Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 940287, 19 pages
doi:10.1155/2012/940287
Research Article
Application of Mawhin’s Coincidence Degree and
Matrix Spectral Theory to a Delayed System
Yong-Hui Xia,1, 2 Xiang Gu,1 Patricia J. Y. Wong,3 and Syed Abbas4
1
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
Center for Applied Mathematics and Theoretical Physics, University of Maribor,
Krekova 2, 2000 Maribor, Slovenia
3
School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798
4
School of Basic Sciences, Indian Institute of Technology Mandi, Mandi 175001, India
2
Correspondence should be addressed to Yong-Hui Xia, yhxia@zjnu.cn
Received 11 August 2012; Accepted 27 August 2012
Academic Editor: Jifeng Chu
Copyright q 2012 Yong-Hui Xia 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.
This paper gives an application of Mawhin’s coincidence degree and matrix spectral theory to
a predator-prey model with M-predators and N-preys. The method is different from that used
in the previous work. Some new sufficient conditions are obtained for the existence and global
asymptotic stability of the periodic solution. The existence and stability conditions are given in
terms of spectral radius of explicit matrices which are much different from the conditions given by
the algebraic inequalities. Finally, an example is given to show the feasibility of our results.
1. Introduction and Motivation
1.1. History and Motivations
Mawhin’s coincidence degree theory has been applied extensively to study the existence of
periodic solutions for nonlinear differential systems e.g. see 1–16 and references therein.
The most important step of applying Mawhin’s degree theory to nonlinear differential
equations is to obtain the priori bounds of unknown solutions to the operator equation
Lx λNx. However, different estimation techniques for the priori bounds of unknown
solutions to the equation Lx λNx may lead to different results. Most of papers obtained
the priori bounds by employing the inequalities:
xt ≤ xξ ω
|ẋt|dt,
0
xξ min xt,
t∈0,ω
ω
xt ≥ x η −
|ẋt|dt,
0
x η max xt.
t∈0,ω
1.1
2
Abstract and Applied Analysis
These inequalities lead to a relatively strong condition given in terms of algebraic inequality
or classic norms see e.g., 3–16. Different from standard consideration, in this paper, we
employ matrix spectral theory to obtain the priori bounds, not the above inequalities. So in
this paper, the existence and stability of periodic solution for a multispecies predator-prey
model is studied by jointly employing Mawhin’s coincidence degree and matrix spectral
theory.
1.2. Model Formulation
One of classical Lotka-Vlterra system is predator-prey models which have been investigated
extensively by mathematicians and ecologist. Many good results have been obtained
for stability, bifurcations, chaos, uniform persistence, periodic solution, almost periodic
solutions. It has been observed that most of works focus on either two or three species model.
There are few paper considering the multispecies model. To model the dynamic behavior of
multispecie predator-prey system, Yang and Rui 17 proposed a predator-prey model with
M-predators and N-preys of the form:
N
M
ẋi t xi t bi t − aik txk t − cil tyl t ,
k1
ẏj t yj t −rj t N
i 1, 2, . . . , N,
l1
1.2
M
djk txk t − ejl tyl t ,
k1
j 1, 2, . . . , M,
l1
where xi t denotes the density of prey species Xi at time t, yj t denotes the density of
predator species Yj at time t. The coefficients bi t, rj t, aik t, cil t, djk t, and ejl t, i, k 1, . . . , N; j, l 1, . . . , M are nonnegative continuous periodic functions defined on t ∈
−∞, ∞. The coefficient bi is the intrinsic growth rate of prey species Xi , rj is the death
rate of the predator species Yj , aik measures the amount of competition between the prey
i, i, k 1, . . . , N, ejl measures the amount of competition between
species Xi and Xk k /
Δ
j, j, k 1, . . . , M, and the constant kij dij /cij denotes
the predator species Yj and Yk k /
the coefficient in conversing prey species Xi into new individual of predator species Yj i 1, . . . , N; j 1, . . . , M. By using the differential inequality, Zhao and Chen 18 improved the
results of Yang and Rui 17. Recently, Xia et al. 19 obtained some sufficient conditions for
the existence and global attractivity of a unique almost periodic solution of the system 1.2.
It is more natural to consider the delay model because most of the species start
interacting after reaching a maturity period. Hence many scholars think that the delayed
models are more realistic and appropriate to be studied than ordinary model. Delayed system
is important also because sometimes time delays may lead to oscillation, bifurcation, chaos,
instability which may be harmful to a system. Inspired by the above argument, Wen 20
considered a periodic delayed multispecies predator-prey system as follows:
⎡
N
ẋi t xi t⎣bi t − aii txi t −
aik txk t − τik −
k1,k /
i
⎡
ẏj t yj t⎣−rj t N
k1
M
⎤
cil tyl t − ηil ⎦,
l1
djk txk t − δjk − ejj tyj t −
M
l1,l /
j
⎤
ejl tyl t − ξjl ⎦,
1.3
Abstract and Applied Analysis
3
where bi t, rj t, aik t, cil t, djk t, ejl t, ejl t i, k 1, 2, . . . , N; j, l 1, 2, . . . , M are
assumed to be continuous ω-periodic functions and the delays τik , δjk , ηil , ξjl are assumed
to be positive constants. The system 1.3 is supplemented with the initial condition:
xi θ φi θ,
yj θ ψi θ,
θ ∈ −τ, 0, φi 0 > 0, ψi 0 > 0,
1.4
where
τ max max τik ,
1≤i,k≤n
max
1≤i≤N,1≤l≤M
ηil , max ξjl ,
1≤j,l≤M
max
1≤j≤M,1≤k≤N
δjk
> 0.
