Existence of Almost-Periodic Solutions for Lotka-Volterra Cooperative

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, [email protected] 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: ⎛ u˙ i 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.


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

4


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 y˙ i 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


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 y, ˙ 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.

6

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


7

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


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 t ξt t ξt 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 t ξt 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


9

we obtain ⎡ y˙ i 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 −y˙ i 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 y˙ i 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 y˙ i 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 ,

10


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 − y˙ i 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 y˙ i 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


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


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: xt ˙ xt r1 t − b1 txt − τ1 t c12 tyt − τ12 t ,

4.1

yt ˙ yt r2 t − b2 tyt − τ2 t c21 txt − τ21 t ,

√ √ √ √ √ √ 2t sin 3t, b1 t 5t, c√ 3t/2, where r1 t√ 2 sin 12 t 2 cos 2t cos √ 2 sin 3t sin √ √ √ √ esin t cos 2t , r2 t 2−sin 2t−sin 3t, b t 2 sin 3t−sin 5t, τ1 t esin 2t sin√5t , τ12 t√ 2 √ √ √ c21 t 2 − cos 2t cos 3t/2, τ2 t esin 2t cos 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.


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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