Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 262461, 17 pages
doi:10.1155/2010/262461
Research Article
Existence of 2n Positive Periodic Solutions to
n-Species Nonautonomous Food Chains with
Harvesting Terms
Yongkun Li1 and Kaihong Zhao1, 2
1
2
Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China
Department of Mathematics, Yuxi Normal University, Yuxi, Yunnan 653100, China
Correspondence should be addressed to Yongkun Li, yklie@ynu.edu.cn
Received 12 November 2009; Accepted 10 January 2010
Academic Editor: Gaston Mandata N’Guerekata
Copyright q 2010 Y. Li and K. Zhao. 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.
By using Mawhin’s continuation theorem of coincidence degree theory and some skills of
inequalities, we establish the existence of at least 2n positive periodic solutions for n-species
nonautonomous Lotka-Volterra type food chains with harvesting terms. An example is given to
illustrate the effectiveness of our results.
1. Introduction
The dynamic relationship between predators and their prey 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. These problems may appear to be simple mathematically
at first; sight, they are, in fact, very challenging and complicated. There are many different
kinds of predator-prey models in the literature. For more details, we refer to 1, 2. 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 persistent
property and positive periodic solution of the system; see 3–13 and the references cited
therein. However, to the best of the authors’ knowledge, to this day, still no scholar study the
n-species nonautonomous case of Food chain predator-prey system with harvesting terms.
Indeed, the exploitation of biological resources and the harvest of population species are
commonly practiced in fishery, forestry, and wildlife management; the study of population
dynamics with harvesting is an important subject in mathematical bioeconomics, which is
related to the optimal management of renewable resources see 14–16. This motivates us
to consider the following n-species nonautonomous Lotka-Volterra type food chain model
with harvesting terms:
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Advances in Difference Equations
ẋ1 t x1 ta1 t − b1 tx1 t − c12 tx2 t − h1 t,
..
.
ẋi t xi t−di t − bi txi t ci,i−1 txi−1 t − ci,i1 txi1 t − hi t,
1.1
..
.
ẋn t xn t−dn t − bn txn t cn,n−1 txn−1 t − hn t,
where i 2, 3, . . . , n − 1, xi t i 1, 2, . . . , n is the ith species population density, a1 t is the
growth rate of the first species that is the only producer in system 1.1, bi t i 1, 2, . . . , n
and hi t i 1, 2, . . . , n stand for the ith species intraspecific competition rate and harvesting
rate, respectively, di t i 2, 3, . . . , n is the death rate of the ith species, ci,i1 t i 1, 2, . . . , n − 1 represents the i 1th species predation rate on the ith species, and ci,i−1 ti 2, 3, . . . , n stands for the transformation rate from the i − 1th species to the ith species. In
addition, the effects of a periodically varying environment are important for evolutionary
theory as the selective forces on systems in a fluctuating environment differ from those
in a stable environment. Therefore, the assumptions of periodicity of the parameters are a
way of incorporating the periodicity of the environment e.g, seasonal effects of weather,
food supplies, mating habits, etc, which leads us to assume that a1 t, bi t, di t, cij t, and
hi t i, j 1, 2, . . . , n are all positive continuous ω-periodic functions.
Since a very basic and important problem in the study of a population growth model
with a periodic environment is the global existence and stability of a positive periodic
solution, which plays a similar role as a globally stable equilibrium does in an autonomous
model, this motivates us to investigate the existence of a positive periodic or multiple positive
periodic solutions for system 1.1. In fact, it is more likely for some biological species to take
on multiple periodic change regulations and have multiple local stable periodic phenomena.
Therefore, it is essential for us to investigate the existence of multiple positive periodic
solutions for population models. Our main purpose of this paper is by using Mawhin’s
continuation theorem of coincidence degree theory 17, to establish the existence of 2n
positive periodic solutions for system 1.1. For the work concerning the multiple existence of
periodic solutions of periodic population models which was done using coincidence degree
theory, we refer to 18–21.
The organization of the rest of this paper is as follows. In Section 2, by employing the
continuation theorem of coincidence degree theory and the skills of inequalities, we establish
the existence of at least 2n positive periodic solutions of system 1.1. In Section 3, an example
is given to illustrate the effectiveness of our results.
2. Existence of at Least 2n Positive Periodic Solutions
In this section, by using Mawhin’s continuation theorem and the skills of inequalities, we
shall show the existence of positive periodic solutions of 1.1. To do so, we need to make
some preparations.
