Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2011, Article ID 380279, 19 pages
doi:10.1155/2011/380279
Research Article
Dynamics of a Nonautonomous Leslie-Gower Type
Food Chain Model with Delays
Hongying Lu and Weiguo Wang
School of Mathematics and Quantitative Economics, Dongbei University of Finance & Economics, Dalian,
Liaoning 116025, China
Correspondence should be addressed to Hongying Lu, hongyinglu543@163.com
Received 11 October 2010; Accepted 26 January 2011
Academic Editor: M. De la Sen
Copyright q 2011 H. Lu and W. Wang. 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.
A nonautonomous Leslie-Gower type food chain model with time delays is investigated. It is
proved the general nonautonomous system is permanent and globally asymptotically stable under
some appropriate conditions. Furthermore, if the system is periodic one, some sufficient conditions
are established, which guarantee the existence, uniqueness, and global asymptotic stability of a
positive periodic solution of the system. The conditions for the permanence, global stability of
system, and the existence, uniqueness of positive periodic solution depend on delays; so, time
delays are profitless.
1. Introduction
Among the relationships between the species living in the same outer environment, the
predator-prey theory plays an important and fundamental role. The dynamic relationship
between predators and their preys has long been and will continue to be one of the dominant
themes in both ecology and mathematical ecology due to its universal existence and
importance 1. Food-chain predator-prey system, as one of the most important predator-prey
system, has been extensively studied by many scholars, many excellent results concerned
with the permanent property and positive periodic solution of the system, see 2–4 and
the references cited therein. Recently, Nindjin and Aziz-Alaoui 5 proposed the following
autonomous delayed predator-prey model with modified Leslie-Gower functional response
a12 x2 t
,
ẋ1 t x1 t a1 − a11 x1 t − τ11 −
x1 t d1
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a21 x1 t − τ21 a23 x3 t
− a22 x2 t − τ22 −
ẋ2 t x2 t −a2 ,
x1 t − τ21 d1
x2 t d2
a33 x3 t − τ33 ,
ẋ3 t x3 t a3 −
x2 t − τ32 d3
1.1
system 1 represents an ecological situation where a prey population x1 is the only food for a
predator x2 . This specialist predator x2 , in turns, serves as the prey of a top-predator x3 . The
interaction between species x2 and its prey x1 has been modeled by the Volterra scheme.
But, the interaction between species x3 and its prey x2 has been modeled by a modified
version of Leslie-Gower scheme. About Leslie-Gower scheme one could refer to 6–13 and
the references cited therein.
In 5, the authors showed that the system is uniformly persistent under some
appropriate conditions and obtained sufficient conditions for global stability of the positive
equilibrium of system 1.
We note that any biological or environment parameters are naturally subject to
fluctuation in time, and if a model is to take into account such fluctuation then the model
must be nonautonomous. On the other hand, time delays occur so often in nature, a number
of models can be formulated as systems of differential equations with time delays see, e.g.,
2, 14, 15 and the references cited therein. Motivated by above considerations, in this paper,
we consider the following general nonautonomous Leslie-Gower type food chain model with
time delays of the form
a12 tx2 t
ẋ1 t x1 t a1 t − a11 tx1 t − τ11 t −
,
x1 t d1 t
a21 tx1 t − τ21 t
a23 tx3 t
− a22 tx2 t − τ22 t −
,
ẋ2 t x2 t −a2 t x1 t − τ21 t d1 t
x2 t d2 t
a33 tx3 t − τ33 t
ẋ3 t x3 t a3 t −
,
x2 t − τ32 t d3 t
1.2
where τii t, i 1, 2, 3 denote the time delays due to negative feedbacks of the prey, specialist
predator and top-predator, respectively. τ21 t is a time delay due to gestation, that is, mature
adult predators can only contribute to the reproduction of predator biomass. τ32 t can be
regarded as a gestation period.
In this paper, for system 1.2 we always assume that for all i, j 1, 2, 3
H1 ai t, aij t, di t are continuous and bounded above and below by positive
constants on 0, ∞, and
H2 τij t are continuous and differentiable bounded functions on 0, ∞, and τ̇ij t is
uniformly continuous with respect to t on 0, ∞ and inft∈0,∞ {1 − τ̇ij t} > 0.
Let τ sup{τij t : t ∈ 0, ∞, i, j 1, 2, 3}, then we have 0 ≤ τ < ∞. Let
σij t t − τij t, then the function σij−1 t is the inverse function of the function σij t,
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3
i, j 1, 2, 3. Motivated by the application of system 1.2 to population dynamics, we assume
that solutions of system 1.2 satisfies the following initial conditions
xi θ φi θ ≥ 0,
θ ∈ −τ, 0, φi 0 > 0,
φi ∈ C−τ, 0, R ,
i 1, 2, 3.
