Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2009, Article ID 317298, 20 pages
doi:10.1155/2009/317298
Research Article
Global Asymptotic Stability of
3-Species Mutualism Models with
Diffusion and Delay Effects
Chang-you Wang,1, 2, 3 Shu Wang,3 and Xiang-ping Yan4
1
College of Mathematics and Physics, Chongqing University of Posts and Telecommunications,
Chongqing 400065, China
2
Key Laboratory of Network control & Intelligent Instrument, (Chongqing University of Posts and
Telecommunications), Ministry of Education, Chongqing 400065, China
3
College of Applied Sciences, Beijing University of Technology, Beijing 100124, China
4
Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China
Correspondence should be addressed to Chang-you Wang, wangcy@cqupt.edu.cn
Received 21 March 2009; Accepted 6 June 2009
Recommended by Leonid Berezansky
In this paper, the Lotka-Volterra 3-species mutualism models with diffusion and delay effects is
investigated. A simple and easily verifiable condition is given to ensure the global asymptotic
stability of the unique positive steady-state solution of the corresponding steady-state problem in
a bounded domain with Neumann boundary condition. Our approach to the problem is based on
inequality skill and the method of the upper and lower solutions for a more general reaction—
diffusion system. Finally, some numerical simulations are given to illustrate our results.
Copyright q 2009 Chang-you Wang et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
Modeling and analysis of the dynamics of biological populations by means of differential
equations is one of the primary concerns in population growth problems. A well-known and
extensively studied class of models in population dynamics is the Lotka-Volterra system
which models the interaction among various species. In the earlier literature most of the
discussions are devoted to coupled systems of two equations cf. 1–11. In the recent years,
attention has been given to reaction-diffusion systems with three population species, and the
main concerns in these works are the prey-predator and competition models with or without
time delays cf. 12–24. The purpose of this paper is to investigate the asymptotic behavior
of the time-dependent solution for a 3-species mutualism model, where the effect of diffusion
and time delays is both taken into consideration by obtaining existence of positive solution
for the corresponding inequalities. The system of equations under consideration is the LotkaVolterra 3-species mutualism model, where the population densities do not move across the
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boundary of a habitat, and time delays may appear in the opposing species; the densities of
three populations u, v,and w are governed by the following coupled equations:
∂u
− d1 L1 u u a1 − c1 u e1 v β1 J2 ∗ v ,
∂t
∂v
− d2 L2 v v a2 b2 u − c2 v e2 w α2 J1 ∗ u β2 J3 ∗ w ,
∂t
∂w
− d3 L3 w w a3 e3 v − c3 w β3 J2 ∗ v
∂t
1.1
x ∈ Ω, t > 0,
with the boundary and initial conditions
∂ux, t ∂vx, t ∂wx, t
0
∂n
∂n
∂n
ux, t η1 x, t t ∈ I1 ,
wx, t η3 x, t
x ∈ ∂Ω, t > 0,
vx, t η2 x, t
t ∈ I2 ,
1.2
t ∈ I3 , x ∈ Ω.
Here Ω is a bounded domain in Rn with boundary ∂Ω, ∂/∂n denotes the outward normal
derivative on ∂Ω, and ηi x, t is a smooth function. The above problem 1.1-1.2 arises in
a simple food chain describing three interacting species in a spatial habitat Ω. For each i 1, 2, 3, ai , ci are positive constants. bi , ei , αi , and βi are nonnegative constants with ei∗ ≡ ei βi >
0, b2∗ ≡ b2 α2 > 0, and its respective diffusion rate is denoted by di . The real number ai
is the net birth rate of the ith species, and ci is its respective intraspecific competition. The
parameters bi , ei , αi and βi are interspecific cooperation. Li is a uniformly elliptic operator in
the form
Li u −
N
i
aj,k x, t
j,k1
N
∂2 u
∂u
i
bj x, t
∂xj ∂xk j1
∂xj
i 1, 2, 3
1.3
cf. 1. The functions Ji ∗ ui , i 1, 2, 3, with u1 , u2 , u3 u, v, w are given either by the
discrete time delay
Ji ∗ ui ui t − τi , x,
1.4
or by the continuous time delay
J i ∗ ui t
−τi
Ji t − sui s, xds,
1.5
and the interval Ii is given by Ii −τi , 0, where τi > 0 is a constant representing the time
delay. It is allowed that the type of time delays and the values of τi may be different for
different ui . This consideration includes various combination of discrete and continuous time
delays for the species u, v, and w.
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3
Throughout the paper we assumed that the function Ji t is piecewise continuous in
R ≡ 0, ∞ and possesses the property
Ji t ≥ 0 for t ≥ 0, Ji t ≡ 0 for t > τi ,
τi
Ji tdt 1 i 1, 2, 3.
1.6
0
The above property implies that for any constant function ui t, x ρi , we have
t
−τi
Ji t − sui s, xds ρi
t ≥ 0.