Let X and Z be real normed vector spaces. Let L : Dom L ⊂ X → Z be a linear
mapping and N : X × 0, 1 → Z be a continuous mapping. The mapping L will be called
a Fredholm mapping of index zero if dim Ker L codim Im L < ∞ and Im L is closed in Z.
Advances in Difference Equations
3
If L is a Fredholm mapping of index zero, then there exist continuous projectors P : X → X
and Q : Z → Z such that Im P Ker L and Ker Q Im L Im I − Q, and X Ker L ⊕
Ker P, Z Im L ⊕ Im Q. It follows that L|Dom L∩Ker P : I − P X → Im L is invertible and
its inverse is denoted by KP . If Ω is a bounded open subset of X, the mapping N is called Lcompact on Ω×0, 1, if QNΩ×0, 1 is bounded and KP I−QN : Ω×0, 1 → X is compact.
Because Im Q is isomorphic to Ker L, there exists an isomorphism J : Im Q → Ker L.
The Mawhin’s continuous theorem 17, page 40 is given as follows.
Lemma 2.1 see 9. Let L be a Fredholm mapping of index zero and let N be L-compact on Ω ×
0, 1. Assume that
a for each λ ∈ 0, 1, every solution x of Lx λNx, λ is such that x /
∈ ∂Ω ∩ Dom L;
b QNx, 0x / 0 for each x ∈ ∂Ω ∩ Ker L;
c degJQNx, 0, Ω ∩ Ker L, 0 /
0.
Then Lx Nx, 1 has at least one solution in Ω ∩ Dom L.
the sake of convenience, we denote f l mint∈0,ω ft, f M maxt∈0,ω ft, f For
ω
1/ω 0 ft dt, respectively; here ft is a continuous ω-periodic function.
For simplicity, we need to introduce some notations as follows:
A±1
l1±
al1
−
M c12
l2
±
al1
−
M c12
l2
2
−
4b1M hM
1
2b1M
aM
1
±
2
− 4bnl hln
aM
1
2bnl
A±i B1±
−
l
ci,i−1
li−1
al1
−
,
A±n
M ci,i1
li1
−
diM
2
al1 − 4bnM hM
±
n
2bnM
−
l
cn,n−1
ln−1
±
,
li±
−
dnM
M ci,i−1
li−1
±
2
− 4bil hli
2bil
−
l
cn,n−1
ln−1
− dnM
2
−
l
M ci,i−1
li−1
− ci,i1
li1 − diM
−
l
ci,i−1
li−1
±
−
l
ci,i−1
li−1
2biM
where i 2, 3, . . . , n.
Throughout this paper, we need the following assumptions:
−
l
M M
l2 > 2 b1M hM
and
c
l
−
d
>
2
bnM hM
H1 al1 − c12
n ;
n
1
n,n−1 n−1
−
l
M li−1
− ci,i1
li1 − diM > 2 biM hM
H2 ci,i−1
i , i 2, 3, . . . , n − 1.
2
2
,
− 4bnM hM
n
,
2bnM
2biM
,
M ci,i−1
li−1
Bi±
±
− 4biM hM
i
− 4biM hM
i
,
,
2.1
4
Advances in Difference Equations
√
Lemma 2.2. Let x > 0, y > 0, z > 0 and x > 2 yz, for the functions fx, y, z x x2 − 4yz/2z and gx, y, z x − x2 − 4yz/2z, the following assertions hold:
1 fx, y, z and gx, y, z are monotonically increasing and monotonically decreasing on the
variable x ∈ 0, ∞, respectively.
2 fx, y, z and gx, y, z are monotonically decreasing and monotonically increasing on the
variable y ∈ 0, ∞, respectively.
3 fx, y, z and gx, y, z are monotonically decreasing and monotonically increasing on the
variable z ∈ 0, ∞, respectively.
Proof. In fact, for all x > 0, y > 0, z > 0, we have
x2 − 4yz
x
∂f
> 0,
∂x
2z x2 − 4yz
x2 − 4yz − x
∂g
< 0,
∂x
2z x2 − 4yz
2 − 4yz
x
−x
x
∂f
< 0,
∂z
2z2 x2 − 4yz
∂g
1
> 0,
∂y
x2 − 4yz
∂f
−1
< 0,
∂y
x2 − 4yz
2 − 4yz
x
x
x
−
∂g
> 0.