1.3
It is well known that by the fundamental theory of functional differential equation
16, one can prove the solution of system 1.2 with initial conditions 1.3 exists and remains
positive for t ≥ 0, we call such a solution the positive solution of system 1.2.
The organization of this paper is as follows. In Section 2, by using comparison theorem
and further developing the analytical technique of 2, 14, we obtain a set of sufficient
conditions, which ensure the permanence of the system 1.2. In Section 3, by constructing
a suitable Lyapunov function, we establish a set of sufficient conditions, which ensure the
global stability of the system 1.2. In Section 4, we will explore the existence and stability of
the solutions of the periodic system 1.2. At last, the conclusion ends with brief remarks.
2. Permanence
In this section, we establish a permanent result for system 1.2.
Here, for any bounded function {ft}
f u lim sup ft ,
t → ∞
f l lim inf ft .
t → ∞
2.1
Definition 2.1. System 1.2 is said to be permanent, if there are positive constants mi and Mi ,
such that each positive solution x1 t, x2 t, x3 t of system 1.2 satisfies
mi ≤ lim inf xi t ≤ lim sup xi t ≤ Mi ,
t → ∞
t → ∞
i 1, 2, 3.
2.2
Theorem 2.2. Assume that H1 and H2 hold assume further that
H3 al1 > au12 M2 /d1l , al21 m1 /m1 d1u > au2 au23 M3 /d2l
hold. Then system 1.2 is permanent.
Proof. From the first equation of the system 1.2 it follows that
ẋ1 t ≤ x1 ta1 t − a11 tx1 t − τ11 t
≤ x1 t au1 − al11 x1 t − τ11 t .
2.3
Let τ sup{τij t : t ∈ 0, ∞, i, j 1, 2, 3}, by H2 , then we have 0 ≤ τ < ∞. Taking
M1 au1 /al11 1 h1 , where 0 < h1 < exp{au1 τ} − 1. Firstly, suppose x1 t is not oscillatory
1 . That is, there exists a T1 > 0, for t > T1 such that
about M
1 ,
x1 t < M
2.4
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or
1 .
x1 t > M
2.5
If 2.4 holds, then our aim is obtained. Suppose 2.5 holds, then for t ≥ T1 τ, we
obtain
ẋ1 t < −h1 au1 x1 t,
2.6
thus x1 t < x1 0 exp−h1 au1 t → 0, as t → ∞, which is contradiction with 2.5. Hence
1 for t > T ∗ . Secondly now assume that
there must exist T1∗ > T1 τ such that x1 t < M
1
1 for t ≥ T1 , that is, there exists a time sequence {tn } such that
x1 t is oscillatory about M
1 with limn → ∞ tn ∞
τ < t1 < t2 < · · · < tn < · · · is a sequence of zeros of x1 tn − M
and x1 tn M1 . Set tn a point where x1 t attends its maximum in tn , tn1 . Thus we get
1 . Then it follows from 2.3 that
x1 tn ≥ x1 tn M
0 ẋ1 t|ttn ≤ x1 tn au1 − al11 x1 tn − τ11 tn
,
2.7
au
≤ l1 .
x1 tn − τ11 tn
a11
2.8
which leads to
Integrating the both sides of 2.3 from tn − τ11 tn to tn , it follows that
tn
x1 tn
ln ≤
au1 − al11 x1 t − τ11 t dt ≤ au1 τ11 tn .
tn −τ11 tn x1 tn − τ11 tn
2.9
From 2.8 and 2.9 we have
au
x1 tn ≤ l1 exp au1 τ .
a11
2.10
Since x1 tn is an arbitrary local maximum of x1 t, we can see that there exists a T2 > T1
such that for all t ≥ T2
x1 t ≤
au1
al11
exp au1 τ : M1 .
2.11
Thus
lim sup x1 t ≤ M1 .
t → ∞
2.12
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5
For t ≥ T2 τ, from 2.11 and the second equation of the system 1.2 it follows that
−al2
ẋ2 t ≤ x2 t
au21 M1
M1 d1l
−
al22 x2 t
2.13
− τ22 t .
Similar to the argument above, from 2.13 we obtain
lim sup x2 t ≤
−al2 au21 M1 / M1 d1l
al22
t → ∞
exp
−al2
au21 M1
M1 d1l
τ
: M2 .
2.14
Similarly, from the third equation of the system 1.2, we have
ẋ3 t ≤ x3 t
au3
al x3 t − τ33 t
− 33
M2 d3u
,
2.15
and so
lim sup x3 t ≤
au3 M2 d3u
al33
t → ∞
exp au3 τ : M3 .