1.5
It is easy to see that for such given initial conditions, the corresponding solution of the system
1.3 remains positive for all t ≥ 0. The purpose of this paper is to obtain some new and
interesting criteria for the existence and global asymptotic stability of periodic solution of the
system 1.3.
1.3. Comparison with Previous Work
To obtain the periodic solutions of the system 1.3, the method used in 20 is based
on employing the differential inequality and Brower fixed point theorem. Different from
consideration taken by 20, our method is based on combining matrix spectral theory
with Mawhin’s degree theory. In our method, we study the global asymptotic stability by
combining matrix’s spectral theory with Lyapunov functional method. The existence and
stability conditions are given in terms of spectral radius of explicit matrices. These conditions
are much different from the sufficient conditions obtained in 20.
1.4. Outline of This Work
The structure of this paper is as follows. In Section 2, some new and interesting sufficient
conditions for the existence of periodic solution of system 1.3 are obtained. Section 3 is
devoted to examining the stability of the periodic solution obtained in the previous section.
In Section 4, some corollaries are presented to show the effectiveness of our results. Finally,
an example is given to show the feasibility of our results.
2. Existence of Periodic Solutions
In this section, we will obtain some sufficient conditions for the existence of periodic solution
of the system 1.3.
2.1. Preliminaries on the Matrix Theory and Degree Theory
For convenience, we introduce some notations, definitions, and lemmas. Throughout this
paper, we use the following notations.
i We always use i, k 1, . . . , N; j, l 1, . . . , M, unless otherwise stated.
4
Abstract and Applied Analysis
ii If ft is a continuous ω-periodic function defined on R, then we denote
f min ft,
t∈0,ω
f max ft,
1
m f ω
t∈0,ω
ω
ftdt.
2.1
0
We use x x1 , . . . , xn T ∈ Rn to denote a column vector, D dij n×n is an n × n matrix, DT
denotes the transpose of D, and En is the identity matrix of size n. A matrix or vector D > 0
resp., D ≥ 0 means that all entries of D are positive resp., nonnegative. For matrices or
vectors D and E, D > E resp., D ≥ E means that D − E > 0 resp., D − E ≥ 0. We denote the
spectral radius of the matrix D by ρD.
If v v1 , v2 , . . . , vn T ∈ Rn , then we have a choice of vector norms in Rn , for instance
v1 , v2 , and v∞ are the commonly used norms, where
⎧
⎫1/2
n
⎨
⎬
,
|vi |2
v2 ⎩ j1
⎭
n
|vi |,
v1 j1
v∞ max|vi |.
1≤i≤n
2.2
We recall the following norms of matrices induced by respective vector norms. For instance
if A aij n×n , the norm of the matrix A induced by a vector norm · is defined by
Ap sup
v∈Rn ,v /0
Avp
vp
sup Avp sup Avp .
vp 1
vp ≤1
In particular one can show that A1 max1≤j≤n
1/2
λmax A A
norm.
T
max. eigenvalue of A A
T
1/2
2.3
n
|aij | column norm, A2 and A∞ max1≤i≤n nj1 |aij | row
i1
Definition 2.1 see 1, 21. Let X, Z be normed real Banach spaces, let L : Dom L ⊂ X → Z
be a linear mapping, and N : X → Z be a continuous mapping. The mapping L is called a
Fredholm mapping of index zero, if dimKer L codim Im L < ∞ and Im L is closed in Z.
If L is a Fredholm mapping of index zero and there exist continuous projectors P : X → X
and Q : Z → Z such that Im P Ker L , Ker Q Im L ImI − Q, then L | dom L ∩ Ker P :
I − P X → Im L is invertible. We denote the inverse of that map by KP . If Ω is an open
bounded subset of X, the mapping N will be called L-compact on Ω if QNΩ is bounded
and KP I − QN : Ω → X is compact. Since Im Q is isomorphic to Ker L, there exists an
isomorphism J : Im Q → Ker L.
Definition 2.2 see 1, 22. Let Ω ⊂ Rn be open and bounded, f ∈ C1 Ω, Rn ∩ CΩ, Rn and
y ∈ Rn /f∂Ω ∪ Nf , that is, y is a regular value of f. Here, Nf {x ∈ Ω : Jf x 0}, the
critical set of f and Jf is the Jacobian of f at x. Then the degree deg{f, Ω, y} is defined by
deg f, Ω, y x∈f −1 y sgn Jf x,
2.4
Abstract and Applied Analysis
with the agreement that
consult Deimling 22.
5
φ 0. For more details about Degree Theory, the reader may
Lemma 2.3 Continuation Theorem 1. Let Ω ⊂ X be an open and bounded set and L be a
Fredholm mapping of index zero and N be L-compact on Ω (i.e., QNΩ is bounded and KP I−QN :
Ω → X is compact). Assume
i for each λ ∈ 0, 1, x ∈ ∂Ω ∩ Dom L, Lx /
λNx;
ii for each x ∈ ∂Ω ∩ Ker L, QNx /
0 and deg{JQN, Ω ∩ Ker L, 0} /
0.
