L], R2) : A0 U 0 U0 S ∈ C([0, L], R2), νU0(0) − δ(U0)0(0) = (U0)0(L) = 0}.
For any (U0, US0) ∈ C([0, L], R2), the mild solution of (2.18)-(2.20) is express as
U (t) US(t) = T (t) U 0 U0 S + Z t 0 T (t − s) R1(s) R2(s) ds, (2.23)
where T (t) is the analytic semigroup generated by A in C([0, L], R2), that is, T (t) is
the solution semigroup generated by (2.9)-(2.11). It is easy to see that (U (t), US(t))
is a τ -periodic solution of (2.21) if and only if (U0, U0
S) = (U (0), US(0)) and (I − T (τ )) U 0 U0 S = Z τ 0 T (τ − s) R1(s) R2(s) ds. (2.24)
By Lemma 2.2 and (2.14), it follows that r(T (τ )) = eλ∗τ < 1. This implies
that I − T (τ ) is invertible, and hence, (2.21) admits a unique τ -periodic solution (U∗(x, t), US∗(x, t)). Let V (x, t) = U (x, t) − U∗(x, t) and VS(x, t) = US(x, t) − US∗(x, t). Then d dt V (t) VS(t) = A V (t) VS(t) , t > 0, (2.25)
By [13, Theorem 4.4.3], there exist M > 0 and δ > 0 such that k T (t) k< M e−δt, t ≥ 0. Hence lim
t→∞(V (t), VS(t)) = (0, 0) uniformly in C([0, L], R
2), that
is, lim
t→∞(V (x, t), VS(x, t)) = (0, 0) uniformly for x ∈ [0, L]. Then limt→∞(U (x, t) −
U (x, t)∗, VS(x, t) − VS∗(x, t)) = (0, 0) uniformly for x ∈ [0, L].
Set W∗(x, t) = U∗(x, t) + R(0)(t) and WS∗(x, t) = US∗(x, t), x ∈ [0, L], and t ≥ 0. It then follows that (W∗(x, t), WS∗(x, t)) is a τ -periodic solution of (2.6)-(2.8). Moreover, for any (W0(·), W0
S(·)) ∈ C([0, L], R2), the unique mild solution
(W (x, t), WS(x, t)) of (2.6)-(2.8) with (W (·, 0), WS(·, 0)) = (W0(·), WS0(·)) satisfies
(2.17).
From [10, Theorem 1 and Remark 1.1], for any (W0(·), W0
S(·)) ∈ C([0, L], R2+),
the unique solution (W (x, t), WS(x, t)) of (2.6)-(2.8) with (W (·, 0), WS(·, 0)) =
(W0(·), W0
S(·)) satisfying
W (x, t) ≥ 0, WS(x, t)) ≥ 0, x ∈ [0, L], t ≥ 0.
It remains to show that
W∗(x, t) > 0, WS∗(x, t) > 0, x ∈ [0, L], t ≥ 0. For any given t ≥ 0, by (2.17), we have
lim
n→∞[(W (x, t + nτ ), WS(x, t + nτ )) − (W ∗
(x, t), WS∗(x, t))] = (0, 0), uniformly for x ∈ [0, L]. Then W∗(x, t) = lim
n→∞W (x, t + nτ ) ≥ 0 and W ∗
S(x, t) =
lim
n→∞WS(x, t + nτ ) ≥ 0 uniformly for x ∈ [0, L]. It is easy to see that R (0)(t
0) > 0,
for some t0 > 0. By the boundary condition of U∗(x, t0) at x = 0, it follows that
U∗(·, t0) ≡/ − R(0)(t0). Thus, W∗(·, t0) = U∗(·, t0) + R(0)(t0) ≡/ 0.
We first show that W∗(x, t) > 0, x ∈ [0, L], t ≥ t0. Suppose that W∗(ˆx, ˆt) = 0,
for some ˆx ∈ [0, L], ˆt ≥ t0. By the first equation of (2.6), it follows that
−∂W ∗ ∂t + δ ∂2W∗ ∂x2 − ν ∂W∗ ∂x − αW ∗ = −αWS∗ ≤ 0 (2.26)
If ˆx ∈ (0, L), then the strong maximum principle ([14, Chapter 3, Theorem 7]) implies that W∗(x, t) ≡ 0 on x ∈ [0, L] and ˆt ≥ t0, which is impossible because
W∗(·, t0) ≡/ 0. Assume that ˆx = 0, that is, W∗(0, ˆt) = 0. Then by [14, Chapter
0, contradicting (2.7). Assume ˆx = L. By a similar argument, it follows that Wx∗(L, ˆt) < 0, contradicting (2.7) again. Thus, W∗(x, t) > 0, x ∈ [0, L], t ≥ t0. By
the τ -periodicity of W∗(x, ·), it follows that W∗(x, t) > 0, x ∈ [0, L], t ≥ 0.
Next, we show that WS∗(x, t) > 0, x ∈ [0, L], t ≥ 0. Suppose that WS∗(˜x, ˜t) = 0, for some ˜x ∈ [0, L], ˜t > 0. By the second equation of (2.6), it follows that
∂WS ∂t (˜x, ˜t) = −α A AS WS(˜x, ˜t) + α A AS W (˜x, ˜t) = α A AS W (˜x, ˜t) > 0. (2.27) Thus, WS∗(˜x, ˜t−δ) < WS∗(˜x, ˜t) = 0 if δ > 0 is sufficiently small. The above inequality contradicts that WS∗(x, t) ≥ 0 uniformly for x ∈ [0, L], t ≥ 0. This completes our proof.
By Lemma 2.1, together with the relation (2.5) and Lemma 2.3, it is easy to see that the following results hold.
Lemma 2.4. Any solution of the system (1.5)-(1.7) exists globally on [0, ∞). More-over, solutions are ultimately bounded and uniformly bounded.
Finally, we see from the relation (2.5) and Lemma 2.3 that the limiting systems of (1.5)-(1.7) take the forms:
∂N1 ∂t = δ ∂2N1 ∂x2 − ν ∂N1 ∂x + α(NS,1− N1) + f1(W ∗ (x, t) − q1N1− q2N2)N1, ∂NS,1 ∂t = −α A AS (NS,1− N1) + f1(WS∗(x, t) − q1NS,1− q2NS,2)NS,1, (2.28) ∂N2 ∂t = δ ∂2N 2 ∂x2 − ν ∂N2 ∂x + α(NS,2− N2) + f2(W ∗ (x, t) − q1N1− q2N2)N2, ∂NS,2 ∂t = −α A AS (NS,2− N2) + f2(WS∗(x, t) − q1NS,1− q2NS,2)NS,2,
in (0, L) × (0, ∞), with boundary conditions νNi(0, t) − δ
∂Ni
∂x (0, t) = 0, ∂Ni
∂x (L, t) = 0, i = 1, 2, (2.29)
and initial conditions
From the biological view of point, the feasible domain D(t) for (2.28)-(2.30) should be
D(t) = {(N1, NS,1, N2, NS,2) ∈ C([0, L], R4+) : q1N1(·) + q2N2(·) ≤ W∗(·, t),
q1NS,1(·) + q2NS,2(·) ≤ WS∗(·, t)}. (2.31)
In the followings, we show that the set D(t) is positively invariant for the solution maps associated with (2.28)-(2.30).
Lemma 2.5. For any φ := (φ1, φ2, φ3, φ4) ∈ D(0), system (2.28)-(2.30) has a
unique mild solution (N1(x, t), NS,1(x, t), N2(x, t), NS,2(x, t)) ∈ D(t), ∀ t ≥ 0,
whenever (N1(x, 0), NS,1(x, 0), N2(x, 0), NS,2(x, 0)) = φ.
Proof. Our proof is similar to that of Lemma 3.1. We will give the proof of Lemma 3.1 in details, so we skip the proof here.
