Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 270104, 14 pages
doi:10.1155/2012/270104
Research Article
Existence of Almost-Periodic
Solutions for Lotka-Volterra Cooperative
Systems with Time Delay
Kaihong Zhao
Department of Applied Mathematics, Kunming University of Science and Technology,
Kunming, Yunnan 650093, China
Correspondence should be addressed to Kaihong Zhao, zhaokaihongs@126.com
Received 13 December 2011; Accepted 29 February 2012
Academic Editor: Hongyong Zhao
Copyright q 2012 Kaihong 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.
This paper considers the existence of positive almost-periodic solutions for almost-periodic LotkaVolterra cooperative system with time delay. By using Mawhin’s continuation theorem of coincidence degree theory, sufficient conditions for the existence of positive almost-periodic solutions
are obtained. An example and its simulation figure are given to illustrate the effectiveness of our
results.
1. Introduction
Lotka-Volterra system is one of the most celebrated models in mathematical biology and population dynamics. In recent years, it has also been found with successful and interesting applications in epidemiology, physics, chemistry, economics, biological science, and other areas
see 1–4. Moreover, in 5, it was shown that the continuous-time recurrent neural networks can be embedded into Lotka-Volterra models by changing coordinates, which suggests
that the existing techniques in the analysis of Lotak-Volterra systems can also be applied to
recurrent neural networks.
Owing to its theoretical and practical significance, Lotka-Volterra system have been
studied extensively see 6–16 and the cites therein. Since biological and environmental
parameters are naturally subject to fluctuation in time, the effects of a periodically varying
environment e.g., seasonal effects of weather, food supplies, mating habits, etc. are
considered as important selective forces on systems in a fluctuating environment. Therefore, on the one hand, models should take into account both the periodically changing
environment and the effects of time delays. However, on the other hand, in fact, it is more
2
Journal of Applied Mathematics
realistic and reasonable to study almost-periodic system than periodic system. Recently, there
are two main approaches to obtain sufficient conditions for the existence and stability of
the almost-periodic solutions of biological models: one is using the fixed point theorem,
Lyapunov functional method, and differential inequality techniques see 17–19; the other
is using functional hull theory and Lyapunov functional method see 14–16. However,
to the best of our knowledge, there are very few published letters considering the almostperiodic solutions for nonautonomous Lotka-Volterra cooperative system with time delay by
applying the method of coincidence degree theory. Motivated by this, in this letter, we apply
the coincidence theory to study the existence of positive almost-periodic solutions for LotkaVolterra cooperative system with time delay as follows:
⎛
u̇i t ui t⎝ri t − bi tui t − τi t n
⎞
cij tuj t − τij t ⎠,
i 1, 2, . . . , n,
1.1
j1,j /i
where ui t stands for the ith species population density at time t ∈ R, ri t is the natural
reproduction rate, bi t represents the inner-specific competition, cij t i / j stands for the
interspecific cooperation, τi t > 0 and τij t > 0 are all continuous almost-periodic functions
on R. Throughout this paper, we always assume that ri t, bi t, and cij t are all nonegative
almost periodic functions with respect to t ∈ R.
The initial condition of 1.1 is of the form
ui s φi s,
i 1, 2, . . . , n,
1.2
where φi s is positive bounded continuous function on −τ, 0 and τ
max1≤i,j≤n supt∈R {|τij t|}.
The organization of the rest of this paper is as follows. In Section 2, we introduce
some preliminary results which are needed in later sections. In Section 3, we establish our
main results for the existence of almost-periodic solutions of 1.1. Finally, an example and its
simulation figure are given to illustrate the effectiveness of our results in Section 4.
2. Preliminaries
To obtain the existence of an almost-periodic solution of system 1.1, we first make the following preparations.
Definition 2.1 see 20. Let ut : R → R be continuous in t. ut is said to be almost-periodic
on R, if, for any > 0, the set Ku, {δ : |ut δ − ut| < , for any t ∈ R} is relatively
dense, that is, for any > 0, it is possible to find a real number l > 0, for any interval with
length l, there exists a number δ δ in this interval such that |ut δ − ut| < , for
any t ∈ R.