∂z
2z2 x2 − 4yz
2.2
By the relationship of the derivative and the monotonicity, the above assertions obviously
hold. The proof of Lemma 2.2 is complete.
Lemma 2.3. Assume that H1 and H2 hold, then we have the following inequalities:
ln li− < ln Bi− < lnA−i < lnAi < lnBi < lnli ,
i 1, 2, . . . , n.
2.3
Proof. Since
l
l
M aM
≥
a
>
a
−
c
l
>
2
b1M hM
1
12 2
1 > 0,
1
1
0 < b1l ≤ b1M ,
0 < hl1 ≤ hM
1 ,
−
l
l
M M ci,i−1
li−1 > ci,i−1
li− > ci,i−1
li−1
− ci,i1
li1 − diM > 0,
0 < bil ≤ biM ,
−
l
M
cn,n−1
ln−1
> cn,n−1
ln−1
0 < hli ≤ hM
i , i 2, 3, . . . , n − 1,
−
l
> cn,n−1
ln−1
− dnM > 2 bnM hM
0 < bnl ≤ bnM ,
n > 0,
0 < hln ≤ hM
n .
2.4
Advances in Difference Equations
5
By assumptions H1 , H2 , Lemma 2.2 and the expressions of A±i , Bi± , and li± , we have
l
l
l
M
M
−
l
M M
M
<
g
a
B
A−1
,
b
,
h
,
b
,
h
<
g
a
−
c
l
,
b
,
h
0 < l1− g aM
1
1
12 2
1
1
1 1
1 1
1
1
M M
l
l
< B1 f al1 , b1M , hM
< f aM
< A1 f al1 − c12
l2 , b1 , hM
1
1
1 , b1 , h1 l1 ,
M l
−
Bi−
li−1 , bil , hli < g ci,i−1
li−1
, biM , hM
0 < li− g ci,i−1
i
l
−
M A−i
< g ci,i−1
li−1
− ci,i1
li1 − diM , biM , hM
i
l
−
M < Ai f ci,i−1
li−1
− ci,i1
li1 − diM , biM , hM
i
l
−
M < f ci,i−1
< Bi f ci,i−1
li−1
, biM , hM
li−1 , bil , hli li ,
i
M
l
−
−
l
−
M
M
B
A−n
ln−1
, bnl , hln < g cn,n−1
ln−1
, bnM , hM
<
g
c
l
,
b
,
h
0 < ln− g cn,n−1
n
n
n
n,n−1 n−1 n
l
−
l
−
M
< Bn f cn,n−1
< f cn,n−1
< An f cn,n−1
ln−1
, bnM , hM
ln−1
, bnM , hM
ln−1
, bnl , hln ln ,
n
n
2.5
where i 2, 3, . . . , n − 1, that is 0 < li− < Bi− < A−i < Ai < Bi < li , i 1, 2, . . . , n. Thus, we
have ln li− < lnBi− < ln A−i < ln Ai < ln Bi < ln li , i 1, 2, . . . , n. The proof of Lemma 2.3 is
complete.
Theorem 2.4. Assume that H1 and H2 hold. Then system 1.1 has at least 2n positive ω-periodic
solutions.
Proof. By making the substitution
xi t exp{ui t},
i 1, 2, . . . , n,
2.6
system 1.1 can be reformulated as
u̇1 t a1 t − b1 teu1 t − c12 teu2 t − h1 te−u1 t ,
..
.
u̇i t −di t − bi teui t ci,i−1 teui−1 t − ci,i1 teui1 t − hi te−ui t ,
..
.
u̇n t −dn t − bn teun t cn,n−1 teun−1 t − hn te−un t ,
where i 2, 3, . . . , n − 1.
2.7
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Advances in Difference Equations
Let
X Z u u1 , u2 , . . . , un T ∈ CR, Rn : ut ω ut
2.8
and define
u n
i1
max |ui t|,
t∈0,ω
u ∈ X or Z.
2.9
Equipped with the above norm · , X and Z are Banach spaces. Let
⎛
a1 t − b1 teu1 t − λc12 teu2 t − h1 te−u1 t
⎞
⎟
⎜
..
⎟
⎜
⎟
⎜
.