Then Lx Nx has at least one solution in Ω ∩ Dom L.
Definition 2.4 see 23, 24. A real n × n matrix A aij is said to be an M-matrix if aij ≤ 0,
i, j 1, 2, . . . , n, i /
j, and A−1 ≥ 0.
Lemma 2.5 see 23, 24. Let A ≥ 0 be an n × n matrix and ρA < 1, then En − A−1 ≥ 0, where
En denotes the identity matrix of size n.
Now we introduce some function spaces and their norms, which will be valid
throughout this paper. Denote
X Ut ut, vtT ∈ C1 R, RNM | Ut ω Ut ∀t ∈ R ,
Z Ut ut, vtT ∈ C R, RNM | Ut ω Ut ∀t ∈ R .
2.5
The norms are given by
|Un t|1 |Un t|0 U̇n t0 ,
|Un t|0 max |Un t|,
t∈0,ω
Un t0 max {|Un t|0 },
1≤n≤NM
Un t1 i 1, 2, . . . , N M,
max {|Un t|1 }.
2.6
1≤n≤NM
Obviously, X and Z, respectively, endowed with the norms · 1 and · 0 are Banach spaces.
2.2. Result on the Existence of Periodic Solutions
Theorem 2.6. Assume that the following conditions hold:
H1 : the system of algebraic equations:
mbi − maii ui −
N
maik uk −
k1,k /
i
m −rj
M
mcil wl 0,
N
M
m ejl wl 0,
m djk uk − m ejj wj −
k1
i 1, 2, . . . , N,
l1
2.7
j 1, 2, . . . , M,
l1,l /j
∗ T
has finite solution u∗1 , . . . , u∗N , w1∗ , . . . , wM
∈ RNM
with u∗ > 0, w∗ > 0;
6
Abstract and Applied Analysis
H2 : ρK < 1, where K PN×N QN×M
MM×N NM×M
PN×N pik
NM×NM
N×N
pik ,
QN×M qil N×M ,
MM×N mjk M×N ,
NM×M njl M×M ,
,
0,
i k,
−1
aik akk , i / k.
qil cil e−1
ll ,
2.8
mjk djk a−1
kk ,
0,
j l,
njl −1
ejl ell , j /
l.
Then system 1.3 has at least one positive ω-periodic solution.
Proof. Note that every solution
Ut ut, vtT u1 t, . . . , uN t, v1 t, . . . , vM tT ∈ X
2.9
of the system 1.3 with the initial condition is positive. By using the following changes of
variables:
ui t ln xi t,
vj t ln yj t,
i 1, 2, . . . , N, j 1, 2, . . . , M,
2.10
the system 1.3 can be rewritten as
u̇i t bi t − aii teui t −
N
aik teuk t−τik −
M
cil tevl t−ηil ,
k1,k /i
i 1, 2, . . . , N,
l1
N
M
v̇j t −rj t djk teuk t−δjk − ejj tevj t −
ejl tevl t−ξjl ,
2.11
j 1, 2, . . . , M.
l1,l /
j
k1
Obviously, system 1.3 has at least one ω-periodic solution which is equivalent to the system
2.11 having at least one ω-periodic solution. To prove Theorem 2.6, our main tasks are
to construct the operators i.e., L, N, P , and Q appearing in Lemma 2.3 and to find an
appropriate open set Ω satisfying conditions i, ii in Lemma 2.3.
For any Ut ∈ X, in view of the periodicity, it is easy to check that
Δi U, t bi t − aii teui t −
N
aik teuk t−τik −
k1,k /i
Δj U, t −rj t N
k1
djk te
uk t−δjk − ejj te
M
cil tevl t−ηil ∈ Z,
l1
vj t
−
M
l1,l /
j
2.12
ejl te
vl t−ξjl ∈ Z.
Abstract and Applied Analysis
7
Now, we define the operators L, N as follows:
L : Dom L ⊂ X −→ Z,
Lut, vt dui t dvj t
,
dt
dt
∈ Z,
!
N : X −→ Z is defined by NU Δi U, t
.
Δj U, t
2.13
Define, respectively, the projectors P : X → X and Q : Z → Z by
PU 1
ω
1
QU ω
ω
U ∈ X,
Utdt,
0
2.14
ω
U ∈ Z.
Utdt,
0
It is obvious that the domain of L in X is actually the whole space, and
Ker L {xt ∈ X | Lxt 0, i.e. ẋt 0} RNM ,
ω
ztdt 0 is closed in Z.
Im L zt ∈ Z |
2.15
0
Moreover, P, Q are continuous operators such that
Im P RN Ker L,
Im L Ker Q ImI − Q,
dim Ker L codim Im L N M < ∞.
2.16
It follows that L is a Fredholm mapping of index zero. Furthermore, the generalized inverse
to L KP : Im L → Dom L ∩ Ker P exists, which is given by
KP y t
1
ysds −
ω
0
ω t
ysdsdt.