Definition 2.2. A solution ut u1 t, u2 t, . . . , un tT of 1.1 is called an almost periodic
solution if and only if for each i 1, 2, . . . , n, ui t is almost periodic.
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3
For convenience, we denote AP R the set of all real-valued, almost-periodic functions
on R and for each j 1, 2, . . . , n, let
1 T
−iλs
fj se ds /
0 ,
λ ∈ R : lim
T →∞T 0
N
mod fj ni λ
i : ni ∈ Z, N ∈ N , λ
i ∈ ∧ fj
∧ fj 2.1
i1
be the set of Fourier exponents and the module of fj , respectively, where fj · is almost
periodic. Suppose fj t, φj is almost periodic in t, uniformly with respect to φj ∈ C−τ, 0, R.
Kj fj , , φj denote the set of -almost periods for fj uniformly with respect to φj ∈
T
C−τ, 0, R. lj denote the length of inclusion interval. Let mfj 1/T 0 fj sds be
the mean value of fj on interval 0, T , where T > 0 is a constant. clearly, mfj depends on
T
T. mfj limT → ∞ 1/T 0 fj sds.
Lemma 2.3 see 20. Suppose that f and g are almost periodic. Then the following statements are
equivalent.
i modf ⊃ modg,
ii for any sequence {t∗n }, if limn → ∞ ft t∗n ft for each t ∈ R, then there exists a
subsequence {tn } ⊆ {t∗n } such that limn → ∞ gt tn ft for each t ∈ R.
Lemma 2.4 see 21. Let u ∈ AP R. Then
t
t−τ
usds is almost periodic.
Let X and Z be Banach spaces. A linear mapping L : domL ⊂ X → Z is called
Fredholm if its kernel, denoted by kerL {X ∈ domL : Lx 0}, has finite dimension and
its range, denoted by ImL {Lx : x ∈ domL}, is closed and has finite codimension. The
index of L is defined by the integer dim KL − codimdomL. If L is a Fredholm mapping
with index 0, then there exist continuous projections P : X → X and Q : Z → Z such
that ImP kerL and kerQ ImL. Then L|domL∩kerP : ImL ∩ kerP → ImL is
bijective, and its inverse mapping is denoted by KP : ImL → domL ∩ kerP . Since kerL
is isomorphic to ImQ, there exists a bijection J : kerL → ImQ. Let Ω be a bounded
open subset of X and let N : X → Z be a continuous mapping. If QNΩ is bounded and
KP I − QN : Ω → X is compact, then N is called L-compact on Ω, where I is the identity.
Let L be a Fredholm linear mapping with index 0 and let N be a L-compact mapping
on Ω. Define mapping F : domL ∩ Ω → Z by F L − N. If Lx /
Nx for all x ∈ domL ∩ ∂Ω,
then by using P, Q, KP , J defined above, the coincidence degree of F in Ω with respect to L is
defined by
degL F, Ω deg I − P − J −1 Q KP I − Q N, Ω, 0 ,
where degg, Γ, p is the Leray-Schauder degree of g at p relative to Γ.
Then The Mawhin’s continuous theorem is given as follows.
2.2
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Lemma 2.5 see 22. Let Ω ⊂ X be an open bounded set and let N : X → Z be a continuous
operator which is L-compact on Ω. Assume
a for each λ ∈ 0, 1, x ∈ ∂Ω ∩ domL, Lx /
λNx;
b for each x ∈ ∂Ω ∩ L, QNx / 0;
c degJNQ, Ω ∩ kerL, 0 /
0.
Then Lx Nx has at least one solution in Ω ∩ domL.
In this paper, since we need some related properties of M-matrix we introduce them
as follows. In addition, A matrix A aij ≥ 0 means that each elements aij ≥ 0.
Definition 2.6 see 23. If a real matrix A aij n×n satisfies the following conditions 1 and
2:
j, i, j 1, 2, . . . , n,
1 aii > 0, i 1, 2, . . . , n, aij ≤ 0, i /
2 A is a positive-definite matrix,
then A is called a M-matrix.