1.7
The same is obviously true if Ji ∗ ui is given by 1.4. It is also assumed that the domain
i
i
Ω is smooth and the coefficients aj,k , bj of Li are smooth functions in Ω cf. 12. In the
special case of the diffusion operator Li ui Di xD2 ui , it suffices to assume that Di x is
strictly positive on Ω and in C1α Ω for some α ∈ 0, 1. We allow Li 0 and without the
corresponding boundary condition for some or all i. In particular, if Li ui ≡ 0 for all i then the
equations in 1.1-1.2 are reduced to the ordinary differential system with time delays
du
u a1 − c1 u e1 v β1 J2 ∗ v
t > 0,
dt
dv
v a2 b2 u − c2 v e2 w α2 J1 ∗ u β2 J3 ∗ w
t > 0,
dt
dw
w a3 e3 v − c3 w β3 J2 ∗ v
dt
ut η1 t t ∈ I1 ,
t > 0,
vt η2 t
wx, t η3 t
1.8
t ∈ I2 ,
t ∈ I3 .
Problem 1.8 and various similar problems have been investigated by many investigators in
the framework of ordinary differential systems cf. 25–28 and references therein. It is to be
noted that if Δui / 0 and αi βi 0 for every i then problem 1.1 is reduced to the following
3-species mutualism models without time delays:
u1t − d1 Δu1 u1 a1 − c1 u1 e1 u2 ,
in ΩT ,
u2t − d2 Δu2 u2 a2 b2 u1 − c2 u2 e2 u3 ,
u3t − d3 Δu3 u3 a3 b3 u2 − c3 u3 ,
u1 x, t u2 x, t u3 x, t 0,
ui x, 0 ui,0 x,
for i 1, 2, 3,
in ΩT ,
in ΩT ,
on ∂ΩT ,
in Ω,
1.9
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Discrete Dynamics in Nature and Society
where the domains ΩT and ∂ΩT are defined as, respectively, Ω × 0, T and ∂Ω × 0, T . Here
Ω is a bounded domain in RN with smooth boundary ∂Ω. T is the maximal existence time
of the solution. ui,0 is a smooth function satisfying the compatibility condition ui,0 x 0 for
x ∈ ∂Ω. The meaning of the parameters of systems 1.9 is as same as those of 1.1-1.2.
Kim and Lin 22 proved that global solutions of 1.9 exist if the intraspecific competitions
are strong; whereas blow-up solutions exist under certain conditions of that the intraspecific
competitions are weak, and 23 obtained the upper bound of blow-up rate for any N and
the lower bound of blow-up rate for N 1. On the other hand, if e2 β2 0 then it reduces
to the 2-species cooperating models, which have been extensively investigated in the current
literature cf. 2, 10, 11 and references therein. Our conclusion about the global asymptotic
stability of a positive steady-state solution to system 1.1-1.2 is directly applicable to the
above special cases see Corollary 3.2.
This work is motivated from the following two-prey one-predator model:
∂u
− L1 u a1 xu 1 − u − b1 v − c1 w − β1 J2 ∗ v − γ1 J3 ∗ w ,
∂t
∂v
− L2 v a2 xv 1 − v − b2 u − c2 w − β2 J1 ∗ u − γ2 J3 ∗ w ,
∂t
∂w
− L3 w a3 xw 1 − w b3 u c3 v β3 J1 ∗ u γ3 J2 ∗ v
∂t
∂u ∂vx, t ∂wx, t
0
∂n
∂n
∂n
ux, t η1 x, t
1.10
x ∈ ∂Ω, t > 0,
t ∈ I1 , vx, t η2 x, t
wx, t η3 x, t
x ∈ Ω, t > 0,
t ∈ I2 ,
t ∈ I3 , x ∈ Ω,
and one-prey two-predator model
∂u
− L1 u a1 xu 1 − u − b1 v − c1 w − β1 J2 ∗ v − γ1 J3 ∗ w ,
∂t
∂v
− L2 v a2 xv 1 − v b2 u − c2 w β2 J1 ∗ u − γ2 J3 ∗ w ,
∂t
∂w
− L3 w a3 xw 1 − w b3 u − c3 v β3 J1 ∗ u − γ3 J2 ∗ v
∂t
∂u ∂vx, t ∂wx, t
0
∂n
∂n
∂n
ux, t η1 x, t
t ∈ I1 ,
wx, t η3 x, t
x ∈ Ω, t > 0,
x ∈ ∂Ω, t > 0,
vx, t η2 x, t
t ∈ I3 , x ∈ Ω
t ∈ I2 ,
1.11
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5
as well as three-species food-chain model
∂u
− L1 u a1 xu 1 − u − b1 v − c1 w − β1 J2 ∗ v − γ1 J3 ∗ w ,
∂t
∂v
− L2 v a2 xv 1 − v b2 u − c2 w β2 J1 ∗ u − γ2 J3 ∗ w ,
∂t
∂w
− L3 w a3 xw 1 − w b3 u c3 v β3 J1 ∗ u γ3 J2 ∗ v
∂t
x ∈ Ω, t > 0,
1.12
∂u ∂vx, t ∂wx, t
0 x ∈ ∂Ω, t > 0,
∂n
∂n
∂n
ux, t η1 x, t
t ∈ I1 ,
wx, t η3 x, t
vx, t η2 x, t
t ∈ I2 ,
t ∈ I3 , x ∈ Ω.
