Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2013, Article ID 580185, 11 pages
http://dx.doi.org/10.1155/2013/580185
Research Article
Global Attractivity of a Periodic Delayed 𝑁-Species Model of
Facultative Mutualism
Ahmadjan Muhammadhaji and Zhidong Teng
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China
Correspondence should be addressed to Zhidong Teng; zhidong@xju.edu.cn
Received 13 November 2012; Accepted 20 January 2013
Academic Editor: Beatrice Paternoster
Copyright © 2013 A. Muhammadhaji and Z. Teng. 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.
Two classes of periodic 𝑁-species Lotka-Volterra facultative mutualism systems with distributed delays are discussed. Based on
the continuation theorem of the coincidence degree theory developed by Gaines and Mawhin and the Lyapunov function method,
some new sufficient conditions on the existence and global attractivity of positive periodic solutions are established.
1. Introduction
Mutualism is the interaction of two species of organisms
that benefits both [1]. In general, mutualism may be either
obligate or facultative. Obligate mutualist may survive only by
association, and facultative mutualist, while benefiting from
the presence of each other, may also survive in the absence
of any of them [2]. As it is well known, in recent years the
nonautonomous and periodic population dynamical systems
are extensively studied. The basic and important studied
questions for these systems are the persistence, permanence,
and extinction of species, global stability of systems and
the existence of positive periodic solutions, positive almost
periodic solutions and strictly positive solutions, and so forth.
Many important and influential results have been established
and can be found in many articles and books. Particularly,
the existence of positive periodic solutions for various type
population dynamical systems has been extensively studied
in [1–16] and the references cited therein.
In [7], the authors studied the following delayed twospecies model of facultative mutualism:
𝑦̇ 1 (𝑡) = 𝑦1 (𝑡) [𝑟1 (𝑡) − 𝑎1 (𝑡) 𝑦1 (𝑡) − 𝑏1 (𝑡) 𝑦1 (𝑡 − 𝜏1 (𝑡))
+ 𝑐1 (𝑡) 𝑦2 (𝑡 − 𝑝1 (𝑡))] ,
𝑦̇ 2 (𝑡) = 𝑦2 (𝑡) [𝑟2 (𝑡) − 𝑎2 (𝑡) 𝑦2 (𝑡) − 𝑏2 (𝑡) 𝑦2 (𝑡 − 𝜏1 (𝑡))
+ 𝑐2 (𝑡) 𝑦1 (𝑡 − 𝑝1 (𝑡))] .
(1)
By using the technique of coincidence degree and the Lyapunov functionals method, the sufficient conditions for the
existence and globally asymptotic stability of positive periodic solutions are obtained for system (1). In [2], the authors
considered the following periodic delayed two-species model
of facultative mutualism:
𝑦̇ 1 (𝑡) = 𝑦1 (𝑡) [𝑟1 (𝑡) − 𝑎1 (𝑡) 𝑦1 (𝑡) + 𝑏1 (𝑡) 𝑦1 (𝑡 − 𝜏1 (𝑡))
+ 𝑐1 (𝑡) 𝑦2 (𝑡 − 𝑝1 (𝑡))] ,
𝑦̇ 2 (𝑡) = 𝑦2 (𝑡) [𝑟2 (𝑡) − 𝑎2 (𝑡) 𝑦2 (𝑡) + 𝑏2 (𝑡) 𝑦2 (𝑡 − 𝜏1 (𝑡))
(2)
+ 𝑐2 (𝑡) 𝑦1 (𝑡 − 𝑝1 (𝑡))] .
By means of the methods of coincidence degree and the
Lyapunov functional, the sufficient conditions for the existence and globally asymptotic stability of positive periodic
solutions are established for system (2). In [12], the following
𝑛-species periodic Lotka-Volterra type competitive systems
2
Discrete Dynamics in Nature and Society
with feedback controls and finite and infinite distributed
delays are discussed:
𝑦̇ 𝑖 (𝑡) = 𝑦𝑖 (𝑡) [𝑟𝑖 (𝑡) − 𝑎𝑖𝑖 (𝑡) 𝑦𝑖 (𝑡)
𝑛
𝜔
− ∑ 𝑎𝑖𝑗 (𝑡) ∫ 𝐾𝑖𝑗 (𝑠) 𝑦𝑗 (𝑡 − 𝑠) 𝑑𝑠
0
𝑗=1, 𝑗 ≠ 𝑖
(3)
𝜔
− 𝛼𝑖 (𝑡) ∫ 𝐻𝑖 (𝑠) 𝑢𝑖 (𝑡 − 𝑠) 𝑑𝑠] ,
0
𝜔
𝑢̇ 𝑖 (𝑡) = −𝜂𝑖 (𝑡) + 𝑎𝑖 (𝑡) ∫ 𝐾𝑖 (𝑠) 𝑦𝑖 (𝑡 − 𝑠) 𝑑𝑠,
where 𝑖 = 1, 2, . . . , 𝑛. By using the technique of coincidence
degree and the Lyapunov functionals method, the sufficient
conditions for the existence and global stability of positive
periodic solutions are obtained for system (3).
Motivated by the above works, in this paper, we investigate the following two classes of 𝑛 species periodic model of
facultative mutualism with finite distributed delays:
𝑥̇ 𝑖 (𝑡) = 𝑥𝑖 (𝑡) [𝑟𝑖 (𝑡) − 𝛼𝑖 (𝑡) 𝑥𝑖 (𝑡)
[
𝑛
𝑚
0
(4)
𝑖 = 1, 2, . . . 𝑛,
0
𝑙=1
−𝜏
𝑚
0
𝑙=1
−𝜏
𝑖 = 1, 2, . . . , 𝑛
(6)
which can be regarded as the passive effect on the growth rate
of a species, and system (5) involves negative feedback terms
𝑚
0
𝑙=1
−𝜏
−∑𝑎𝑖𝑖𝑙 (𝑡) ∫ 𝑘𝑖𝑖𝑙 (𝑠) 𝑥𝑖 (𝑡 + 𝑠) 𝑑𝑠,
𝑖 = 1, 2, . . . , 𝑛
(7)
which are due to gestation. In this paper, we always assume
the following:
(H1) 𝑟𝑖 (𝑡) are 𝜔-periodic continuous functions with
𝜔
∫0 𝑟𝑖 (𝑡)𝑑𝑡 > 0 (𝑖 = 1, 2, . . . , 𝑛); 𝛼𝑖 (𝑡) (𝑖 = 1, 2, . . . , 𝑛)
and 𝑎𝑖𝑗𝑙 (𝑡) (𝑖, 𝑗 = 1, 2, . . . , 𝑛; 𝑙 = 1, 2, . . . , 𝑚)
are positive 𝜔-periodic continuous functions;
𝑘𝑖𝑗𝑙 (𝑠) (𝑖, 𝑗 = 1, 2, . . . , 𝑛; 𝑙 = 1, 2, . . . , 𝑚) are nonnegative integrable functions on [−𝜏, 0] satisfying
0
∫−𝜏 𝑘𝑖𝑗𝑙 (𝑠)𝑑𝑠 = 1.