Lemma 2.7 see 23. If matrix A aij n×n is a M-matrix, then A−1 exists and its every element
is nonnegative.
Lemma 2.8. Suppose that matrix A aij n×n is a M-matrix, then AX ≤ B implies X ≤ A−1 B.
Proof. In fact, there exists a nonnegative positive vector ε0 ε1 , ε2 , . . . , εn T ∈ Rn such that
AX − B ε0 0, 0, . . . , 0T which imply that X − A−1 B A−1 ε0 0, 0, . . . , 0T . According to
Lemma 2.4, there exists at least one positive element in the every row of A−1 , which imply
A−1 ε0 ≥ 0, 0, . . . , 0T . Thus, we obtain X ≤ A−1 B.
3. Main Result
In this section, we state and prove our main results of our this paper. By making the substitution
ui t eyi t ,
3.1
i 1, 2, . . . , n,
Equation 1.1 can be reformulated as
ẏi t ri t − bi teyi t−τii t n
cij teyj t−τij t ,
i 1, 2, . . . , n.
j1,i /
j
3.2
The initial condition 1.2 can be rewritten as follows:
yi s ln φi s : ψi s,
i 1, 2, . . . , n.
3.3
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5
Set X Z V1 ⊕ V2 , where
T
V1 yt y1 t, y2 t, . . . , yn t ∈ CR, Rn : yi t ∈ AP R ,
mod yi t ⊂ modHi t, ∀λ
i ∈ ∧ yi t satisfies
λi > β, i 1, 2, . . . , n ,
V2 yt ≡ h1 , h2 , . . . , hn T ∈ Rn ,
n
Hi t ri t − bi teψi −τii 0 cij teψj −τij 0
j1,i /j
3.4
and ψ· is defined as 3.3, i 1, 2, . . . , n, β > 0 is a given constant. For y y1 , y2 , . . . , yn T ∈
Z, define y max1≤i≤n supt∈R |yi t|.
Lemma 3.1. Z is a Banach space equipped with the norm · .
{k}
{k}
{k}
Proof. If y{k} ⊂ V1 and y{k} y1 , y2 , . . . , yn T converges to y y1 , y2 , . . . , y n T , that
{k}
is, yj
→ y j , as k → ∞, j 1, 2, . . . , n. Then it is easy to show that yj ∈ AP R and
| ≤ β, we have that
mody ∈ modHj . For any |λ
j
j
1
T →∞T
T
lim
0
{k}
yj te−iλj t dt 0,
j 1, 2, . . . , n;
3.5
therefore,
1
lim
T →∞T
T
0
y j te−iλj t dt 0,
j 1, 2, . . . , n,
3.6
which implies y ∈ V1 . Then it is not difficult to see that V1 is a Banach space equipped with
the norm · . Thus, we can easily verify that X and Z are Banach spaces equipped with the
norm · . The proof of Lemma 3.1 is complete.
Lemma 3.2. Let L : X → Z, Ly ẏ, then L is a Fredholm mapping of index 0.
Proof. Clearly, L is a linear operator and kerL V2 . We claim that ImL V1 . Firstly, we
{1}
suppose that zt z1 t, z2 t, . . . , zn tT ∈ ImL ⊂ Z. Then there exist z{1} t z1 t,
{1}
{1}
{2} {2}
{2}
z2 t, . . . , zn tT ∈ V1 and constant vector z{2} z1 , z2 , . . . , zn T ∈ V2 such that
zt z{1} t z{2} ,
3.7
{1}
3.8
that is,
{2}
zi t zi t zi ,
i 1, 2, . . . , n.
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Journal of Applied Mathematics
{1}
From the definition of zi t and zi t, we can easily see that
t
t−τ
{2}
zi sds and
t
t−τ
{1}
zi sds are
almost-periodic functions. So we have zi ≡ 0, i 1, 2, . . . , n, then z{2} ≡ 0, 0, . . . , 0T , which
implies zt ∈ V1 , that is ImL ⊂ V1 .