2.16
Condition H3 of Theorem 2.2 also implies that we could choose ε > 0 small enough such
that
al1 >
au12 M2 ε
d1l
,
al21 m1 − ε
au23 M3 ε
u
>
a
,
2
m1 − ε d1u
d2l
2.17
hence, for ε > 0 satisfies 2.17, from 2.12, 2.14, and 2.16, we know that there exists
T3 > T2 τ such that for i 1, 2, 3 and t ≥ T3
xi t ≤ Mi ε.
2.18
From the first equation and 2.18 it follows that for t ≥ T3 τ,
a12 tx2 t
ẋ1 t x1 t a1 t − a11 tx1 t − τ11 t −
x1 t d1 t
u
a12 M2 ε
l
u
.
≥ x1 t a1 − a11 M1 ε −
d1l
2.19
Note that au1 /al11 ≤ M1 implies that
al1 − au11 M1 ε −
au12 M2 ε
d1l
≤ al1 − au11 M1 ε ≤ au1 − al11 M1 ε ≤ 0.
Now we consider the following two cases.
2.20
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Case 1. Suppose al1 − au11 M1 ε − au12 M2 ε/d1l 0, then for t ≥ T3 τ, from the positivity
of the solution and 2.19 it follows that
ẋ1 t 0.
2.21
Then from 2.12 and 2.21 it follows that limt → ∞ x1 t : p1 < al1 /au11 , then there exists
T4 > T3 τ such that for t ≥ T4
x1 t < p1 al
al1 /au11 − p1
< u1 < M1 .
2
a11
2.22
From the positivity of the solution, 2.19 and 2.22 it follows that
al1 /au11 p1 au12 M2 ε
−
ẋ1 t ≥ x1 t
−
2
d1l
au12 M2 ε
l
u
> x1 t a1 − a11 M1 ε −
d1l
al1
0,
au11
t ≥ T4 τ,
x1 t ≥ x1 T4 τ exp
al1
−
au11
2.23
al1 /au11 p1 au12 M2 ε
−
2
d1l
t − T4 τ ,
then we can see that x1 t → ∞ as t → ∞, which is contradiction with 2.12. Hence we
have limt → ∞ x1 t ≥ al1 /au11 , which implies that there exists T4∗ > T4 τ such that x1 t ≥
al1 /2au11 for t ≥ T4∗ .
Case 2. Suppose al1 − au11 M1 ε − au12 M2 ε/d1l < 0, from 2.19, for t ≥ T3 τ, it follows
that
ẋ1 t ≥ x1 t
al1
−
au12 M2 ε
d1l
−
au11 x1 t
− τ11 t .
2.24
Let
m
1 al1 − au12 M2 ε/d1l
1 − σ1 ,
au11
2.25
where 0 < σ1 < 1 − exp{al1 − au11 M1 ε − au12 M2 ε/d1l τ}.
1 . That is, there exists a T5 > 0, for t > T5
Firstly, suppose x1 t is not oscillatory about m
such that
x1 t > m
1,
2.26
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7
or
2.27
x1 t < m
1.
If 2.26 holds, then our aim is obtained. Suppose 2.27 holds, then for t ≥ T5 τ, we
obtain
ẋ1 t ≥ σ1
al1
−
au12 M2 ε
2.28
x1 t.
d1l
1 for t > T5∗ , which is a contradiction.
thus there must exist T5∗ > T5 τ such that x1 t > m
Hence, 2.27 could not hold. Secondly now assume that x1 t is oscillatory about m
1 for
t ≥ T3 τ, that is, there exists a time sequence {tn } such that τ < t1 < t2 < · · · < tn < · · · is a
1 with limn → ∞ tn ∞ and x1 tn m
1 . Set tn be a point where
sequence of zeros of x1 tn − m
1 . Then it follows from
x1 t attends its minimum in tn , tn1 . Thus, we get x1 tn ≤ x1 tn m
2.24 that
0 ẋ1 t|ttn
au12 M2 ε
l
u
≥ x1 tn
− a11 x1 tn − τ11 tn
a1 −
,
d1l
2.29
al − au M2 ε/dl
12
1
.
≥ 1
x1 tn − τ11 tn
au11
2.30
which implies that
Integrating 2.24 on the interval tn − τ11 tn , tn , we have
tn
x1 tn
au12 M2 ε
l
u
a1 −
− a11 x1 t − τ11 t dt
ln ≥
tn −τ11 tn d1l
x1 tn − τ11 tn
≥
tn
tn −τ11 tn al1
−
al1
−
au11 M1
au12 M2 ε
d1l
ε −
−
au11 M1
au12 M2 ε
d1l
ε dt
τ11 tn .