𝑥𝑖 (𝑡) = 𝜙𝑖 (𝑠) ,
− ∑𝑎𝑖𝑖𝑙 (𝑡) ∫ 𝑘𝑖𝑖𝑙 (𝑠) 𝑥𝑖 (𝑡 + 𝑠) 𝑑𝑠
𝑛
𝑚
From the viewpoint of mathematical biology, in this paper
for systems (4) and (5) we only consider the solution with the
following initial condition:
𝑥̇ 𝑖 (𝑡) = 𝑥𝑖 (𝑡) [𝑟𝑖 (𝑡) − 𝛼𝑖 (𝑡) 𝑥𝑖 (𝑡)
[
𝑚
In systems (4) and (5), we have that 𝑥𝑖 (𝑡) (𝑖 = 1, 2, . . . 𝑛)
represent the density of 𝑛 species 𝑥𝑖 (𝑖 = 1, 2, . . . 𝑛) at time
𝑡, respectively; 𝑟𝑖 (𝑡) (𝑖 = 1, 2, . . . 𝑛) represent the intrinsic
growth rate of species 𝑥𝑖 (𝑖 = 1, 2, . . . 𝑛) at time 𝑡, respectively;
𝛼𝑖 (𝑡) (𝑖 = 1, 2, . . . , 𝑛) represent the intrapatch restriction
density of species 𝑥𝑖 (𝑖 = 1, 2, . . . 𝑛) at time 𝑡, respectively;
𝑎𝑖𝑗𝑙 (𝑡) (𝑙 = 1, 2, . . . , 𝑚, 𝑖 ≠ 𝑗, 𝑖, 𝑗 = 1, 2, . . . , 𝑛) represent the
mutualism coefficients between 𝑛 species 𝑥𝑖 (𝑖 = 1, 2, . . . 𝑛) at
time 𝑡, respectively, while 𝜏 ≥ 0 is a constant and 𝜏 may be
+∞. System (4) involves positive feedback terms
∑𝑎𝑖𝑖𝑙 (𝑡) ∫ 𝑘𝑖𝑖𝑙 (𝑠) 𝑥𝑖 (𝑡 + 𝑠) 𝑑𝑠,
0
+ ∑ ∑𝑎𝑖𝑗𝑙 (𝑡) ∫ 𝑘𝑖𝑗𝑙 (𝑠) 𝑥𝑗 (𝑡 + 𝑠) 𝑑𝑠]
−𝜏
𝑗=1 𝑙=1
]
2. Preliminaries
∀𝑠 ∈ [−𝜏, 0] , 𝑖 = 1, 2, . . . , 𝑛,
(8)
where 𝜙𝑖 (𝑠) (𝑖 = 1, 2, . . . , 𝑛) are nonnegative continuous functions defined on [−𝜏, 0] satisfying 𝜙𝑖 (0) > 0 (𝑖 = 1, 2, . . . , 𝑛).
In this paper, for any 𝜔-periodic continuous function 𝑓(𝑡)
we denote the following:
0
+ ∑ ∑𝑎𝑖𝑗𝑙 (𝑡) ∫ 𝑘𝑖𝑗𝑙 (𝑠) 𝑥𝑗 (𝑡 + 𝑠) 𝑑𝑠]
−𝜏
𝑗 ≠ 𝑖 𝑙=1
]
𝑖 = 1, 2, . . . 𝑛.
(5)
By using the technique of coincidence degree developed
by Gaines and Mawhin in [17] and the Lyapunov functional
method, we will establish some new sufficient conditions
which guarantee that the system has at least one positive
periodic solution and is globally attractive.
The organization of this paper is as follows. In the next
section we will present some basic assumptions and main
definitions and lemmas. In Section 3, conditions for the
existence and global attractivity of positive periodic solution.
In Section 4, two examples are given to illustrate that our
main results are applicable. In the final section, we will discuss
what we study in this paper and what we had in this paper.
𝑓𝐿 = min 𝑓 (𝑡) ,
𝑡∈[0,𝜔]
𝑓𝑀 = max 𝑓 (𝑡) ,
𝑡∈[0,𝜔]
1 𝜔
𝑓 = ∫ 𝑓 (𝑡) 𝑑𝑡.
𝜔 0
(9)
In order to obtain the existence of positive 𝜔-periodic
solutions of systems (4) and (5), we will use the continuation
theorem developed by Gaines and Mawhin in [17]. For the
reader’s convenience, we will introduce the continuation
theorem in the following.
Let 𝑋 and 𝑍 be two normed vetor spaces. Let 𝐿 : Dom 𝐿 ⊂
𝑋 → 𝑍 be a linear operator and 𝑁 : 𝑋 → 𝑍 be a
continuous operator. The operator 𝐿 is called a Fredholm
operator of index zero, if dim Ker 𝐿 = codim Im 𝐿 < ∞ and
Im 𝐿 is a closed set in 𝑍. If 𝐿 is a Fredholm operator of index
Discrete Dynamics in Nature and Society
3
zero, then there exist continuous projectors 𝑃 : 𝑋 → 𝑋
and 𝑄 : 𝑍 → 𝑍 such that Im 𝑃 = Ker 𝐿 and Im 𝐿 =
Ker 𝑄 = Im(𝐼 − 𝑄). It follows that 𝐿 | Dom 𝐿 ∩ Ker 𝑃 :
Dom 𝐿∩Ker 𝑃 → Im 𝐿 is invertible and its inverse is denoted
by 𝐾𝑃 and denote by 𝐽 : Im 𝑄 → Ker 𝐿 an isomorphism of
Im 𝑄 onto Ker 𝐿. Let Ω be a bounded open subset of 𝑋, we
say that the operator 𝑁 is 𝐿-compact on Ω, where Ω denotes
the closure of Ω in 𝑋, if 𝑄𝑁(Ω) is bounded and 𝐾𝑃 (𝐼 − 𝑄)𝑁 :
Ω → 𝑋 is compact.
Lemma 1 (see [17]). Let 𝐿 be a Fredholm operator of index zero
and let 𝑁 be 𝐿-compact on Ω. If
(a) for each 𝜆 ∈ (0, 1) and 𝑥 ∈ 𝜕Ω ∩ Dom 𝐿, 𝐿𝑥 ≠ 𝜆𝑁𝑥;
(b) for each 𝑥 ∈ 𝜕Ω ∩ Ker 𝐿, 𝑄𝑁𝑥 ≠ 0;
Then, system (4) is rewritten in the following form:
𝑢̇ 𝑖 (𝑡) = 𝑟𝑖 (𝑡) − 𝛼𝑖 (𝑡) exp {𝑢𝑖 (𝑡)}
𝑚
0
𝑙=1
−𝜏
+ ∑𝑎𝑖𝑖𝑙 (𝑡) ∫ 𝑘𝑖𝑖𝑙 (𝑠) exp {𝑢𝑖 (𝑡 + 𝑠)} 𝑑𝑠
𝑛
𝑚
(14)
0
+ ∑ ∑𝑎𝑖𝑗𝑙 (𝑡) ∫ 𝑘𝑖𝑗𝑙 (𝑠) exp {𝑢𝑗 (𝑡 + 𝑠)} 𝑑𝑠,
−𝜏
𝑗 ≠ 𝑖 𝑙=1
𝑖 = 1, 2, . . . , 𝑛.
In order to apply Lemma 1 to system (14), we introduce the normed vector spaces 𝑋 and 𝑍 as follows. Let
𝐶(𝑅, 𝑅𝑛 ) denote the space of all continuous function 𝑢(𝑡) =
(𝑢1 (𝑡), 𝑢2 (𝑡), . . . , 𝑢𝑛 (𝑡)) : 𝑅 → 𝑅𝑛 . We take
𝑋 = 𝑍 = {𝑢 (𝑡) ∈ 𝐶 (𝑅, 𝑅𝑛 ) : 𝑢 (𝑡)
(c) deg{𝐽𝑄𝑁, Ω ∩ Ker 𝐿, 0} ≠ 0,
then the operator equation 𝐿𝑥 = 𝑁𝑥 has at least one solution
lying in Dom 𝐿 ∩ Ω.
is an 𝜔-periodic function}
(15)
with norm
𝑛
󵄨
󵄨
‖𝑢‖ = ∑ max 󵄨󵄨󵄨𝑢𝑖 (𝑡)󵄨󵄨󵄨 .
𝑡∈[0,𝜔]
3. Main Results
Now, for the convenience of statements, we denote the
function
𝑚
𝑎𝑖𝑗 (𝑡) = ∑𝑎𝑖𝑗𝑙 (𝑡) ,
𝑖, 𝑗 = 1, 2, . . . , 𝑛.
It is obvious that 𝑋 and 𝑍 are the Banach spaces.
We define a linear operator 𝐿 : Dom 𝐿 ⊂ 𝑋 → 𝑍 and a
continuous operator 𝑁 : 𝑋 → 𝑍 as follows:
𝐿𝑢 (𝑡) = 𝑢̇ (𝑡) ,
(10)
𝑙=1
𝑁𝑢 (𝑡) = (𝑁𝑢1 (𝑡) , 𝑁𝑢2 (𝑡) , . . . , 𝑁𝑢𝑛 (𝑡)) ,
The following theorem is about the existence and global
attractivity of positive periodic solutions of system (4).
