EXISTENCE OF POSITIVE ALMOST PERIODIC SOLUTIONS FOR

Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 157, pp. 1–13. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

EXISTENCE OF POSITIVE ALMOST PERIODIC SOLUTIONS FOR DELAY LOTKA-VOLTERRA COOPERATIVE SYSTEMS KAIHONG ZHAO, JUQING LIU

Abstract. In this article, we study a Lotka-Volterra cooperative system of equations with time-varying delays and distributed delays. By using Mawhin’s continuation theorem of coincidence degree theory, we obtain sufficient conditions for the existence of positive almost periodic solutions. Also we present an example to illustrate our results.

1. Introduction The well-known Lotka-Volterra model for ecological population modeling was proposed by Lotka [11] and Volterra [25], and have been extensively investigated. In recent years, it has found applications in epidemiology, physics, chemistry, economics, biological science, and other areas (see [1, 6, 7]). Due to their theoretical and practical significance, Lotka-Volterra systems have an extensive literature; see for example the references in this article. Since biological and environmental parameters are naturally subject to fluctuation in time, periodically and almost periodically varying effects are important selective forces for biological systems. Models should take into account the seasonality or periodic changing conditions [9, 10, 13, 14, 15, 20, 23]. However, models are more realistic when considering almost periodic conditions. In both cases, models should include the effects of time delays. There are many works on the study of the Lotka-Volterra type periodic systems [12, 13, 14, 15, 27, 28]. However, there are only a few articles on the existence of almost periodic solutions. Recently, by using: almost periodic functions, contraction mappings, fixed point theory, appropriate Lyapunov functionals, and almost periodic functional hull theory, some authors have published works in the theory on almost periodic systems [2, 3, 16, 17, 18, 19, 20, 22, 26, 30]. Motivated by this, in this article, we apply the coincidence theory to study the existence of positive almost periodic solutions for the delay Lotka-Volterra type

2000 Mathematics Subject Classification. 34K14, 92D25. Key words and phrases. Lotka-Volterra cooperative system; almost-periodic solution; coincidence degree; delay equation. c

2013 Texas State University - San Marcos. Submitted December 4, 2012. Published July 8, 2013. 1

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system 

u˙ i (t) = ui (t) ri (t) − Fi t, ui (t) − fi t, ui (t − τii (t)) Z 0 n X
µii (t, s)ui (t + s)ds + − Gij t, uj (t) −σii

+

n X

j=1,j6=i n Z 0 X


gij t, uj (t − τij (t)) +

j=1

j=i,j6=i

(1.1)

 µij (t, s)uj (t + s)ds ,

−σij

where i = 1, 2, . . . , n, ui (t) stands for the ith species population density at time t ∈ R, ri (t) is the natural reproduction rate, Fi , fi and µii represent the innerspecific competition, Gij , gij and µij (i 6= j) stand for the interspecific cooperation, τij (t) > 0 are all continuous almost periodic functions on R, µij (t, s) are positive almost periodic functions on R × [−σij , 0] and continuous with respect to t ∈ R and integrable with respect to s ∈ [−σij , 0], where σij are nonnegative constants, R0 moreover −σij µij (t, s)ds = 1, i, j = 1, 2, . . . , n. Throughout this paper, we always assume that ri , Fi , fi , Gij and gij are nonnegative almost periodic functions with respect to t ∈ R and satisfy the following conditions for each i, j = 1, 2, . . . , n ∂fi (t, x) ∂Gij (t, x) ∂gij (t, x) ∂Fi (t, x) > 0, > 0, > 0, > 0, ∂x ∂x ∂x ∂x and for each t ∈ R, there exist positive constants αi , βi , γij , δij such that Fi (t, αi ) = 0,

fi (t, βi ) = 0,

Gij (t, γij ) = 0,

gij (t, δij ) = 0.