⎟
⎜
⎟
⎜
ui t
ui−1 t
ui1 t
−ui t ⎟
−λd
t
−
b
te
c
te
−
λc
te
−
h
te
Nu, λ ⎜
i
i
i,i−1
i,i1
i
⎟
⎜
⎟
⎜
⎟
⎜
.
..
⎟
⎜
⎠
⎝
−λdn t − bn teun t cn,n−1 teun−1 t − hn te−un t
,
2.10
n×1
ω
where i 2, 3, . . . , n − 1, Lu u̇ dut/dt. We put P u 1/ω 0 ut dt, u ∈ X; Qz ω
ω
1/ω 0 zt dt, z ∈ Z. Thus it follows that Ker L Rn , ImL {z ∈ Z : 0 zt dt 0} is
closed in Z, dim Ker L n codim ImL, and P, Q are continuous projectors such that
ImP Ker L,
Ker Q ImL Im I − Q.
2.11
Hence, L is a Fredholm mapping of index zero. Furthermore, the generalized inverse to L
KP : Im L → Ker P Dom L is given by
t
1
KP z zs ds −
ω
0
ω s
zs ds.
0
0
2.12
Advances in Difference Equations
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Then
⎞
1 ω
⎜ ω F1 s, λds ⎟
0
⎟
⎜
⎟
⎜
..
⎟ ,
QNu, λ ⎜
2.13
.
⎟
⎜
⎟
⎜ ω
⎠
⎝1
Fn s, λds
ω 0
n×1
⎛t
⎞
ω t
ω
t
1
1
−
F1 s, λ dsdt F1 s, λds ⎟
⎜ F1 s, λ ds −
ω 0 0
2 ω
⎜ 0
⎟
0
⎜
⎟
..
⎜
⎟
KP I − QNu, λ ⎜
⎟ ,
.
⎜
⎟
ω t
⎜ t
⎟
ω
⎝
⎠
t
1
1
−
Fn s, λ ds −
Fn s, λ dsdt Fn s, λds
ω 0 0
2 ω
0
0
n×1
⎛
2.14
where
⎛
a1 s − b1 seu1 s − λc12 seu2 s − h1 se−u1 s
⎞
⎜
⎟
..
⎜
⎟
⎜
⎟
.
⎜
⎟
⎜
⎟
ui s
ui−1 s
ui1 s
−ui s ⎟
⎜
Fu, λ ⎜−λdi s − bi se
ci,i−1 se
− λci,i1 se
− hi se
⎟
⎜
⎟
⎜
⎟
.
..
⎜
⎟
⎝
⎠
−λdn s − bn seun s cn,n−1 seun−1 s − hn se−un s
.
2.15
n×1
Obviously, QN and KP I −QN are continuous. It is not difficult to show that KP I −QNΩ
is compact for any open bounded set Ω ⊂ X by using the Arzela-Ascoli theorem. Moreover,
QNΩ is clearly bounded. Thus, N is L-compact on Ω with any open bounded set Ω ⊂ X.
In order to use Lemma 2.1, we have to find at least 2n appropriate open bounded
subsets of X. Corresponding to the operator equation Lu λNu, λ, λ ∈ 0, 1, we have
u̇1 t λ a1 t − b1 teu1 t − λc12 teu2 t − h1 te−u1 t ,
..
.
u̇i t λ −λdi t − bi teui t ci,i−1 teui−1 t − λci,i1 teui1 t − hi te−ui t ,
..
.
u̇n t λ −λdn t − bn teun t cn,n−1 teun−1 t − hn te−un t ,
2.16
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Advances in Difference Equations
where i 2, 3, . . . , n − 1. Assume that u ∈ X is an ω-periodic solution of system 2.16 for
some λ ∈ 0, 1. Then there exist ξi , ηi ∈ 0, ω such that ui ξi maxt∈0,ω ui t, ui ηi mint∈0,ω ui t, i 1, 2, . . . , n. It is clear that u̇i ξi 0, u̇i ηi 0, i 1, 2, . . . , n. From this and
2.16, we have
a1 ξ1 − b1 ξ1 eu1 ξ1 − λc12 ξ1 eu2 ξ1 − h1 ξ1 e−u1 ξ1 0,
..
.
−λdi ξi − bi ξi eui ξi ci,i−1 ξi eui−1 ξi − λci,i1 ξi eui1 ξi − hi ξi e−ui ξi 0,
2.17
..
.