Theorem 2. Suppose that assumption (H1) holds and there
exists a constant 𝜇𝑖 > 0 (𝑖 = 1, 2, . . . , 𝑛) such that
𝑛
𝑚
}
{
min 𝜇𝑖 𝛼𝑖 (𝑡) − ∑ ∑𝜇𝑗 ∫ 𝑎𝑗𝑖𝑙 (𝑡 − 𝑠) 𝑘𝑗𝑖𝑙 (𝑠) 𝑑𝑠}
𝑡∈[0,𝜔] {
−𝜏
𝑗=1 𝑙=1
}
{
=: 𝛿𝑖 > 0,
(16)
𝑖=1
where
𝑁𝑢𝑖 (𝑡) = 𝑟𝑖 (𝑡) − 𝛼𝑖 (𝑡) exp {𝑢𝑖 (𝑡)}
𝑚
0
𝑙=1
−𝜏
+ ∑𝑎𝑖𝑖𝑙 (𝑡) ∫ 𝑘𝑖𝑖𝑙 (𝑠) exp {𝑢𝑖 (𝑡 + 𝑠)} 𝑑𝑠
0
𝑛
𝑚
(18)
0
+ ∑ ∑𝑎𝑖𝑗𝑙 (𝑡) ∫ 𝑘𝑖𝑗𝑙 (𝑠) exp {𝑢𝑗 (𝑡 + 𝑠)} 𝑑𝑠,
(11)
𝑗 ≠ 𝑖 𝑙=1
−𝜏
𝑖 = 1, 2, . . . , 𝑛,
𝑖 = 1, 2, . . . , 𝑛.
Further, we define continuous projectors 𝑃 : 𝑋 → 𝑋 and
𝑄 : 𝑍 → 𝑍 as follows:
and the algebraic equation
𝑛
𝑟𝑖 − 𝛼𝑖 V𝑖 + ∑ 𝑎𝑖𝑗 V𝑗 = 0,
(17)
𝑖 = 1, 2, . . . , 𝑛
𝑗=1
(12)
𝑃𝑢 (𝑡) =
1 𝜔
∫ 𝑢 (𝑡) 𝑑𝑡,
𝜔 0
𝑄V (𝑡) =
1 𝜔
∫ V (𝑡) 𝑑𝑡.
𝜔 0
(19)
𝜔
has a unique positive solution. Then, system (4) has a positive
𝜔-periodic solution which is globally attractive.
Proof. We firstly consider the existence of positive periodic
solutions of system (4). For system (4), we introduce new
variables 𝑢𝑖 (𝑡) (𝑖 = 1, 2, . . . , 𝑛) such that
𝑥𝑖 (𝑡) = exp {𝑢𝑖 (𝑡)} ,
𝑖 = 1, 2, . . . , 𝑛.
(13)
We easily see that Im 𝐿 = {V ∈ 𝑍 : ∫0 V(𝑡)𝑑𝑡 = 0} and Ker 𝐿 =
𝑅𝑛 . It is obvious that Im 𝐿 is closed in 𝑍 and dimKer𝐿 = 𝑛.
Since for any V ∈ 𝑍 there are unique V1 ∈ 𝑅𝑛 and V2 ∈ Im 𝐿
with
V1 =
1 𝜔
∫ V (𝑡) 𝑑𝑡,
𝜔 0
V2 (𝑡) = V (𝑡) − V1 ,
(20)
such that V(𝑡) = V1 +V2 (𝑡), we have co dim Im 𝐿 = 𝑛. Therefore,
𝐿 is a Fredholm mapping of index zero. Furthermore, the
4
Discrete Dynamics in Nature and Society
generalized inverse (to 𝐿) 𝐾𝑝 : Im 𝐿 → Ker 𝑃 ∩ Dom 𝐿 is
given in the following form:
𝑡
𝐾𝑝 V (𝑡) = ∫ V (𝑠) 𝑑𝑠 −
0
1 𝜔 𝑡
∫ ∫ V (𝑠) 𝑑𝑠 𝑑𝑡.
𝜔 0 0
(21)
integrating system (24) with the interval [0, 𝜔], we obtain the
following:
𝜔
∫ [𝑟𝑖 (𝑡) − 𝛼𝑖 (𝑡) exp {𝑢𝑖 (𝑡)}
0
[
For convenience, we denote 𝐹(𝑡) = (𝐹1 (𝑡), 𝐹2 (𝑡), . . . , 𝐹𝑛 (𝑡)) as
follows:
𝑚
0
𝑙=1
−𝜏
+ ∑𝑎𝑖𝑖𝑙 (𝑡) ∫ 𝑘𝑖𝑖𝑙 (𝑠) exp {𝑢𝑖 (𝑡 + 𝑠)} 𝑑𝑠
𝑛
𝐹𝑖 (𝑡) = 𝑟𝑖 (𝑡) − 𝛼𝑖 (𝑡) exp {𝑢𝑖 (𝑡)}
𝑚
0
𝑙=1
−𝜏
𝑚
0
+ ∑ ∑𝑎𝑖𝑗𝑙 (𝑡) ∫ 𝑘𝑖𝑗𝑙 (𝑠) exp {𝑢𝑗 (𝑡 + 𝑠)} 𝑑𝑠] 𝑑𝑡 = 0,
−𝜏
𝑗 ≠ 𝑖 𝑙=1
]
𝑖 = 1, 2, . . . 𝑛.
(25)
+ ∑𝑎𝑖𝑖𝑙 (𝑡) ∫ 𝑘𝑖𝑖𝑙 (𝑠) exp {𝑢𝑖 (𝑡 + 𝑠)} 𝑑𝑠
𝑛
𝑚
(22)
0
+ ∑ ∑𝑎𝑖𝑗𝑙 (𝑡) ∫ 𝑘𝑖𝑗𝑙 (𝑠) exp {𝑢𝑗 (𝑡 + 𝑠)} 𝑑𝑠,
−𝜏
𝑗 ≠ 𝑖 𝑙=1
Consequently,
𝑖 = 1, 2, . . . 𝑛.
𝜔
∫ [𝛼𝑖 (𝑡) exp {𝑢𝑖 (𝑡)}
0
[
Thus, we have
𝑡
0
0
𝑙=1
−𝜏
𝑛
𝐾𝑝 (𝐼 − 𝑄) 𝑁𝑢 (𝑡) = 𝐾𝑝 𝐼𝑁𝑢 (𝑡) − 𝐾𝑝 𝑄𝑁𝑢 (𝑡)
= ∫ 𝐹 (𝑠) 𝑑𝑠 −
𝑚
− ∑𝑎𝑖𝑖𝑙 (𝑡) ∫ 𝑘𝑖𝑖𝑙 (𝑠) exp {𝑢𝑖 (𝑡 + 𝑠)} 𝑑𝑠
1 𝜔
𝑄𝑁𝑢 (𝑡) = ∫ 𝐹 (𝑡) 𝑑𝑡,
𝜔 0
0
− ∑ ∑𝑎𝑖𝑗𝑙 (𝑡) ∫ 𝑘𝑖𝑗𝑙 (𝑠) exp {𝑢𝑗 (𝑡 + 𝑠)} 𝑑𝑠] 𝑑𝑡
−𝜏
𝑗 ≠ 𝑖 𝑙=1
]
(23)
1 𝜔 𝑡
∫ ∫ 𝐹 (𝑠) 𝑑𝑠 𝑑𝑡
𝜔 0 0
= 𝑟𝑖 𝜔,
𝜔
1 𝑡
+ ( − ) ∫ 𝐹 (𝑠) 𝑑𝑠.
2 𝜔 0
𝑚
𝑖 = 1, 2, . . . 𝑛.
From the continuity of 𝑢(𝑡) = (𝑢1 (𝑡), 𝑢2 (𝑡), . . . , 𝑢𝑛 (𝑡)), there
exist constants 𝜉𝑖 , 𝜂𝑖 ∈ [0, 𝜔] (𝑖 = 1, 2, . . . , 𝑛) such that
From formulas (23), we easily see that 𝑄𝑁 and 𝐾𝑝 (𝐼 − 𝑄)𝑁
are continuous operators. Furthermore, it can be verified that
𝐾𝑝 (𝐼 − 𝑄)𝑁(Ω) is compact for any open bounded set Ω ⊂
𝑋 by using Arzela-Ascoli theorem and 𝑄𝑁(Ω) is bounded.
Therefore, 𝑁 is 𝐿-compact on Ω for any open bounded subset
Ω ⊂ 𝑋.
Now, we reach the position to search for an appropriate
open bounded subset Ω for the application of the continuation theorem (Lemma 1) to system (4).