On the other hand, if ut u1 t, u2 t, . . . , un tT ∈ V1 \ {0}, then we have
t
u sds ∈ AP R, j 1, 2, . . . , n. If λ
j /
0, then we obtain
0 j
1
lim
T →∞T
T t
0
uj sds e−iλj t dt 0
1
1
lim
T
→
∞
T
i λj
T
uj te−iλj t dt,
j 1, 2, . . . , n.
3.9
0
It follows that
∧
t
uj sds − m
t
0
uj sds
∧ uj t ,
j 1, 2, . . . , n,
3.10
0
hence
t
usds − m
t
usds
0
∈ V1 ⊂ X.
3.11
0
t
t
Note that 0 usds − m 0 usds is the primitive of ut in X, we have ut ∈ ImL, that is,
V1 ⊂ ImL. Therefore, ImL V1 .
Furthermore, one can easily show that ImL is closed in Z and
dim kerL n co dim ImL;
3.12
therefore, L is a Fredholm mapping of index 0. The proof of Lemma 3.2 is complete.
y
y
y
Lemma 3.3. Let N : X → Z, Ny G1 , G2 , . . . , Gn T , where
Gi ri t − bi tyi t−τi t n
y
cij teyj t−τij t ,
i 1, 2, . . . , n.
j1,i /
j
3.13
Set
P : X −→ Z,
Q : Z −→ Z,
T
P y m y1 , m y2 , . . . , m yn
,
Qz mz1 , mz2 , . . . , mzn T .
3.14
Then N is L-compact on Ω, where Ω is an open bounded subset of X.
Proof. Obviously, P and Q are continuous projectors such that
Im P kerL,
ImL kerQ.
3.15
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It is clear that I − QV2 {0, 0, . . . , 0}, I − QV1 V1 . Hence
ImI − Q V1 ImL.
3.16
Then in view of
ImP kerL,
ImL kerQ ImI − Q,
3.17
we obtain that the inverse KP : ImL → kerP ∩ domL of LP exists and is given by
KP z t
zsds − m
t
zsds .
0
3.18
0
Thus,
y T
y y
QNy m G1 , m G2 , . . . , m Gn
T
KP I − QNy f y1 − Q f y1 , f y2 − Q f y2 , . . . , f yn − Q f yn
,
3.19
where
f yi t
0
y y
Gi − m Gi ds,
i 1, 2, . . . , n.
3.20
Clearly, QN and I − QN are continuous. Now we will show that KP is also continuous. By assumptions, for any 0 < < 1 and any compact set φi ⊂ C−τ, 0, R, let
li i be the length of the inclusion interval of Ki Hi , i , φi , i 1, 2, . . . , n. Suppose that
{zk t} ⊂ ImL V1 and zk t zk1 t, zk2 t, . . . , zkn tT uniformly converges to zt t
z1 t, z2 t, . . . , zn tT , that is zki → zi , as k → ∞, i 1, 2, . . . , n. Because of 0 zk sds ∈ Z,
t
k 1, 2, . . . , n, there exists σi 0 < σi < i such that Ki Hi , σi , φi ⊂ Ki 0 zki sds, σi , φi ,
i 1, 2, . . . , n. Let li σi be the length of the inclusion interval of Ki Hi , σi , φi and
li max{li i , li σi },
i 1, 2, . . . , n.
3.21
It is easy to see that li is the length of the inclusion interval of Ki Hi , σi , φi and Ki Hi , i , φi ,
t
i 1, 2, . . . , n. Hence, for any t ∈
/ 0, li , there exists ξt ∈ Ki Hi , σi , φi ⊂ Ki 0 zki sds, σi , φi 8
Journal of Applied Mathematics
such that t ξt ∈ 0, li , i 1, 2, . . . , n. Hence, by the definition of almost periodic function we
have
t
t
k
k
z sds max sup zi sds
1≤i≤n t∈R 0
0
tξt
tξt
t
t
k
k
k
k
≤ max sup zi sds max sup zi sds −
zi sds zi sds
1≤i≤n t∈0,l 0
1≤i≤n t/
0
0
0
∈ 0,li i
tξt
t
t
≤ 2max sup zki sds max sup zki sds −
zki sds
1≤i≤n t∈0,l 0
1≤i≤n t/
0
∈ 0,li 0
i
li
k
≤ 2max zi sds max i .