−λdn ξi − bn ξn eun ξn cn,n−1 ξn eun−1 ξn − hn ξn e−un ξn 0
a1 η1 − b1 η1 eu1 η1 − λc12 η1 eu2 η1 − h1 η1 e−u1 η1 0,
..
.
−λdi ηi − bi ηi eui ηi ci,i−1 ηi eui−1 ηi − λci,i1 ηi eui1 ηi − hi ηi e−ui ηi 0,
2.18
..
.
−λdn ηi − bn ηn eun ηn cn,n−1 ηn eun−1 ηn − hn ηn e−un ηn 0,
where i 2, 3, . . . , n − 1.
On one hand, according to 2.17, we have
u1 ξ1 b1l e2u1 ξ1 − aM
hl1 ≤ b1 ξ1 e2u1 ξ1 − a1 ξ1 eu1 ξ1 h1 ξ1 −λc12 ξ1 eu1 ξ1 u2 ξ1 < 0,
1 e
2.19
namely,
u1 ξ1 hl1 < 0,
b1l e2u1 ξ1 − aM
1 e
2.20
which implies that
ln l1− < u1 ξ1 < ln l1 ,
b2l e2u2 ξ2 hl2 < b2 ξ2 e2u2 ξ2 λc23 ξ2 eu2 ξ2 u3 ξ2 λd2 ξ2 eu2 ξ2 h2 ξ2 M u2 ξ2 l1 e
;
c21 ξ2 eu2 ξ2 u1 ξ2 < c21
2.21
2.22
that is,
M u2 ξ2 b2l e2u2 ξ2 − c21
l1 e
hl2 < 0,
2.23
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which implies that
ln l2− < u2 ξ2 < ln l2 .
2.24
By deducing for i 3, 4, . . . , n − 1, we obtain
bil e2ui ξi hli < bi ξi e2ui ξi λci,i1 ξi eui ξi ui1 ξi λdi ξi eui ξi hi ξi M ci,i−1 ξi eui ξi ui−1 ξi < ci,i−1
li−1 eui ξi ,
2.25
namely,
M bil e2ui ξi − ci,i−1
li−1 eui ξi hli < 0,
2.26
which implies that
ln li− < ui ξi < ln li ,
i 3, 4, . . . , n − 1,
bnl e2un ξi hln < bn ξn e2un ξn λdn ξn eun ξn hn ξn M
cn,n−1 ξn eun ξn un−1 ξn < cn,n−1
ln−1
eun ξn ,
2.27
2.28
namely,
M
bnl e2un ξn − cn,n−1
ln−1
eun ξn hln < 0,
2.29
ln ln− < un ξn < ln ln .
2.30
which implies that
In view of 2.20, 2.23, 2.26, and 2.29, we have
ln li− < ui ξi < ln li ,
i 1, 2, . . . , n.
2.31
i 1, 2, . . . , n.
2.32
From 2.18, one can analogously obtain
ln li− < ui ηi < ln li ,
By 2.30 and 2.31, we get
ln li− < ui ηi < ui ξi < ln li ,
i 1, 2, . . . , n.
2.33
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Advances in Difference Equations
On the other hand, in view of 2.17, we have
2u1 ξ1 a1 ξ1 eu1 ξ1 − h1 ξ1 −b1M e2u1 ξ1 al1 eu1 ξ1 − hM
1 ≤ −b1 ξ1 e
M u1 ξ1 λc12 ξ1 eu1 ξ1 u2 ξ1 < c12
l2 e
,
2.34
namely,
M b1M e2u1 ξ1 − al1 − c12
l2 eu1 ξ1 hM
1 > 0,
2.35
which implies that
ln A1 < u1 ξ1 or
u1 ξ1 < ln A−1 ,
2.36
l −
c21
l1 < c21 ξ2 eu1 ξ2 b2 ξ2 eu2 ξ2 h2 ξ2 e−u2 ξ2 λd2 ξ2 λc23 ξ2 eu3 ξ2 −u2 ξ2 M < b2M eu2 ξ2 hM
d2M c23
l3 ,
2 e
2.37
that is,
l −
M b2M e2u2 ξ2 − c21
l1 − c23
l3 − d2M eu2 ξ2 hM
2 > 0,
2.38
ln A2 < u2 ξ2 2003or u2 ξ2 < ln A−2 .