3
Single species growth
In this section, we investigate the single population model. Mathematically, it means that we set (N1, NS,1) = (0, 0) or (N2, NS,2) = (0, 0) in the model system
(1.5)-(1.7). In order to simplify notations, we drop all subscripts in the remaining equations and then consider
∂R ∂t = δ ∂2R ∂x2 − ν ∂R ∂x − qf (R)N + α(RS− R), ∂N ∂t = δ ∂2N ∂x2 − ν ∂N ∂x + α(NS− N ) + f (R)N, ∂RS ∂t = −α A AS (RS− R) − qf (RS)NS, (3.1) ∂NS ∂t = −α A AS (NS− N ) + f (RS)NS, 0 < x < L, t > 0,
with boundary conditions
νR(0, t) − δ∂R ∂x(0, t) = νR (0)(t), ∂R ∂x(L, t) = 0, νN (0, t) − δ∂N ∂x(0, t) = ∂N ∂x(L, t) = 0, t > 0, (3.2)
and initial conditions
R(x, 0) = R0(x) ≥ 0, N (x, 0) = N0(x) ≥ 0, 0 < x < L,
RS(x, 0) = R0S(x) ≥ 0, NS(x, 0) = NS0(x) ≥ 0, (3.3)
where R(0)(t) satisfies (1.4). By similar arguments as in Section 2, it follows that
the limiting system of (3.1)-(3.3) takes the following form: ∂N ∂t = δ ∂2N ∂x2 − ν ∂N ∂x + α(NS− N ) + f (W ∗ (x, t) − qN )N, ∂NS ∂t = −α A AS (NS− N ) + f (WS∗(x, t) − qNS)NS, 0 < x < L, t > 0, (3.4)
with boundary conditions νN (0, t) − δ∂N
∂x(0, t) = 0, ∂N
∂x(L, t) = 0, t > 0, (3.5)
and initial conditions
N (x, 0) = N0(x) ≥ 0, NS(x, 0) = NS0(x) ≥ 0, 0 < x < L. (3.6)
From the biological view of point, the feasible domain Λ(t) for (3.4)-(3.6) should be
Λ(t) = {(N, NS) ∈ C([0, L], R2+) : qN (·) ≤ W ∗
(·, t), qNS(·) ≤ WS∗(·, t)}. (3.7)
Next, we prove some basic properties of the set Λ(t).
Lemma 3.1. For any φ := (φ1, φ2) ∈ Λ(0), system (3.4)-(3.6) has a unique mild
solution (N (x, t), NS(x, t)) with (N (x, 0), NS(x, 0)) = φ and (N (x, t), NS(x, t)) ∈
Λ(t), ∀ t ≥ 0.
Proof. Let T (t) be the semigroup generated by ( ∂N ∂t = δ ∂2N ∂x2 − ν ∂N ∂x − αN, 0 < x < L, t > 0, νN (0, t) − δ∂N∂x(0, t) = 0, ∂N∂x(L, t) = 0, and TS(t)φ2 = e −α A
AStφ2. From the system (3.4)-(3.6) with the initial condition (N (x, 0), NS(x, 0)) = φ, we have ( N (·, t, φ) = T (t)φ1+ Rt 0 T (t − θ)[αNS(·, θ) + f (W ∗(·, θ) − qN (·, θ))N (·, θ)]dθ, NS(·, t, φ) = TS(t)φ2+ Rt 0 TS(t − θ)[ αA ASN (·, θ) + f (W ∗ S(·, θ) − qNS(·, θ))NS(·, θ)]dθ.
It follows that (3.4)-(3.6) can be written as the following integral equation u(t) = T(t)φ + Z t 0 T(t − θ)B(θ, ·, u(θ))dθ, (3.8) where u(t) = N (t) NS(t) , T(t) = T (t) 0 0 TS(t) , and for any v := (v, vs) ∈ Λ(t), B(t, ·, v) is defined by
B(t, ·, v) := B1(t, ·, v) B2(t, ·, v) = αvs+ F (t, ·, v) αAA Sv + G(t, ·, vs) , with F (t, ·, v) = f (W∗(·, t) − qv)v and G(t, ·, vs) = f (WS∗(·, t) − qvs)vs.
We first show that B(t, ·, v) is quasi-monotone on Λ(t) in the sense that lim h→0+ 1 hd(v − w + h(B(t, ·, v) − B(t, ·, w)); C([0, L], R 2 +)) = 0,
for all v := (v, vs), w := (w, ws) ∈ Λ(t) with w(x) ≤ v(x), x ∈ [0, L]. By the
mean-value theorem and the fact that v, w ∈ Λ(t), it is easy to see that there is a constant δ > 0 such that B(t, ·, v) − B(t, ·, w) =
F (t, ·, v) − F (t, ·, w) + α(vs− ws) G(t, ·, vs) − G(t, ·, ws) + αAA S(v − w) ≥ −δ(v − w) + α(vs− ws) −δ(vs− ws) + αAA S(v − w) . Hence, for any h > 0 satisfying hδ < 1, it follows that
(v − w) + h(B(t, ·, v) − B(t, ·, w)) ≥ (1 − δh)(v − w) + hα(vs− ws) (1 − δh)(vs− ws) + hαAA S(v − w) ≥ 0. Thus, [10, Corallary 5] and the discussions above complete the proof.
By Lemma 3.1, we can define solution maps Ψt : Λ(0) → Λ(t) associated with
(3.4)-(3.6) by
Ψt(P ) = (N (·, t, P ), NS(·, t, P )), ∀P := (N0(·), NS0(·)) ∈ Λ(0), t ≥ 0. (3.9)
For convenience, we let
Y+ = Λ(0), Y0 = Y+\{(0, 0)}, ∂Y0 := Y+\Y0 = {(0, 0)}.
Since one equation in (3.4)-(3.6) has no diffusion term, its solution map Ψt is
not compact. Due to the lack of compactness, we need to impose the following condition:
α A
AS
> f (WS∗(x, t)), ∀ x ∈ [0, L], t ≥ 0. (3.10) Remark 3.1. As in [4, Remark 3.3], if the Monod function f (R) := µmaxR
Kµ+R satisfies µmax < α
A AS
, (3.11)
then it follows that
f (WS∗(x, t)) = µmax WS∗(x, t) Kµ+ WS∗(x, t) < µmax < α A AS ,
that is, (3.10) holds. We note that conditions (3.11) means that the cross-section of the storage zone is small or the exchange rate is large.
Recall that the Kuratowski measure of noncompactness (see [3]), κ, is defined by
κ(B) := inf{r : B has a finite cover of diameter < r}, (3.12) for any bounded set B. We set κ(B) = ∞ whenever B is unbounded. It is easy to see that B is precompact(i.e., ¯B is compact) if and only if κ(B) = 0.
Lemma 3.2. Let (3.10) hold. Then Ψτ is κ-contracting in the sense that
lim
n→∞κ(Ψ n
τB) = 0
for any bounded set B ⊂ Y+.
Proof. For the sake of convenience, we let (u, v) := (N, NS) and rewrite (3.4)-(3.6)
as ∂u ∂t = d ∂2u ∂x2 − ν ∂u ∂x+ m(t, x, u, v), ∂v ∂t = g(t, x, u, v), x ∈ (0, L), t > 0,
νu(0, t) − d∂u∂x(0, t) = 0, ∂u∂x(L, t) = 0, u(·, 0) = φ1(·), v(·, 0) = φ2(·),
where m(t, x, u, v) = α(v − u) + f (W∗(x, t) − qu)u and g(t, x, u, v) = −αAA
S(v − u) + f (WS∗(x, t) − qv)v. For any φ(·) = (φ1(·), φ2(·)) ∈ Y+, the solution maps associated
with system (3.13) are defined by
Ψt(φ) = (u(·, t, φ), v(·, t, φ)), ∀ φ ∈ Y+, t ≥ 0. Define ˜ Λ = {(t, x, u, v) ∈ R4+ : x ∈ [0, L], qu ≤ W ∗(x, t), qv ≤ W∗ S(x, t)}.