Pao 29 obtained some simple and easily verifiable conditions for the existence and global
asymptotic stability of a positive steady-steady solution for each of the above three-model
problems.
Based on the above results, we are mainly interested in studying the asymptotic
behavior of the solution of 1.1-1.2. The corresponding steady-state problem is given by
−d1 L1 u u a1 − c1 u e1 v β1 v
x ∈ Ω,
−d2 L2 v v a2 b2 u − c2 v e2 w α2 u β2 w
−d3 L3 w w a3 e3 v − c3 w β3 v
x ∈ Ω,
x ∈ Ω,
1.13
∂u ∂v ∂w
0 x ∈ ∂Ω.
∂n ∂n
∂n
It is clear that the above system has always the trivial solution 0, 0, 0 and various forms
of constant semitrivial solutions, that is, constant nonnegative solutions with at least one
component zero and one component positive. Our aim is to prove the existence and global
asymptotic stability of a positive constant steady-state solution in the mutualism model with
respect to nonnegative initial perturbations. This global asymptotic stability result implies
that system 1.1-1.2 is permanent, the trivial and all forms of semitrivial solutions are
unstable, and the nonuniform steady-state solution does not exist.
This paper is arranged as follows. In Section 2, we give some preliminary results for
a more general time-delayed reaction-diffusion system. The main results and their proofs
are given in Section 3. Some numerical simulations are shown in Section 4 to illustrate our
theoretical analysis.
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4
3.5
u-axes
3
2.5
2
1.5
1
0.5
0
8
x- 6 4
ax
es 2
0 −1
0
1
2
3 4 5
t-axes
6
7
8
2
3
5
4
t-axes
6
7
8
2
3
5
4
t-axes
6
7
8
a
3
2.5
v-axes
2
1.5
1
0.5
0
10
xax 5
es
0 −1
0
1
b
4
3.5
w-axes
3
2.5
2
1.5
1
0.5
0
10
xax 5
es
0 −1
0
1
c
Figure 1: The solution of system 4.1.
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2. Some Preliminary Results
To prove the main results in this paper we first give some preliminary results for a more
general time-delayed parabolic system in the form
∂ui
− Li ui fi x, U, J ∗ U
∂t
∂ui
0
∂n
x ∈ Ω, t > 0,
2.1
x ∈ ∂Ω, t > 0,
ui x, t ηi x, t
x ∈ Ω, t ∈ Ii , i 1, 2, . . . , N,
where U u1 , u2 , . . . , uN , J ∗ U J1 ∗ u1 , J2 ∗ u2 , . . . , JN ∗ uN , and for each i 1, 2, . . . , N, fi ·, U, J ∗ U is, in general, a nonlinear function of U and J ∗ U, and Li is a
uniformly elliptic operator in the form of 1.3. The components Ji ∗ ui of J ∗ U are given
by 1.4 for i 1, 2, . . . , n0 and by 1.5 for i n0 1, . . . , N, where n0 is a nonnegative integer.
The case n0 0 corresponds to 1.5 for every i 1, 2, . . . , N.
By writing U and V in the split forms U ≡ ui , uai , ubi and V ≡ vci , vdi ,
respectively, where ai , bi , ci , and di are nonnegative integers satisfying
ai bi N − 1,
ci di N
i 1, 2, . . . , N,
2.2
and wδi denotes a vector with δi components of w, we write fi x, U, V in the form
fi x, U, V fi x, ui , uai , ubi , vci , vdi
for every i 1, 2, . . . , N.
2.3
Definition 2.1. We say that the vector function F·, U, V ≡ f1 ·, U, V , f2 ·, U, V ,
. . . , fN ·, U, V possesses a mixed quasimonotone property in a subset S of RN , if
for each i there exist nonnegative integers ai , bi , ci , and di satisfying 2.2 such that
fi ·, ui , uai , ubi , vci , vdi is nondecreasing in uai , vci and is nonincreasing in
ubi , vdi for every U, V in S cf. 1, 12. In particular, if bi di 0 for all i then F·, U, V is said to be quasimonotone nondecreasing in S. It is easily seen that the reaction functions in
the models 1.1 are quasimonotone nondecreasing in RN
.
Definition 2.2. For quasimonotone nondecreasing functions F·, U, V we call a pair of smooth
≡ ≡ 2 , . . . , u
N , U
u1 , u
2 , . . . , u
N coupled upper and lower solutions of
functions U
u1 , u
2.1 if U > U and if
∂
ui
J ∗U
− Li u
i ≥ fi x, U,
∂t
∂
ui
J ∗U
− Li u
i ≤ fi x, U,
∂t
∂
ui
∂
ui
≥0≥
∂n
∂n
i x, t
u
i x, t ≥ ηi x, t ≥ u
x ∈ Ω, t > 0,
x ∈ Ω, t > 0,
x ∈ ∂Ω, t > 0,
x ∈ Ω, t ∈ Ii , i 1, 2, . . . , N,
2.4
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Discrete Dynamics in Nature and Society
≡
where inequality between vectors is in the componentwise sense. It is obvious that if U
≡ δ ≡ δ1 , δ2 , . . . , δN are constant vectors, then the above
M ≡ M1 , M2 , . . . , MN and U
inequalities become
fi x, M, M ≤ 0
fi x, δ, δ ≥ 0
Mi ≥ ηi x, t ≥ δi
x ∈ Ω,
2.5
x ∈ Ω,
x ∈ Ω, t ∈ Ii ,
i 1, 2, . . . , N.