2.39
which implies that
By deducing for i 3, 4, . . . , n − 1, we obtain
l
−
ci,i−1
li−1
< ci,i−1 ξi eui−1 ξi bi ξi eui ξi hi ξi e−ui ξi λdi ξi λci,i1 ξi eui1 ξi −ui ξi M < biM eui ξi hM
diM ci,i1
li1 ,
i e
2.40
that is,
l
−
M biM e2ui ξi − ci,i−1
li−1
− ci,i1
li1 − diM eui ξi hM
i > 0,
2.41
which implies that
ln Ai < ui ξi or
ui ξi < ln A−i ,
i 3, 4, . . . , n − 1,
l
−
cn,n−1
ln−1
< cn,n−1 ξn eun−1 ξn bn ξn eun ξn hn ξn e−un ξn λdn ξn −un ξn dnM ,
< bnM eun ξn hM
n e
2.42
2.43
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namely,
l
−
bnM e2un ξn − cn,n−1
ln−1
− dnM eun ξn hM
n > 0,
2.44
which implies that
ln An < un ξn or
un ξn < ln A−n .
2.45
It follows from 2.35, 2.38, 2.41, and 2.44 that
ln Ai < ui ξi ui ξi < ln A−i ,
or
i 1, 2, . . . , n.
2.46
From 2.18, one can analogously obtain
ln Ai < ui ηi
or
ui ηi < ln A−i ,
i 1, 2, . . . , n.
2.47
By 2.32, 2.45, 2.46, and Lemma 2.3, we get
ln Ai < ui ηi < ui ξi < ln li
ln li− < ui ηi < ui ξi < ln A−i ,
or
i 1, 2, . . . , n,
2.48
which implies that, for all t ∈ R,
ln Ai < ui t < ln li
or
ln li− < ui t < ln A−i ,
i 1, 2, . . . , n.
2.49
For convenience, we denote
Gi ln li− , ln A−i ,
Hi ln Ai , ln li ,
i 1, 2, . . . , n.
2.50
Clearly, li± i 1, 2, . . . , n and A±i i 1, 2, . . . , n are independent of λ. For each i 1, 2, . . . , n, we choose an interval between two intervals Gi and Hi , and denote it as Δi , then
define the set
u u1 , u2 , . . . , un T ∈ X : ui t ∈ Δi , t ∈ R, i 1, 2, . . . , n .
2.51
Obviously, the number of the above sets is 2n . We denote these sets as Ωk , k j. Thus
1, 2, . . . , 2n . Ωk , k 1, 2, . . . , 2n are bounded open subsets of X, Ωi ∩ Ωj φ, i /
Ωk k 1, 2, . . . , 2n satisfies the requirement a in Lemma 2.1.
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Advances in Difference Equations
Now we show that b of Lemma 2.1 holds; that is, we prove when u ∈ ∂Ωk ∩ Ker L 0, 0, . . . , 0T , k 1, 2, . . . , 2n . If it is not true, then when u ∈ ∂Ωk ∩
∂Ωk ∩ Rn , QNu, 0 /
n
Ker L ∂Ωk ∩ R , i 1, 2, . . . , 2n , constant vector u u1 , u2 , . . . , un T , with u ∈ ∂Ωk , k 1, 2, . . . , 2n , satisfies
ω
a1 t − b1 teu1 − h1 te−u1 dt 0,
0
..
.
ω
−bi teui ci,i−1 teui−1 − hi te−ui ds 0,
2.52
0
..
.
ω
−bn teun cn,n−1 teun−1 − hn te−un ds 0,
0
where i 2, 3, . . . , n − 1. In view of the mean value theorem of calculous, there exist n points
ti i 1, 2, . . . , n such that
a1 t1 − b1 t1 eu1 − h1 t1 e−u1 0,
..
.
−bi ti eui ci,i−1 ti eui−1 − hi ti e−ui 0,
2.53
..
.
−bn tn eun cn,n−1 tn eun−1 − hn tn e−un 0,
where i 2, 3, . . . , n − 1. Following the argument of 2.20–2.47, from 2.52, we obtain
ln li− < ui < ln Bi− < ln A−i
or
ln Ai < ln Bi < ui < lnli ,
i 1, 2, . . . , n.
2.54
Then u belongs to one of Ωk ∩ Rn , k 1, 2, . . . , 2n . This contradicts the fact that u ∈ ∂Ωk ∩
Rn , k 1, 2, . . . , 2n . This proves that b in Lemma 2.1 holds.