0
2.17
0
Then QN : X → Z and KP I − QN : X → X are defined by
⎛
⎞
1 ω
Δi U, tdt ⎟
⎜ω
⎜
⎟
0
⎜
⎟,
QNU ⎜ ⎟
ω
⎝1
⎠
Δj U, tdt
ω 0
ω t
t
! ω
1
1
t
KP I − QNx NUsds −
−
NUsdsdt −
NUsds.
ω 0 0
ω 2
0
0
2.18
8
Abstract and Applied Analysis
Clearly, QN and KP I−QN are continuous. By using the generalized Arzela-Ascoli theorem,
it is not difficult to prove that KP I − QNΩ is relatively compact in the space X, · 1 .
The proof of this step is complete.
Then, in order to apply condition i of Lemma 2.3, we need to search for an
appropriate open bounded subset Ω, denoted by
Ω Un t ∈ X | |Un t|1 |Un t|0 U̇n t0 < hn .
2.19
Specifically, our aim is to find an appropriate hn . Corresponding to the operator equation
Lx λNx for each λ ∈ 0, 1, we have
⎡
N
u̇i t λ⎣bi t − aii teui t −
aik teuk t−τik −
M
k1,k /
i
⎤
cil tevl t−ηil ⎦,
l1
2.20
⎤
N
M
v̇j t λ⎣−rj t djk teuk t−δjk − ejj tevj t −
ejl tevl t−ξjl ⎦.
⎡
l1,l /j
k1
Since Ut ∈ X, each Un t, n 1, 2, . . . , N M, as components of Ut, is continuously
differentiable and ω-periodic. In view of continuity and periodicity, there exists ti ∈ 0, ω
such that ui ti maxt∈0,ω |ui t|, i 1, 2, . . . , N, and there also exists tNj ∈ 0, ω such that
vj tNj maxt∈0,ω |vj t|, j 1, 2, . . . , M. Accordingly, u̇i ti 0, v̇j tNj 0, and we get
N
bi ti − aii ti eui ti −
aik ti euk ti −τik −
k1,k /i
−rj tNj
M
cil ti evl tNj −ηil 0,
l1
N
M
ejl tNj evl tNj −ξjl 0.
djk tNj euk ti −δjk − ejj tNj evj tNj −
2.21
l1,l /j
k1
That is,
N
aii ti eui ti bi ti −
M
aik ti euk ti −τik −
k1,k /i
ejj tNj e
vj tNj −rj tNj N
cil ti evl tNj −ηil ,
l1
djk tNj e
uk ti −δjk k1
−
M
ejl tNj e
2.22
vl tNj −ξjl .
l1,l /
j
Note that uk tk maxt∈0,ω |uk t| and vl tNl maxt∈0,ω |vl t|, which implies
uk ti ≤ uk tk ,
vl tNj ≤ vl tNl ,
uk ti − τik ≤ uk tk ,
vl tNj − ηil ≤ vl tNl ,
uk ti − δjk ≤ uk tk ;
vl tNj − ξjl ≤ vl tNl .
2.23
Abstract and Applied Analysis
9
It follows that
aii eui ti ≤ aii ti eui ti N
M
u
t
−τ
v
t
−η
bi ti −
aik ti e k i ik − cil ti e l Nj il k1,k /
i
l1
≤ bi N
aik euk ti −τik M
cil evl tNj −ηil k1,k /
i
≤ bi N
l1
aik euk tk k1,k /
i
ejj evj tNj M
cil evl tNl ,
l1
≤ ejj tNj evj tNj N
M
uk ti −δjk vl tNj −ξjl −rj tNj djk tNj e
−
ejl tNj e
k1
l1,l /
j
≤ rj N
M
djk euk ti −δjk ejl evl tNj −ξjl l1,l /
j
k1
≤ rj 2.24
N
M
djk euk tk ejl evl tNl .
l1,l /
j
k1
Let
aii eui ti zi ti ,
ejj evj tNj zj tNj .
2.25
Using 2.25, the inequalities 2.24 become
N
zi ti ≤ bi k1,k /
i
zj tNj
aik a−1
kk zk tk M
l1
cil e−1
l tNl ,
ll z
N
M
z
ejl e−1
l tNl ,
≤ r j djk a−1
t
k
k
kk
ll z
2.26
l1,l /
j
k1
or
N
zi ti −
k1,k /i
zj tNj −
N
k1
aik a−1
kk zk tk −
djk a−1
kk zk tk −
M
cil e−1
l tNl ≤ bi ,
ll z
l1
M
l1,l /
j
2.27
ejl e−1
l tNl ll z
≤ rj ,
10
Abstract and Applied Analysis
which implies
⎞
⎞ ⎛
z1 t1 ,
b1 ,
⎜ . . . , ⎟ ⎜. . . ,⎟
⎟
⎜
⎟ ⎜
!
⎟
⎜ z t , ⎟ ⎜
−QN×M
⎜ N N ⎟ ⎜bN ,⎟
×⎜
⎟,
⎟≤⎜
EM×M − NM×M NM×NM ⎜ z1 tN1 , ⎟ ⎜ r 1 , ⎟
⎟
⎜
⎟ ⎜
⎝ . . . , ⎠ ⎝. . . ,⎠
zM tNM rM
⎛
EN×N − PN×N
−MM×N
2.28
where
PN×N pik
N×N
pik ,
QN×M qil N×M ,
MM×N mjk M×N ,
NM×M njl M×M ,
0,
i k,
aik a−1
,
kk
i/
k,
qil cil e−1
ll ,
mjk djk a−1
kk ,
0,
j l,
njl −1
ejl ell , j /
l.