From (3.10), it is easy to see that there exists a real number r > 0 such that ∂g(t, x, u, v)
∂v ≤ −r < 0, ∀ (t, x, u, v) ∈ ˜Λ. (3.14)
By (3.14) and the same arguments as in Lemma 4.1, it then follows that Ψτ is
κ-contracting.
Theorem 3.1. Ψτ admits a global attractor on Y+ provided that (3.10) holds.
Proof. By Lemma 3.2, it follows that Ψτ is κ-contracting on Y+. Further, Lemma 2.4
implies that Ψτ is point dissipative on Y+ and that the positive orbits of bounded
subsets of Y+for Ψ
τare bounded. By Theorem 2.6 in [11], Ψτ has a global attractor
that attracts each bounded set in Y+.
Note that (0, 0) is a solution of (3.4)-(3.6). Linearizing system (3.4)-(3.6) at (0, 0), we have ( ∂N ∂t = δ ∂2N ∂x2 − ν ∂N ∂x + α(NS− N ) + f (W ∗ (x, t))N, ∂NS ∂t = −α A AS(NS − N ) + f (W ∗ S(x, t))NS, 0 < x < L, t > 0, (3.15)
with boundary conditions (3.5) and initial conditions (3.6).
Substituting N (x, t) = e−µtφ1(x, t) and NS(x, t) = e−µtφ2(x, t), we obtain the
associated eigenvalue problem ∂φ1 ∂t = δ ∂2φ 1 ∂x2 − ν ∂φ1 ∂x + α(φ2− φ1) + f (W ∗(x, t))φ 1+ µφ1, t > 0, x ∈ (0, L), ∂φ2 ∂t = −α A As(φ2 − φ1) + f (W ∗ S(x, t))φ2+ µφ2, t > 0, x ∈ (0, L), νφ1(0, t) − δ∂φ∂x1(0, t) = ∂φ∂x1(L, t) = 0, t > 0, φ1, φ2 are τ -periodic in t. (3.16)
Lemma 3.3. Let (3.10) hold. Then the eigenvalue problem (3.16) has a principal eigenvalue, denoted by µ∗, with an associated eigenvector φ∗ = (φ∗1, φ∗2) 0. Proof. Let (u, v) := (N, NS) and rewrite (3.15) as
∂u ∂t = d ∂2u ∂x2 − ν ∂u ∂x+ E1(t, x, u, v), ∂v ∂t = −r(x, t)v + E2(u), x ∈ (0, L), t > 0,
νu(0, t) − d∂u∂x(0, t) = 0, ∂u∂x(L, t) = 0 u(x, 0) = φ1(x), v(x, 0) = φ2(x),
(3.17)
where E1(t, x, u, v) = α(v − u) + f (W∗(x, t))u, E2(u) = αAA
Su and r(x, t) = α
A AS − f (WS∗(x, t)). By (3.10), it follows that
r(x, t) ≥ r0, ∀ x ∈ [0, L], t ≥ 0, and r0 is a positive number. (3.18)
Let C := C([0, L], R2). For every initial value functions φ = (φ
1, φ2) ∈ C, one
may use comparison theorem to show that the solution (u(x, t, φ), v(x, t, φ)) ∈ C, ∀ t ≥ 0. Thus, the linear semigroup Πt: C → C associated with the linear system
(3.17) is defined by
Πt(φ) = (u(x, t, φ), v(x, t, φ)), ∀ φ ∈ C, t ≥ 0.
We first show that for each t > 0, Πt is an κ-contraction on C in the sense
that κ(ΠtB) ≤ e−r0tκ(B) for any bounded set B in C, where κ is the Kuratowski
measure of noncompactness as defined in (3.12).
Let T1(t) be the analytic semigroup on C([0, L], R) generated by
∂u ∂t = d ∂2u ∂x2 − ν ∂u ∂x
subject to the above boundary condition and T2(t)φ2 = e− Rt
0r(·,η)dηφ2, ∀φ2 ∈ C([0, L], R). Obviously, T (t) = (T1(t), T2(t)) is a linear semigroup on C.
Define a linear operator
L(t)φ = (0, T2(t)φ2), ∀φ = (φ1, φ2) ∈ C, (3.19)
and a nonlinear operator Q(t)φ = (u(·, t, φ),
Z t
0
where
u(·, t, φ) = T1(t)φ1+
Z t
0
T1(t − s)E1(s, ·, u(·, s, φ), v(·, s, φ))ds.
It is easy to see that
Πt(φ) = L(t)φ + Q(t)φ, ∀φ ∈ C, t ≥ 0.
By (3.18) and (3.19), it follows that sup φ∈C kL(t)φk kφk ≤ supφ∈C ke−Rt 0r(·,η)dηφ2k kφk ≤ supφ∈C ke−r0tφ 2k kφk ≤ e −r0t, and hence kL(t)k ≤ e−r0t.
By the boundedness of Πt and the compactness of T1(t) for t > 0, it follows
that Q(t) : C → C is compact for each t > 0. For any bounded set B in C, there holds κ(Q(t)B) = 0 since Q(t)B is precompact, and consequently,
κ(ΠtB) ≤ κ(L(t)B) + κ(Q(t)B) ≤ kL(t)kκ(B) ≤ e−r0tκ(B), ∀ t > 0.
Thus, Πt is an κ-contraction on C with a contracting function e−r0t.
From the discussions above, it is easy to see that the Poincar´e map Πτ generated
by (3.17) is κ-condensing in the sense that
κ(ΠτB) < κ(B), for any bounded set B in C with κ(B) > 0.
Note that (3.17) is a cooperative system. By a generalized Krein-Rutman Theorem (see [12] or [8, Lemma 2.2]) and [5, Chapter II.14], it then follows that (3.16) has a principal eigenvalue, denoted by µ∗, with an associated eigenvector φ∗ = (φ∗1, φ∗2) 0.
Theorem 3.2. Assume that (3.10) holds. For any P := (N0(·), N0
S(·)) ∈ Y+, let
(N (x, t), NS(x, t)) be the solution of (3.4)-(3.6). Then the following statements are
valid.
(1) If µ∗ > 0, then lim
t→∞k (N (x, t), NS(x, t)) k∞= 0 uniformly for x ∈ [0, L];
(2) If µ∗ < 0, then (3.4)-(3.6) admit a unique positive τ -periodic solution (N∗(x, t), NS∗(x, t)) and for any (N0(·), NS0(·)) ∈ Y0, we have
lim
t→∞k (N (x, t), NS(x, t))−(N ∗
Proof. In the case where µ∗ > 0, it follows that lim
t→∞k (v(x, t, P ), vS(x, t, P )) k∞= 0
uniformly for x ∈ [0, L] , ∀ P ∈ Y+, where (v(x, t, P ), v
S(x, t, P )) is the unique
solution of (3.15) with (v(x, 0, P ), vS(x, 0, P )) = P. For convenience, we rewrite
the reaction terms in (3.4) as follows: F (t, x, N, NS) = α(NS− N ) + f (W∗(x, t) − qN )N −α A AS(NS− N ) + f (W ∗ S(x, t) − qNS)NS . (3.20)
It is easy to see that F (t, x, θN, θNS) θF (t, x, N, NS), ∀ 0 < θ < 1, (N, NS) ∈
Y+. Clearly, the solution (N (x, t, P ), N
S(x, t, P )) of (3.4)-(3.6) satisfies ∂N ∂t ≤ δ ∂2N ∂x2 − ν ∂N ∂x + α(NS− N ) + f (W ∗ (x, t))N, ∂NS ∂t ≤ −α A AS (NS− N ) + f (WS∗(x, t))NS, 0 < x < L, t > 0, (3.21)
that is, (N (x, t, P ), NS(x, t, P )) < (v(x, t, P ), vS(x, t, P )). Thus, we have
lim
t→∞k (N (x, t, P ), NS(x, t, P )) k∞= 0 uniformly for x ∈ [0, L] , ∀ P ∈ Y +.
