Journal of Inequalities in Pure and
Applied Mathematics
EXISTENCE AND GLOBAL ATTRACTIVITY OF PERIODIC SOLUTIONS
IN N -SPECIES FOOD-CHAIN SYSTEM WITH TIME DELAYS
QIMING LIU AND HAIYUN ZHOU
Department of Mathematics
Shijiazhuang Institute of Mechanical Engineering
No. 97 Heping West Road
Shijiazhuang 050003, Hebei
P.R. CHINA.
EMail: llqm2002@yahoo.com.cn
volume 4, issue 5, article 89,
2003.
Received 21 August, 2003;
accepted 10 October, 2003.
Communicated by: R.P. Agarwal
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
113-03
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Abstract
A delayed periodic n-species simple food-chain system with Holling type-II functional response is investigated. By means of Gaines and Mawhin’s continuation theorem of coincidence degree theory and by constructing appropriate
Lyapunov functionals, sufficient conditions are obtained for the existence and
global attractivity of positive periodic solutions of the system.
2000 Mathematics Subject Classification: 34K13, 92D25.
Key words: Time delay, Periodic solution, Global attractivity.
Contents
1
Introducion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Existence of Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 6
3
Global Attractivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
References
Existence and Global
Attractivity of Periodic
Solutions in n-Species
Food-Chain System with Time
Delays
Qiming Liu and Haiyun Zhou
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J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
1.
Introducion
The traditional Lotka-Volterra type predator-prey model with Michaelis-Menten
or Holling type II functional response has received great attention from both
theoretical and mathematical biologists, and has been well studied (see, for
example, [1] – [12]). Up to now, most of the works on Lotka-Volterra type
predator-prey models with Michaelis-Menten or Holling type II functional responses have dealt with autonomous population systems. The analysis of these
models has been centered around the coexistence of populations and the local
and global stability of equilibria. We note that any biological or environmental
parameters are naturally subject to fluctuation in time. As Cushing [13] pointed
out, it is necessary and important to consider models with periodic ecological
parameters or perturbations which might be quite naturally exposed (for example, those due to seasonal effects of weather, food supply, mating habits,
hunting or harvesting seasons, etc.). Thus, the assumption of periodicity of the
parameters is a way of incorporating the periodicity of the environment.
It has been widely argued and accepted that for various reasons, time delay
should be taken into consideration in modelling, we refer to the monographs of
Cushing [14], Gopalsamy [15], Kuang [16], and MacDonald [17] for general
delayed biological systems and to Beretta and Kuang [18], Gopalsamy [19, 20],
He [21], Wang and Ma [22], and the references cited therein for studies on delayed biological systems. In general, delay differential equations exhibit much
more complicated dynamics than ordinary differential equations since a time
delay could cause a stable equilibrium to become unstable and cause the population to fluctuate. Time delay due to gestation is a common example, because
generally the consumption of prey by the predator throughout its past history
Existence and Global
Attractivity of Periodic
Solutions in n-Species
Food-Chain System with Time
Delays
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J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
governs the present birth rate of the predator. Therefore, more realistic models
of population interactions should take into account the effect of time delays.
The main purpose of this paper is to discuss the combined effects of the
periodicity of the ecological and environmental parameters and time delays due
to gestation and negative feedback on the dynamics of an n-species food-chain
model with Michaelis-Menten or Holling type II functional responses. To do
so, we consider the following delay differential equations

a12 (t)x2 (t)



,
ẋ1 (t) = x1 (t) r1 (t) − a11 (t)x1 (t − τ11 ) −


1 + m1 x1 (t)





aj,j−1 (t)xj−1 (t − τj,j−1 )



ẋj (t) = xj (t) −rj (t) +


1 + mj−1 xj−1 (t − τj,j−1 )



aj,j+1 (t)xj+1 (t)
− ajj (t)xj (t − τjj ) −
, 1 < j < n,
(1.1)

1 + mj+1 xj (t)





an,n−1 (t)xn−1 (t − τn,n−1 )


ẋn (t) = xn (t) −rn (t) +



1 + mn−1 xn−1 (t − τn,n−1 )