2.29
Set D b1 , . . . , bN , r 1 , . . . , r M T . It follows from 2.28 and H2 that
E − Kz1 t1 , . . . , zN tN , z1 tN1 , . . . , zM tNM T ≤ D.
2.30
In view of ρK < 1 and Lemma 2.5, we get ENM − K−1 ≥ 0. Let
T
N, h
N1 , . . . , h
NM : E − K−1 D ≥ 0.
1, . . . , h
H h
2.31
Using 2.30 and 2.31, we get
z1 t1 , . . . , zN tN , z1 tN1 , . . . , zM tNM T ≤ H,
2.32
or
i,
zi ti ≤ h
i 1, 2, . . . , N,
Nj ,
zj tNj ≤ h
j 1, 2, . . . , M.
2.33
Then
ui ti ≤ ln
i
h
,
aii
Nj
h
vj tNj ≤ ln
,
ejj
2.34
Abstract and Applied Analysis
11
which implies
(
Nj
i
h
h
.
|un t|0 max |un t| max ui ti , vj tNj max ln , ln
aii
ejj
t∈0,ω
t∈0,ω
2.35
On the other hand, it follows from 2.31 that
E − KH D,
or H KH D,
2.36
that is
N
i h
k pik h
k1,k /
i
M
Nl bi ,
qil h
l1
2.37
N
M
k Nl r j .
mjk h
njl h
Nj
h
l1,l /
j
k1
Estimating 2.20, by using 2.25, 2.33, and 2.37, we have
N
M
aik teuk t−τik − cil tevl t−ηil |u̇i t|0 λbi t − aii teui t −
k1,k /
i
l1
N
M
≤ bi aii eui t aik euk t−τik cil evl t−ηil 0
0
k1,k /
i
N
bi aii eui ti aik euk tk k1,k /
i
bi aii a−1
ii zi ti N
bi zi ti M
N
N
k1,k /
i
k1,k /
i
i h
i 2h
i,
h
cil evl tNl aik a−1
kk zk tk aik a−1
kk zk tk k pik h
0
l1
k1,k /
i
i ≤ bi h
l1
M
l1
Nl
qil h
M
l1
cil e−1
l tNl ll z
M
cil e−1
l tNl ll z
l1
0
12
Abstract and Applied Analysis
N
M
vl t−ξjl v̇j t λ−rj t djk teuk t−δjk − ejj tevj t −
e
te
jl
0
k1
l1,l /
j
0
≤ rj N
M
djk euk t−δjk ejl evl t−ξjl ejj evj t 0
k1
rj N
djk euk tk N
k1
rj N
k1
≤ rj N
M
0
ejl evl tNl ejj evj tNj l1,l /
j
k1
rj 0
l1,l /j
djk a−1
kk zk tk djk a−1
kk zk tk k mjk h
M
M
l1,l /
j
M
l1,l /
j
ejl e−1
l tNl ejj e−1
ll z
jj zj tNj
ejl e−1
l tNl zj tNj
ll z
Nj
Nl h
njl h
l1,l /
j
k1
Nj h
Nj 2h
Nj .
h
2.38
The above relations imply
Nj .
i , 2h
|u̇n t|0 max |u̇n t| max u̇i t, v̇j t max 2h
t∈0,ω
t∈0,ω
2.39
Further, it follows form the definition of norm that
Nj
i
h
h
|un t|1 |un t|0 |u̇n t|0 max ln , ln
aii
ejj
(
i , 2h
Nj .
max 2h
2.40
Let us set the following:
Nj
i
h
h
hn max ln , ln
aii
ejj
(
i , 2h
Nj d,
max 2h
2.41
where d is any positive constant.
Then for any solution of Lx λNx, we have |un t|1 |un t|0 |u̇n t|0 < hn for all
n 1, 2, . . . , N M. Obviously, hn are independent of λ and the choice of Ut. Consequently,
by taking this hn , the open subset Ω satisfies that Ω ∩ Dom L, that is, the open subset Ω
satisfies the assumption i of Lemma 2.3.
Now in the last step of the proof, we need to verify that for the given open bounded
set Ω obtained in Step 2, the assumption ii of Lemma 2.3 also holds. That is, for each U ∈
∂Ω ∩ Ker L, QNU /
0 and deg{JQN, Ω ∩ Ker L, 0} /
0.
Abstract and Applied Analysis
13
Take U ∈ ∂Ω ∩ Ker L. Then, in view of Ker L RNM , U is a constant vector in RNM ,
denoted by U u1 , . . . , uN , v1 , . . . , vM T and with the property
|Un | |Un |0 |Un |1 hn .
2.42
By operating U by QN gives
⎞
N
M
ui
uk
vl
−
ma
−
mc
−
ma
mb
e
e
e
i
ii
ik
il
⎟
⎜
⎟
⎜
k1,k l1
/i
QNUn ⎜ N ⎟.