1≤i≤n
1≤i≤n 0
3.22
t
From this inequality, we can conclude that 0 zsds is continuous, where zt z1 t, z2 t,
. . . , zn tT ∈ ImL. Consequently, KP and KP I − QNy are continuous.
t
From 3.22, we also have 0 zsds and KP I −QNy also are uniformly bounded in Ω.
Further, it is not difficult to verify that QNΩ is bounded and KP I−QNy is equicontinuous
in Ω. By the Arzela-Ascoli theorem, we have immediately concluded that KP I − QNΩ is
compact. Thus N is L-compact on Ω. The proof of Lemma 3.3 is complete.
Theorem 3.4. Assume that the following conditions H1 and H2 hold:
H1 ei : mri t > 0, mbi t > 0, mcij t > 0, i, j 1, 2, . . . , n;
H2 D is a positive-definite matrix, where dii mbi t, dij mcij t, i / j, i, j 1, 2, . . . , n,
⎛
d11
⎜
⎜ −d21
⎜
D⎜
⎜ ..
⎜ .
⎝
−dn1
−d12 · · · −d1n
⎞
⎟
d22 · · · −d2n ⎟
⎟
⎟.
..
..
.. ⎟
.
.
. ⎟
⎠
−dn2 · · · dnn
3.23
Then 1.1 has at least one positive almost periodic solution.
Proof. To use the continuation theorem of coincidence degree theorem to establish the existence of a solution of 3.2, we set Banach space X and Z the same as those in Lemma 3.1
and set mappings L, N, P , Q the same as those in Lemmas 3.2 and 3.3, respectively. Then we
can obtain that L is a Fredholm mapping of index 0 and N is a continuous operator which is
L-compact on Ω.
Now, we are in the position of searching for an appropriate open, bounded subset Ω
for the application of the continuation theorem. Corresponding to the operator equation
Ly λNy,
λ ∈ 0, 1,
3.24
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9
we obtain
⎡
ẏi t λ⎣ri t − bi teyi t−τi t n
⎤
cij teyj t−τij t ⎦,
i 1, 2, . . . , n.
3.25
j1,i /j
Assume that yt y1 t, y2 t, . . . , yn tT ∈ X is a solution of 3.25 for some λ ∈ 0, 1.
Denote Mi supt∈R {yi t}, Mi inft∈R {yi t}.
On the one hand, by 3.25, we derive
⎡
⎤
n
−ẏi t ≤ λ⎣ri t − bi teyi t−τi t cij teyj t−τij t ⎦,
i 1, 2, . . . , n.
3.26
j1,i /
j
On the both sides of 3.26, integrating from 0 to T and applying the mean value theorem of
integral calculus, we have
⎡
0 ≤ λ⎣mri t − mbi teyi ξi −τi ξi n
⎤
m cij t eyj ηij −τij ηij ⎦
j1,i /
j
m ẏi t ,
3.27
i 1, 2, . . . , n,
where ξi ∈ 0, T , ηij ∈ 0, T , i, j 1, 2, . . . , n. In the light of 3.27, we get for i 1, 2, . . . , n,
⎡
⎤
n
m cij t eyj ηij −τij ηij ⎦ m ẏi t .
λ mbi teyi ξi −τi ξi ≤ λ⎣mri t 3.28
j1,i /j
On the both sides of 3.28, taking the supremum with respect to ξi , ηij and letting T → ∞,
we obtain
mbi teMi ≤ mri t n
!
m cij t eMj ,
3.29
i 1, 2, . . . , n.