−ann (t)xn (t − τnn ) ,

with initial conditions
Existence and Global
Attractivity of Periodic
Solutions in n-Species
Food-Chain System with Time
Delays
Qiming Liu and Haiyun Zhou
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xj (s) = φj (s), s ∈ [−τ, 0], φj (0) > 0, j = 1, 2, . . . , n.
Close
In system (1.1), xi (t) denotes the density of the ith population, respectively, i =
1, . . . , n. τj,j−1 (j = 2, . . . , n) are time delays due to gestation, that is, mature
adult predators can only contribute to the reproduction of predator biomass.
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(1.2)
Page 4 of 36
J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
τii ≥ 0 denotes the delay due to negative feedback of the species xi . τ =
max{τij , 1 ≤ i, j ≤ n}; ri (t), aij (t) (i, j = 1, 2, . . . , n) are positively periodic
continuous functions with common period ω > 0 and mi (i = 1, 2, . . . , n − 1)
are positive constants.
It is well known by the fundamental theory of functional differential equations [23] that system (1.1) has a unique solution x(t) = (x1 , x2 , . . . , xn ) satisfying initial conditions (1.2). It is easy to verify that solutions of system (1.1)
corresponding to initial conditions (1.2) are defined on [0, +∞) and remain positive for all t ≥ 0. In this paper, the solution of system (1.1) satisfying initial
conditions (1.2) is said to be positive.
The organization of this paper is as follows. In the next section, by using
Gaines and Mawhin’s continuation theorem of coincidence degree theory, sufficient conditions are established for the existence of positive periodic solutions
of system (1.1) with initial conditions (1.2). In Section 3, by constructing suitable Lyapunov functionals, sufficient conditions are derived for the uniqueness
and global attractivity of positive periodic solutions of system (1.1).
Existence and Global
Attractivity of Periodic
Solutions in n-Species
Food-Chain System with Time
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Qiming Liu and Haiyun Zhou
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J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
2.
Existence of Periodic Solutions
In this section, by using Gaines and Mawhin’s continuation theorem of coincidence degree theory, we show the existence of positive periodic solutions of
system (1.1) with initial conditions (1.2). In order to prove our existence result,
we need the following notations.
Let X, Y be real Banach spaces, let L : Dom L ⊂ X → Y be a linear
mapping, and N : X → Y be a continuous mapping. The mapping L is called
a Fredholm mapping of index zero if dim Ker L = codim Im L < +∞ and
Im L is closed in Y . If L is a Fredholm mapping of index zero and there exist
continuous projectors P : X → X, and Q : Y → Y such that Im P = Ker L,
Ker Q = Im L = Im(I − Q), then the restriction LP of L to Dom L ∩ Ker P :
(I − P )X → Im L is invertible. Denote the inverse of LP by KP . If Ω is an
open bounded subset of X, the mapping N will be called L−compact on Ω̄ if
QN (Ω̄) is bounded and KP (I − Q)N : Ω̄ → X is compact. Since Im Q is
isomorphic to Ker L, there exists an isomorphism J : Im Q → Ker L.
For convenience of use, we introduce the continuation theorem of coincidence degree theory (see [24, p. 40]) as follows.
Lemma 2.1. Let Ω ⊂ X be an open bounded set. Let L be a Fredholm mapping
of index zero and N be L-compact on Ω̄. Assume
(a) For each λ ∈ (0, 1), x ∈ ∂Ω∩ Dom L, Lx 6= λN x;
(b) For each x ∈ ∂Ω ∩ Ker L, QN x 6= 0;
(c) deg{JQN, Ω ∩ Ker L, 0} =
6 0.
Existence and Global
Attractivity of Periodic
Solutions in n-Species
Food-Chain System with Time
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Qiming Liu and Haiyun Zhou
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Then Lx = N x has at least one solution in Ω̄∩ Dom L.
J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
In what follows we shall also need the following notations:
Z
1 ω
¯
f=
f (t)dt, f L = min f (t), f M = max f (t),
t∈[0,ω]
t∈[0,ω]
ω 0
where f is a continuous ω-periodic function.
Theorem 2.2. System (1.1) with initial conditions (1.2) has at least one strictly
positive ω-periodic solution provided that
aj+1,j > mj rj+1 , aj,j−1 > mj−1 Hj , 2 ≤ j ≤ n,
a11 H2
r1 > K 1 +
e2r1 ω ,
a21 − m1 H2
(H1)
(H2)
Qiming Liu and Haiyun Zhou
where
(2.1)


(a21 /m1 ) − r2 2(a21 /m1 )ω


K1 = a12
,
H n = rn
e


a22






ajj Hj+1
e2aj,j−1 ω/mj−1 ,
2 ≤ j ≤ n − 1,
H j = Kj +

a
−
m
H
j+i,j
j j+1







aj+1,j − mj rj+1 2aj+1,j ω/mj


e
, 2 ≤ j ≤ n − 1.
 Kj = rj + aj,j+1
mj ajj
Proof. Let
(2.2)
Existence and Global
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yi (t) = ln[xi (t)], i = 1, . . . , n.
Page 7 of 36
J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
On substituting (2.2) into system (1.1), it follows

a12 (t)ey2 (t)

y1 (t−τ11 )

ẏ1 (t) = r1 (t) − a11 (t)e
−
,


1 + m1 ey1 (t)





aj,j−1 (t)eyj−1 (t−τj,j−1 )



ẏ
(t)
=
−r
(t)
+
− ajj (t)eyj (t−τjj )
j

 j
1 + mj−1 eyj−1 (t−τj,j−1 )
(2.3)
aj,j+1 (t)eyj+1 (t)


−
,
j = 2, . . . .n − 1,

yj (t)

1
+
m
e
j








a
(t)eyn−1 (t−τn,n−1 )

 ẏn (t) = −rn (t) + n,n−1
− ann (t)eyn (t−τnn ) .
1 + mn−1 eyn−1 (t−τn,n−1 )
It is easy to see that if system (2.3) has one ω−periodic solution (y1∗ (t), . . . , yn∗ (t))T ,
then
x∗ (t) = (x∗1 (t), . . . , x∗n (t))T = (exp[y1∗ (t)], . . . , exp[yn∗ (t)])T
is a positive ω-periodic solution of system (1.1). Therefore, to complete the
proof, it suffices to show that system (2.3) has one ω-periodic solution.
Take
X = Y = {(y1 (t), . . . , yn (t))T ∈ C(R, Rn ) : yi (t + ω) = yi (t), i = 1, . . . , n}
and
k(y1 (t), . . . , yn (t))T k =
n
X
i=1
Existence and Global
Attractivity of Periodic
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Food-Chain System with Time
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max |yi (t)|,
t∈[0,ω]
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J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
here | · | denotes L∞ −norm.. It is easy to verify that X and Y are Banach spaces
with the norm k · k. Set
L : Dom L ∩ X,
T
L(y1 (t), . . . , yn (t)) =
dy1 (t)
dyn (t)
,...,
dt
dt
T
,
where Dom L = {(y1 (t), . . . , yn (t))T ∈ C 1 (R, Rn )} and N : X → X,


r1 (t) − a11 (t)ey1 (t−τ11 ) −
..
.
a12 (t)ey2 (t)
1+m1 ey1 (t)




y1 (t)


 ..  


 .  
y1 j−1(t−τj,j−1 )
yj+1 (t)


 
aj,j−1 (t)e
aj,j+1 (t)e
yj (t−τjj )
−a
(t)e
−
N yj (t) =−rj (t)+ 1+m
.
jj
yj−1 (t−τj,j−1 )
1+mj eyj (t) 
j−1 e
 .  

 ..  
..


.


yn−1 (t−τn,n−1 )
y( t)
−rn (t) + an,n−1 (t)eyn−1 (t−τn,n−1 ) − ann (t)eyn (t−τnn )
1+mn−1 e
Define two projectors P and Q as



 
y1
y1
 .. 
 ..  
 . 
 .  



 
P  yj  = Q  yj  = 
 . 
 .  
 .. 
 ..  
yn
yn
Existence and Global
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Food-Chain System with Time
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1
ω
Rω

y1 (t)dt
..

.

R

1 ω
y
(t)dt
,
ω 0 j

..