Corresponding to the operator equation 𝐿𝑢(𝑡) = 𝜆𝑁𝑢(𝑡)
with parameter 𝜆 ∈ (0, 1), we have
𝑢𝑖 (𝜉𝑖 ) = max 𝑢𝑖 (𝑡) ,
𝑡∈[0,𝜔]
𝑖 = 1, 2, . . . , 𝑛,
(24)
where 𝐹𝑖 (𝑡) (𝑖 = 1, 2, . . . , 𝑛) is given in (22).
Assume that 𝑢(𝑡) = (𝑢1 (𝑡), 𝑢2 (𝑡), . . . , 𝑢𝑛 (𝑡)) ∈ 𝑋 is a
solution of system (24) for some parameter 𝜆 ∈ (0, 1). By
𝑢𝑖 (𝜂𝑖 ) = min 𝑢𝑖 (𝑡) ,
𝑡∈[0,𝜔]
(27)
𝑖 = 1, 2, . . . , 𝑛.
From (26) and (27), we obtain
𝜔
∫ 𝛼𝑖 (𝑡) exp {𝑢𝑖 (𝜉𝑖 )} 𝑑𝑡 ≥ 𝑟𝑖 𝜔,
0
𝑢̇ 𝑖 (𝑡) = 𝜆𝐹𝑖 (𝑡) ,
(26)
𝑖 = 1, 2, . . . , 𝑛.
(28)
Therefore, we further have
𝑢𝑖 (𝜉𝑖 ) ≥ ln (
𝑟𝑖
),
𝛼𝑖
𝑖 = 1, 2, . . . , 𝑛.
(29)
Discrete Dynamics in Nature and Society
5
𝜔
For each 𝑖, 𝑗 = 1, 2, . . . , 𝑛 and 𝑙 = 1, 2, . . . , 𝑚, we have
+ ⋅ ⋅ ⋅ + ∫ [(𝛼𝑛 (𝑡)
0
𝜔
0
0
−𝜏
∫ 𝑎𝑖𝑗𝑙 (𝑡) ∫ 𝑘𝑖𝑗𝑙 (𝑠) exp {𝑢𝑗 (𝑡 + 𝑠)} 𝑑𝑠 𝑑𝑡
0
𝑠+𝜔
−𝜏 𝑠
0
(30)
−𝜏 0
0
−𝜏
0
0
−𝜏
−𝜏
𝑚
0
= ∫ [𝛼1 (𝑡) − ∑ (∫ 𝑎11𝑙 (𝑡 − 𝑠) 𝑘11𝑙 (𝑠) 𝑑𝑠
0
−𝜏
𝑙=1
[
= ∫ ∫ 𝑎𝑖𝑗𝑙 (V − 𝑠) 𝑘𝑖𝑗𝑙 (𝑠) exp {𝑢𝑗 (V)} 𝑑𝑠 𝑑V
𝜔
0
× exp {𝑢𝑗 (𝑡)}] 𝑑𝑡
𝜔
0
𝑚
𝑗 ≠ 𝑛 𝑙=1
= ∫ ∫ 𝑎𝑖𝑗𝑙 (V − 𝑠) 𝑘𝑖𝑗𝑙 (𝑠) exp {𝑢𝑗 (V)} 𝑑V 𝑑𝑠
𝜔
−𝜏
− ∑ ∑ (∫ 𝑎𝑛𝑗𝑙 (𝑡 − 𝑠) 𝑘𝑛𝑗𝑙 (𝑠) 𝑑𝑠)
𝑎𝑖𝑗𝑙 (V − 𝑠) 𝑘𝑖𝑗𝑙 (𝑠) exp {𝑢𝑗 (V)} 𝑑V 𝑑𝑠
𝜔
𝑙=1
𝑛
−𝜏 0
0
0
× exp {𝑢𝑛 (𝑡)}
𝜔
= ∫ ∫ 𝑎𝑖𝑗𝑙 (𝑡) 𝑘𝑖𝑗𝑙 (𝑠) exp {𝑢𝑗 (𝑡 + 𝑠)} 𝑑𝑡 𝑑𝑠
=∫ ∫
𝑚
−∑ (∫ 𝑎𝑛𝑛𝑙 (𝑡 − 𝑠) 𝑘𝑛𝑛𝑙 (𝑠)) 𝑑𝑠)
𝑛
= ∫ (∫ 𝑎𝑖𝑗𝑙 (𝑡 − 𝑠) 𝑘𝑖𝑗𝑙 (𝑠) 𝑑𝑠) exp {𝑢𝑗 (𝑡)} 𝑑𝑡.
0
+ ∑ ∫ 𝑎𝑗1𝑙 (𝑡 − 𝑠) 𝑘𝑗1𝑙 (𝑠) 𝑑𝑠)]
𝑗 ≠ 1 −𝜏
]
× exp {𝑢1 (𝑡)} 𝑑𝑡
Hence, from (26) we further obtain
𝜔
𝑚
0
0
𝑙=1
−𝜏
+ ∫ [𝛼2 (𝑡) − ∑ (∫ 𝑎22𝑙 (𝑡 − 𝑠) 𝑘22𝑙 (𝑠) 𝑑𝑠
𝑚
𝜔
0
𝑛
∫ [(𝛼𝑖 (𝑡) − ∑ (∫ 𝑎𝑖𝑖𝑙 (𝑡 − 𝑠) 𝑘𝑖𝑖𝑙 (𝑠) 𝑑𝑠)) exp {𝑢𝑖 (𝑡)}
0
−𝜏
𝑙=1
[
𝑛
𝑚
= 𝑟𝑖 𝜔,
𝑗 ≠ 2 −𝜏
0
− ∑ ∑ (∫ 𝑎𝑖𝑗𝑙 (𝑡 − 𝑠) 𝑘𝑖𝑗𝑙 (𝑠) 𝑑𝑠) exp {𝑢𝑗 (𝑡)}] 𝑑𝑡
−𝜏
𝑗 ≠ 𝑖 𝑙=1
]
𝑖 = 1, 2, . . . , 𝑛.
(31)
0
+ ∑ ∫ 𝑎𝑗2𝑙 (𝑡 − 𝑠)
× 𝑘𝑗2𝑙 (𝑠) 𝑑𝑠) ]
× exp {𝑢2 (𝑡)} 𝑑𝑡
𝜔
+ ⋅ ⋅ ⋅ + ∫ [𝛼𝑛 (𝑡)
0
Consequently,
𝑚
0
𝑙=1
−𝜏
− ∑ (∫ 𝑎𝑛𝑛𝑙 (𝑡 − 𝑠) 𝑘𝑛𝑛𝑙 (𝑠) 𝑑𝑠
𝑛
𝑚
𝜔
∫ [(𝛼1 (𝑡) − ∑ (∫ 𝑎11𝑙 (𝑡 − 𝑠) 𝑘11𝑙 (𝑠) 𝑑𝑠)) exp {𝑢1 (𝑡)}
0
−𝜏
𝑙=1
[
𝑛
𝑚
0
+ ∑ ∫ 𝑎𝑗𝑛𝑙 (𝑡 − 𝑠)
0
𝑗 ≠ 𝑛 −𝜏
× 𝑘𝑗𝑛𝑙 (𝑠) 𝑑𝑠)]
0
− ∑ ∑ (∫ 𝑎1𝑗𝑙 (𝑡 − 𝑠) 𝑘1𝑗𝑙 (𝑠) 𝑑𝑠) exp {𝑢𝑗 (𝑡)}] 𝑑𝑡
−𝜏
𝑗 ≠ 1 𝑙=1
]
𝜔
𝜔
𝑚
0
0
𝑙=1
−𝜏
+ ∫ [(𝛼2 (𝑡) − ∑ (∫ 𝑎22𝑙 (𝑡 − 𝑠) 𝑘22𝑙 (𝑠) 𝑑𝑠))
𝑛
× exp {𝑢𝑛 (𝑡)} 𝑑𝑡
𝑚
0
×exp {𝑢2 (𝑡)}− ∑ ∑ (∫ 𝑎2𝑗𝑙 (𝑡−𝑠) 𝑘2𝑗𝑙 (𝑠) 𝑑𝑠)
𝑗 ≠ 2 𝑙=1
× exp {𝑢𝑗 (𝑡)}] 𝑑𝑡
−𝜏
= ∫ [𝛼1 (𝑡)
0
[
𝑚
𝑛
0
− ∑ (∫ [𝑎11𝑙 (𝑡 − 𝑠) 𝑘11𝑙 (𝑠)+ ∑ 𝑎𝑗1𝑙 (𝑡 − 𝑠)
−𝜏
𝑗 ≠ 1
𝑙=1
[
× 𝑘𝑗1𝑙 (𝑠) ] 𝑑𝑠)]
]
]
6
Discrete Dynamics in Nature and Society
× exp {𝑢1 (𝑡)} 𝑑𝑡
From (35), we further obtain
𝜔
𝑚
0
0
𝑙=1
−𝜏
+ ∫ [𝛼2 (𝑡) − ∑ (∫ [𝑎22𝑙 (𝑡 − 𝑠) 𝑘22𝑙 (𝑠)
𝑢𝑖 (𝜂𝑖 ) ≤ ln (
𝑛
+ ∑ 𝑎𝑗2𝑙 (𝑡 − 𝑠)
∑𝑛𝑖=1 𝑟𝑖
),
𝛿𝑖
𝑖 = 1, 2, . . . , 𝑛.