Notice that the boundary inequalities in 2.4 are trivially satisfied. For a given pair of
constant vectors M, δ satisfying 2.5 we set
S ≡ {U ∈ CΩ : δ ≤ U ≤ M},
2.6
where CΩ denotes the set of continuous functions on Ω. In the following we give our basic
hypotheses on F·, U, V with V ≡ J ∗ U.
H The function F·, U, V is quasimonotone nondecreasing for U, V ∈ S, and for each
i,fi ·, U, V satisfies the Lipschitz condition
fi x, U, V − fi x, U , V ≤ Ki U − U V − V U, U , V, V ∈ S ,
2.7
where Ki is a positive constant, and w |w1 | |w2 | · · · |wN | for w ∈ RN | · | is a maximum
norm.
It is clear that if there exists a positive constant vector M M1 , M2 , . . . , MN and a
zero vector o 0, 0, . . . , 0 such that
fi x, M, M ≤ 0 x ∈ Ω,
fi x, o, o ≥ 0
x ∈ Ω, i 1, 2, . . . , N,
2.8
M, U
o are coupled upper and lower solutions of 2.1 whenever Mi ≥
then the pairs U
ηi x, t ≥ 0. By an application of 12, Theorem 2.1 we have the following global existence
result.
≡ δ ≡ δ1 , δ2 , . . . , δN be a pair of constant
≡ M ≡ M1 , M2 , . . . , MN , U
Lemma 2.3. Let U
vectors satisfying 2.5 and M ≥ δ, and let hypothesis (H) hold. Then for any ηx, t in S, problem
2.1 has a unique global solution Ux, t ∈ S for all t ≥ 0. In particular, if condition 2.8 holds
for every M ≥ o then problem 2.1 has a unique bounded nonnegative global solution whenever
ηx, t ≥ o.
Discrete Dynamics in Nature and Society
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4
3.5
3
u-axes
2.5
2
1.5
1
0.5
0
8
x- 6 4
ax
es 2 0
−10 0
10
20
30 40 50 60
t-axes
70 80
20
30 40 50 60
t-axes
70 80
20
30 40 50 60
t-axes
70 80
a
3
2.5
v-axes
2
1.5
1
0.5
0
8
x- 6 4
ax
es 2 0
−10 0
10
b
4
3.5
w-axes
3
2.5
2
1.5
1
0.5
0
8
x- 6 4
ax
es 2 0
−10 0
10
c
Figure 2: The solution of system 4.2.
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In system 2.1 if fi is given in the form
fi x, U, V ui fi∗ x, U, V ,
i 1, 2, . . . , N,
2.9
x ∈ Ω, i 1, 2, . . . , N.
2.10
M, U
o satisfy 2.8 whenever
then the pairs U
fi∗ x, M, M ≤ 0
Moreover, by the positivity lemma for the linear scalar parabolic problem
∂Wi
− Li Wi Wi fi∗ x, U, V ∂t
∂Wi
0
∂n
Wi x, 0 ηi x, 0
x ∈ Ω, t > 0,
x ∈ ∂Ω, t > 0,
2.11
x ∈ Ω, i 1, 2, . . . , N,
where U ≡ Ux, t is the solution of 2.1, the solution Wi ≡ Wi x, t is positive in 0, ∞ × Ω
whenever ηi x, 0 ≥ 0 and not identical 0 cf. 1. Since for each i the solution component
ui ≡ ui x, t is also a solution of 2.11 the uniqueness property of Wi ensures that ui x, t Wi x, t > 0 in 0, ∞ × Ω. This observation leads to the following.
Lemma 2.4. Let fi x, U, V be given by 2.9 and satisfy hypothesis (H), and let M ≡
M1 , M2 , . . . , MN be a constant vector satisfying 2.10. Then for any nontrivial nonnegative
ηi x, t ≤ Mi problem 2.1 has a unique bounded positive solution ux, t in 0, ∞ × Ω.