For the case where µ∗ < 0, we first prove the following claim.
Claim. Zero is a uniform weak repeller for (3.4)-(3.6) in the sense that there exists δ > 0 such that lim sup
t→∞
k Ψt(P ) k≥ δ, ∀ P := (N0(·), NS0(·)) ∈ Y0.
Indeed, let µ be the principal eigenvalue of
∂φ1 ∂t = δ ∂2φ 1 ∂x2 − ν ∂φ1 ∂x + α(φ2− φ1) + (f (W ∗(x, t)) − )φ 1+ µφ1, t > 0, x ∈ (0, L), ∂φ2 ∂t = −α A As(φ2− φ1) + (f (W ∗ S(x, t)) − )φ2 + µφ2, t > 0, x ∈ (0, L) νφ1(0, t) − δ∂φ∂x1(0, t) = ∂φ∂x1(L, t) = 0, t > 0, φ1, φ2 are τ -periodic in t, (3.22) with a positive eigenfunction φ∗(x, t) = (φ∗1, φ∗2) 0. Since lim
→0µ = µ
∗, we can
fix a sufficiently small number > 0 such that µ < 0. It is easy to see that
lim N →0f (W ∗ (x, t) − qN ) = f (W∗(x, t)) and lim NS→0 f (WS∗(x, t) − qNS) = f (WS∗(x, t)),
uniformly for x ∈ [0, L] and t ≥ 0. Thus, we can choose δ > 0 such that
for any x ∈ [0, L], t ≥ 0, and (N, NS) ∈ [0, δ] × [0, δ].
Suppose, by contradiction, there exists P0 ∈ Y0 such that lim sup t→∞
k Ψt(P0) k<
δ, thus, there exists t0 > 0 such that k Ψt(P0) k< δ, ∀ t ≥ t0. It then follows that
Ψt(P0) = (N (x, t, P0), NS(x, t, P0)) satisfy ∂N ∂t ≥ δ ∂2N ∂x2 − ν ∂N ∂x + α(NS− N ) + [f (W ∗(x, t)) − ]N, ∂NS ∂t ≥ −α A AS (NS− N ) + [f (WS∗(x, t)) − ]NS, (3.23)
for any 0 < x < L, t ≥ t0. Note that (N, NS) := e −µtφ∗ (x, t) is a solution of ∂N ∂t = δ ∂2N ∂x2 − ν ∂N ∂x + α(N S− N ) + [f (W ∗ (x, t)) − ]N, ∂NS ∂t = −α A AS (NS − N) + [f (W∗ S(x, t)) − ]N S, 0 < x < L, t > 0,(3.24)
with boundary conditions (3.5). Since Ψt0(P0) = (N (x, t0, P0), NS(x, t0, P0)) 0 for all x ∈ [0, L], there exists a > 0 such that
Ψt0(P0) > aφ
∗
(x, t0) for all x ∈ [0, L].
By the comparison theorem, it follows that
Ψt(P0) > aφ∗(x, t0)e−µ(t−t0) , ∀ t ≥ t0, x ∈ [0, L].
Since µ < 0, we see that Ψt(P0) is unbounded, a contradiction.
By the claim above, Ψτ is weakly uniformly persistent with respect to (Y0, ∂Y0).
Since Ψτ admits a global attractor on Y+, it follows from [20, Theorem 1.3.3] that
Ψτ is uniformly persistent with respect to (Y0, ∂Y0) in the sense that there exists
η > 0 such that lim inf
t→∞ k Ψt(P ) k≥ η, ∀ P ∈ Y0.
Note that Ψτ is κ-contracting, point dissipative and uniformly persistent. It
follows from [11, Theorem 3.8] that Ψτ : Y0 → Y0 admits a global attractor A0. It
is easy to see that Ψτ is strongly monotone, and strictly subhomogeneous in the
sense that Ψτ(θN, θNS) θΨτ(N, NS), ∀ (N, NS) 0, θ ∈ (0, 1). Since A0 ⊂ Y0
and A0 = Ψτ(A0), we further have A0 ⊂ Int(C([0, L], R2+)). It then follows from
[20, Theorem 2.3.2] with K = A0 that Ψτ has a fixed point e 0 such that
A0 = {e}. This implies that e is globally attractive for Ψτ in Y0. Consequently,
Remark 3.2. By Theorem 3.2 and the method of chain transitive sets, as illustrated in Section 5, we can also obtain a threshold type result on the global dynamics of the single species model (3.1)-(3.3).
4
Two species competition
In this section, we study the global dynamics of the limiting system (2.28)-(2.30). Let λ1(a(x, t), b(x, t)) be the principal eigenvalue of the following eigenvalue
prob-lem: ∂ϕ ∂t = δ ∂2ϕ ∂x2 − ν ∂ϕ ∂x + α(ψ − ϕ) + a(x, t)ϕ + λϕ, t > 0, x ∈ (0, L), ∂ψ ∂t = −α A As(ψ − ϕ) + b(x, t)ψ + λψ, t > 0, x ∈ (0, L), νϕ(0, t) − δ∂ϕ∂x(0, t) = ∂ϕ∂x(L, t) = 0, t > 0, ϕ, ψ are τ -periodic in t. (4.1)
Theorem 3.2 can be applied to either of the two systems obtained from (2.28)-(2.30) by setting one of the two ordered pairs (N1, NS,1) or (N2, NS,2) to be (0, 0).
Let
µ∗1 := λ1(f1(W∗(x, t)), f1(WS∗(x, t))) and µ ∗
2 := λ1(f2(W∗(x, t)), f2(WS∗(x, t))).
(4.2) Then we conclude that the system (2.28)-(2.30) has the following results:
(i) Trivial solution ˆ0 := (0, 0, 0, 0) always exists;
(ii) Semi-trivial solution (N1∗(x, t), NS,1∗ (x, t), 0, 0) exists provided that µ∗1 < 0; (iii) Semi-trivial solution (0, 0, N2∗(x, t), NS,2∗ (x, t)) exists provided that µ∗2 < 0; (iv) There may be additional τ -periodic solutions as well and these must be
pos-itive.
Here, (Ni∗(x, t), NS,i∗ (x, t)) denotes the unique positive τ -periodic solution of (3.4)-(3.6) resulting from putting f = fi and q = qi. The two organisms can coexist if a
positive τ -periodic solution exists.
Recall that D(t) is the feasible domain for (2.28)-(2.30) and it is defined in (2.31). From Lemma 2.5, we may define the solution maps Φt : D(0) → D(t)
associated with (2.28)-(2.30) by
where P := (N0
1(·), NS,10 (·), N20(·), NS,20 (·)) ∈ D(0).
Let K = C([0, L], R2
+) × (−C([0, L], R2+)) and denote its induced order by ≤K.
Thus, the solution map Φt is monotone [15] with respect to the partial order ≤K.
Note that Φτ : D(0) → D(τ ) = D(0) and for the Poincar´e map S := Φτ, we have
Sn(P ) = Φ
nτ(P ), ∀ n ∈ Z.
For convenience, let Y+ = D(0), Y
0 := {(N1, NS,1, N2, NS,2) ∈ Y+: (N1, NS,1) 6=
(0, 0) and (N2, NS,2) 6= (0, 0)} and ∂Y0 := Y+\Y0.