M
⎠
⎝
m −rj m djk euk − m ejj evj −
m ejl evl
⎛
2.43
l1,l /
j
k1
It is easy to obtain that QNUn and deg{JQN, Ω ∩ Ker L, 0} / 0, where deg· is the Brouwer
degree and J is the identity mapping since Im Q Ker L. We have shown that the open subset
Ω ⊂ X satisfies all the assumptions of Lemma 2.3. Hence, by Lemma 2.3, the system 2.11
has at least one positive ω-periodic solution in Dom L ∩ Ω. By 2.10, the system 1.3 has at
least one positive ω-periodic solution. This completes the proof of Theorem 2.6.
3. Globally Asymptotic Stability
Under the assumption of Theorem 2.6, we know that system 1.3 has at least one positive
∗
∗
t, y1∗ t, . . . , yM
tT . The aim of this
ω-periodic solution, denoted by X ∗ t x1∗ t, . . . , xN
section is to derive a set of sufficient conditions which guarantee the existence and global
asymptotic stability of the positive ω-periodic solution X ∗ t.
Before the formal analysis, we recall some facts which will be used in the proof.
Lemma 3.1 see 25. Let f be a nonnegative function defined on 0, ∞ such that f is integrable
on 0, ∞ and is uniformly continuous on 0, ∞. Then limt → ∞ ft 0.
Lemma 3.2 see 23, 24. Let K Γij n×n be a matrix with nonpositive off-diagonal elements. K
is an M-matrix if and only if there exists a positive diagonal matrix ξ diagξ1 , ξ2 , . . . , ξn such that
ξi aii >
ξj aij ,
j/
i
i 1, 2, . . . , n.
3.1
Theorem 3.3. Assume that all the assumptions in Theorem 2.6 hold. Then system 1.3 has a unique
positive ω-periodic solution X ∗ t which is globally asymptotically stable.
Proof. Let Xt xt, ytT x1 t, . . . , xN t, y1 t, . . . , yM tT be any positive solution
of system 1.3. It is easy to see that ρKT ρK < 1. Thus, in view of Lemma 2.5 and
Definition 2.4, E − KT is an M-matrix, where E denotes an identity matrix of size N M.
Therefore, by Lemma 3.2, there exists a diagonal matrix L diagα1 , . . . , αN , β1 , . . . , βM with
14
Abstract and Applied Analysis
positive diagonal elements such that the product E − KT L is strictly diagonally dominant
with positive diagonal entries, namely,
N
αi aii >
αk aki M
k1,k /i
N
βj ejj >
i 1, . . . , N,
βl dli ,
l1
αk ckj M
3.2
βl elj ,
j 1, . . . , M.
l1,l /
j
k1
Now, we define a Lyapunov function V t as follows:
⎡
N
V t αi ⎣ln xi t − ln xi∗ t i1
M t
l1
M
N
t
K1,K /
i
t−τik
⎤
cil s ηil yl t − yl∗ tds⎦
t−ηil
⎡
N t
∗
⎣
βj ln yj t − ln yj t j1
K1
M t
aik s τik xk t − xk∗ tds
l1,l /j
ejl
t−ξjl
3.3
djk
s δjk xk t − x∗ tds
k
t−δjk
⎤
s ξjl yl t − yl∗ tds⎦,
t ≥ t0 .
Calculating the upper right derivative of V t and using 3.2, we get
⎡
N
N
D V t ≤
αi ⎣ − aii txi t − xi∗ t aik t τik xk t − xk∗ t
k1,k /i
i1
⎤
M
cil t ηil yl t − yl∗ t⎦
l1
⎡
M
N
βj ⎣ − ejj tyj t − yj∗ t djk t δjk xk t − xk∗ t
j1
k1
M
l1,l /
j
ejl
⎤
t ξjl yl t − yl∗ t⎦
Abstract and Applied Analysis
⎡
N
N
≤ − αi ⎣ − aii xi t − xi∗ t aik xk t − xk∗ t
15
k1,k /
i
i1
⎤
M
cil yl t − yl∗ t⎦
l1
⎡
M
N
βj ⎣ − ejj yj t − yj∗ t djk xk t − xk∗ t
j1
k1
⎤
ejl yl t − y∗ t⎦
M
l1,l /
j
l
⎛
⎞
N
N
M
− ⎝αi aii −
αk aki − βl dli ⎠xi t − xi∗ t
k1,k /i
i1
l1
⎛
⎞
M
N
M
− ⎝βj ejj − αk ckj −
βl elj ⎠yj t − yj∗ t
j1
k1
l1,l /
j
⎧
⎫
N M
⎨
⎬
x t − x∗ t yj t − y∗ t ,
−c
i
j
⎩ i1 i
⎭
j1
3.4
where
⎧
⎫
N
M
N
M
⎨
⎬
c min αi aii −
αk aki − βl dli , βj ejj − αk ckj −
βl elj > 0.
⎩
⎭
k1,k i
l1
k1
l1,l j
/
3.5
/
It follows from 3.4 that D V t ≤ 0. Obviously, the zero solution of 1.3 is Lyapunov stable.
On the other hand, integrating 3.4 over t0 , t leads to
⎤
⎡
t N M
V t − V t0 −c ⎣ xi s − xi∗ s yj s − yj∗ s⎦ds,
t0
or
i1
⎤
⎡
t N M
V t c ⎣ xi s − xi∗ s yj s − yj∗ s⎦ds V t0 < ∞,
t0
i1
t 0,
3.6
t t0 .