(1.2)

(1.3)

The initial condition of (1.1) is of the form ui (s) = φi (s), i = 1, 2, . . . , n,

(1.4)

where φi (s) is positive bounded continuous function on [−τ, 0] and τ = max1≤i,j≤n {supt∈R : τij (t)|, σij }. 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 is 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 ([5]). Let u(t) : R → R be continuous in t. u(t) is said to be almost periodic on R, if, for any  > 0, the set K(u, ) = {δ : |u(t + δ) − u(t)| < , 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 |u(t + δ) − u(t)| < , for any t ∈ R. Definition 2.2. A solution u(t) = (u1 (t), u2 (t), . . . , un (t))T 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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For convenience, we denote AP (R) is the set of all real valued, almost periodic functions on R. For each j = 1, 2, . . . , n, let Z T  e e ∈ R : lim 1 ∧(fj ) = λ fj (s)e−iλs ds 6= 0 , T →∞ T 0 mod(fj ) =

N X

ni λei : ni ∈ Z, N ∈ N + , λei ∈ ∧(fj )



i=1

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). Let Kj (fj , , φj ) denote the set of -almost periods for fj uniformly with respect to φj ∈ C([−τ, 0], R). lj () denote the length of inclusion RT interval. m(fj ) = T1 0 fj (s)ds be the mean value of fj on interval [0, T ], where RT T > 0 is a constant. clearly, m(fj ) depends on T . m[fj ] = limT →∞ T1 0 fj (s)ds. Lemma 2.3 ([5]). Suppose that f and g are almost periodic. Then the following statements are equivalent (i) mod(f ) ⊃ mod(g), (ii) for any sequence {t∗n }, if limn→∞ f (t + t∗n ) = f (t) for each t ∈ R, then there exists a subsequence {tn } ⊆ {t∗n } such that limn→∞ g(t + tn ) = f (t) for each t ∈ R. Rt Lemma 2.4 ([4]). Let u ∈ AP (R). Then t−τ u(s)ds is almost periodic. Let X and Z be Banach spaces. A linear mapping L : dom(L) ⊂ X → Z is called Fredholm if its kernel, denoted by ker(L) = {X ∈ dom(L) : Lx = 0}, has finite dimension and its range, denoted by Im(L) = {Lx : x ∈ dom(L)}, is closed and has finite codimension. The index of L is defined by the integer dim K(L) − codim dom(L). If L is a Fredholm mapping with index zero, then there exists continuous projections P : X → X and Q : Z → Z such that Im(P ) = ker(L) and ker(Q) = Im(L). Then L|dom(L)∩ker(P ) : Im(L) ∩ ker(P ) → Im(L) is bijective, and its inverse mapping is denoted by KP : Im(L) → dom(L)∩ker(P ). Since ker(L) is isomorphic to Im(Q), there exists a bijection J : ker(L) → Im(Q). Let Ω be a bounded open subset of X and let N : X → Z be a continuous mapping. If QN (Ω) is bounded and KP (I − Q)N : 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 zero and let N be a L-compact mapping on Ω. Define mapping F : dom(L) ∩ Ω → Z by F = L − N . If Lx 6= N x for all x ∈ dom(L) ∩ ∂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 deg(g, Γ, p) is the Leray-Schauder degree of g at p relative to Γ. Then the Mawhin’s continuous theorem is reads as follows. Lemma 2.5 ([8]). 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 ∩ dom(L), Lx 6= λN x; (b) for each x ∈ ∂Ω ∩ L, QN x 6= 0; (c) deg(JN Q, Ω ∩ ker(L), 0) 6= 0.

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Then Lx = N x has at least one solution in Ω ∩ dom(L). 3. Existence of positive almost-periodic solutions In this section, we state and prove our main results of our this paper. By making the substitution ui (t) = eyi (t) , i = 1, 2, . . . , n, Equation (1.1) can be reformulated as Z 0 y˙ i (t) = ri (t) − Fi (t, eyi (t) ) − fi (t, eyi (t−τii (t)) ) − µii (t, s)eyi (t+s) ds −σii

+

−

n X

Gij (t, eyj (t) ) +

j=1,j6=i Z 0 n X

n X

gij (t, eyj (t−τij (t)) )

(3.1)

j=1,i6=j

µij (t, s)eyj (t+s) ds,

i = 1, 2, . . . , n.