3.30
j1,j /i
that is,
dii eMi −
n
j1,j /
i
dij eMj ≤ ei ,
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Journal of Applied Mathematics
Equation 3.30 can be written by the following matrix form
⎛
d11
⎞⎛ M1 ⎞ ⎛ ⎞
e
e1
⎟ ⎜ ⎟
⎟⎜
⎜
⎟
· · · −d2n ⎟
e2 ⎟
e M2 ⎟ ⎜
⎟⎜
⎟
⎟ ⎜
⎜
⎟⎜
.
≤
⎜ . ⎟ ⎜.⎟
..
.. ⎟
⎜ . ⎟ ⎜.⎟
⎟
⎜
⎟
.
. ⎟
.
⎠⎝ . ⎠ ⎝ ⎠
en
· · · dnn
e Mn
−d12 · · · −d1n
⎜
⎜ −d21 d22
⎜
⎜
⎜ ..
..
⎜ .
.
⎝
−dn1 −dn2
3.31
By Lemma 2.8 and assumption, we obtain
⎛
⎞
⎛ ⎞ ⎛ ⎞
H1
e1
⎜
⎟
⎜ ⎟ ⎜ ⎟
⎜ M2 ⎟
⎜
⎜
⎟
⎟
⎜e ⎟
⎜ e2 ⎟ ⎜H2 ⎟
⎜
⎟
−1 ⎜ ⎟
⎜
⎜ . ⎟ ≤ D ⎜ . ⎟ : ⎜ . ⎟
⎟,
⎜ . ⎟
⎜ .. ⎟ ⎜ .. ⎟
⎜ . ⎟
⎝ ⎠ ⎝ ⎠
⎝
⎠
Mn
en
Hn
e
e M1
3.32
which imply that
Mi ≤ ln Hi ,
3.33
i 1, 2, . . . , n.
On the two sides of 3.28, taking the infimum with respect to ξi , ηij , and letting T → ∞, we
obtain
Mi ≤ ln Hi ,
i 1, 2, . . . , n.
3.34
On the other hand, according to 3.25, we derive
λri t − ẏi t < λbi teyi t−τi t ,
i 1, 2, . . . , n.
3.35
On the both sides of 3.35, integrating from 0 to T and using the mean value theorem of
integral calculus, we get
λmri t < m ẏi t λmbi teyi ζi −τi ζi ,
i 1, 2, . . . , n,
3.36
where ζi ∈ 0, T , i 1, 2, . . . , n. On the both sides of 3.36, take the supremum and infimum
with respect to ζi , respectively, and let T → ∞, then we have for i 1, 2, . . . , n,
mri t < mbii teMi ,
mri t < mbii teMi ,
3.37
namely,
e Mi >
ei
mri t
,
mbi t dii
e Mi >
ei
mri t
mbi t dii
3.38
Journal of Applied Mathematics
11
which imply that
Mi > ln
ei
,
dii
Mi > ln
ei
.
dii
3.39
Combining with 3.33, 3.34, and 3.39, we derive for all t ∈ R, i 1, 2, . . . , n,
"
#
$
%
ei
ei
min ln
< Mi ≤ yi t ≤ Mi ≤ ln Hi < max ln Hi 1.
≤ ln
1≤i≤n
1≤i≤n
dii
dii
3.40
Denote M max{|min1≤i≤n {lnei /dii }|, |max1≤i≤n {ln Hi } 1|}. Clearly, M is independent of
λ. Take
T
Ω y y1 , y2 , . . . , yn ∈ X : y < M .
3.41
It is clear that Ω satisfies the requirement a in Lemma 2.5. When y ∈ ∂Ω ∩ kerL, y y1 , y2 , . . . , yn T is a constant vector in Rn with y M. Then
QNy mG1 , mG2 , . . . , mGn T ,
y ∈ X,
3.42
where
Gi ri t − bi teyi n
cij teyj ,
i 1, 2, . . . , n,
j1,j /
i
y
mGi mri t − mbii te i n
dij eyj ,
n
!
m cij t eyj ei − dii eyi
j1,j /
i
3.43
i 1, 2, . . . , n.
j1,j /
i
If QNy 0, 0, . . . , 0T , then we have
⎛
d11
⎜
⎜ −d21
⎜
⎜
⎜ ..