.
R
1 ω
y (t)dt
ω 0 n
0


y1
 .. 
 . 


 yj  ∈ X.
 . 
 .. 
yn
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J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
It is clear that
Ker L = {x| x ∈ X, x = h, h ∈ Rn },
Z ω
Im L = {y| y ∈ Y,
y(t)dt = 0} is closed in Y,
0
and
dim KerL = codim ImL = n.
Therefore, L is a Fredholm mapping of index zero. It is easy to show that P and
Q are continuous projectors such that
Im P = Ker L,
Ker Q = Im L = Im(I − Q).
Furthermore, the inverse KP of LP exists and is given by KP : Im L →
Dom L ∩ Ker P ,
Z t
Z Z
1 ω t
KP (y) =
y(s)ds −
y(s)dsdt.
ω 0 0
0
Then QN : X → Y and KP (I − Q)N : X → X read


i
R h
a12 (t)ey2 (t)
1 ω
y1 (t−τ11 )
r
(t)
−
a
(t)e
−
dt
1
11
ω 0
1+m1 ey1 (t)




..


.
 R h

i
yj+1 (t)
y1 j−1(t−τj,j−1 )
1 ω

aj,j−1 (t)e
aj,j+1 (t)e
yj (t−τjj )
−
dt ,
QN x=ω 0 −rj (t)+ 1+m eyj−1 (t−τj,j−1 ) −ajj (t)e
yj (t)
1+m
e
j−1
j




..


.
h
i


yn−1 (t−τn,n−1 )
R
an,n−1 (t)e
1 ω
yn (t−τnn )
−r
(t)
+
−
a
(t)e
dt
nn
n
yn−1 (t−τn,n−1 )
ω 0
Existence and Global
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1+mn−1 e
J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
Z ω
Z Z
1 ω t
1
t
KP (I−Q)N x =
N x(s)ds−
−
N x(s)dsdt−
N x(s)ds.
ω 0 0
ω 2
0
0
Clearly, QN and KP (I − Q)N are continuous.
In order to apply Lemma 2.1, we need to search for an appropriate open,
bounded subset Ω.
Corresponding to the operator equation Lx = λN x, λ ∈ (0, 1), we obtain

i
h
a12 (t)ey2 (t)
y1 (t−τ11 )


ẏ
(t)
=
λ
r
(t)
−
a
(t)e
−
,
1
1
11

1+m1 ey1 (t)


h

y
(t−τ
)


 ẏj (t) = λ −rj (t) + aj,j−1 (t)ey j−1(t−τ j,j−1)
j,j−1
1+mj−1 e j−1
i
(2.4)
aj,j+1 (t)eyj+1 (t)
yj (t−τjj )


−ajj (t)e
−
, 1 < j < n,

1+mj eyj (t)


h
i

y
(t−τ
)