Finally, in order to show that c in Lemma 2.1 holds, we only prove that for
u ∈ ∂Ωk ∩ Ker L ∂Ωk ∩ Rn , k 1, 2, . . . , 2n , then it holds that deg{JQNu, 0, Ωk ∩
0. To this end, we define the mapping φ : Dom L × 0, 1 → X by
Ker L, 0, 0, . . . , 0T } /
φ u, μ μQNu, 0 1 − μ Gu;
2.55
Advances in Difference Equations
13
here μ ∈ 0, 1 is a parameter and Gu is defined by
⎛ ω
−u1 s
u1 s
⎞
− h1 se
a1 s − b1 se
ds ⎟
⎜
⎟
⎜
0
⎟
⎜
..
⎟
⎜
.
⎟
⎜
⎟
⎜ ⎟
⎜ ω
⎟
⎜
M Gu ⎜
ci,i−1
li−1 − bi seui s − hi se−ui s ds ⎟
⎟
⎜ 0
⎟
⎜
..
⎟
⎜
⎟
⎜
.
⎟
⎜
⎜ ω ⎟
⎠
⎝
M
cn,n−1
ln−1
− bn seun s − hn se−un s ds
0
,
2.56
n×1
where i 2, 3, . . . , n − 1. We show that for u ∈ ∂Ωk ∩ Ker L ∂Ωk ∩ Rn , k 1, 2, . . . , 2n , μ ∈
0, 1, then it holds that φu, μ /
0, 0, . . . , 0T . Otherwise, parameter μ and constant vector
T
n
u u1 , u2 , . . . , un ∈ R satisfy φu, μ 0, 0 . . . , 0T , that is,
0μ
ω
a1 s − b1 se − h1 se
u1
−u1
ω
ds 1 − μ
a1 s − b1 seu1 − h1 se−u1 ds,
0
0
..
.
0μ
ω
ci,i−1 se
ui−1
− bi se − hi se
ui
−ui
ds 1 − μ
0
ω
0
M ci,i−1
li−1 − bi seui − hi se−ui ds,
..
.
0μ
ω
cn,n−1 seun−1 − bn seun − hn se−un ds
0
1−μ
ω
0
M
cn,n−1
ln−1
− bn seun − hn se−un ds,
2.57
where i 2, 3, . . . , n − 1. In view of the mean value theorem of calculous, there exist n points
ti ∈ 0, ω i 1, 2, . . . , n such that
a1 t1 − b1 t1 eu1 − h1 t1 e−u1 0,
..
.
M M ci,i−1
li−1 − bi ti eui − hi ti e−ui μ ci,i−1
li−1 − ci,i−1 ti eui−1 ,
..
.
M
M
cn,n−1
ln−1
− bn tn eun − hn tn e−un μ cn,n−1
ln−1
− ci,n−1 tn eun−1 ,
2.58
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Advances in Difference Equations
where i 2, 3, . . . , n − 1. Following the argument of 2.20–2.47, from 2.57, we obtain
ln li− < ui < ln Bi− < ln A−i
or
ln Ai < ln Bi < ui < ln li ,
i 1, 2, . . . , n.
2.59
given that u belongs to one of Ωk ∩ Rn , k 1, 2, . . . , 2n . This contradicts the fact that u ∈
0, 0, . . . , 0T holds. Note that the system of the
∂Ωk ∩ Rn , k 1, 2, . . . , 2n . This proves φu, μ /
following algebraic equations:
a1 t1 − b1 t1 ex1 − h1 t1 e−x1 0,
..
.
M ci,i−1
li−1 − bi ti exi − hi ti e−xi 0,
2.60
..
.
M
cn,n−1
ln−1
− bn tn exn − hn tn e−xn 0
has 2n distinct solutions since H1 and H2 hold, x1∗ , x2∗ , . . . , xn∗ ln x1 , ln x2 , . . . , ln xn ,
2
M
a1 t1 ±
a1 t1 − 4b1 t1 h1 t1 /2b1 t1 , x±k
ck,k−1
lk−1
±
where x1±
2
M
ck,k−1
lk−1
− 4bk tk hk tk /2bk tk k 2, 3, . . . , n, xi xi− or xi xi , i 1, 2, . . . , n.
Similar to the proof of Lemma 2.3, it is easy to verify that
ln li− < ln xi− < ln Bi− < ln A−i < ln Ai < ln Bi < ln xi < ln li ,
i 1, 2, . . . , n.