⎜ .
⎝
⎛ ⎞
e1
⎟⎜ y ⎟ ⎜ ⎟
⎜ 2⎟ ⎜ ⎟
· · · −d2n ⎟
⎟⎜ e ⎟ ⎜ e2 ⎟
⎟⎜ ⎟ ⎜ ⎟,
⎜ .. ⎟ ⎜ .. ⎟
..
.. ⎟
⎜ ⎟ ⎜ ⎟
.
. ⎟
⎠⎝ . ⎠ ⎝ . ⎠
e yn
en
· · · dnn
−d12 · · · −d1n
d22
..
.
−dn1 −dn2
⎞⎛
e y1
⎞
3.44
which imply that yi ln Hi , i 1, 2, . . . , n. Thus, y y1 , y2 , . . . , yn T ∈ Ω, this contradicts the
fact that y ∈ ∂Ω ∩ kerL. Therefore, QNy /
0, 0, . . . , 0T , which implies that the requirement
b in Lemma 2.5 is satisfied. If necessary, we can let M be greater such that yT QNy < 0, for
12
Journal of Applied Mathematics
any y ∈ ∂Ω ∩ kerL. Furthermore, take the isomorphism J : ImQ → kerL, Jz ≡ z and let
Φγ; y −γy 1 − γJQNy, then for any y ∈ ∂Ω ∩ kerL, yT Φγ; y < 0, we have
$
%
deg{JQN, Ω ∩ kerL, 0} deg −y, Ω ∩ kerL, 0 /
0.
3.45
So, the requirement c in Lemma 2.5 is satisfied. Hence, 3.2 has at least one almost-periodic
solution in Ω, that is, 1.1 has at least one positive almost periodic solution. The proof is
complete.
4. An Example and Simulation
Consider the following two species cooperative system with time delay:
ẋt xt r1 t − b1 txt − τ1 t c12 tyt − τ12 t ,
4.1
ẏt yt r2 t − b2 tyt − τ2 t c21 txt − τ21 t ,
√
√
√
√
√
√
2tsin 3t, b1 t
5t, c√
3t/2,
where r1 t√ 2sin
12 t 2cos 2tcos
√ 2sin 3tsin
√
√
√
√
esin tcos 2t , r2 t 2−sin
2t−sin
3t,
b
t
2sin
3t−sin
5t,
τ1 t esin 2tsin√5t , τ12 t√
2
√
√
√
c21 t 2 − cos 2t cos 3t/2, τ2 t esin 2tcos 5t , τ21 t esin t−cos 2t . Since
e1 mr1 t 2,
d11 mb1 t 2,
d12 mc12 t 1,
e2 mr1 t 2,
2 −1
D
,
−1 2
d22 mb1 t 2,
2 −1
det
3 > 0,
−1 2
d21 mc12 t 1,
1 2 1
−1
,
D 3 1 2
4.2
then, the matrix D is positive definite, and
e1
e2
ln
ln
0,
d11
d22
H1
H2
D
−1
"
"
#
ei ,
M max min ln
1≤i≤n
d ii
e1
e2
2
2
,
ln H1
ln H2
#
$
%
max ln H 1 1 ln 2,
i
ln 2
ln 2
,
4.3
1≤i≤n
T
Ω y y1, y2 , . . . , yn ∈ X : y < 1 In 2 .
Therefore, all conditions of Theorem 3.4 are satisfied. By Theorem 3.4, system 4.1 has one
positive almost-periodic solution. The resulting numerical simulation is depicted in Figure 1.
Journal of Applied Mathematics
13
9
8
7
6
5
4
3
2
1
0
−1
0
20
40
60
80
100
t
x
y
Figure 1
Acknowledgments
This work is supported by the National Natural Sciences Foundation of People’s Republic
of China under Grant no. 11161025, Yunnan Province natural scientific Research Fund
Project no. 2011FZ058, and Yunnan Province education department scientific Research
Fund Project no. 2001Z001.
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