2.31
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From 2.30 and 2.31, we get that
al − au M2 ε/dl
12
1
x1 tn ≥ 1
exp
au11
al1
−
au11 M1
ε −
au12 M2 ε
d1l
τ .
2.32
Since x1 tn is an arbitrary local minimum of x1 t, we can have that there exists a
1 such that for all t ≥ T6∗
q1 ≤ m
2.33
x1 t ≥ q1 ,
where
al − au12 M2 /d1l
exp
0 < q1 ≤ 1
au11
al1
−
au11 M1
−
au12 M2
d1l
τ .
2.34
Taken m1 min{al1 /2au11 , q1 }, thus, we have
lim inf x1 t ≥ m1 .
t → ∞
2.35
Thus, for ε > 0 satisfies 2.17, it follows from 2.35 that there exists large enough T6 such
that for t ≥ T6
x1 t ≥ m1 − ε.
2.36
For t ≥ T6 τ, by using 2.36, from the second equation of system 1.2 it follows that
a21 tx1 t − τ21 t
a23 tx3 t
ẋ2 t x2 t −a2 t − a22 tx2 t − τ22 t −
x1 t − τ21 t d1 t
x2 t d2 t
al m1 − ε
au23 M3 ε
u
≥ x2 t −au2 21
−
a
ε
−
,
M
2
22
m1 − ε d1u
d2l
al21 m1 − ε
au23 M3 ε
u
−
− a22 tx2 t − τ22 t ;
ẋ2 t ≥ x2 t −a2 m1 − ε d1u
d2l
2.37
from 2.37, by a procedure similar to the discussion above, we can verify that
lim inf x2 t ≥ m2 ,
t → ∞
2.38
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where
−au2 al21 m1 / m1 d1u
, q2 ,
m2 min
2au22
−au2 al21 m1 / m1 d1u − au23 M3 /d2l
al21 m1
au23 M3
u
u
0 < q2 ≤
exp
−a2 − a22 M2 −
τ .
au22
m1 d1u
d2l
2.39
From 2.38, we see that there exists large enough T7 such that for t ≥ T7
x2 t ≥ m2 − ε.
2.40
Substituting 2.18 to the last equation of system 1.2, it follows that
ẋ3 t ≥ x3 t
al3
ẋ3 t ≥ x3 t
al3
−
−
au33 M3 ε
d3l
au33 x3 t − τ33 t
d3l
,
2.41
;
from 2.41, similar to the argument of 2.35, we also have
2.42
lim inf x3 t ≥ m3 ,
t → ∞
where
al3 d3l
m3 min
, q3 ,
2au33
al3 d3l
au33 M3
l
τ .
a3 −
0 < q3 ≤ u exp
a33
d3l
2.43
Consequently, 2.12, 2.14, 2.16, 2.35, 2.38, and 2.42 show that under the assumption
H1 –H3 , for any positive solution x1 t, x2 t, x3 t of system 1.2, one has
mi ≤ lim inf xi t ≤ lim sup xi t ≤ Mi ,
t → ∞
t → ∞
i 1, 2, 3,
2.44
where mi and Mi , i 1, 2, 3 are independent of the solution of system 1.2, thus system 1.2
is permanent. This completes the proof of Theorem 2.2.
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3. Global Stability
Now we study the global stability of the positive solution of system 1.2. We say a positive
solution of system 1.2 is globally asymptotically stable if it attracts all other positive solution
of the system.
The following lemma is from 17, and will be employed in establishing the global
stability of positive solution of system 1.2.
Lemma 3.1. Let h be a real number and f be a nonnegative function defined on h; ∞ such that f
is integrable on h; ∞ and is uniformly continuous on h; ∞, then limt → ∞ ft 0.