(36)
𝑗 ≠ 2
× 𝑘𝑗2𝑙 (𝑠)] 𝑑𝑠)]
On the other hand, directly from system (14) we have
× exp {𝑢2 (𝑡)} 𝑑𝑡
𝜔
󵄨
󵄨
∫ 󵄨󵄨󵄨𝑢̇ 𝑖 (𝑡)󵄨󵄨󵄨 𝑑𝑡
0
𝜔
+ ⋅ ⋅ ⋅ + ∫ [𝛼𝑛 (𝑡)
0
[
𝜔
𝑚
󵄨
󵄨
≤ ∫ [󵄨󵄨󵄨𝑟𝑖 (𝑡)󵄨󵄨󵄨 + 𝛼𝑖 (𝑡) exp {𝑢𝑖 (𝑡)}
0
[
0
− ∑ (∫ [𝑎𝑛𝑛𝑙 (𝑡 − 𝑠) 𝑘𝑛𝑛𝑙 (𝑠)
−𝜏
𝑙=1
[
𝑚
0
𝑙=1
−𝜏
+ ∑𝑎𝑖𝑖𝑙 (𝑡) ∫ 𝑘𝑖𝑖𝑙 (𝑠) exp {𝑢𝑖 (𝑡 + 𝑠)} 𝑑𝑠
𝑛
+ ∑ 𝑎𝑗𝑛𝑙 (𝑡 − 𝑠)
𝑗 ≠ 𝑛
𝑛
× 𝑘𝑗𝑛𝑙 (𝑠)] 𝑑𝑠)]
]
]
𝑖=1
(32)
𝜔
󵄨
󵄨
= ∫ 󵄨󵄨󵄨𝑟𝑖 (𝑡)󵄨󵄨󵄨 𝑑𝑡
0
From the assumptions of Theorem 2, we can obtain
0
𝑚
0
𝑙=1 −𝜏
0
× exp {𝑢𝑖 (𝑡)} 𝑑𝑡
𝑛
𝑚
0
+ ∫ ( ∑ ∑ ∫ 𝑎𝑖𝑗𝑙 (𝑡 − 𝑠) 𝑘𝑖𝑗𝑙 (𝑠) 𝑑𝑠)
0
𝑗 ≠ 𝑖
× 𝑘𝑗𝑖𝑙 (𝑠)] 𝑑𝑠)]
]
𝑛
𝜔
+ ∑ 𝑎𝑗𝑖𝑙 (𝑡 − 𝑠)
(33)
𝑗 ≠ 𝑖 𝑙=1 −𝜏
× exp {𝑢𝑗 (𝑡)} 𝑑𝑡
𝜔
𝜔
󵄨
󵄨
≤ ∫ 󵄨󵄨󵄨𝑟𝑖 (𝑡)󵄨󵄨󵄨 𝑑𝑡 + ∫ 𝛼𝑖 (𝑡) exp {𝑢𝑖 (𝑡)} 𝑑𝑡
0
0
× exp {𝑢𝑖 (𝑡)} 𝑑𝑡
≤ ∑𝑟𝑖 𝜔,
𝜔
+ ∫ [𝛼𝑖 (𝑡) + ∑ ∫ 𝑎𝑖𝑖𝑙 (𝑡 − 𝑠) 𝑘𝑖𝑖𝑙 (𝑠) 𝑑𝑠]
∫ [𝛼𝑖 (𝑡) − ∑ (∫ [𝑎𝑖𝑖𝑙 (𝑡 − 𝑠) 𝑘𝑖𝑖𝑙 (𝑠)
0
−𝜏
𝑙=1
[
𝑛
−𝜏
× exp {𝑢𝑗 (𝑡 + 𝑠)} 𝑑𝑠] 𝑑𝑡
]
𝑛
𝑚
0
𝑗 ≠ 1 𝑙=1
× exp {𝑢𝑛 (𝑡)} 𝑑𝑡 = ∑𝑟𝑖 𝜔.
𝜔
𝑚
+ ∑ ∑𝑎𝑖𝑗𝑙 (𝑡) ∫ 𝑘𝑖𝑗𝑙 (𝑠)
𝑚
𝜔
𝑙=1
0
𝑀
+ ∑𝑎𝑖𝑖𝑙
∫ exp {𝑢𝑖 (𝑡)} 𝑑𝑡
𝑖 = 1, 2, . . . , 𝑛.
𝑖=1
𝑛
𝑚
𝜔
𝑀
+ ∑ ∑𝑎𝑖𝑗𝑙
∫ exp {𝑢𝑗 (𝑡)} 𝑑𝑡
Hence,
𝜔
𝑛
𝛿𝑖 ∫ exp {𝑢𝑖 (𝑡)} 𝑑𝑡 ≤ ∑𝑟𝑖 𝜔,
0
𝑗 ≠ 𝑖 𝑙=1
𝑖 = 1, 2, . . . , 𝑛.
(34)
𝑖=1
𝑛
𝑛 𝑚
󵄨 󵄨
𝑀 ∑𝑖=1 𝑟𝑖 𝜔
≤ 󵄨󵄨󵄨𝑟𝑖 󵄨󵄨󵄨𝜔 + ∑ ∑𝑎𝑖𝑗𝑙
𝛿𝑖
𝑗=1 𝑙=1
Consequently,
∑𝑛 𝑟 𝜔
∫ exp {𝑢𝑖 (𝑡)} 𝑑𝑡 ≤ 𝑖=1 𝑖 ,
𝛿𝑖
0
𝜔
+ 𝛼𝑖𝑀
𝑖 = 1, 2, . . . , 𝑛.
(35)
0
∑𝑛𝑖=1 𝑟𝑖 𝜔
,
𝛿𝑖
𝑖 = 1, 2, . . . , 𝑛.
(37)
Discrete Dynamics in Nature and Society
7
From (36) and (37), we have, for any 𝑡 ∈ [0, 𝜔],
𝜔
󵄨
󵄨
𝑢𝑖 (𝑡) ≤ 𝑢𝑖 (𝜂𝑖 ) + ∫ 󵄨󵄨󵄨𝑢̇ 𝑖 (𝑡)󵄨󵄨󵄨 𝑑𝑡 ≤ ln (
0
+
𝑛
𝑛 𝑚
𝑀 ∑𝑖=1 𝑟𝑖 𝜔
∑ ∑𝑎𝑖𝑗𝑙
𝛿𝑖
𝑗=1 𝑙=1
+
∑𝑛𝑖=1 𝑟𝑖
𝛿𝑖
where
󵄨 󵄨
) + 󵄨󵄨󵄨𝑟𝑖 󵄨󵄨󵄨𝜔
∑𝑛 𝑟 𝜔
𝛼𝑖𝑀 𝑖=1 𝑖
𝛿𝑖
=: 𝑀𝑖 ,
𝑓𝑢𝑖 𝑗 = − (𝛼𝑖 − 𝑎𝑖𝑗 ) exp {𝑢𝑗∗ } ,
(38)
𝑓𝑢𝑖 𝑗 = 𝑎𝑖𝑗 exp {𝑢𝑗∗ } ,
𝑖 = 𝑗,
𝑖 ≠ 𝑗, 𝑖, 𝑗 = 1, 2, . . . , 𝑛.
(46)
𝑖 = 1, 2, . . . , 𝑛.