 ẏn (t) = λ −rn (t) + an,n−1 (t)ey n−1(t−τ n,n−1) − ann (t)eyn (t−τnn ) .
n−1
n,n−1
Z
t
Existence and Global
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Food-Chain System with Time
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Qiming Liu and Haiyun Zhou
1+mn−1 e
Suppose that (y1 (t), . . . , yn (t))T ∈ X is a solution of system (2.4) for some
λ ∈ (0, 1). Integrating system (2.4) over [0, ω] gives
Z ω
Z ω
Z ω
a12 (t)ey2 (t)
y1 (t−τ11 )
(2.5)
a11 (t)e
dt +
dt =
r1 (t)dt,
y1 (t)
0
0 1 + m1 e
0
Z
(2.6)
0
ω
aj,j−1 (t)eyj−1 (t−τj,j−1 )
dt
1 + mj−1 eyj−1 (t−τj,j−1 )
Z ω
Z ω
aj,j+1 (t)eyj+1 (t)
=
rj (t)dt +
dt
1 + mj eyj (t)
0
0
Z ω
+
ajj (t)eyj (t−τjj ) dt, j = 2, 3, . . . , n − 1,
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0
J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
Z
ω
ω
Z
yn (t−τnn )
rn (t)dt +
(2.7)
0
ann (t)e
0
Z
dt =
0
ω
an,n−1 (t)eyn−1 (t−τn,n−1 )
dt.
1 + mn−1 eyn−1 (t−τn,n−1 )
It follows from (2.5)-(2.7) that
Z ω
Z ω
a12 (t)ey2 (t) y1 (t−τ11 )
(2.8)
|ẏ1 (t)|dt = λ
−
dt
r1 (t) − a11 (t)e
1 + m1 ey1 (t) 0
0
Z ω
a12 (t)ey2 (t)
y1 (t−τ11 )
dt
≤
r1 (t) + a11 (t)e
+
1 + m1 ey1 (t)
0
∆
= 2r1 ω = d1 ,
Existence and Global
Attractivity of Periodic
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Food-Chain System with Time
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Qiming Liu and Haiyun Zhou
Z
(2.9)
0
ω
ω
yj−1 (t−τj,j−1 )
−rj (t) + aj,j−1 (t)e
|ẏj (t)|dt = λ
1 + mj−1 eyj−1 (t−τj,j−1 )
0
aj,j+1 (t)eyj+1 (t) yj (t−τjj )
−ajj (t)e
−
dt
1 + mj eyj (t) Z ω
aj,j−1 (t)eyj−1 (t−τj,j−1 )
≤
rj (t) +
1 + mj−1 eyj−1 (t−τj,j−1 )
0
aj,j+1 (t)eyj+1 (t)
yj (t−τjj )
+ajj (t)e
+
dt
1 + mj eyj (t)
aj,j−1 ∆
≤2
ω = dj , j = 2, . . . , n − 1,
mj−1
Z
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J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
Z
(2.10)
0
ω
|ẏn (t)|dt
Z ω
an,n−1 (t)eyn−1 (t−τn,n−1 )
yn (t−τnn ) =λ
−
a
e
−r
(t)
+
nn
n
dt
1 + mn−1 eyn−1 (t−τn,n−1 )
0
Z ω
an,n−1 (t)eyn−1 (t−τn,n−1 )
yn (t−τnn )
+ ann e
≤
rn (t) +
dt
1 + mn−1 eyn−1 (t−τ32 )
0
an,n−1 ∆
ω = dn .
≤2
mn−1
Since (y1 (t), . . . , yn (t))T ∈ X, there exists ti , Ti such that
yi (ti ) = min yi (t), yi (Ti ) = max yi (t),
t∈[0,ω]
We derive from (2.5) that
Z ω
y1 (t−τ11 )
a11 (t)e
Qiming Liu and Haiyun Zhou
Z
dt ≤
0
ω
r1 (t)dt,
0
which implies
y1 (t1 ) ≤ ln
r1 ∆
= ρ1 .
a11
This, together with (2.8), leads to
y1 (t) ≤ y1 (t1 ) +
ω
|ẏ1 (t)|dt
0
≤ ln
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Z
(2.11)
i = 1, . . . , n.
t∈[0,ω]
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r1
+ 2r1 ω.
a11
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J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
It follows from (2.6) and (2.7) that
Z ω
Z
yj (t−τjj )
ajj (t)e
dt ≤
0
ω
0
aj,j−1 (t)eyj−1 (t−τj,j−1 )
dt − rj ω
1 + mj−1 eyj−1 (t−τj,j−1 )
aj,j−1
≤
ω − rj ω,
mj−1
which implies
aj,j−1 − mj−1 rj ∆
= ρj .
mj−1 ajj
This, together with (2.8) and (2.9), leads to
Z ω
(2.12)
yj (t) ≤ yj (tj ) +
|ẏj (t)|dt
yj (tj ) ≤ ln
Qiming Liu and Haiyun Zhou
0
≤ ln
aj,j−1
aj,j−1 − mj−1 rj
ω, j = 2, . . . , n.
+2
mj−1
mj−1 ajj
From (2.5) and (2.12) we obtain
(2.13)
a11 ey1 (T1 ) ≥ r1 −
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ω
ω
Z
a12 (t)ey2 (t) dt
0
(a21 /m1 ) − r2 2(a21 /m1 )ω
e
a22
= r 1 − K1 ,
≥ r1 − a12
that is
(2.14)
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y1 (T1 ) ≥ ln
r 1 − K1 ∆
= δ1 .
a11
Page 14 of 36
J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
It follows from (2.8) and (2.14) that
Z ω
r 1 − K1
− 2r1 ω.
(2.15)
y1 (t) ≥ y1 (T1 ) −
|ẏ1 (t)|dt ≥ ln
a11
0
We obtain from (2.6) and (2.15) that
Z
Z
1 ω a21 (t)ey1 (t−τ21 )
1 ω
y2 (T2 )
≥
dt − r2 −
(2.16) a22 e
a23 (t)ey3 (t) dt
ω 0 1 + m1 ey1 (t−τ21 )
ω 0
≥
=
1
a21 ( r1a−K
)e−2r1 ω
11
1
1 + m1 ( r1a−K
)e−2r1 ω
11
− K2
a21 (r1 − K1 )
− K2
a11 e2r1 ω + m1 (r1 − K1 )
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If ∆1 − K2 > 0 then
∆1 − K2
,
y2 (T2 ) ≥ ln
a22
Z
y2 (t) ≥ y2 (T2 ) −
ω
|ẏ2 (t)|dt
0
≥ ln
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this, together with (2.9), leads to
(2.18)
Qiming Liu and Haiyun Zhou
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∆
= ∆ 1 − K2 .
(2.17)
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∆1 − K2
a21
− 2 ω.
a22
m1
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J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
It follows from (2.7) and (2.18) that
Z
Z ω
1 ω a32 (t)ey2 (t−τ32 )
y3 (T3 )
a33 e
≥
dt − r3 −
a34 (t)eu4 (t) dt
ω 0 1 + m2 ey2 (t−τ32 )
0
≥
≥
a32 (∆1 − K2 )e−2(a21 /m1 )ω /a22
−2(a21 /m1 )ω /a
1 + mM
22
2 (∆1 − K2 )e
a32 (∆1 − K2 )
a22 e2(a21 /m1 )ω + m2 (∆1 − K2 )
− K3
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− K3
∆
= ∆ 2 − K3 .
Qiming Liu and Haiyun Zhou
If ∆2 − K3 > 0, together with (2.9), it follows
(2.19)
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∆2 − K3
a32 ∆
a21
y3 (t) ≥ ln
− 2 ω = δ2 − 2 ω.
a33
m2
m1
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Similarly, we have
ajj eyj (Tj ) ≥
aj,j−1 (∆j−2 − Kj−1 )
2[a
/mj−2 ]ω + m
j−1,j−2
aj−1,j−1 e
j−1 (∆j−2
− Kj−1 )
− Kj
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∆
= ∆j−1 − Kj .
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J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
If ∆j−1 − Kj > 0, we have
Z
ω
yj (t) ≥ yj (Tj ) −
(2.20)
|ẏj (t)|dt
0
≥ ln
∆
aj,j−1
∆j−1 − Kj
ω,
−2
mj−1
ajj
= δj − 2
aj,j−1
ω,
mj−1
where
(2.21)
aj+1,j (∆j−1 − Kj )
∆j =
ajj e
2aj,j−1
ω
mj−1
,
3 ≤ j ≤ n − 1.
+ mj (∆j−1 − Kj )
From (2.7) and (2.20), we have
Existence and Global
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Qiming Liu and Haiyun Zhou
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yn (Tn )
ann e
≥ ∆n−1 − rn .
If ∆n−1 − rn > 0, together with (2.10), it follows that
Z ω
(2.22)
yn (t) ≥ yn (Tn ) −
|ẏn (t)|dt
0
≥ ln
∆n−1 − rn
an,n−1
−2
ω
ann
mn−1
an,n−1
∆
ω.
= δn − 2
mn−1
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J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
We note that (2.15), (2.18) – (2.20) and (2.22) hold if the following are true:
(2.23)
r1 > K1 , ∆j > Kj+1 (j = 1, 2, . . . , n − 2),
∆n−1 > rn .
We now show that assumptions (H1) and (H2) imply (2.23).
Let (H1) and (H2) hold. Then we have
r1 > K 1 +
a11 H2 e2r1 ω
,
a21 − m1 r2
a21 > m1 H2 ,
Existence and Global
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which implies
r1 > K1 , ∆1 > H2 .
Noting that a32 − m2 r3 > 0, a32 − m2 H3 > 0, we have
Qiming Liu and Haiyun Zhou
r1 > K1 , ∆1 > K2 , ∆2 > H3 ,
which, together with a43 − m3 r4 > 0, a43 − m3 H4 > 0, leads to
r1 > K1 , ∆1 > K2 , ∆2 > K3 , ∆3 > H4 .
Similarly, we have
r1 > K 1 ,
∆l > Kl+1
(1 ≤ l ≤ j − 2),
∆j−1 > Hj ,
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which, together with aj+1,j > mj rj+1 , aj+1,j > mj Hj+1 , implies
r1 > K 1 ,
∆l > Kl+1
(1 ≤ l ≤ j − 1),
∆j > Hj+1 .
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J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
Finally, by a similar argument we obtain
r1 > K 1 ,
∆l > Kl+1
(1 ≤ l ≤ n − 2),
∆n−1 > rn .
From what has been discussed above, we finally derive that
(2.24)
∆
max |yi (t)| ≤ max{|ρi | + di , |δi | + di } = Bi ,
i = 1, . . . , n.
t∈[0,ω]
P
Clearly, Bi (i = 1, 2, . . . , n) are independent of λ. Denote B = ni=1 Bi + B0 ,
here B0 is taken sufficiently large such that each solution (v1∗ , v2∗ , . . . , vn∗ )T of
the system of algebraic equations