Theorem 3.2. In addition to H1 –H3 , assume further that
H4 there exist constants λi > 0 such that
lim inf{Ai t} > 0,
t → ∞
i 1, 2, 3,
3.1
where
−1 −1 −1
a12 tM2 a11 σ11 t M1 σ11 σ11 t
−
a11 t −
a11 sds
−1 d12 t
1 − τ̇11 σ11
t σ11−1 t
A1 t λ1
σ11−1 t
a12 tM2
a11 sds
− a1 t a11 tM1 d1 t
t
−1 −1 σ −1 σ −1 t
22
21
a21 σ21
a21 σ21
t
t M2
a22 sds ,
−1 −1 −1 −1 −1
d1 σ21
d1 σ21 t 1− τ̇21 σ21
t 1− τ̇21 σ21
t
t
σ21
t
− λ2
A2 t λ2
−1 −1 −1
t M2 σ22 σ22 t
a23 tM3 a22 σ22
a22 sds
−
a22 t −
−1
−1
d22 t
1 − τ̇22 σ22
t
σ22
t
σ22−1 t
a23 t
a22 sds
M3
− a2 t a22 tM2 a21 t d2 t
t
− λ1 a12 t
− λ3
1
d1 t
σ −1 t
11
a11 sds
t
−1 −1 2
σ −1 σ −1 t
33
32
a33 σ32
a33 σ32
t M3
t M3
a33 s
ds ,
−1 −1 2 −1 −1 −1
d3 s
d32 σ32
d3 σ32 t 1− τ̇32 σ32
t 1− τ̇32 σ32
t
t
σ32
t
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A3 t λ3
11
σ33−1 t
a33 t
a33 s
a33 tM3
− a3 t ds
M2 d3 t
d3 t
d3 s
t
−1 σ −1 σ −1 t
33
33
a33 σ33
t M3
a33 s
ds
− −1 −1 −1
d3 s
d3 σ33 t 1 − τ̇33 σ33
t
σ33
t
− λ2 a23 t
1
d2 t
σ −1 t
22
a22 sds .
t
3.2
Then for any positive solutions x1 t, x2 t, x3 t and x1∗ t, x2∗ t, x3∗ t of system 1.2, one has
lim xi t − xi∗ t 0,
t → ∞
i 1, 2, 3.
3.3
Proof. For two arbitrary nontrivial solutions x1 t, x2 t, x3 t and x1∗ t, x2∗ t, x3∗ t of
system 1.2, we have from Theorem 2.2 that there exist positive constants T > T7 and Mi ,
mi i 1, 2, 3 such that for all t ≥ T and i 1, 2, 3
mi ≤ xi t ≤ Mi .
3.4
V11 t ln x1 t − ln x1∗ t.
3.5
We define
Calculating the upper right derivative of V11 t along the solution of system 1.2, for t ≥ T τ,
it follow that
ẋ1 t ẋ1∗ t
− ∗
sgn x1 t − x1∗ t
D V11 t x1 t x1 t
sgn x1 t − x1∗ t
a12 tx2∗ t
a12 tx2 t
∗
× −a11 t x1 t − τ11 t − x1 t − τ11 t −
x1 t d1 t x1∗ t d1 t
sgn x1 t − x1∗ t
a12 t
x2 t − x2∗ t
× −a11 t x1 t − x1∗ t −
x1 t d1 t
a12 tx2∗ t
∗
x1 t − x1∗ t
x1 t d1 t x1 t d1 t
t
∗
ẋ1 u − ẋ1 u du
a11 t
t−τ11 t
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a12 t x2 t − x∗ t
≤ −a11 tx1 t − x1∗ t 2
d1 t
a12 tM2 x1 t − x1∗ t
2
d1 t
t
∗
a11 t
ẋ u − ẋ1 u du.
t−τ11 t 1
3.6
On substituting 1.2 into 3.6, we derive that
a12 t x2 t − x∗ t a12 tM2 x1 t − x∗ t
D V11 t ≤ −a11 tx1 t − x1∗ t 2
1
2
d1 t
d1 t
t
a11 t
a u − a11 ux1∗ u − τ11 u
t−τ11 t 1
a12 ud1 ux2∗ u
∗
x1 u − x1∗ u
x1 u d1 u x1 u d1 u
− a11 ux1 u x1 u − τ11 u − x1∗ u − τ11 u
a12 ux1 u −
x2 u − x2∗ u du.
x1 u d1 u
−
3.7
It follows 3.4 and 3.7 that for t > T τ
a12 t x2 t − x∗ t
D V11 t ≤ −a11 tx1 t − x1∗ t 2
d1 t
a12 tM2 x1 t − x1∗ t
2
d1 t
t
a12 uM2 x1 u − x1∗ u
a11 t
a1 u a11 uM1 d1 u
t−τ11 t
a11 uM1 x1 u − τ11 u − x1∗ u − τ11 u
a12 ux2 u − x2∗ u du.
3.8
Let σ11 t t − τ11 t, by H2 , we can obtain the inverse function of the function σ11 t
−1
t.
denoted by σ11
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13
Define
V12 t σ −1 t t
11
σ11 s
t
a11 s
a12 uM2 x1 u − x1∗ u
× a1 u a11 uM1 d1 u
a11 uM1 x1 u − τ11 u − x∗ u − τ11 u
3.9
1
a12 ux2 u − x2∗ u du ds.
We obtain from 3.8 and 3.9 for t > T τ
a12 tM2
D V11 t V̇12 t ≤ − a11 t −
d12 t
σ −1 t
11
x1 t − x∗ t a12 t x2 t − x∗ t
2
1
d1 t
a11 sds
t
a12 tM2 x1 t − x1∗ t
a1 t a11 tM1 d1 t
a11 tM1 x1 t − τ11 t − x1∗ t − τ11 t
a12 tx2 t − x2∗ t .