Since two equations in (2.28)-(2.30) have no diffusion terms, its solution maps are not compact. So we require the following conditions in this section:
α A
AS
> fi(WS∗(x, t)), ∀x ∈ [0, L], t ≥ 0, i = 1, 2. (4.4)
For convenience, we let (ui, vi) := (Ni, NS,i), i = 1, 2, and define
mi(t, x, u1, u2, v1, v2) = α(vi− ui) + fi(W∗(x, t) − q1u1− q2u2)ui, i = 1, 2, and gi(t, x, u1, u2, v1, v2) = −α A AS (vi− ui) + fi(WS∗(x, t) − q1v1− q2v2)vi, i = 1, 2.
Then (2.28)-(2.30) can be rewritten as ∂ui ∂t = d ∂2u i ∂x2 − ν ∂ui ∂x + mi(t, x, u1, u2, v1, v2), ∂vi ∂t = gi(t, x, u1, u2, v1, v2), x ∈ (0, L), t > 0,
νui(0, t) − d∂u∂xi(0, t) = 0, ∂u∂xi(L, t) = 0
ui(x, 0) = φi, vi(x, 0) = ψi, i = 1, 2.
(4.5)
Let u := (u1, u2) and v := (v1, v2), and define
D = {(t, x, u, v) ∈ R6
+ : x ∈ [0, L], q1u1+ q2u2 ≤ W∗(x, t), q1v1+ q2v2 ≤ WS∗(x, t)}.
With the assumption (4.4), it is easy to see that whenever αAA
S is sufficiently large, there exists a constant r > 0 such that
xT ∂g(t, x, u, v) ∂v
x ≤ −rxTx, ∀ x ∈ R2, (t, x, u, v) ∈ D, (4.6) where g(t, x, u, v) := (g1(t, x, u1, u2, v1, v2), g2(t, x, u1, u2, v1, v2)).
Remark 4.1. As in [4, Remark 3.3], we choose fi(R) =
µmax,iR
Kµ,i+R and assume that
α A AS > µmax,i + 1 2 q2 q1 ·µmax,1 Kµ,1 +q1 q2 · µmax,2 Kµ,2 · W∗S, ∀ i = 1, 2, (4.7) where W∗S := maxx∈[0,L], t∈[0,τ ]WS∗(x, t). It then follows that (4.7) implies (4.4)
and (4.6). Biologically, two conditions in (4.7) mean that the cross-section of the storage zone is sufficiently small or the exchange rate is sufficiently large.
Lemma 4.1. Let (4.4) and (4.6) hold. Then the map Φτ is κ-contracting in the
sense that
lim
n→∞κ(Φ n
τ(B)) = 0
for any bounded set B ⊂ Y+, where κ is the Kuratowski measure of noncompactness
as defined in (3.12).
Proof. Let B be a given bounded subset in Y+. We first show that Φ
tis
asymptot-ically compact on B in the sense that for any sequences ϕn ∈ B and tn→ ∞, there
exist subsequences ϕnk and tnk → ∞ such that Φtnk(ϕnk) converges in C([0, L], R
4)
as k → ∞. Note that the family of functions {Φtn(ϕn)(x)}n≥1is uniformly bounded on [0, 1] for all n ≥ 1. In view of the Arzela-Ascoli theorem, it suffices to prove that {Φtn(ϕn)(x)}n≥1 is equicontinuous on x ∈ [0, L] for all n ≥ 1.
Let (un(x, t), vn(x, t)) = Φt(ϕn)(x), ∀ ϕn∈ Y+, t ≥ 0, x ∈ [0, L]. For
simplic-ity, we define un(x, t) := un(x, t + tn) and vn(x, t) := vn(x, t + tn), ∀ t ≥ −tn, x ∈
[0, L]. Clearly, (un(x, 0), vn(x, 0)) = Φtn(ϕn)(x), ∀ n ≥ 1, x ∈ [0, L]. Note that un(x, t) and vn(x, t) are uniformly bounded, ∀ n ≥ 1, x ∈ [0, L], t ≥ 0 (see also
Lemma 2.4).
For each i = 1, 2, we define fi(s) = 0 for all s ≤ 0 so that fi(s) is a continuous
function on R, and hence, g(t, x, u, v) is a continuous function on R+× [0, L] × R4.
By a direct computation, we see that for all t ≥ −tn, x, y ∈ [0, L], there holds
∂ ∂t h (vn(x, t) − vn(y, t)) T · (vn(x, t) − vn(y, t)) i = 2 (vn(x, t) − vn(y, t))T · ∂ ∂t(vn(x, t) − vn(y, t)) = 2 (vn(x, t) − vn(y, t)) T · [g(t + tn, x, un(x, t), vn(x, t)) − g(t + tn, y, un(y, t), vn(y, t))] . (4.8)
In the case where WS∗(y, t+tn) ≤ WS∗(x, t+tn), we have (t+tn, x, un(x, t), vn(y, t)) ∈
D, and hence,
[g(t + tn, x, un(x, t), vn(x, t)) − g(t + tn, y, un(y, t), vn(y, t))]
= [g(t + tn, x, un(x, t), vn(x, t)) − g(t + tn, x, un(x, t), vn(y, t))]
+ [g(t + tn, x, un(x, t), vn(y, t)) − g(t + tn, y, un(y, t), vn(y, t))] .
= Z 1 0 ∂g(t + tn, x, un(x, t), vn(y, t) + η(vn(x, t) − vn(y, t)) ∂v dη · [vn(x, t) − vn(y, t)]
+ [g(t + tn, x, un(x, t), vn(y, t)) − g(t + tn, y, un(y, t), vn(y, t))] . (4.9)
Set
hn(t, x, y) := kg(t + tn, x, un(x, t), vn(y, t)) − g(t + tn, y, un(y, t), vn(y, t))k.
It then follows from (4.6), (4.8) and (4.9) that there exists a real number M > 0 such that
∂
∂tkvn(x, t) − vn(y, t)k
2 ≤ −2rkv
n(x, t) − vn(y, t)k2 + M hn(t, x, y). (4.10)
In the case where WS∗(y, t+tn) ≥ WS∗(x, t+tn), we have (t+tn, y, un(y, t), vn(x, t)) ∈
D. By exchanging the positions of x and y in (4.9) and (4.10), we then obtain ∂
∂tkvn(x, t) − vn(y, t)k
2 ≤ −2rkv
n(x, t) − vn(y, t)k2 + M hn(t, y, x). (4.11)
Define Hn(t, x, y) := hn(t, x, y) + hn(t, y, x). It then follows from (4.10) and (4.11)
that ∂ ∂tkvn(x, t) − vn(y, t)k 2 ≤ −2rkv n(x, t) − vn(y, t)k2+ M Hn(t, x, y) (4.12) for all t ≥ −tn, x, y ∈ [0, L].
By the constant variation formula and the comparison argument, we obtain kvn(x, t)−vn(y, t)k2 ≤ e−2r(t−s)kvn(x, s)−vn(y, s)k2+M
Z t
s
e−2r(t−θ)Hn(θ, x, y)dθ,
(4.13) for all t ≥ s ≥ −tn. Letting t = 0 and s = −tn in (4.13), we further have
kvn(x, 0)−vn(y, 0)k2 ≤ e−2rtnkvn(x, −tn)−vn(y, −tn)k2+M
Z 0
−tn
and hence, kvn(x, tn) − vn(y, tn)k2 ≤ e−2rtnkvn(x, 0) − vn(y, 0)k2+ M Z 0 −tn e2rθHn(θ, x, y)dθ, (4.14) for all n ≥ 1, x, y ∈ [0, L].
Note that (un(x, 0), vn(x, 0)) = ϕn and ϕn ∈ B, for all n ≥ 1 and x ∈ [0, L],
and that {un(x, tn)}n≥1 is equicontinuous on [0, L] for all n ≥ 1. Thus, it suffices
to prove that {vn(x, tn)}n≥1 is equicontinuous on [0, L] for all n ≥ 1 in the sense
that for any > 0, there exists δ > 0 such that
kvn(x, tn) − vn(y, tn)k < , ∀ n ≥ 1, ∀ x, y ∈ [0, L] with | x − y |< δ.