Z ω
1
a12 (t)ev2

v
1

r
−
a
e
−
dt = 0,
 1
11

ω 0 1 + m1 ev1



Z



1 ω aj,j−1 (t)evj−1


dt − ajj evj
 −rj +
ω 0 1 + mj−1 evj−1
Z
(2.25)
1 ω aj,j+1 (t)evj+1



dt = 0, 2 ≤ j ≤ n − 1
−


ω 0 1 + mj evj



Z


1 ω an,n−1 (t)evn−1


 −rn +
dt − ann evn = 0,
ω 0 1 + mn−1 evn−1
Pn
∗
satisfies k(v1∗ , v2∗ , . . . , vn∗ )T k =
i=1 |vi | < B (if it exists). Now, we take
Ω = {(y1 , y2 , .., yn )T ∈ X : k(y1 , y2 , . . . , yn )T k < B}. Thus, the condition (a)
)T ∈ ∂Ω ∩ Ker L = ∂Ω ∩ Rn ,
of Lemma 2.1 is satisfied. When (y1 , y2 , . . . , ynP
(y1 , y2 , . . . , yn )T is a constant vector in Rn with ni=1 |yi | = B. If system (2.25)
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J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
has solutions, then


R ω a12 (t)ey2
r1 − a11 ey1 − ω1 0 1+m
dt
y
 
 
1
1e


y1
0
.


..
 ..  
  .. 
yj−1
yj+1
R
R
 .
 .  
  −rj + ω1 0ω aj,j−1 (t)eyj−1 dt−ajj eyj − ω1 0ω aj,j+1 (t)eyj dt  
1+m
e
1+m
e
QN yj =
6= 0 .
j−1
j
 .  
 .
..
  .. 
 ..  
.


y
n−1
R


yn
0
1 ω an,n−1 (t)e
yn
dt
−
a
e
−rn + ω 0
nn
1 + mn−1 eyn−1
If system (2.25) does not have a solution, then we can directly derive

  
0
y1
 ..   .. 
QN  .  6=  .  .
0
yn
Qiming Liu and Haiyun Zhou
Thus, the condition (b) in Lemma 2.1 is satisfied.
In the following, we will prove that the condition (c) in Lemma 2.1 is satisfied. To this end, we define φ : Dom L × [0, 1] → X by
φ(y1 , . . . , yn , µ)

r1 − a11 ey1
..
.



y
R
1 ω aj,j−1 (t)e j−1

y
=  −rj + ω 0 1+mj−1 eyj−1 dt − ajj e j

..

.

R
y
1 ω an,n−1 (t)e n−1
−rn + ω 0 1+mn−1 eyn−1 dt − ann eyn


− ω1
a12 (t)ey2
dt
0 1+m1 ey1
Rω






R


+µ  − ω1 0ω






..
.
Existence and Global
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



yj+1

aj,j+1 (t)e
,
dt 
1+mj eyj


..

.
0
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J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
where µ is a parameter. WhenP(y1 , y2 , . . . , yn )T ∈ ∂Ω ∩ Rn , (y1 , y2 , . . . , yn )T is
a constant vector in Rn with ni=1 |yi | = B. We will show that when
(y1 , y2 , . . . , yn )T ∈ ∂Ω ∩ Ker L, φ(y1 , y2 , . . . , yn , µ) 6= 0.
P
Otherwise, there is a constant vector (y1 , . . . , yn )T ∈ Rn with ni=1 |yi | = B
satisfying φ(y1 , y2 , . . . , yn , µ) = 0, that is
Z
1 ω a12 (t)ey2
y1
dt = 0,
r1 − a11 e − µ
ω 0 1 + m1 ey1
1
− rj +
ω
Z
0
ω
aj,j−1 (t)eyj−1
dt − ajj eyj
y
j−1
1 + mj−1 e
Z
1 ω aj,j+1 (t)eyj+1
−µ
dt = 0,
ω o 1 + mj eyj
and
Qiming Liu and Haiyun Zhou
2≤j ≤n−1
Z
1 ω an,n−1 (t)eyn−1
−rn +
dt − ann eyn = 0.
ω 0 1 + mn−1 eyn−1
By some similar arguments in (2.11), (2.12), (2.14), (2.15), (2.17), (2.18), (2.20)
and (2.22) we can show that
|yi | ≤ max{|δi |, |ρi |}, i = 1, 2, . . . , n.
Thus
n
X
i=1
|yi | ≤
n
X
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max{|ρi |, |δi |} < B,
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J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
which is leads to a contradiction. Using the property of topological degree and
taking J = I : Im Q → Ker L, (y1 , y2 , . . . , yn )T → (y1 , y2 , . . . , yn )T , we have
deg(JQN (y1 , . . . , yn )T , Ω ∩ Ker L, (0, . . . , 0)T )
= deg(φ(y1 , . . . , yn , 1), Ω ∩ Ker L, (0, . . . , 0)T )
= deg(φ(y1 , . . . , yn , 0), Ω ∩ Ker L, (0, . . . , 0)T )
Z
1 ω aj,j−1 (t)eyj−1
y1
= deg
(r1 − a11 e , . . . , −rj +
dt − ajj eyj , . . . ,
ω 0 1 + mj−1 eyj−1
1
−rn +
ω
Z
0
ω
an,n−1 (t)eyn−1
dt − ann eyn
1 + mn−1 eyn−1
T
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!
, Ω ∩ Ker L, (0, . . . , 0)T
.
Under assumptions (H1) – (H2), one can easily show that the following system
of algebraic equations