×
3.10
We now define
V1 t V11 t V12 t V13 t,
3.11
where
V13 t M1
t
t−τ11 t
σ −1 σ −1 l
11
11
−1
σ11
l
−1 a11 sa11 σ11
l −1 x1 l − x1∗ lds dl.
1 − τ̇11 σ11 l
It then follows from 3.10 that for t > T τ
a12 tM2 ∗
a12 t
D V1 t ≤ − a11 t −
x
−
x
t
t
1
1
d12 t
σ −1 t
11
1
×
a11 sds x2 t − x2∗ t
d1 t
t
3.12
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Discrete Dynamics in Nature and Society
σ11−1 t
a12 tM2
a11 sdsx1 t − x1∗ t
a1 t a11 tM1 d1 t
t
−1 σ −1 σ −1 t
a11 σ11 t M1 11 11
a11 sdsx1 t − x1∗ t.
−1 −1
1 − τ̇11 σ11 t σ11 t
3.13
Similarly, we define
V2 t V21 t V22 t V23 t,
3.14
where
V22 t t
t−τ21 t
V21 t ln x2 t − ln x2∗ t,
−1 a21 σ21
s
−1 −1 x1 s − x1∗ sds
d1 σ21 s 1 − τ̇21 σ21 s
σ −1 t t
22
t
σ22 s
a22 s
a23 uM3 x2 u − x2∗ u
a2 u a22 uM2 a21 u d2 u
a23 ux3 u − x3∗ u a22 uM2 x2 u − τ22 u − x2∗ u − τ22 u
a21 uM2 ∗
x1 u − τ21 u − x1 u − τ21 u du ds,
d1 u
−1 t
σ −1 σ −1 l
22
22
a22 sa22 σ22
l V23 t M2
−1 x2 l − x2∗ lds dl
−1
1 − τ̇22 σ22 l
t−τ22 t σ22 l
−1 t
σ −1 σ −1 l
22
21
a22 sa21 σ21
l
M2
−1 −1 x1 l − x1∗ lds dl.
−1
d1 σ21 l 1 − τ̇21 σ21 l
t−τ21 t σ21
l
3.15
×
Calculating the upper right derivative of V2 t along solutions of 1.2, we derive that for
t>T τ
a23 tM3 D V2 t ≤ − a22 t −
x2 t − x2∗ t
2
d2 t
σ −1 t
22
1
a22 sds x3 t − x3∗ t
a23 t
d2 t
t
−1 a21 σ t
−1 21 −1 x1 t − x1∗ t
d1 σ21 t 1 − τ̇21 σ21 t
Discrete Dynamics in Nature and Society
a23 tM3
a2 t a22 tM2 a21 t d2 t
×
σ −1 t
15
a22 sdsx2 t − x2∗ t
22
t
−1 −1 −1
a22 σ22
t M2 σ22 σ22 t
a22 sdsx2 t − x2∗ t
−1 1 − τ̇22 σ22 t σ22−1 t
−1 σ −1 σ −1 t
22
21
a21 σ21
t M2
−1 a22 sdsx1 t − x1∗ t.
−1 −1
d1 σ21 t 1 − τ̇21 σ21 t
σ21
t
3.16
Similarly, we define
V3 t V31 t V32 t V33 t,
3.17
where
V31 t ln x3 t − ln x3∗ t,
V32 t M3
t
t−τ32 t
−1 a33 σ32
s
−1 −1 x2 s − x2∗ sds
2
d3 σ32 s 1 − τ̇32 σ32 s
σ −1 t t
33
σ33 s
t
a33 s
d3 s
×
a3 u a33 uM3 x3 u − x3∗ u
d3 u
a33 uM3 x3 u − τ33 u − x3∗ u − τ33 u
d3 u
a33 uM32 ∗
x2 u − τ32 u − x u − τ32 u du ds,
2
d32 u
V33 t M3
t
σ −1 σ −1 l
33
t−τ33 t
M32
33
−1
σ33
l
t
t−τ32 t
−1 a33 sa33 σ33
l
−1 −1 x3 l − x3∗ lds dl
d3 sd3 σ33 l 1 − τ̇33 σ33 l
σ −1 σ −1 l
33
32
−1
σ32
l
−1 a33 sa33 σ32
l
−1 −1 x2 l − x2∗ lds dl.