Suppose, by contradiction, that there exist an 0 > 0, nk → ∞, xk, yk ∈ [0, L] with
| xk− yk |< 1k such that kvnk(xk, tnk) − vnk(yk, tnk)k ≥ 0, ∀ k ≥ 1. Letting x = xk, y = yk and n = nk in (4.14), we then obtain
20 ≤ lim sup k→∞ kvnk(xk, tnk) − vnk(yk, tnk)k 2 ≤ M · lim sup k→∞ Z 0 −tnk e2rθHnk(θ, xk, yk)dθ. (4.15) Note that for each θ ≤ 0, there exists a large integer n0 > 0 such that the sequence
of functions {un(x, θ) = un(x, θ + tn)}n≥n0 is equicontinuous on [0, L], and that g(t, x, u, v) is uniformly continuous in (t, x, u, v) ∈ [0, ∞) × [0, L] × H, where H is any given compact subset of R4+. Since limk→∞kunk(xk, θ) − unk(yk, θ)k = 0, it follows that for any given θ ≤ 0, we have limk→∞Hnk(θ, xk, yk) = 0. Using Fatou’s lemma in (4.15), we then obtain
20 ≤ M · Z 0 −∞ e2rθlim sup k→∞ Hnk(θ, xk, yk)dθ = 0, a contradiction. Consequently, Φt is asymptotically compact on B.
Now we consider the omega limit set of B for the Poincar´e map S := Φτ on
Y+, which is defined as ω(B) = {ϕ ∈ Y+ : lim
k→∞S nk(ϕ
k) = ϕ for some sequences ϕk ∈ B and nk→ ∞}.
From what we proved for Φt and the fact that Sn = Φnτ, ∀n ≥ 0, we easily see
that Sn is asymptotically compact on B in the sense that for any sequences ϕ k ∈ B
and nk → ∞, there exist subsequences, which we label as ϕk and nk → ∞, such
that Snk(ϕ
k) converges in C([0, L], R4) as k → ∞. It then follows that ω(B) is a
nonempty, compact, and invariant set for S in Y+, and ω(B) attracts B (see, e.g., the proof of [17, Lemma 23.1 (2)] for continuous-time semiflows). In view of [11, Lemma 2.1 (b)], we have
κ(Sn(B)) ≤ κ(ω(B)) + δ(Sn(B), ω(B)) = δ(Sn(B), ω(B)) → 0 as n → ∞. This completes the proof.
Recall that λ1(a(x, t), b(x, t)) is the principal eigenvalue of the eigenvalue
prob-lem (4.1). By the similar arguments as in Lemma 3.3, it follows that
η1∗ := λ1(f2(W∗(x, t) − q1N1∗(x, t)), f2(WS∗(x, t) − q1NS,1∗ (x, t))) exists, (4.16)
with an associated eigenfunction
(ϕ∗1(x, t), ψ∗1(x, t)) 0. (4.17)
Similarly,
η2∗ := λ1(f1(W∗(x, t) − q2N2∗(x, t)), f1(WS∗(x, t) − q2NS,2∗ (x, t))) exists, (4.18)
with an associated eigenfunction
(ϕ∗2(x, t), ψ∗2(x, t)) 0. (4.19)
For P1, P2 ∈ Y+ with P1 K P2, we define type−K order intervals
[P1, P2]K = {P ∈ Y+ : P1 ≤K P ≤K P2},
and
[[P1, P2]]K = {P ∈ Y+ : P1 K P K P2}.
Let ω(P ) be the omega limit set of the map S := Φτ.
Theorem 4.1. Let (4.4) and (4.6) hold. Then Φτ admits a global attractor on Y+.
Proof. By Lemma 4.1, it follows that Φτ is κ-contracting on Y+. By Lemma 2.4,
it follows that Φτ is point dissipative on Y+ and positive orbits of bounded subsets
of Y+ for S = Φ
τ are bounded. By [11, Theorem 2.6], Φτ has a global attractor
Let E1 = (N1∗(x, 0), N ∗ S,1(x, 0), 0, 0), E2 = (0, 0, N2∗(x, 0), N ∗ S,2(x, 0)) and ˆ0 = (0, 0, 0, 0).
Theorem 4.2. Let (4.4) and (4.6) hold, and assume that µ∗i < 0 and ηi∗ < 0, i = 1, 2. Then system (2.28)-(2.30) admits two positive τ -periodic solutions
E−(x, t) := (N1(x, t), NS,1(x, t), N2(x, t), NS,2(x, t)),
E+(x, t) := (N1(x, t), NS,1(x, t), N2(x, t), NS,2(x, t)), (4.20)
such that E−(·, t) ≤K E+(·, t) for all t ≥ 0, and for any P ∈ Y+0, the solution map
Φt(P ) of (2.28)-(2.30) satisfies
lim
t→∞d(Φt(P ), [E
−(·, t), E+(·, t)]
K) = 0.
Furthermore, for any compact internal chain transitive set I of the Poincar´e map S with I ∈/ {ˆ0}, {E1}, {E2} , we have I ⊂ [E−(·, 0), E+(·, 0)]K.
Proof. It is easy to see that the stability of the semi-trivial solution E1(x, t) :=
(N1∗(x, t), NS,1∗ (x, t), 0, 0) for (2.28)-(2.30) is determined by the sign of η1∗. More pre-cisely, E1(x, t) is unstable if η1∗ < 0. Similarly, E2(x, t) := (0, 0, N2∗(x, t), NS,2∗ (x, t))
is unstable if η2∗ < 0. By the theory of abstract competitive systems (see, e.g., [6, Theorem A and Corollary 1]), it follows that (2.28)-(2.30) admits two positive τ -periodic solution E−(x, t) and E+(x, t) with E−(x, t) ≤
K E+(x, t) for x ∈ [0, L],
t ≥ 0 such that the dynamics of (2.28)-(2.30) stated in the Theorem 4.2 hold. Recall that S = Φτ : Y+ → Y+ is the Poincar´e map associated with (2.28)-(2.30). Thus, limn→∞d(Sn(P ), [E−(·, 0), E+(·, 0)]K) = 0, ∀P ∈ Y0. Hence,
S : Y+ →
Y+ is uniformly persistence with respect to Y0. By [11,
The-orem 3.7], it follows that S : Y0 → Y0 has a global attractor A0. Clearly,
A0 ⊂ [E−(·, 0), E+(·, 0)]K. By [20, Theorem 1.3.1 and Remark 1.3.1], it then
fol-lows that there exists δ > 0 such that for any compact internal chain transitive set I of S with I ∈/ {ˆ0}, {E1}, {E2} , we have infP ∈Id(P, ∂Y0) > δ. This implies
5
Dynamics of the full system
In this section, we discuss the global dynamics of the full system (1.5)-(1.7). Rewrite (1.5)-(1.7) as ∂N1 ∂t = δ ∂2N1 ∂x2 − ν ∂N1 ∂x + α(NS,1− N1) + f1(W − q1N1− q2N2)N1, ∂NS,1 ∂t = −α A AS (NS,1− N1) + f1(WS− q1NS,1− q2NS,2)NS,1, ∂N2 ∂t = δ ∂2N2 ∂x2 − ν ∂N2 ∂x + α(NS,2− N2) + f2(W − q1N1− q2N2)N2, ∂NS,2 ∂t = −α A AS (NS,2− N2) + f2(WS− q1NS,1− q2NS,2)NS,2, (5.1) ∂W ∂t = δ ∂2W ∂x2 − ν ∂W ∂x + αWS− αW, ∂WS ∂t = −α A AS WS+ α A AS W, in (0, L) × (0, ∞), with boundary conditions
νNi(0, t) − δ ∂Ni ∂x (0, t) = 0, ∂Ni ∂x (L, t) = 0, i = 1, 2, νW (0, t) − δ∂W ∂x (0, t) = νR (0)(t), ∂W ∂x (L, t) = 0, (5.2)
and nonnegative initial functions, where W (x, t) and WS(x, t) are defined as in
(2.5). Let Σ = {(N1, NS,1, N2, NS,2, W, WS) ∈ C([0, L], R+6) : q1N10(·) + q2N20(·) ≤ W (·), q1NS,10 (·) + q2NS,20 (·) ≤ WS(·) on [0, L]}, and Σ0 = {(N1, NS,1, N2, NS,2, W, WS) ∈ Σ : (N1, NS,1) 6= (0, 0) and (N2, NS,2) 6= (0, 0)}.