To investigate the asymptotic behavior of the solution of 2.1, we consider the special
case
fi x, U, V gi U, V ,
i 1, 2, . . . , N,
2.12
where gi U, V is not explicitly dependent on x. The corresponding steady-state problem of
2.1 is given by
−Li ui gi U, V ∂ui
0
∂n
x ∈ Ω,
x ∈ ∂Ω, , i 1, 2, . . . , N,
2.13
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11
Assume that GU, V ≡ g1 U, V , . . . , gN U, V is quasimonotone nondecreasing in S. Then
by using U
{U
m
0
M, U0 δ as a pair of coupled initial iterations we construct two sequences
} ≡ {u1 , . . . , uN }, {Um } ≡ {u1 , . . . , uN } from the linear iteration process
m
Ki ui
m
Ki ui
−Li ui
−Li ui
m−1
m−1
m−1
m−1
m−1
,
g i u1
, . . . , uN , J ∗ u1
, . . . , J ∗ uN
m−1
m−1
m−1
m−1
m−1
,
g i u1
, . . . , uN , J ∗ u1
, . . . , J ∗ uN
m
Ki ui
m
Ki ui
m
∂ui
∂n
2.14
m
∂ui
∂n
0 i 1, 2, . . . , N,
m
where Ki is the Lipschitz constant in 2.7. It is clear that the sequences {U }, {Um } are
well defined. Since the initial iterations in 2.14 are the constant vectors M, δ and gi U, V is
independent of x whenever U and V are constants we conclude from the uniqueness property
m
m
of solution for linear boundary-value problems that the solutions ui , ui of 2.14 are
constants and are given by
m
ui
m
ui
1 m−1
m−1
m−1
m−1
,
g i u1
, . . . , uN , u1
, . . . , uN
Ki
1 m−1
m−1
m−1
m−1
m−1
ui
g i u1
, . . . , uN , u1
, . . . , uN
i 1, 2, . . . , N.
Ki
m−1
ui
2.15
It is easy to show from the quasimonotone nondecreasing property of GU, V that the
m
m
sequences ui , ui possess the monotone property
δ ≤ Um ≤ Um1 ≤ U
m1
≤U
m
≤M
m 1, 2, . . .
2.16
cf. 12. The above property implies that the constant limits
lim U
m→∞
m
ρ,
lim Um ρ
m→∞
2.17
exist and satisfy the relation δ ≤ ρ ≤ ρ ≤ M. Letting m → ∞ in 2.15 shows that ρ and ρ
satisfy the equations
gi ρ, ρ 0,
gi ρ, ρ 0
i 1, 2, . . . , N.
2.18
It is easy to verify that the limits ρ and ρ are maximal and minimal solutions of systems
2.13 in S, respectively, furthermore, if ρ ρ≡ ρ∗ then ρ∗ is the unique solution in S cf.
1, 12. In the latter case, the time-dependent solution of 2.1 with fi x, U, V given by
2.12 converges to ρ∗ as t → ∞ cf. 12. To summarize the above conclusions we have the
following.
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Discrete Dynamics in Nature and Society
Lemma 2.5. Let the conditions in Lemma 2.3 be satisfied with respect to the function given by
2.12.Then
m
0
i the sequences {U }, {Um } given by 2.14 with U M, U0 δ are constant
functions and converge monotonically to their respective constant limits ρ and ρ that satisfy
2.18;
m
Um , U
Um
ii for any m ∈ N, if Um ≤ ηx, t ≤ U , then the constant pairs U
are also coupled upper and lower solutions of 2.1, and the problem 2.1 has a unique
positive solution Ux, t such that Um ≤ Ux, t ≤ U
m
and ρ ≤ ux, t ≤ ρ, as t →
∞ x ∈ Ω;
iii if ρ ρ≡ ρ∗ then ρ∗ is the unique solution of 2.13 in S, and for any initial function
ηx, t ∈ S the corresponding solution Ux, t of 2.1 converges to ρ∗ as t → ∞.
In Lemma 2.5 the convergence of the time-dependent solution Ux, t to ρ∗ is for the
class of initial functions in S. For arbitrary nontrivial nonnegative initial functions we have
the following result from 12.
Lemma 2.6. Let the conditions in Lemma 2.5 be satisfied, and let U ≡ u1 , u2 , . . . , uN be the solution
of 2.1 corresponding to an arbitrary nontrivial nonnegative ηx, t. Assume that ρ ρ≡ ρ∗ and
there exists t∗ > 0 such that
δi ≤ ui x, t ≤ Mi
for t∗ − τi ≤ t ≤ t∗ , x ∈ Ω, i 1, 2, . . . , N.
2.19
Then Ux, tconverges to ρ∗ as t → ∞.
To prove the existence and uniqueness of constant positive solution of the steady-state
problem 1.13 we also need the following result.
Lemma 2.7. Let ai > 0, bi > 0, ci > 0 i 1, 2, 3. If a1 b2 c3 − a2 b1 c3 − a1 b3 c2 > 0 then the
inequalities
a1 x − b1 y > 0,
−a2 x b2 y − c2 z > 0,
2.20
−b3 y c3 z > 0
have a positive solutionρ1 , ρ2 , ρ3 .