3.7
j1
j1
Noting that V t 0, it follows that
⎤
⎡
t N M
⎣ xi s − xi∗ s yj s − yj∗ s⎦ds V t0 < ∞,
c
t0 i1
j1
t t0 .
3.8
16
Abstract and Applied Analysis
Therefore, by Lemma 3.1, it is not difficult to conclude that
lim Xi t − Xi∗ t 0.
t → ∞
3.9
Theorem 3.3 follows.
4. Corollaries and Remarks
In order to illustrate some features of our main results, we will present some corollaries and
remarks in this section. From the proofs of Theorems 2.6 and 3.3, one can easily deduce the
following corollary.
Corollary 4.1. In addition to H1 , further suppose that E − K or E − KT is an M-matrix. Then
system 1.3 has a unique positive ω-periodic solution which is globally asymptotically stable.
Now recall that for a given matrix K, its spectral radius ρK is equal to the minimum
of all matrix norms of K, that is, for any matrix norm · , ρK ≤ K. Therefore, we have
the following corollary.
Corollary 4.2. In addition to H1 , if one further supposes that there exist positive constants ξi , i 1, 2, . . . , n, ηj , j 1, 2, . . . , m such that one of the following inequalities holds.
m
ξk−1 ni1,i / k ξi aik ηj djk }, max1≤l≤m {e−1
ηl−1 ni1 ξi cli 1 max{max1≤k≤n {a−1
l1
kk
ll
m
j1,j /
l ηj ejl }} < 1, or equivalently, for all k 1, . . . , n, l 1, 2, . . . , n,
ξk akk >
ηl ell >
n
ξi aik i1,i /
k
n
ξi cli m
ηj djk ,
l1
m
4.1
ηj ejl .
j1,j /
l
i1
nm nm
ξi−1 ξj kij 2 < 1, where K kij nm×nm has been defined in Theorem 2.6.
m
−1 n
−1 −1 n
3 max{max1≤i≤n {a−1
i ξk aki l1 ηl dli }, max1≤j≤m {ejj ηj k1 ξk ckj ii ξi k1,k /
m
l1,l /
j ηl elj }} < 1, or equivalently, for all i 1, . . . , n, j 1, 2, . . . , n,
2
i1
j1
ξi aii >
n
ξk |aki | k1,k /
i
ηj ejj >
m
ηl |dli |,
l1
n
m
ξk ckj ηl elj .
k1
4.2
l1,l /
j
Then system 1.3 has a unique positive ω-periodic solution which is globally asymptotically stable.
Proof. For any matrix norm · and any nonsingular matrix S, KS S−1 KS also
defines a matrix norm. Let us denote D diagξ1 , ξ2 , . . . , ξn , then the conditions 1.2 and
1.3 correspond to the column norms and Frobenius norm of matrix DKD−1 , respectively.
Abstract and Applied Analysis
17
Asymptotic behavior of the periodic solution
18
16
14
x
12
10
8
6
4
2
0
0
5
10
15
20
25
t
x1 (t)
x2 (t)
y1 (t)
Figure 1: Asymptotic behavior of system 5.1 with initial values x1 0, x2 0, y1 0 1, 1, 1,
0.3, 0.3, 0.3, 7, 7, 7, respectively, t ∈ 0, 25.
Condition 2.10 corresponds to the row norms of DKT D−1 and note that ρDKT D−1 ρDKD−1 . Now Corollary 4.2 follows immediately.
5. Example
In this section, an example and its simulations are presented to illustrate the feasibility and
effectiveness of our results.
Example 5.1. Consider the following periodic predator-prey model with 2-predators and 1prey:
)
*
1
1
ẋ1 t x1 t 7 sin t − x1 t − x2 t − y1 t − 1 ,
4
10
)
*
1
1
ẋ2 t x2 t 7 cos t − x1 t − x2 t − y1 t ,
5.1
4
4
)
*
1
1
1
3
ẏ1 t y1 t − 1 cos t x1 t − 1 x2 t − y1 t .
20
2
4
2
Simple computation leads to
⎛
0
⎜
⎜
⎜
K ⎜a−1
a
⎜ 11 21
⎝
a−1
22 a12
0
−1
a−1
11 d11 a22 d12
⎞
−1
e11
c11
⎛
⎜0
⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜1
−1
e11
c21 ⎟ ⎜
⎟ ⎜4
⎠ ⎜
⎜
⎝3
0
2
⎞
1 1
4 5⎟
⎟
⎟
⎟
⎟
0 0 ⎟.
⎟
⎟
⎟
1 ⎠
0
4
5.2
18
Abstract and Applied Analysis
By using mathematica, we see that ρK 0.633982 < 1. Thus, the system 5.1 has a periodic
solution which is globally asymptotically stable. Figure 1 shows the asymptotic behavior of
the periodic solution.
Remark 5.2. In this example, one can observe that though the spectral ρK < 1, the matrix
norms of the matrix K are all bigger than 1. For instance, the column norm: is
⎧
⎨
3
⎫
⎬
3 1
aij 0 > 1.