Further, from (29) and (37), we have, for any 𝑡 ∈ [0, 𝜔],
From the assumption of Theorem 2, we have
𝜔
󵄨
󵄨
𝑢𝑖 (𝑡) ≥ 𝑢𝑖 (𝜉𝑖 ) − ∫ 󵄨󵄨󵄨𝑢̇ 𝑖 (𝑡)󵄨󵄨󵄨 𝑑𝑡
0
≥ ln (
𝑟𝑖
󵄨 󵄨
) − 󵄨󵄨󵄨𝑟𝑖 󵄨󵄨󵄨𝜔 −
𝛼𝑖
− 𝛼𝑖𝑀
𝑛
𝑀 ∑𝑖=1 𝑟𝑖 𝜔
∑ ∑𝑎𝑖𝑗𝑙
𝛿𝑖
𝑗=1 𝑙=1
𝑛
𝑚
∑𝑛𝑖=1 𝑟𝑖 𝜔
=: 𝑁𝑖 ,
𝛿𝑖
(39)
𝑖 = 1, 2, . . . , 𝑛.
Therefore, from (38) and (39) we have
󵄨 󵄨 󵄨 󵄨
󵄨
󵄨
max 󵄨󵄨󵄨𝑢𝑖 (𝑡)󵄨󵄨󵄨 ≤ max {󵄨󵄨󵄨𝑀𝑖 󵄨󵄨󵄨 , 󵄨󵄨󵄨𝑁𝑖 󵄨󵄨󵄨} =: 𝐵𝑖 ,
𝑡∈[0,𝜔]
deg {𝐽𝑄𝑁, Ω ∩ Ker 𝐿, (0, 0, . . . , 0)} ≠ 0.
(41)
where
𝑛
𝑗 ≠ 𝑖
(42)
𝑖 = 1, 2, . . . , 𝑛.
We consider the following algebraic equation:
𝑛
𝑟𝑖 − (𝛼𝑖 − 𝑎𝑖𝑖 ) V𝑖 + ∑ 𝑎𝑖𝑗 V𝑗 = 0,
𝑖 = 1, 2, . . . , 𝑛.
𝑗 ≠ 𝑖
(47)
(40)
It can be seen that the constants 𝐵𝑖 (𝑖 = 1, 2, . . . , 𝑛) are
independent of parameter 𝜆 ∈ (0, 1).
For any 𝑢 = (𝑢1 , 𝑢2 , . . . , 𝑢𝑛 ) ∈ 𝑅𝑛 , from (18) we can obtain
𝑄𝑁𝑢 = 𝑟𝑖 − (𝛼𝑖 − 𝑎𝑖𝑖 ) exp {𝑢𝑖 } + ∑ 𝑎𝑖𝑗 exp {𝑢𝑗 } ,
𝑓𝑢12 ⋅ ⋅ ⋅ 𝑓𝑢1𝑛 󵄨󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨
𝑓𝑢22 ⋅ ⋅ ⋅ 𝑓𝑢2𝑛 󵄨󵄨󵄨󵄨 ≠ 0.
󵄨
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 󵄨󵄨󵄨
𝑛
𝑛 󵄨󵄨
𝑓𝑢2 ⋅ ⋅ ⋅ 𝑓𝑢𝑛 󵄨󵄨
From this, we finally have
𝑖 = 1, 2, . . . , 𝑛.
𝑄𝑁𝑢 = (𝑄𝑁𝑢1 , 𝑄𝑁𝑢2 , . . . , 𝑄𝑁𝑢𝑛 ) ,
󵄨󵄨𝑓1
󵄨󵄨 𝑢1
󵄨󵄨
󵄨󵄨 2
󵄨󵄨𝑓
󵄨󵄨 𝑢1
󵄨󵄨
󵄨󵄨 ⋅ ⋅ ⋅
󵄨󵄨 𝑛
󵄨󵄨𝑓𝑢1
(43)
(48)
This shows that Ω satisfies condition (c) of Lemma 1.
Therefore, system (14) has a 𝜔-periodic solution 𝑢∗ (𝑡) =
(𝑢1∗ (𝑡), 𝑢2∗ (𝑡), . . . , 𝑢𝑛∗ (𝑡)) ∈ Ω. Further, from (13), system (4)
has a positive 𝜔-periodic solution 𝑥∗ (𝑡) = (𝑥1∗ (𝑡), 𝑥2∗ (𝑡), . . . ,
𝑥𝑛∗ (𝑡)).
Next, we will consider the global attractivity of positive
periodic solutions 𝑥∗ (𝑡) = (𝑥1∗ (𝑡), 𝑥2∗ (𝑡), . . . , 𝑥𝑛∗ (𝑡)) of system
(4). Choose positive constants 𝑚𝑖 > 0, 𝑀𝑖 > 0 such that
𝑚𝑖 ≤ 𝑥𝑖∗ (𝑡) ≤ 𝑀𝑖 ,
𝑖 = 1, 2, . . . , 𝑛.
(49)
From the assumption of Theorem 2, the equation has a
unique positive solution V∗ = (V1∗ , V2∗ , . . . , V𝑛∗ ). Hence,
the equation 𝑄𝑁𝑢 = 0 has a unique solution 𝑢∗ =
(𝑢1∗ , 𝑢2∗ , . . . , 𝑢𝑛∗ ) = (ln V1∗ , ln V2∗ , . . . , ln V𝑛∗ ) ∈ 𝑅𝑛 .
Choosing constant 𝐵 > 0 large enough such that |𝑢1∗ | +
∗
|𝑢2 | + ⋅ ⋅ ⋅ + |𝑢𝑛∗ | < 𝐵 and 𝐵 > 𝐵1 + 𝐵2 + ⋅ ⋅ ⋅ + 𝐵𝑛 , we define a
bounded open set Ω ⊂ 𝑋 as follows:
From the assumption of Theorem 2, there exists constant 𝛽 >
0 such that for all 𝑡 ≥ 0 we have
Ω = {𝑢 ∈ 𝑋 : ‖𝑢‖ < 𝐵} .
Let (𝑥1 (𝑡), 𝑥2 (𝑡), . . . , 𝑥𝑛 (𝑡)) be any solution of system (4), we
define Lyapunov function as follows:
(44)
It is clear that Ω satisfies conditions (a) and (b) of Lemma 1.
On the other hand, by direct calculating we can obtain
deg {𝐽𝑄𝑁, Ω ∩ Ker 𝐿, (0, 0, . . . , 0)}
󵄨󵄨𝑓1
󵄨󵄨 𝑢1
󵄨󵄨
󵄨󵄨 2
󵄨
= sgn 󵄨󵄨󵄨𝑓𝑢1
󵄨󵄨
󵄨󵄨󵄨 ⋅ ⋅𝑛⋅
󵄨󵄨𝑓
󵄨 𝑢1
𝑓𝑢12 ⋅ ⋅ ⋅ 𝑓𝑢1𝑛 󵄨󵄨󵄨󵄨
󵄨󵄨
󵄨
2
2 󵄨󵄨󵄨
𝑓𝑢2 ⋅ ⋅ ⋅ 𝑓𝑢𝑛 󵄨󵄨 ,
󵄨
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 󵄨󵄨󵄨
𝑛
𝑛 󵄨󵄨
𝑓𝑢2 ⋅ ⋅ ⋅ 𝑓𝑢𝑛 󵄨󵄨
𝛿𝑖 ≥ 𝛽 > 0,
𝑖 = 1, 2, . . . , 𝑛.
(50)
󵄨
󵄨
𝑉𝑖 (𝑡) = 𝜇𝑖 󵄨󵄨󵄨ln 𝑥𝑖∗ (𝑡) − ln 𝑥𝑖 (𝑡)󵄨󵄨󵄨
𝑛
(45)
𝑚
0
𝑡
−𝜏
𝑡+𝑠
+ ∑ ∑𝜇𝑗 ∫ 𝑘𝑖𝑗𝑙 (𝑠) ∫ 𝑎𝑖𝑗𝑙 (𝜃 − 𝑠)
𝑗=1 𝑙=1
󵄨
󵄨
× 󵄨󵄨󵄨󵄨𝑥𝑗∗ (𝜃) − 𝑥𝑗 (𝜃)󵄨󵄨󵄨󵄨 𝑑𝜃 𝑑𝑠.