 r1 − a11 u1 = 0,




aj,j−1 uj−1

−rj +
− ajj uj = 0,
2≤j ≤n−1
(2.26)
1 + mj−1 uj−1





an,n−1 un−1

 −rn +
− ann un = 0,
1 + mn−1 un−1
has a unique solution (u∗1 , . . . , u∗n )T which satisfies u∗i > 0, i = 1, . . . , n.
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J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
A direct calculation shows that
deg(JQN (y1 , y2 , . . . , yn )T , Ω ∩ Ker L, (0, 0, . . . , 0)T )
( n
)
Y
= sgn
(−aii ) = (−1)n 6= 0.
i=1
Finally, it is easy to show that the set {KP (I − Q)N u|u ∈ Ω̄} is equicontinuous and uniformly bounded. By using the Arzela-Ascoli theorem, we see that
KP (I − Q)N : Ω → X is compact. Moreover, QN (Ω̄) is bounded. Consequently, N is L−compact.
By now we have proved that Ω satisfies all the requirements in Lemma 2.1.
Hence, system (2.3) has at least one ω-periodic solution. As a consequence,
system (1.1) has at least one positive ω-periodic solution. This completes the
proof.
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3.
Global Attractivity
We now proceed to a discussion on the global attractivity of the positive ω−periodic
solution of system (1.1). It is immediate that if any positive periodic solution
of system (1.1) is globally attractive, then it is in fact unique. We first derive
certain upper and lower bound estimates for solutions of (1.1) – (1.2).
Lemma 3.1. Let x(t) = (x1 (t), x2 (t), . . . , xn (t))T denote any positive solution
of system (1.1) with initial conditions (1.2). Then there exists a T > 0 such that
0 < xi ≤ Mi f or t > T, i = 1, 2, . . . , n,
(3.1)
where
(3.2)
Qiming Liu and Haiyun Zhou
r1M r1M τ1
e
,
aL11
L
aM
j,j−1 /mj−1 − rj [aj,j−1 M/mj−1 −rjL ]τjj
Mj =
e
, j = 2, 3, . . . , n.
aLjj
M1 =
The proof of Lemma 3.1 is similar to that of Lemma 2.1 in [22], we therefore
omit it here.
We now formulate the global attractivity of the positive ω−periodic solutions
of system (1.1) as follows.
Theorem 3.2. In addition to (H1) – (H2), assume further that
(H3)
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lim inf Ai (t) > 0, i = 1, 2, . . . , n,
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t→∞
J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
where
(3.3)
A1 (t) = a11 (t) − m1 a12 (t)M2 − a21 (t + τ21 )
Z
t+τ11
− [r1 (t) + a11 (t)M1 + a12 (t)M2 ]
t
Z t+2τ11
− a11 (t + τ1 )M1
a11 (s)ds
t+τ11
Z t+τ21 +τ22
− a21 (t + τ21 )M2
a21 (s)ds,
a11 (s)ds
t+τ21
(3.4)
Aj (t) = ajj (t) − mj aj,j+1 (t)Mj+1 − aj+1,j (t + τj+1,j )
Z
aj1 ,j (t) t+τj−1,j−1
−
aj−1,j−1 (s)ds
mj−1 t
aj,j−1 (t)
+ ajj (t)Mj + aj,j+1 (t)Mj+1
− rj (t) +
mj−1
Z t+τjj
Z t+2τjj
×
ajj (s)ds − ajj (t + τjj )Mj
ajj (s)ds
t
t+τjj
Z
t+τj+1,j +τj+1,j+1
− aj+1,j (t + τj+1,j )Mj+1
aj+1,j+1 (s)ds,
t+τj+1,j
for 2 ≤ j ≤ n − 1, and
(3.5) An (t) = ann (t) − an−1,n (t)
Z t+τnn
an,n−1 (t)
− rn (t) +
+ ann (t)Mn
ann (s)ds
mn−1
t
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Z
t+2τnn
− ann (t + τnn )Mn
t+τnn
an−1,n (t)
ann (s)ds −
mn−1
Z
t+τn−1,n−1
an−1,n−1 (s)ds.
t
Then system (1.1) has a unique positive ω−periodic solution x∗ (t) =
(x∗1 (t), . . . , x∗n (t))T which is globally attractive.
Proof. Due to the conclusion of Theorem 2.2, we need only to show the global
attractivity of the positive periodic solution of (1.1). Let x∗ (t) = (x∗1 (t), . . . , x∗n (t))T
be a positive ω− periodic solution of (1.1), and y(t) = (y1 (t), . . . , yn (t))T be
any positive solution of system (1.1) – (1.2). It follows from Lemma 3.1 that
there exist positive constants T and Mi (defined by (3.2)) such that for all t ≥ T ,
(3.6)
0 < x∗i (t) ≤ Mi , 0 < yi (t) ≤ Mi , i = 1, 2, . . . , n.
Existence and Global
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Qiming Liu and Haiyun Zhou
We define
(3.7)
V11 (t) = | ln x∗1 (t) − ln y1 (t)|.
Calculating the upper right derivative of V11 (t) along solution of (1.1), it follows
for t ≥ T that
∗
ẋ1 (t) ẏ1 (t)
+
(3.8)
−
sgn(x∗1 (t) − y1 (t))
D V11 =
x∗1 (t) y1 (t)
∗
= sgn(x1 (t) − y1 (t)) − a11 (t)(x∗1 (t − τ11 ) − y1 (t − τ11 ))
a12 (t)x∗2 (t)
a12 (t)y2 (t)
−
+
1 + m1 x∗1 (t) 1 + m1 y1 (t)
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J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
a12 (t)(x∗2 (t) − y2 (t))
=
− y1 (t)) −a11 (t)(x∗1 (t) − y1 (t)) −