Proof. With the inequalities 2.20 exists a positive solution if and only if there exist positive
constants λ1 , λ2 , λ3 such that the following equations:
a1 x − b1 y λ1 ,
−a2 x b2 y − c2 z λ2 ,
−b3 y c3 z λ3
2.21
Discrete Dynamics in Nature and Society
have a positive solution. Let A x
y
a1 −b1 0
, we have |A| > 0. From the Cramer theorem,
−a2 b2 −c2
the solutions of 2.21 are as follows:
λ1 13
b2
−c2
−b3
c3
0 −b3 c3
− λ2 −b1
0
−b3
c3
−b
1
λ3 b2
−c2
|A|
λ1 b2 c3 − b3 c2 λ2 b1 c3 λ3 b1 c3
,
|A|
−a2 −c2 a 0
a
1
1
λ2 − λ3 −λ1 0
c
−a
3
0
c3
2
0
−c2
|A|
λ1 a2 c3 λ2 a1 c3 λ3 a1 c2
,
|A|
−a
a −b a
b2 1 2
1
1
λ1 − λ2 λ3 0
−bc3
0
−b3
0
z
|A|
0
2.22
−b1
−b3
λ1 a2 b3 λ2 a1 b3 λ3 a1 b2 − a2 b1 .
|A|
Let λ1 λ3 1, λ2 max{1 − b1 c3 − b2 c3 b3 c2 /b1 c3 , 1 − a1 b2 a2 b1 − a2 b3 /a1 b3 , 1},
then with the equations 2.21 exists a positive solution ρ1 , ρ2 , ρ3 . Therefore the inequalities
2.20 have a positive solution.
3. The Main Theorems and Proof
To prove the main results in this paper, we apply Lemmas 2.3–2.7 for the models in 1.11.2 with U u, v, w by constructing a suitable pair of constant upper and lower solutions
M and U
δ. This is equivalent to show that M and δ satisfy condition 2.5 for the
U
reaction functions fi x, U, V .
Theorem 3.1. Let ei∗ ≡ ei βi > 0 i 1, 2, 3 and b2∗ ≡ b2 α2 > 0. If c1 c2 c3 − c1 e2∗ e3∗ − e1∗ b2∗ c3 > 0,
then the following statements hold true.
i The steady-state problem 1.9 has a unique constant positive solution.
ii For any nontrivial nonnegative η1 , η2 , η3 , problems 1.1-1.2 have a unique nonnegative global solution u, v, w, and the solution is uniformly boundary in Ω × 0, ∞.
iii The positive constant solution ρ∗ of the steady-state problem 1.9 is globally asymptotic
stable.
14
Discrete Dynamics in Nature and Society
7
6
6
5
4
v-axes
u-axes
5
4
3
3
2
2
1
1
0
8
x- 6 4
ax
es 2 0
−1
0
1
2
4
3
t-axes
5
8
7
6
0
8
x- 6 4
ax
es 2 0
−1
2
1
0
a
4
3
t-axes
5
6
7
8
b
9
8
7
w-axes
6
5
4
3
2
1
0
8
x- 6 4
ax
es 2 0
−1
0
1
2
4
3
t-axes
5
6
7
8
c
Figure 3: The solution of system 4.3.
Proof. iSince the boundary conditions are Neumann type, we seek a pair of positive constant
vectors M M1 , M2 , M3 , δ δ1 , δ2 , δ3 satisfying
M1 a1 − c1 M1 e1∗ M2 ≤ 0 ≤ δ1 a1 − c1 δ1 e1∗ δ2 ,
M2 a2 b2∗ M1 − c2 M2 e2∗ M3 ≤ 0 ≤ δ2 a2 b2∗ δ1 − c2 δ2 e2∗ δ3 ,
M3 a3 e3∗ M2 − c3 M3 ≤ 0 ≤ δ3 a3 e3∗ δ2 − c3 δ3 ,
∂Mi
∂δi
≤0≤
∂n
∂n
i 1, 2, 3.
3.1
Discrete Dynamics in Nature and Society
15
4
3.5
u-axes
3
2.5
2
1.5
1
0.5
0
10
xax 5
es
0 −1
0
1
2
4
3
t-axes
5
6
7
8
2
4
3
t-axes
5
6
7
8
2
4
3
t-axes
5
6
7
8
a
4
3.5
w-axes
3
2.5
2
1.5
1
0.5
80
x- 6 4
ax
es 2 0
−1
0
1
b
4
3.5
w-axes
3
2.5
2
1.5
1
0.5
0
8
x- 6 4
ax
es 2 0
−1
0
1
c
Figure 4: The solution of system 4.4.
16
Discrete Dynamics in Nature and Society
It is clear that the boundary conditions are satisfied, and from the positive property of Mi and
δi that condition 3.1 is satisfied if
a1 − c1 M1 e1∗ M2 ≤ 0 ≤ a1 − c1 δ1 e1∗ δ2 ,
a2 b2∗ M1 − c2 M2 e2∗ M3 ≤ 0 ≤ a2 b2∗ δ1 − c2 δ2 e2∗ δ3 ,
a3 e3∗ M2 − c3 M3 ≤ 0 ≤ a3 e3∗ δ2 − c3 δ3 .
3.2
We construct the following inequalities:
c1 x − e1∗ y > 0,
−b2∗ x c2 y − e2∗ z > 0,
3.3
−e3∗ y c3 z > 0.