(51)
8
Discrete Dynamics in Nature and Society
Calculating the upper right derivation of 𝑉𝑖 (𝑡) along system
(4) for 𝑖 = 1, 2, . . . , 𝑛, we have
𝐷+ 𝑉𝑖 (𝑡) = sign (𝑥𝑖∗ (𝑡) − 𝑥𝑖 (𝑡))
𝑚𝑖 exp {−
× [−𝜇𝑖 𝛼𝑖 (𝑡) (𝑥𝑖∗ (𝑡) − 𝑥𝑖 (𝑡))
𝑛
𝑚
which, together with (49), leads to
𝑉 (0)
} ≤ 𝑥𝑖 (𝑡)
𝜇𝑖
≤ 𝑀𝑖 exp {
0
+ ∑ ∑𝜇𝑗 𝑎𝑖𝑗𝑙 (𝑡) ∫ 𝑘𝑖𝑗𝑙 (𝑠)
−𝜏
𝑗=1 𝑙=1
× (𝑥𝑗∗ (𝑡 + 𝜃) − 𝑥𝑗 (𝑡 + 𝜃)) 𝑑𝑠]
𝑛
𝑚
0
󵄨
󵄨
+ ∑ ∑𝜇𝑗 ∫ 𝑎𝑖𝑗𝑙 (𝑡 − 𝑠) 𝑘𝑖𝑗𝑙 (𝑠) 𝑑𝑠 󵄨󵄨󵄨󵄨𝑥𝑗∗ (𝑡) − 𝑥𝑗 (𝑡)󵄨󵄨󵄨󵄨
−𝜏
𝑗=1 𝑙=1
𝑛
𝑚
0
− ∑ ∑𝜇𝑗 𝑎𝑖𝑗𝑙 (𝑡) ∫ 𝑘𝑖𝑗𝑙 (𝑠)
−𝜏
𝑗=1 𝑙=1
󵄨
󵄨
× 󵄨󵄨󵄨󵄨𝑥𝑗∗ (𝑡 + 𝜃) − 𝑥𝑗 (𝑡 + 𝜃)󵄨󵄨󵄨󵄨 𝑑𝑠
󵄨
󵄨
≤ −𝜇𝑖 𝛼𝑖 (𝑡) 󵄨󵄨󵄨𝑥𝑖∗ (𝑡) − 𝑥𝑖 (𝑡)󵄨󵄨󵄨
𝑛
𝑚
𝑉 (0)
},
𝜇𝑖
𝑖 = 1, 2, . . . , 𝑛,
(59)
and, hence, ∑𝑛𝑖=1 |𝑥𝑖∗ (𝑡) − 𝑥𝑖 (𝑡)| ∈ 𝐿1 [0, +∞). From the
boundedness of 𝑥𝑖∗ (𝑡) and (58), it follows that 𝑥𝑖 (𝑡) (𝑖 =
1, 2, . . . , 𝑛) are bounded for 𝑡 ≥ 0. It is obvious that both 𝑥𝑖 (𝑡)
and 𝑥𝑖∗ (𝑡) satisfy the equations of system (4), then by system
(4) and the boundedness of 𝑥𝑖 (𝑡) and 𝑥𝑖∗ (𝑡), we know that the
derivatives 𝑥̇ 𝑖 (𝑡) and 𝑥̇ ∗𝑖 (𝑡) are bounded. Furthermore, we can
obtain that 𝑥̇ ∗𝑖 (𝑡) − 𝑥̇ 𝑖 (𝑡) (𝑖 = 1, 2, . . . , 𝑛) and their derivatives
remain bounded on [0, +∞). Therefore ∑𝑛𝑖=1 |𝑥𝑖∗ (𝑡) − 𝑥𝑖 (𝑡)| is
uniformly continuous on [0, +∞). Thus, from (56), we have
𝑛
󵄨
󵄨
lim ∑ 󵄨󵄨󵄨𝑥𝑖∗ (𝑡) − 𝑥𝑖 (𝑡)󵄨󵄨󵄨 = 0.
𝑡 → +∞
0
󵄨
󵄨
+ ∑ ∑𝜇𝑗 ∫ 𝑎𝑖𝑗𝑙 (𝑡 − 𝑠) 𝑘𝑖𝑗𝑙 (𝑠) 𝑑𝑠 󵄨󵄨󵄨󵄨𝑥𝑗∗ (𝑡) − 𝑥𝑗 (𝑡)󵄨󵄨󵄨󵄨 .
−𝜏
(60)
𝑖=1
𝑗=1 𝑙=1
(52)
Therefor,
Further, we define a Lyapunov function as follows:
𝑛
𝑉 (𝑡) = ∑𝑉𝑖 (𝑡) .
(53)
𝑖=1
Calculating the upper right derivation of 𝑉(𝑡), from (52) we
finally can obtain, for all 𝑡 ≥ 0,
𝑛
󵄨
󵄨
𝐷+ 𝑉 (𝑡) ≤ −∑𝛿𝑖 󵄨󵄨󵄨𝑥𝑖∗ (𝑡) − 𝑥𝑖 (𝑡)󵄨󵄨󵄨 .
(54)
𝑖=1
Integrating from 0 to 𝑡 on both sides of (54) and by (50)
produces
𝑡
𝑛
0
𝑖=1
󵄨
󵄨
𝑉 (𝑡) + 𝛽 ∫ (∑ 󵄨󵄨󵄨𝑥𝑖∗ (𝑠) − 𝑥𝑖 (𝑠)󵄨󵄨󵄨) 𝑑𝑠 ≤ 𝑉 (0) ,
𝑡 ≥ 0,
(55)
then
𝑡
lim (𝑥𝑖∗ (𝑡) − 𝑥𝑖 (𝑡)) = 0,
𝑡 → +∞
𝑉 (0)
󵄨
󵄨
,
∫ (∑ 󵄨󵄨󵄨𝑥𝑖∗ (𝑠) − 𝑥𝑖 (𝑠)󵄨󵄨󵄨) 𝑑𝑠 ≤
𝛽
0
𝑖=1
𝑡 ≥ 0.
(61)
This completes the proof of Theorem 2.
From the proof of Theorem 2, on the existence and global
attractivity of positive periodic solutions of system (5), we
have the following result.
Corollary 3. Suppose that assumption (H1) holds and there
exists a constant 𝜌𝑖 > 0 (𝑖 = 1, 2, . . . , 𝑛) such that
𝑛 𝑚
0
{
}
min {𝜌𝑖 𝛼𝑖 (𝑡) − ∑ ∑𝜌𝑗 ∫ 𝑎𝑗𝑖𝑙 (𝑡 − 𝑠) 𝑘𝑗𝑖𝑙 (𝑠) 𝑑𝑠}
𝑡∈[0,𝜔]
−𝜏
𝑗=1 𝑙=1
{
}
=: 𝜆 𝑖 > 0,
𝑛
𝑖 = 1, 2, . . . , 𝑛.
(62)
𝑖 = 1, 2, . . . , 𝑛,
(56)
and the algebraic equation
By the definition of 𝑉(𝑡) and (53), we have
𝑛
󵄨
󵄨
∑𝜇𝑖 󵄨󵄨󵄨ln 𝑥𝑖∗ (𝑡) − ln 𝑥𝑖 (𝑡)󵄨󵄨󵄨 ≤ 𝑉 (𝑡) ≤ 𝑉 (0) ,
𝑛
𝑡 ≥ 0.
(57)
𝑖=1
𝑟𝑖 − (𝛼𝑖 + 𝑎𝑖𝑖 ) V𝑖 + ∑ 𝑎𝑖𝑗 V𝑗 = 0,
𝑗 ≠ 𝑖
𝑖 = 1, 2, . . . , 𝑛,
(63)
Therefore, for 𝑖 = 1, 2, . . . , 𝑛 we have
󵄨
󵄨
𝜇𝑖 󵄨󵄨󵄨ln 𝑥𝑖∗ (𝑡) − ln 𝑥𝑖 (𝑡)󵄨󵄨󵄨 ≤ 𝑉 (0) ,
𝑡 ≥ 0,
(58)
has a unique positive solution. Then, system (5) has a positive
𝜔-periodic solution which is globally attractive.