1 + m1 x∗1 (t)
Z
t
m1 a12 (t)y2 (t)(x∗1 (t) − y1 (t))
∗
+
+ a11 (t)
(ẋ1 (u) − ẏ1 (u))du
(1 + m1 x∗1 (t))(1 + m1 y1 (t))
t−τ11
≤ −a11 (t)|x∗1 (t) − y1 (t)| + a12 |x∗2 (t) − y2 (t)| + m1 a12 (t)y2 (t)|x∗1 (t) − y1 (t)|
Z t
∗
+ a11 (t) (ẋ1 (u) − ẏ1 (u))du
sgn(x∗1 (t)
t−τ11
∗
−a11 (t)|x1 (t) − y1 (t)|
Z t
+ a12 (t)|x∗2 (t) − y2 (t)| + m1 a12 (t)y2 (t)|x∗1 (t) − y1 (t)|
∗
+ a11 (t) x1 (u) r1 (u)
t−τ11
a12 (u)x∗2 (u)
∗
− a11 (u)x1 (u − τ11 ) −
1 + m1 x∗1 (u)
a12 (u)y2 (u)
− y1 (u) r1 (u) − a11 (u)y1 (u − τ11 ) −
du
1 + m1 y1 (u)
∗
∗
= −a11 (t)|x1 (t) − y1 (t)| + a12 (t)|x2 (t) − y2 (t)| + m1 a12 (t)y2 (t)|x∗1 (t) − y1 (t)|
Z t + a11 (t) r1 (u) − a11 (u)y1 (u − τ11 )
t−τ11
a12 (u)y2 (u)
(x∗1 (t) − y1 (t))
−
(1 + m1 x∗1 (u))(1 + m1 y1 (u))
− a11 (u)x∗1 (u)(x∗1 (u − τ11 ) − y1 (u − τ11 ))
a12 (u)x∗1 (u) ∗
∗
−
(x
(u)
−
y
(u))
du
2
1 + m1 x∗1 (u) 2
=
Existence and Global
Attractivity of Periodic
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J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
≤ −a11 (t)|x∗1 (t) − y1 (t)| + a12 (t)|x∗2 (t) − y2 (t)| + m1 a12 (t)y2 (t)|x∗1 (t) − y1 (t)|
Z t + a11 (t) [r1 (u) + a11 (u)y1 (u − τ11 ) + a12 (u)y2 (u)]|x∗1 (u) − y1 (u)|
t−τ11
a12 (u) ∗
∗
∗
+ a11 (u)x1 (u)|x1 (u − τ11 ) − y1 (u − τ11 )| +
|x2 (u) − y2 (u)| du .
m1
We derive from (3.6) and (3.8) that for t ≥ T + τ
(3.9) D+ V11 (t)
≤ −a11 (t)|x∗1 (t) − y1 (t)| + a12 (t)|x∗2 (t) − y2 (t)|
+ m1 a12 (t)M2 |x∗1 (t) − y1 (t)|
Z t + a11 (t) [r1 (u) + a11 (u)M1 + a12 (u)M2 ]|x∗1 (u) − y1 (u)|
+
Qiming Liu and Haiyun Zhou
t−τ11
a11 (u)M1 |x∗1 (u
− τ11 ) − y1 (u − τ11 )|
a12 (u) ∗
+
|x2 (u) − y2 (u)| du .
m1
(3.10) V12 (t)
Z t+τ11 Z
=
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Define
t
Existence and Global
Attractivity of Periodic
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Food-Chain System with Time
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t
a11 (s) [r1 (u) + a11 (u)M1 + a12 (u)M2 ]|x∗1 (u) − y1 (u)|
s−τ11
+
a11 (u)M1 |x∗1 (u
− τ11 ) − y1 (u − τ11 )|
a12 (u) ∗
+
|x2 (u) − y2 (u)| duds.
m1
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J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
It follows from (3.9) and (3.10) that for t ≥ T + τ
(3.11) D+ (V11 (t) + V12 (t))
≤ −a11 (t)|x∗1 (t) − y1 (t)| + a12 (t)|x∗2 (t) − y2 (t)| + m1 a12 (t)M2 |x∗1 (t) − y1 (t)|
Z t+τ11
+
a11 (s)ds [r1 (t) + a11 (t)M1 + a12 (t)M2 ]|x∗1 (t) − y1 (t)|
t
a12 (t) ∗
∗
+ a11 (t)M1 |x1 (t − τ11 ) − y1 (t − τ11 )| +
|x2 (t) − y2 (t)| .
m1
We now define
V1 (t) = V11 (t) + V12 (t) + V13 (t),
(3.12)
Existence and Global
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Qiming Liu and Haiyun Zhou
where
Z
(3.13)
t
Z
l+2τ11
V13 (t) = M1
t−τ11
a11 (s)a11 (l + τ11 )|x∗1 (l) − y1 (l)|dsdl.
l+τ11
It then follows from (3.11) – (3.13) that for t ≥ T + τ
(3.14) D+ V1 (t)
≤ −a11 (t)|x∗1 (t) − y1 (t)| + a12 (t)|x∗2 (t) − y2 (t)| + m1 a12 (t)M2 |x∗1 (t) − y1 (t)|
Z t+τ11
+
a11 (s)ds [r1 (t) + a11 (t)M1 + a12 (t)M2 ]|x∗1 (t) − y1 (t)|
t
Z t+2τ11
a12 (t) ∗
|x2 (t) − y2 (t)| + a11 (t + τ11 )M1
|x∗1 (t) − y1 (t)|.
+
m1
t+τ11
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J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
Define
(3.15) Vj (t) = | ln x∗j (t) − ln yj (t)|
Z t
+
aj,j−1 (s + τj,j−1 )|x∗j−1 (s) − yj−1 (s)|ds
t−τj,j−1
t+τjj Z t
ajj (s) rj (u)
Z
+
t
s−τjj
aj,j−1 (u)
+ ajj (u)Mj + aj,j+1 (u)Mj+1 |x∗j (u) − yj (u)|
+
mj−1
+ ajj (u)Mj |x∗j (u − τjj ) − yj (u − τjj )|
+ aj,j−1 (u)Mj |x∗j−1 (u − τj,j−1 ) − yj−1 (u − τj,j−1 )|
aj,j+1 (u) ∗
+
|xj+1 (u) − yj+1 (u))| duds
mj
Z t Z l+2τjj
+ Mj
ajj (s)ajj (l + τjj )|x∗j (l) − yj (l)|dsdl
t−τjj
Z t
l+τjj
Z l+τj,j−1 +τjj
+ Mj
×
ajj (s)aj,j−1 (l + τj,j−1 )
t−τj,j−1 l+τj,j−1
|x∗j−1 (l) − yj−1 (l)|dsdl,
j = 2, 3, . . . , n − 1.
Existence and Global
Attractivity of Periodic
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(3.16) Vn (t) =
| ln x∗n (t)
Z t
− ln yn (t)|
+
an,n−1 (s + τn,n−1 )|x∗n−1 (s) − yn−1 (s)|ds
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t−τn,n−1
J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
t+τnn
t
+
ann (s) rn (u)
t
s−τnn
an,n−1 (u)
+
+ ann (u)Mn |x∗n (u) − yn (u)|
mn−1
+ Mn ann (u)|x∗n (u − τnn ) − yn (u − τnn )|
Z
+
Z
Mn an,n−1 (u)|x∗n−1 (u