K1 max a−1
jj
⎭
1≤j≤3 ⎩
2 4
i1,i /
j
5.3
Acknowledgments
This work was supported by NNSFC no. 11271333 and no. 11171090. The first author also
acknowledges the supports by the foundation of Slovene human resources development and
scholarship fund. He thanks Professor Valery G. Romanovski and Professor Marko Robnik
for warm hospitality during his stay at CAMTP in University of Maribor as a Research
Fellow.
References
1 R. E. Gaines and J. L. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, Springer, Berlin,
Germany, 1977.
2 J. K. Hale and J. Mawhin, “Coincidence degree and periodic solutions of neutral equations,” Journal
of Differential Equations, vol. 15, pp. 295–307, 1974.
3 J. Mawhin, “Leray-Schauder continuation theorems in the absence of a priori bounds,” Topological
Methods in Nonlinear Analysis, vol. 9, no. 1, pp. 179–200, 1997.
4 A. Capietto, J. Mawhin, and F. Zanolin, “Continuation theorems for periodic perturbations of
autonomous systems,” Transactions of the American Mathematical Society, vol. 329, no. 1, pp. 41–72,
1992.
5 Z. Zhang and J. Luo, “Multiple periodic solutions of a delayed predator-prey system with stage
structure for the predator,” Nonlinear Analysis, vol. 11, no. 5, pp. 4109–4120, 2010.
6 Z. Zhang and Z. Hou, “Existence of four positive periodic solutions for a ratio-dependent predatorprey system with multiple exploited or harvesting terms,” Nonlinear Analysis, vol. 11, no. 3, pp.
1560–1571, 2010.
7 Y. Li, “Periodic solutions of a periodic delay predator-prey system,” Proceedings of the American
Mathematical Society, vol. 127, no. 5, pp. 1331–1335, 1999.
8 M. Fan, Q. Wang, and X. Zou, “Dynamics of a non-autonomous ratio-dependent predator-prey
system,” Proceedings of the Royal Society of Edinburgh A, vol. 133, no. 1, pp. 97–118, 2003.
9 H.-F. Huo, “Periodic solutions for a semi-ratio-dependent predator-prey system with functional
responses,” Applied Mathematics Letters, vol. 18, no. 3, pp. 313–320, 2005.
10 W. Ding and M. Han, “Dynamic of a non-autonomous predator-prey system with infinite delay and
diffusion,” Computers & Mathematics with Applications, vol. 56, no. 5, pp. 1335–1350, 2008.
11 Y. Xia, J. Cao, and S. S. Cheng, “Periodic solutions for a Lotka-Volterra mutualism system with several
delays,” Applied Mathematical Modelling, vol. 31, no. 9, pp. 1960–1969, 2007.
12 Y. Xia and M. Han, “Multiple periodic solutions of a ratio-dependent predator-prey model,” Chaos,
Solitons & Fractals, vol. 39, no. 3, pp. 1100–1108, 2009.
13 J. Chu and M. Li, “Positive periodic solutions of Hill’s equations with singular nonlinear
perturbations,” Nonlinear Analysis A, vol. 69, no. 1, pp. 276–286, 2008.
14 J. Chu and J. J. Nieto, “Impulsive periodic solutions of first-order singular differential equations,”
Bulletin of the London Mathematical Society, vol. 40, no. 1, pp. 143–150, 2008.
Abstract and Applied Analysis
19
15 J. Wang, Y. Zhou, and W. Wei, “Impulsive problems for fractional evolution equations and optimal
controls in infinite dimensional spaces,” Topological Methods in Nonlinear Analysis, vol. 38, no. 1, pp.
17–43, 2011.
16 J. Wang, Y. Zhou, and M. Medved, “Qualitative analysis for nonlinear fractional differential equations
via topological degree method,” Topological Methods in Nonlinear Analysis. In press.
17 P. Yang and X. Rui, “Global attractivity of the periodic Lotka-Volterra system,” Journal of Mathematical
Analysis and Applications, vol. 233, no. 1, pp. 221–232, 1999.
18 J. Zhao and W. Chen, “Global asymptotic stability of a periodic ecological model,” Applied Mathematics
and Computation, vol. 147, no. 3, pp. 881–892, 2004.
19 Y. Xia, F. Chen, A. Chen, and J. Cao, “Existence and global attractivity of an almost periodic ecological
model,” Applied Mathematics and Computation, vol. 157, no. 2, pp. 449–475, 2004.
20 X. Z. Wen, “Global attractivity of a positive periodic solution of a multispecies ecological competitionpredator delay system,” Acta Mathematica Sinica, vol. 45, no. 1, pp. 83–92, 2002.
21 D. Guo, J. Sun, and Z. Liu, Functional Method in Nonlinear Ordinary Differential Equations, Shangdong
Scientific Press, Shandong, China, 2005.
22 K. Deimling, Nonlinear Functional Analysis, Springer, New York, NY, USA, 1985.
23 J. P. LaSalle, The Stability of Dynamical System, SIAM, Philadelphia, Pa, USA, 1976.
24 A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Science, Academic Press, New
York, NY, USA, 1929.
25 I. Barbălat, “Systems d’equations differentielle d’oscillations nonlineaires,” Revue Roumaine de
Mathématique Pures et Appliquées, vol. 4, pp. 267–270, 1959.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014