Discrete Dynamics in Nature and Society
9
4. Two Examples
Example 5. Next, we consider the following delayed system:
Example 4. First, we consider the following delayed system:
𝑥̇ 1 (𝑡) = 𝑥1 (𝑡) [2 + cos (𝑡) −
𝑥̇ 1 (𝑡) = 𝑥1 (𝑡) [2 + cos (𝑡) − (6 + cos (𝑡)) 𝑥1 (𝑡)
+
3 + 2 cos (𝑡) 0
∫ 𝑘111 (𝑠) 𝑥1 (𝑡 + 𝑠) 𝑑𝑠
2
−𝜏
+
2 + cos (𝑡) 0
∫ 𝑘111 (𝑠) 𝑥1 (𝑡 + 𝑠) 𝑑𝑠
3
−𝜏
+
3 + 2 cos (𝑡) 0
∫ 𝑘121 (𝑠) 𝑥2 (𝑡 + 𝑠) 𝑑𝑠
4
−𝜏
+
2 + cos (𝑡) 0
∫ 𝑘121 (𝑠) 𝑥2 (𝑡 + 𝑠) 𝑑𝑠
3
−𝜏
+
2 + cos (𝑡) 0
∫ 𝑘131 (𝑠) 𝑥3 (𝑡 + 𝑠) 𝑑𝑠] ,
2
−𝜏
+
2 + cos (𝑡) 0
∫ 𝑘131 (𝑠) 𝑥3 (𝑡 + 𝑠) 𝑑𝑠] ,
3
−𝜏
𝑥̇ 2 (𝑡) = 𝑥2 (𝑡) [2 + cos (𝑡) −
41 + 40 cos (𝑡)
𝑥2 (𝑡)
100
0
𝑥̇ 2 (𝑡) = 𝑥2 (𝑡) [2 + cos (𝑡) − (5 + cos (𝑡)) 𝑥2 (𝑡)
+ (2 + cos (𝑡)) ∫ 𝑘221 (𝑠) 𝑥2 (𝑡 + 𝑠) 𝑑𝑠
−𝜏
0
+
2 + cos (𝑡)
∫ 𝑘221 (𝑠) 𝑥2 (𝑡 + 𝑠) 𝑑𝑠
7
−𝜏
+
9 + 6 cos (𝑡) 0
∫ 𝑘211 (𝑠) 𝑥1 (𝑡 + 𝑠) 𝑑𝑠
8
−𝜏
+
3 + cos (𝑡) 0
∫ 𝑘211 (𝑠) 𝑥1 (𝑡 + 𝑠) 𝑑𝑠
4
−𝜏
+
3 + 2 cos (𝑡) 0
∫ 𝑘231 (𝑠) 𝑥3 (𝑡 + 𝑠) 𝑑𝑠] ,
4
−𝜏
+
3 + cos (𝑡) 0
∫ 𝑘231 (𝑠) 𝑥3 (𝑡 + 𝑠) 𝑑𝑠] ,
4
−𝜏
𝑥̇ 3 (𝑡) = 𝑥3 (𝑡) [2 + cos (𝑡) −
5 + 4 cos (𝑡)
𝑥3 (𝑡)
20
0
+ (4 + cos (𝑡)) ∫ 𝑘331 (𝑠) 𝑥3 (𝑡 + 𝑠) 𝑑𝑠
𝑥̇ 3 (𝑡) = 𝑥3 (𝑡) [2 + cos (𝑡) − (6 + cos (𝑡)) 𝑥3 (𝑡)
−𝜏
+
3 + cos (𝑡)
∫ 𝑘331 (𝑠) 𝑥3 (𝑡 + 𝑠) 𝑑𝑠
4
−𝜏
+
5 + 4 cos (𝑡) 0
∫ 𝑘311 (𝑠) 𝑥1 (𝑡 + 𝑠) 𝑑𝑠
8
−𝜏
+
4 + cos (𝑡) 0
∫ 𝑘311 (𝑠) 𝑥1 (𝑡 + 𝑠) 𝑑𝑠
5
−𝜏
+
+
3 + cos (𝑡) 0
∫ 𝑘321 (𝑠) 𝑥2 (𝑡 + 𝑠) 𝑑𝑠] .
5
−𝜏
(64)
6 + 3 cos (𝑡) 0
∫ 𝑘321 (𝑠) 𝑥2 (𝑡 + 𝑠) 𝑑𝑠] .
8
−𝜏
(68)
0
Corresponding to system (4), 𝑛 = 3, 𝑚 = 1, 𝜔 = 2𝜋, by direct
calculation, we can get
𝜎1 ≈ 2,
𝜎2 ≈ 1.6,
𝜎3 ≈ 2.2,
Corresponding to system (4), 𝑛 = 3, 𝑚 = 1, 𝜔 = 2𝜋, by direct
calculation we can get
𝜎1 ≈ −5.2,
𝑎22 V1 − 𝑎21 V2 − 𝑎23 V3 = 𝑟2 ,
(66)
𝑎22 V1 − 𝑎21 V2 − 𝑎23 V3 = 𝑟2 ,
(70)
𝑎33 V1 − 𝑎31 V2 − 𝑎32 V3 = 𝑟3 ,
V1 = −0.1622,
V2 = 0.7007,
V3 = 1.2717.
(71)
Clearly, the conditions of Theorem 2 do not hold.
From Figure 1 we can see that system (68) has no globally
attractive positive periodic solution.
where
V3 ≈ 0.8819.
(69)
where
𝑎33 V1 − 𝑎31 V2 − 𝑎32 V3 = 𝑟3 ,
V2 ≈ 0.5625,
𝜎3 ≈ −7.2,
𝑎11 V1 − 𝑎12 V2 − 𝑎13 V3 = 𝑟1 ,
(65)
𝑎11 V1 − 𝑎12 V2 − 𝑎13 V3 = 𝑟1 ,
𝜎2 ≈ −6.1,
and the following equations have a unique positive solutions:
and the following equations have unique positive solutions:
V1 ≈ 0.1944,
5 + 4 cos (𝑡)
𝑥1 (𝑡)
20
(67)
It is clear that all the conditions of Theorem 2 hold. Hence,
system (64) has a positive periodic solution which is globally
attractive.
Remark 6. From these two examples, we can see that if
the conditions of Theorem 2 hold, then the system has a
globally attractive positive periodic solution. If the conditions
of Theorem 2 do not hold, then the system has no globally
attractive positive periodic solution.
10
Discrete Dynamics in Nature and Society
7000
6000
𝑥 2 (𝑡)
5000
4000
3000
2000
1000
0
0
500
900
1000
𝑥1 (𝑡)
1500
2000
800
relatively simple. The most critical thing in the using of the
theorem is the calculation of topological degree, that is, the
condition (c) of the theorem.
In this paper, motivated by [2, 7] of Liu et al. we propose
two classes of periodic 𝑁-species Lotka-Volterra facultative
mutualism systems with distributed delays. By applying
the continuation theorem of the coincidence degree theory
developed by Gaines and Mawhin and the Lyapunov function method, we easily obtain sufficient conditions for the
existence and global attractivity of positive periodic solutions
of the system. From Theorem 2 and Corollary 3, we can see
that the distributed time delays have effect on the existence
and global attractivity of positive periodic solutions, and
conditions (11) and (62) are very crucial to find the criteria
for globally attractive positive periodic solutions. Further,
the conditions (11) and (62) indicate that the undelayed
intraspecific competition dominates the delayed intraspecific
reproduction, and the intraspecific competition is more
significant than the interspecific cooperation.
𝑥 3 (𝑡)
700
600
Acknowledgments
500
This work was supported by the National Natural Science
Foundation of China (Grants nos. 11271312, 11261056, and
11261058), the China Postdoctoral Science Foundation (Grant
no. 20110491750), and the Natural Science Foundation of
Xinjiang (Grant no. 2011211B08).
400
300
200
100
0
References
0
500
900
1000
𝑥1 (𝑡)
1500
2000
800
700
𝑥3 (𝑡)
600
500
400
300
200
100
0
0
1000
2000
3000
4000
𝑥 2 (𝑡)
5000
6000
7000
Figure 1: Numerical simulation for system (68). Here, we take the
initial value 𝑥0 = (𝑥10 , 𝑥20 , 𝑥30 ) = (1, 1.5, 2).
5. Conclusions
Mawhin’s continuation theorem is a powerful tool for studying the existence of periodic solutions of periodic highdimensional time-delayed problems. When dealing with
time-delayed problem, it is very convenient and the result is
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