2.61
Therefore, x1∗ , x2∗ , . . . , xn∗ uniquely belongs to the corresponding Ωk . Since Ker L ImQ, we
can take J I. A direct computation gives, for k 1, 2, . . . , 2n ,
deg JQNu, 0, Ωk ∩ Ker L, 0, 0, . . . , 0T
deg φu, 1, Ωk ∩ Ker L, 0, 0, . . . , 0T
deg φu, 0, Ωk ∩ Ker L, 0, 0, . . . , 0T
⎡ ⎛
⎞⎤
n
hi ti
⎟⎥
⎢ ⎜
sign⎣ ⎝−bi ti xi∗ ∗ ⎠⎦.
x
i
i1
2.62
Advances in Difference Equations
15
M li−1 − bi ti xi∗ − hi ti /xi∗ 0 i 2, 3, . . . , n,
Since a1 t1 − b1 t1 x1∗ − h1 t1 /x1∗ 0, ci,i−1
then
deg JQNu, 0, Ωk ∩ Ker L, 0, 0, . . . , 0T
n
∗
M M ∗
sign
a1 t1 − 2b1 t1 x1
ci,i−1 li−1 − 2bi xi
±1, k 1, 2, . . . , 2n .
2.63
i2
So far, we have proved that Ωk k 1, 2, . . . , 2n satisfies all the assumptions in
Lemma 2.1. Hence, system 2.7 has at least 2n different ω-periodic solutions. Thus, by 2.6
system 1.1 has at least 2n different positive ω-periodic solutions. This completes the proof
of Theorem 2.4.
In system 1.1, if ci,i−1 t ≥ 0 i 2, 3, . . . , n, cj,j1 t ≥ 0 j 1, 2, . . . , n − 1, and
a1 t > 0, di t > 0, bi t > 0, hi t > 0 are continuous periodic functions, then similar to the
proof of Theorem 2.4, one can prove the following
Theorem 2.5. Assume that H1 and H2 hold. Then system 1.1 has at least 2n positive ω-periodic
solutions.
Remark 2.6. In Theorem 2.5, ci−1,i t 0 means that the ith species does not prey the i − 1th
species, thus ci,i−1 t 0. That is to say, there is no relationship between the ith species and
the i − 1th species.
3. Illustrative Examples
Example 3.1. Consider the following three-species food chain with harvesting terms:
4 sin t
9 cos t
xt − c12 tyt −
,
ẋt xt 3 sin t −
10
20
3 cos t 5 cos t
2 cos t
ẏt yt −
−
yt c21 txt − c23 tzt −
,
10
10
5
3 sin 2t 8 sin 2t
8 cos 2t
−
zt c32 tyt −
.
żt zt −
10
10
10
3.1
In this case, a1 t 3 sin t, b1 t 4 sin t/10, h1 t 9 cos t/20, d2 t 3 cos t/10, b2 t 5 cos t/10, h2 t 2 cos t/5, d3 t 3 sin 2t/10, b3 t 8 sin 2t/10, and h3 t 8 cos 2t/10. Since
l1±
aM
1 ±
aM
1
2
2b1l
− 4b1l hl1
√
20 ± 2 97
,
3
3.2
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Advances in Difference Equations
taking c21 t ≡ 14/5l1− , then we have
l2± M l1 ±
c21
M l1
c21
2
− 4b2l hl2
2b2l
7l1 ±
2
2
49 l1 − 2 l1−
2l1−
3.3
.
Take c32 t ≡ 12/5l2− , then
l3±
M c32
l2 ±
M c32
l2
2
− 4b3l hl3
2b3l
12l2 ±
12l2
2
7l2−
2
− 7l2−
.
3.4
Take c12 t ≡ 1/2l2 , c23 t ≡ 1/l3 , then
M al1 − c12
l2 3
> 1 2 b1M hM
1 ,
2
l −
l1
c21
−
M c23
l3
−
d2M
l −
c32
l2 − d3M 2 >
9
2 b3M hM
3 ,
5
7 6
> 2 b2M hM
2 .
5 5
3.5
Therefore, all conditions of Theorem 2.4 are satisfied. By Theorem 2.4, system 3.1 has at least
eight positive 2π-periodic solutions.
Acknowledgment
This work is supported by the National Natural Sciences Foundation of People’s Republic of
China under Grant no. 10971183.
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