⎛
c1 −e1∗ 0
Let A ⎝ −b2∗
0
c2 −e2∗
−e3∗
⎞
⎠, we have |A| > 0. Since e∗ ≡ ei βi > 0 i 1, 2, 3 and b∗ ≡ b2 α2 > 0,
2
i
c3
from the Lemma 2.7 the inequalities 3.3 have a positive solution λ1 , λ2 , λ3 . By choosing
M M1 , M2 , M3 P λ1 , P λ2 , P λ3 , δ δ1 , δ2 , δ3 γλ1 , γλ2 , γλ3 , where P and γ are
a sufficiently large and small positive constant, respectively, we see that all the inequalities
in 3.2 are satisfied. That is, M and δ is a pair of positive constant upper-lower solution
of 1.1-1.2. By Lemma 2.5 the sequences governed by 2.14 with the reaction function
f1 , f2 , f3 given by 1.1 converge monotonically, respectively, to some constant limits ρ ≡
ρ1 , ρ2 , ρ3 , ρ ≡ ρ , ρ , ρ that satisfy
1
2
3
ρ1 a1 − c1 ρ1 e1∗ ρ2 0,
ρ
ρ2 a2 b2∗ ρ1 − c2 ρ2 e2∗ ρ3 0,
ρ
ρ3 a3 e3∗ ρ2 − c3 ρ3 0,
ρ
1
2
3
a1 − c1 ρ e1∗ ρ
1
2
0,
a2 b2∗ ρ − c2 ρ e2∗ ρ
1
a3 e3∗ ρ − c3 ρ
2
2
3
3
0,
3.4
0.
Let ρi ρi − ρ i 1, 2, 3. Then by the positivity of ρi and ρ , a subtraction of the above pairs
i
i
of equations, leads to
c1 ρ1 − e1∗ ρ2 0,
−b2∗ ρ1 c2 ρ2 − e2∗ ρ3 0,
3.5
−e3∗ ρ2 c3 ρ3 0.
Since these equations are equivalent to Aρ 0, where A is coefficient matrix and ρ is the
column vector ρ1 , ρ2 , ρ3 T , we conclude from the nonsingular property of A that ρ o. This
proves ρ ρ≡ ρ∗ and ρ∗ is the unique positive solution of 1.13 in S.
Discrete Dynamics in Nature and Society
17
iiIt is obvious that the pairs M M1 , M2 , M3 P λ1 , P λ2 , P λ3 , δ δ1 , δ2 , δ3 0, 0, 0 satisfy all the equalities in 3.1. By Lemma 2.3 and the arbitrariness of P, problems
1.1-1.2 have a unique bounded nonnegative global solution u, v, w for any nontrivial
nonnegative η 1,η 2,η 3.
iiiLemma 2.5 ensures that for any initial function ηx, t ≡ η1 x, t, η2 x, t, η3 x, t
with γλ1 , λ2 , λ3 ≡ δ ≤ ηx, t ≤ M ≡ P λ1 , λ2 , λ3 , where P and γare a
sufficiently large and small positiveconstants, respectively, the corresponding solution
Ux, t ≡ ux, t, vx, t, wx, t of 1.1-1.2 converges to ρ∗ as t → ∞. To show
this convergence property for an arbitrary nontrivial nonnegative ηx, t, we observe
from Lemma 2.4, the arbitrary smallness of γ and the arbitrary largeness of P that
there exists t0 > 0 such that δ ≤ Ux, t ≤ M on t0 , ∞ × Ω. The upper and
lower bounds of Ux, t show that there exists t∗ > 0 such that condition 2.19 is
satisfied for U ≡ u1 , u2 , u3 u, v, w. The conclusion of Ux, t → ρ∗ as t →
∞ follows from Lemma 2.6. We next consider the stability of Ux, t. We observe from
Lemma 2.5 that for any Uρ∗ , ε there exists δ > 0 such that Ux, t ∈ U∗ ≡
Uρ∗ , δ ⊆ Um , U
proof.
m
⊂ Uρ∗ , ε as initial function ηx, t ∈ U∗ . This completes the
Corollary 3.2. All the results of Theorem 3.1 hold true if di 0 for some or all i. In particular, these
results hold true for the ordinary differential system 1.8. They are also true if the reaction functions
involve no time delays.
Proof. This follows from the argument in the proof of Theorem 3.1 by letting di 0 and
without the corresponding boundary condition and by letting τi 0 i 1, 2, 3, respectively.
Remark 3.3. Our model and result are different from the existence ones such as those of Pao
29 and Kim and Lin 22, 23. In some sense, we enrich the results of the 3-specics LotkaVolterra reaction-diffusion systems.