2
d3 sd3 σ32 l 1 − τ̇32 σ32 l
3.18
16
Discrete Dynamics in Nature and Society
Calculating the upper right derivative of V3 t along solutions of 1.2, we derive that for
t>T τ
a33 t x3 t − x3∗ t
M2 d3 t
−1 a33 σ32
t
M3 2 −1 −1 x2 t − x2∗ t
d3 σ32 t 1 − τ̇32 σ32 t
D V3 t ≤ −
σ33−1 t
a33 s a33 tM3
a3 t ds x3 t − x3∗ t
d3 t
d3 s
t
−1 σ −1 σ −1 t
33
33
a33 σ t M3
a33 s ds x3 t − x3∗ t
−1 33
−1 −1
d3 s
d3 σ33 t 1 − τ̇33 σ33 t
σ33
t
3.19
−1 2
σ −1 σ −1 t
33
32
a33 σ32
t M3
a33 s ds x2 t − x2∗ t.
2 −1 −1 −1
d
d3 σ32 t 1 − τ̇32 σ32 t
3 s
σ32 t
We now define a Lyapunov functional as
V t 3
λi Vi t.
3.20
i1
It then follows from 3.13, 3.16, 3.19, and 3.20 that for t > T τ
3
D V t ≤ − Ai txi t − xi∗ t.
3.21
i1
By the hypothes is H4 , there exist enough small positive constants αi , i 1, 2, 3 and a large
enough constant T ∗ ≥ T τ, such that for all i 1, 2, 3 and t ≥ T ∗
Ai t ≥ αi > 0.
3.22
Integrating both sides of 3.21 on interval T ∗ , t
V t 3 t
i1
T∗
Ai sxi s − xi∗ sds ≤ V T ∗ .
3.23
It follows from 3.22 and 3.23 that
V t t
3
xi s − x∗ sds ≤ V T ∗ .
αi
i
i1
T∗
3.24
Discrete Dynamics in Nature and Society
17
Therefore, V t is bounded on T ∗ , ∞ and also
t
T∗
xi s − x∗ sds < ∞,
i
i 1, 2, 3.
3.25
By Theorem 2.2, we know that |xi t − xi∗ t|, i 1, 2, 3 are bounded on T ∗ , ∞. On the other
hand, it is easy to see that ẋi t, i 1, 2, 3 are bounded for t ≥ T ∗ . Therefore, |xi t − xi∗ t|,
i 1, 2, 3 are uniformly continuous on T ∗ , ∞. By Lemma 3.1, one can conclude that
lim xi t − xi∗ t 0,
t → ∞
i 1, 2, 3.
3.26
This completes the proof of Theorem 3.2.
Remark 3.3. In the proof of the stability of system 1.2, we construct the Lyapunov functional,
which need the condition H2 . Similar method and conditon can be found in the 2, 14.
4. Existence and Stability of the Positive Periodic Solutions
In this section, we suppose that all the coefficients in system 1.2 are continuous and positive
ω-periodic functions, then the system 1.2 is an ω-periodic system for this case.
We let the following denote the unique solution of periodic system 1.2 for initial
value Z0 {x10 , x20 , x30 }:
Z t, Z0 x1 t, Z0 , x2 t, Z0 , x3 t, Z0 ,
for t > 0,
Z 0, Z0 Z0 .
4.1
Now define Poincâre transformation A : R3 → R3 is
A Z0 Z ω, Z0 .
4.2
In this way, the existence of periodic solution of system 1.2 will be equal to the existence of
the fixed point A.
Theorem 4.1. Assume that the conditions of H1 –H3 hold, then system 1.2 with initial
condition 1.3 has at least one positive ω-periodic solution.
Proof. If assumption H1 –H3 are satisfied, then from Theorem 2.2 we have that there exist
positive constants T > T7 and Mi , mi i 1, 2, 3 such that for all t ≥ T and i 1, 2, 3
mi ≤ xi t ≤ Mi .
4.3
K {x1 t, x2 t, x3 t | mi ≤ xi t ≤ Mi , i 1, 2, 3},
4.4
Let
18
Discrete Dynamics in Nature and Society
then the compact region K ⊂ R3 is a positive invariant set of system 1.2, and K is also a
close bounded convex set. So we have
Z0 ∈ K ⇒ Z t, Z0 ∈ K,
4.5
also Zω, Z0 ∈ K, thus AK ⊂ K. The operator A is continuous because the solution is
continuous about the initial value. Using the fixed point theorem of Brower, we can obtain
that A has at least one fixed point in K, then there exists at least one strictly positive ωperiodic solution of system 1.2. This ends the proof of Theorem 4.1.
By constructing similar Lyapunov functional to those of Theorem 3.2, and using
Theorem 4.1, we have the following theorem.
Theorem 4.2. Assume that the conditions of H1 –H4 hold, then system 1.2 has a unique positive
ω-periodic solution which is globally asymptotically stable.