Z
t
− τn,n−1 ) − yn−1 (u − τn,n−1 )| duds
l+2τnn
Z
ann (s)ann (l + τnn )|xn (l) − yn (l)|dsdl
+ Mn
t−τnn
t
Z
l+τnn
Z l+τn,n−1 +τnn
+ Mn
ann (s)an,n−1 (l + τn,n−1 ))
t−τn,n−1
×
l+τn,n−1
|x∗n−1 (l) −
Qiming Liu and Haiyun Zhou
yn−1 (l)|dsdl.
Finally, we define
V (t) =
n
X
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Vi (t).
i=1
We derive from (3.14) – (3.16) that for t ≥ T + τ
n
X
dV (t)
(3.17)
≤−
Ai (t)|x∗i (t) − yi (t)|,
dt
i=1
where Ai (t)(i = 1, . . . , n) are as defined in (3.3) – (3.5).
By hypothesis (H3), there exist constants αi > 0 (i = 1, . . . , n) and T ∗ ≥
T + τ , such that
(3.18)
Existence and Global
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Ai (t) ≥ αi > 0 for t ≥ T ∗ .
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Integrating both sides of (3.17) on [T ∗ , t], we derive
(3.19)
n Z
X
V (t) +
t
T∗
i=1
Ai (t)|x∗i (s) − yi (s)|ds ≤ V (T ∗ ).
It follows from (3.18) and (3.19) that
V (t) +
n
X
Z
t
αi
i=1
T∗
|x∗i (s) − yi (s)|ds ≤ V (T ∗ ) for t ≥ T ∗
Rt
Therefore, V (t) is bounded on [T ∗ , ∞] and also T ∗ |x∗i (s)−yi (s)|ds < ∞, i =
1, . . . , n. On the other hand, by Lemma 3.1, |x∗i (t) − yi (t)| are bounded on
[T ∗ , ∞). According to system (1.1), we see that ẋ∗i (t) and ẏi (t) are also bounded.
Hence, |x∗i (t) − yi (t)| (i = 1, . . . , n) are uniformly continuous on [T ∗ , ∞). By
Barbalat’s Lemma (see [15]), we can conclude that
lim |x∗i (t) − yi (t)| = 0, i = 1, . . . , n.
Existence and Global
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t→+∞
The proof is complete.
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J. Ineq. Pure and Appl. Math. 4(5) Art. 89, 2003
References
[1] C.S. HOLLING, The components of predation as revealed by a study of
small mammal predation of the European pine sawfly, Canad. Entomol.,
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[3] C.S. HOLLING, The functional response of predators to prey density and
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[4] A.D. BAZYKIN, Structural and dynamic stability of model predator-prey
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KUZNETZOV, The influence of predator saturation effect and competition among predators on predator-prey system dynamics, Ecological Modelling, 14 (1981), 39–57.
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[7] J. HAINZL, Stability and Hopf bifurcation in a predator-prey system with
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Existence and Global
Attractivity of Periodic
Solutions in n-Species
Food-Chain System with Time
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[9] R.E. KOOIJ AND A. ZEGELING, Qualitative properties of twodimensional predator-prey systems, Nonlinear Anal., 29 (1997), 693–
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Existence and Global
Attractivity of Periodic
Solutions in n-Species
Food-Chain System with Time
Delays
[13] J.M. CUSHING, Periodic time-dependent predator-prey system, SIAM J.
Appl. Math., 32 (1977), 82–95.
Qiming Liu and Haiyun Zhou
[14] J.M. CUSHING, Integro-differential Equations and Delay Models in Population Dynamics, Springer-Verlag, Heidelberg, 1977.
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[15] K. GOPALSAMY, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht/ Norwell,
MA, 1992.
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[17] N. MACDONALD, Time Lags in Biological Models, Springer-Verlag, Heidelberg, 1978.
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[18] E. BERETTA AND Y. KUANG, Convergence results in a well-known delayed predator-prey system, J. Math. Anal. Appl., 204 (1996), 840–853.
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(1983), 295–309.
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[23] J. HALE, Theory of Functional Differential Equations, Springer-Verlag,
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[24] R.E. GAINES AND J.L. MAWHIN, Coincidence Degree and Nonlinear
Differential Equations, Springer-Verlag, Berlin, 1977.
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Attractivity of Periodic
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Food-Chain System with Time
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