Lemma 5.1. If ˜P ∈ Σ, then the solution of (5.1)-(5.2) through ˜P satisfies
Proof. Let R(x, t) = W (x, t) − (q1N1(x, t) + q2N2(x, t)) and RS(x, t) = WS(x, t) −
(q1NS,1(x, t)+q2NS,2(x, t)). Then (R(x, t), N1(x, t), NS,1(x, t), RS(x, t), N2(x, t), NS,2(x, t))
satisfies (1.5)-(1.7). By Lemma 2.1, it follows that
(R(x, t, ˜P ), N1(x, t, ˜P ), NS,1(x, t, ˜P ), RS(x, t, ˜P ), N2(x, t, ˜P ), NS,2(x, t, ˜P )) ≥ 0, ∀ t ≥ 0.
This completes our proof.
Let ˜Φt : Σ → Σ be the solution maps associated with (5.1)-(5.2). We denote
the Poincar´e map
˜
S : Σ → Σ by ˜S = ˜Φτ. (5.3)
Then ˜Sn( ˜P ) = ˜Φ
nτ( ˜P ), ∀n ∈ Z+. For any ˜P ∈ Σ, let ˜ω( ˜P ) be the omega limit set
of ˜P for ˜S. Then we have the following observation.
Lemma 5.2. Let (4.4) and (4.6) hold. Then for any ˜P ∈ Σ, ˜ω( ˜P ) is a nonempty, compact and invariant set for ˜S.
Proof. It suffices to prove that the forward orbit ˜Sn( ˜P ) is asymptotically compact
in the sense that for any sequence nk → ∞, there exists a subsequence, which we
label as nk → ∞, such that ˜Snk( ˜P ) converges in C([0, L], R6) as k → ∞. Let
(u, v) = (u1, u2, v1, v2) := (N1, N2, NS,1, NS,2) and (u(x, t), v(x, t), W (x, t), WS(x, t)) := ˜Φt( ˜P )(x), ∀ t ≥ 0, x ∈ [0, L]. By Lemma 2.3, we have lim t→∞((W (x, t), WS(x, t)) − (W ∗(x, t), W∗ S(x, t))) = (0, 0) (5.4)
uniformly for x ∈ [0, L]. Note that for any given sequence tn → ∞, the sequence
of functions {(u(x, tn), v(x, tn), W (x, tn), WS(x, tn))}n≥1 is uniformly bounded on
[0, L] for all n ≥ 1. Further, {u(x, tn)}n≥1 and {(W (x, tn), WS(x, tn))}n≥1 are
equicontinuous on [0, L] for all n ≥ 1. It is easy to see that (u(x, t), v(x, t)) satisfies (4.5) with mi(t, x, u1, u2, v1, v2) and gi(t, x, u1, u2, v1, v2) replaced, respectively, by
ˆ mi(t, x, u1, u2, v1, v2) = α(vi− ui) + fi(W (x, t) − q1u1− q2u2)ui, i = 1, 2, ˆ gi(t, x, u1, u2, v1, v2) = −α A AS (vi − ui) + fi(WS(x, t) − q1v1− q2v2)vi, i = 1, 2.
Let ˆg(t, x, u, v) = (ˆg1(t, x, u1, u2, v1, v2), ˆg2(t, x, u1, u2, v1, v2)). Using the same
con-tinuous extensions of f1(s) and f2(s) onto R as in the proof of Lemma 4.1, we then
see from (5.4) that
lim
t→∞kˆg(t, x, u, v) − g(t, x, u, v)k = 0, (5.5)
uniformly for (x, u, v) ∈ [0, L] × H, where H is any given compact subset of R4 +
and g(t, x, u, v) is defined as in Section 4. By (4.4), (4.6) and (5.4), it follows that there exist two positive numbers a and t0 such that
xT ∂ ˆg(t, x, u, v) ∂v x ≤ −axTx, ∀ x ∈ R2, (t, x, u, v) ∈ W, (5.6) where W = {(t, x, u, v) ∈ R6 +: t ≥ t0, x ∈ [0, L], q1u1+q2u2 ≤ W (x, t), q1v1+q2v2 ≤ WS(x, t)}.
By using (5.4), (5.5) and (5.6), and a slight modification of the proof in Lemma 4.1, we can show that the sequence {v(x, tn)}n≥1is equicontinuous on [0, L] for all n ≥ 1.
Thus, ˜Φt( ˜P ) is asymptotically compact, and hence, so is ˜Sn( ˜P ).
Now we are ready to prove the main result of this section.
Theorem 5.1. Let (4.4) and (4.6) hold, and assume that µ∗i < 0 and ηi∗ < 0, i = 1, 2. Then system (5.1)-(5.2) admits at least one positive τ -periodic solutions. More-over, if
˜
Φt( ˜P ) := (N1(·, t, ˜P ), NS,1(·, t, ˜P ), N2(·, t, ˜P ), NS,2(·, t, ˜P ), W (·, t, ˜P ), WS(·, t, ˜P ))
is the solution of (5.1)-(5.2) through ˜P ∈ Σ0, then
lim t→∞d (N1(·, t, ˜P ), NS,1(·, t, ˜P ), N2(·, t, ˜P ), NS,2(·, t, ˜P )), [E−(·, t), E+(·, t)]K = 0, and lim t→∞((W (·, t), WS(·, t)) − (W ∗ (·, t), WS∗(·, t))) = (0, 0).
Proof. Let ˜P ∈ Σ0 be given, and let ˜ω := ˜ω( ˜P ). It is easy to see from Theorem
4.2 that (E+(x, t), W∗(x, t), WS∗(x, t)) and (E−(x, t), W∗(x, t), WS∗(x, t)) are positive τ -periodic solutions of system (5.1)-(5.2). By Lemma 2.3, we further have
lim
t→∞((W (x, t), WS(x, t)) − (W ∗
uniformly for x ∈ [0, L]. It then follows that for any P0 ∈ C([0, L], R4
+) with
(P0, W0, W0
S) ∈ ˜ω, there holds (W0, WS0) = (W
∗(·, 0), W∗
S(·, 0)). Thus, there exists
a set I ⊂ C([0, L], R4+) such that ˜ω = I × {(W∗(·, 0), WS∗(·, 0))}.
Since Σ is closed, it follows from Lemma 5.1 that ˜ω ⊂ Σ. For any given (N1, NS,1, N2, NS,2) ∈ I, we have (N1, NS,1, N2, NS,2, W∗(·, 0), WS∗(·, 0)) ∈ ˜ω ⊂ Σ.
By the definition of Σ, it follows that (N1, NS,1, N2, NS,2) ∈ D(0) := Y+. Thus,
I ⊂ Y+.
By Lemma 5.2 and [20, Lemma 1.2.10], ˜ω is a compact, invariant and in-ternal chain transitive set for ˜S = ˜Φτ. Moreover, if P0 ∈ C([0, L], R4+) with
(P0, W0, W0 S) ∈ ˜ω, there holds ˜ S |ω˜ (P0, W0, WS0) = (S(P 0), W∗(·, 0), W∗ S(·, 0)), where S(P0) = Φ
τ(P0) is Poincar´e map associated with (2.28)-(2.30) on Y+. It
then follows that I is a compact, invariant and internal chain transitive set for S = Φτ : Y+ → Y+.