4. Numerical Simulations
In this section, we give numerical simulations supporting our theoretical analysis. As an
example, we consider system 1.1 with different diffusion rates di , birth rate ai , and time
delays τi , that is, the following systems:
∂u
1
1
− uxx ut, x 1 − ut, x vt, x vt − 0.1, x
2π > x > 0, t > 0,
∂t
2
2
∂v
1
1
− vxx vt, x 1 ut, x − 3vt, x wt, x,
∂t
2
2
1
1
ut − 0.1, x wt − 0.1, x
2π > x > 0, t > 0,
2
2
∂w
1
1
− wxx wt, x 1 vt, x − w vt − 0.1, x
2π > x > 0, t > 0,
∂t
2
2
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Discrete Dynamics in Nature and Society
∂u ∂v ∂w
0
∂n ∂n
∂n
t > 0, x 0, 2π,
π ut, x vt, x wt, x 1 10t 1 − sin x t, x ∈ −0.1, 0 × 0, 2π. 4.1
2
1
1
∂u
− uxx ut, x 1 − ut, x vt, x vt − 0.01, x
2π > x > 0, t > 0,
∂t
2
2
∂v
1
1
− vxx vt, x 1 ut, x − 3vt, x wt, x,
∂t
2
2
1
1
ut − 0.01, x wt − 0.01, x
2π > x > 0, t > 0,
2
2
4.2
∂w
1
1
− wxx wt, x 1 vt, x − w vt − 0.01, x
2π > x > 0, t > 0,
∂t
2
2
∂u ∂v ∂w
0 t > 0, x 0, 2π,
∂n ∂n
∂n
π ut, x vt, x wt, x 1 t 1 − sin x t, x ∈ −0.01, 0 × 0, 2π.
2
1
1
∂u
− uxx ut, x 1 − ut, x vt, x vt − 0.1, x
2π > x > 0, t > 0,
∂t
2
2
∂v
1
1
− vxx vt, x 2 ut, x − 3vt, x wt, x,
∂t
2
2
1
1
ut − 0.1, x wt − 0.1, x
2π > x > 0, t > 0,
2
2
∂w
1
1
− wxx wt, x 3 vt, x − w vt − 0.1, x
2π > x > 0, t > 0,
∂t
2
2
4.3
∂u ∂v ∂w
0 t > 0, x 0, 2π,
∂n ∂n
∂n
π ut, x vt, x wt, x 1 10t 1 − sin x t, x ∈ −0.1, 0 × 0, 2π.
2
1
1
∂u
− uxx ut, x 1 − ut, x vt, x vt − 0.1, x
2π > x > 0, t > 0,
∂t
2
2
∂v
1
1
− 2vxx vt, x 1 ut, x − 3vt, x wt, x,
∂t
2
2
1
1
ut − 0.1, x wt − 0.1, x
2π > x > 0, t > 0,
2
2
∂w
1
1
− 3wxx wt, x 1 vt, x − w vt − 0.1, x
2π > x > 0, t > 0,
∂t
2
2
∂u ∂v ∂w
0
∂n ∂n
∂n
t > 0, x 0, 2π,
π ut, x vt, x wt, x 1 10t 1 − sin x t, x ∈ −0.1, 0 × 0, 2π.
2
4.4
Discrete Dynamics in Nature and Society
19
By using the classical implicit format solving the partial differential equations and the
method of steps for differential difference equations and employing the software package
MATLAB7.0, we can solve the numerical solutions of systems 4.1, 4.2, 4.3, and 4.4
which are shown respectively in Figures 1, 2, 3, and 4.
5. Conclusions
It is well known that the analysis of stability for a system of delay reaction-diffusion
multispecies model is quite difficult since the reaction among multispecies is more complex.
Therefore, the works on this subject are very rare. A detailed analysis on the stability for a
two-prey one-predator model, one-prey two-predator model, and three-species food-chain
model with delay and diffusion was given by Pao 29, and he obtained some simple and
easily verifiable conditions for the existence and global asymptotic stability of a positive
steady-state solution for each of the three model problems.
In this paper, based on the ideas of Pao 12, we have considered a delay cooperative
three-species system with Neumann boundary condition. It is shown that the system
has a positive equilibrium under some certain conditions. We have obtained the similar
conclusions to those of Pao 29. More precisely, we have obtained the following results.
a The steady-state problem 1.9 has a unique constant positive solution if c1 c2 c3 −
c1 e2∗ e3∗ − e1∗ b2∗ c3 > 0.
b For any nontrivial nonnegativ η1 , η2 , η3 , problems 1.1-1.2 have a unique
nonnegative global solution u, v, w, and the solution is uniformly boundary in Ω × 0, ∞.
c The positive constant solution ρ∗ of the steady-state problem 1.9 is globally
asymptotic stable.
The condition c1 c2 c3 − c1 e2∗ e3∗ − e1∗ b2∗ c3 > 0 involves only the reaction rate constants,
which shows that the diffusion rates di , the birth rate ai , and time delays τi do not bring effect
on permanence of one species as well as contribution to its extinction. The result of global
asymptotic stability implies that the three-species model system coexists, is permanent, and
the trivial and all semitrivial solutions are unstable.
Acknowledgments
The authors are grateful to the referee for her/his comments. This work is supported
by Science and Technology Study Project of Chongqing Municipal Education Commission
Grant no. KJ 080511 of China, Natural Science Foundation Project of CQ CSTC Grant
no. 2008BB7415 and 2007BB2450 of China, Foundation Project of Doctor Graduate Student
Innovation of Beijing University of Technology of China, the NSFC Grant nos.10471009, and
BSFC Grant no. 1052001 of China.
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