5. Concluding Remarks
In this paper, a nonautonomous Leslie-Gower type food chain model with time delays is
investigated, which is based on the Holling type II and a Leslie-Gower modified functional
response. By using comparison theorem, we prove the system is permanent under some
appropriate conditions. Further, by constructing the suitable Lyapunov functional, we show
that the system is globally asymptotically stable under some appropriate conditions. If the
system is periodic one, some sufficient conditions are established, which guarantee the
existence, uniqueness and global asymptotic stability of a positive periodic solution of the
system. Our results have showed that the permanence, global stability of system and the
existence, uniqueness of positive periodic solution depend on delays, and so time delays are
profitless.
Acknowledgments
The authors are grateful to the Associate Editor, Manuel De la Sen, and two referees for
a number of helpful suggestions that have greatly improved our original submission. This
work is supported by the National Natural Science Foundation of China no. 70901016 and
Excellent Talents Program of Liaoning Educational Committee no. 2008RC15.
References
1 A. A. Berryman, “The origins and evolution of predator-prey theory,” Ecology, vol. 73, no. 5, pp. 1530–
1535, 1992.
2 X. Meng, J. Jiao, and L. Chen, “Global dynamics behaviors for a nonautonomous Lotka-Volterra
almost periodic dispersal system with delays,” Nonlinear Analysis. Theory, Methods & Applications, vol.
68, no. 12, pp. 3633–3645, 2008.
3 R. Xu and M. A. J. Chaplain, “Persistence and global stability in a delayed predator-prey system with
Michaelis-Menten type functional response,” Applied Mathematics and Computation, vol. 130, no. 2-3,
pp. 441–455, 2002.
Discrete Dynamics in Nature and Society
19
4 R. Xu, M. A. J. Chaplain, and F. A. Davidson, “Periodic solutions for a delayed predator-prey model
of prey dispersal in two-patch environments,” Nonlinear Analysis. Real World Applications, vol. 5, no.
1, pp. 183–206, 2004.
5 A. F. Nindjin and M. A. Aziz-Alaoui, “Persistence and global stability in a delayed Leslie-Gower type
three species food chain,” Journal of Mathematical Analysis and Applications, vol. 340, no. 1, pp. 340–357,
2008.
6 M. A. Aziz-Alaoui, “Study of a Leslie-Gower-type tritrophic population model,” Chaos, Solitons and
Fractals, vol. 14, no. 8, pp. 1275–1293, 2002.
7 M. A. Aziz-Alaoui and M. Daher Okiye, “Boundedness and global stability for a predator-prey model
with modified Leslie-Gower and Holling-type II schemes,” Applied Mathematics Letters, vol. 16, no. 7,
pp. 1069–1075, 2003.
8 A. F. Nindjin, M. A. Aziz-Alaoui, and M. Cadivel, “Analysis of a predator-prey model with
modified Leslie-Gower and Holling-type II schemes with time delay,” Nonlinear Analysis. Real World
Applications, vol. 7, no. 5, pp. 1104–1118, 2006.
9 C. S. Holling, “The functional response of predator to prey density and its role in mimicry and
population regulation,” Memoirs of the Entomological Society of Canada, vol. 45, pp. 1–60, 1965.
10 P. H. Leslie, “Some further notes on the use of matrices in population mathematics,” Biometrika, vol.
35, pp. 213–245, 1948.
11 P. H. Leslie and J. C. Gower, “The properties of a stochastic model for the predator-prey type of
interaction between two species,” Biometrika, vol. 47, pp. 219–234, 1960.
12 C. Letellier and M. A. Aziz-Alaoui, “Analysis of the dynamics of a realistic ecological model,” Chaos,
Solitons and Fractals, vol. 13, no. 1, pp. 95–107, 2002.
13 R. K. Upadhyay and S. R. K. Iyengar, “Effect of seasonality on the dynamics of 2 and 3 species preypredator systems,” Nonlinear Analysis. Real World Applications, vol. 6, no. 3, pp. 509–530, 2005.
14 X. Meng and L. Chen, “Periodic solution and almost periodic solution for a nonautonomous LotkaVolterra dispersal system with infinite delay,” Journal of Mathematical Analysis and Applications, vol.
339, no. 1, pp. 125–145, 2008.
15 M. De la Sen, “Stability of impulsive time-varying systems and compactness of the operators mapping
the input space into the state and output spaces,” Journal of Mathematical Analysis and Applications, vol.
321, no. 2, pp. 621–650, 2006.
16 J. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 2nd edition, 1977.
17 I. Barbălat, “Systems d’equations differential d’oscillations nonlinearities,” Revue Roumaine De
Mathematiques Pures et Appliquees, vol. 4, no. 2, pp. 267–270, 1959.
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