By Theorem 4.2, if we can prove I ∈/ {ˆ0}, {E1}, {E2} , then we must have
I ⊂ [E−(·, 0), E+(·, 0)]
K, and hence,
˜
ω ⊂ [E−(x, 0), E+(x, 0)]K× {(W∗(x, 0), WS∗(x, 0))},
which implies that solutions of (5.1)-(5.2) have the asymptotic behavior as stated in the theorem. So it remains to prove that I ∈/ {ˆ0}, {E1}, {E2} . We only prove
the claim that I 6= {E1} since other two claims can be proved in a similar way.
Suppose, by contradiction, that I = {E1}, then
˜
ω( ˜P ) = (N1∗(·, 0), NS,1∗ (·, 0), 0, 0, W∗(·, 0), WS∗(·, 0)) := ˜E. Thus, we have limn→∞( ˜S)n( ˜P ) = ˜E, or equivalently,
lim t→∞[ ˜Φt( ˜P ) − (N ∗ 1(·, t), N ∗ S,1(·, t), 0, 0, W ∗(·, t), W∗ S(·, t)] = 0. Since η1∗ := λ1 f2(W∗(x, t) − q1N1∗(x, t)), f2(WS∗(x, t) − q1NS,1∗ (x, t)) < 0, it fol-lows that there exists a sufficiently small > 0 such that ([5, Lemma15.7])
η1 := λ1 f2(W∗(x, t) − q1N1∗(x, t)) − , f2(WS∗(x, t) − q1NS,1∗ (x, t)) − < 0.
There also exists n0 = n0() > 0 such that
f2(W (x, t) − q1N1(x, t) − q2N2(x, t)) > f2(W∗(x, t) − q1N1∗(x, t)) − ,
for all x ∈ [0, L] and t ≥ n0τ . Therefore, we obtain ∂N2 ∂t = δ ∂2N2 ∂x2 − ν ∂N2 ∂x + α(NS,2− N2) + f2(W (x, t) − q1N1− q2N2)N2, > δ∂ 2N 2 ∂x2 − ν ∂N2 ∂x + α(NS,2− N2) + [f2(W ∗ (x, t) − q1N1∗(x, t)) − ] N2, ∂NS,2 ∂t = −α A AS (NS,2− N2) + f2(WS(x, t) − q1NS,1− q2NS,2)NS,2, (5.7) > −α A AS (NS,2− N2) + [f2(WS∗(x, t) − q1NS,1∗ ) − ]NS,2,
for all x ∈ [0, L] and t ≥ n0τ . Let (ϕ(x, t), ψ(x, t)) 0 be the eigenfunction
corresponding to η1, that is, ∂ϕ ∂t = δ ∂2ϕ ∂x2 − ν ∂ϕ ∂x + α(ψ− ϕ) + [f2(W ∗(x, t) − q 1N1∗(x, t)) − ] ϕ+ η1ϕ, ∂ψ ∂t = −α A As(ψ− ϕ) +f2(W ∗ S(x, t) − q1NS,1∗ (x, t)) − ψ+ η1ψ, t > 0, x ∈ (0, L), νϕ(0, t) − δ∂ϕ∂x(0, t) = ∂ϕ∂x(L, t) = 0, t > 0, ϕ, ψ are τ -periodic in t. (5.8) Since ˜P ∈ Σ0, it follows that (N2(x, 0), NS,2(x, 0)) 6= (0, 0), and hence,
(N2(x, t), NS,2(x, t)) 0, ∀t > 0.
In particular, N2(·, n0τ ) 0 and NS,2(·, n0τ ) 0 in Σ. Thus, there exists δ :=
δ(, E1) > 0 such that N2(·, n0τ ) ≥ δϕ(·, n0τ ) = δϕ(·, 0) and NS,2(·, n0τ ) ≥
δψ(·, n0τ ) = δψ(·, 0). By the comparison theorem, it follows that
N2(x, t) ≥ δe−η 1tϕ (x, t) and NS,2(x, t) ≥ δe−η 1tψ (x, t), ∀t ≥ n0τ. In particular, N2(x, nτ ) ≥ δe−η 1nτϕ (x, nτ ) and NS,2(x, nτ ) ≥ δe−η 1nτψ (x, nτ ), ∀n ≥ n0, that is, N2(x, nτ ) ≥ δe−η 1nτϕ (x, 0) and NS,2(x, nτ ) ≥ δe−η 1nτψ (x, 0), ∀n ≥ n0,
which contradicts that limn→∞(N2(x, nτ ), NS,2(x, nτ )) = (0, 0).
Acknowledgment. Xiao-Qiang Zhao would like to thank the National Center for Theoretical Science, National Tsing-Hua University, Taiwan for its financial support and kind hospitality during his visit there.
References
[1] Mary Ballyk, LE Dung, D. A. Jones and Hal L. Smith, Effects of random motility on microbial growth and competition in a flow reactor, SIAM J. Appl. Math., 59 (1998), pp. 573–596.
[2] J. V. Baxley and S. B. Robinson, Coexistence in the unstirred chemostat, Appl. Math. Computation, 89 (1998), pp. 41–65.
[3] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1988. [4] James P. Grover, Sze-Bi Hsu and Feng-Bin Wang, Competition and
coexistence in flowing habitats with a hydraulic storage zone, Mathematical Biosciences, 222 (2009), pp. 42–52.
[5] P. Hess, Periodic-parabolic Boundary Value Problem and Positivity, Pitman Res. Notes Math., 247, Longman Scientific and Technical, 1991.
[6] S. B. Hsu, H.L. Smith and P. Waltman, Competitive exclusion and co-existence for competitive system on ordered Banach space, Tran. Amer. Math. Soc., 348 (1996), pp. 4083–4094.
[7] S. B. Hsu, and P. Waltman, On a system of reaction-diffusion equations arising from competition in an unsirred chemostat, SIAM J. Appl. Math., 53 (1993), pp. 1026–1044.
[8] J. Jiang, X. Liang, and X.-Q. Zhao, Saddle point behavior for monotone semiflows and reaction-diffusion models, J. Diff. Eqns. (2004), pp. 313–330. [9] C. M. Kung and B. Baltzis, The growth of pure and simple microbial
competitors in a moving distributed medium, Math. Biosci. 111(1992), pp. 295-313.
[10] R. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. of A.M.S., 321 (1990), pp. 1–44.
[11] P. Magal, and X. -Q. Zhao, Global attractors and steady states for uni-formly persistent dynamical systems, SIAM. J. Math. Anal 37 (2005), pp. 251-275.
[12] R. D. Nussbaum, Eigenvectors of nonlinear positive operator and the linear Krein-Rutman theorem, in: E. Fadell, G. Fournier (Eds.), Fixed Point Theory, Lecture Notes in Mathematics, Vol. 886, Springer, New York/Berlin, (1981), pp. 309-331.
[13] A. Pazy, Semigroups of linear operators and application to partial differential equations, Springer-Verlag, 1983.
[14] M. H. Protter and H. F. Weinberger, Maximum Principles in Differ-ential Equations, Springer-Verlag, 1984.
[15] H. L. Smith, Monotone Dynamical Systems:An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr 41, American Mathematical Society Providence, RI, 1995.
[16] H. L. Smith and P. E. Waltman, The Theory of the Chemostat, Cambridge Univ. Press, 1995.
[17] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002.
[18] H. L. Smith and X.-Q. Zhao, Dynamics of a periodically pulsed bio-reactor model, J. Diff. Eq. 155 (1999), pp. 368-404.
[19] F. Zhang and X.-Q. Zhao, Asymptotic behaviour of a reaction-diffusion model with a quiescent stage, Proc. R. Soc. A., 463 (2007), pp. 1029–1043. [20] X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, New York,