−σij

j=1,j6=i

The initial condition (1.4) can be rewritten as yi (s) = ln φi (s) =: ψi (s),

i = 1, 2, . . . , n.

(3.2)

Set X = Z = V1 ⊕ V2 , where n V1 = y(t) = (y1 (t), y2 (t), . . . , yn (t))T ∈ C(R, Rn ) : yi (t) ∈ AP (R), o mod(yi (t)) ⊂ mod(Hi (t)), ∀λei ∈ ∧(yi (t)), |λei | > β, i = 1, 2, . . . , n , n o V2 = y(t) ≡ (h1 , h2 , . . . , hn )T ∈ Rn Hi (t) = ri (t) − Fi (t, eψi (t) ) − fi (t, eψi (−τii (0)) ) −

Z

0

µii (t, s)eψi (s) ds

−σii n n Z X   X + Gij (t, eψj (t) ) + gij (t, eψj (−τij (0)) ) −

µij (t, s)eψj (s) ds

−σij

j=1

j=1,j6=i

0

and ψ(·) is defined as (3.2), i = 1, 2, . . . , n, β > 0 is a given constant. For y = (y1 , y2 , . . . , yn )T ∈ Z, define kyk = max1≤i≤n supt∈R |yi (t)|. Lemma 3.1. Z is a Banach space equipped with the norm k · k. {k}

{k}

{k}

Proof. Let y {k} = (y1 , y2 , . . . , yn )T ⊂ V1 , and let y {k} converge to y = {k} (y 1 , y 2 , . . . , y n )T ; that is, yj → y j , as k → ∞, j = 1, 2, . . . , n. Then it is easy to e | ≤ β, we have that show that y ∈ AP (R) and mod(y ) ∈ mod(Hj ). For any |λ j

j

lim

T →∞

1 T

Z

T

{k}

yj

j

(t)e−iλj t dt = 0, e

j = 1, 2, . . . , n;

0

therefore, Z 1 T e lim y j (t)e−iλj t dt = 0, j = 1, 2, . . . , n, T →∞ T 0 which implies y ∈ V1 . Then it is not difficult to see that V1 is a Banach space equipped with the norm k · k. Thus, we can easily verify that X and Z are Banach spaces equipped with the norm k · k. The proof is complete. 

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Lemma 3.2. Let L : X → Z, Ly = y, ˙ then L is a Fredholm mapping of index zero. Proof. Clearly, L is a linear operator and ker(L) = V2 . We claim that Im(L) = V1 . Firstly, we suppose that z(t) = (z1 (t), z2 (t), . . . , zn (t))T ∈ Im(L) ⊂ Z. Then there {1} {1} {1} exist z {1} (t) = (z1 (t), z2 (t), . . . , zn (t))T ∈ V1 and constant vector z {2} = {2} {2} {2} T (z1 , z2 , . . . , zn ) ∈ V2 such that z(t) = z {1} (t) + z {2} ; that is, {1}

zi (t) = zi

{2}

(t) + zi

,

i = 1, 2, . . . , n.

{1} zi (t),

Rt From the definition of zi (t) and we can easily see that t−τ zi (s)ds and Rt {1} {2} z (s)ds are almost periodic function. So we have zi ≡ 0, i = 1, 2, . . . , n, t−τ i then z {2} ≡ (0, 0, . . . , 0)T , which implies z(t) ∈ V1 , that is Im(L) ⊂ V1 . On the other hand, if u(t) = (u1 (t), u2 (t), . . . , un (t))T ∈ V1 \{0}, then we have Rt ej 6= 0, then we obtain u (s)ds ∈ AP (R), j = 1, 2, . . . , n. If λ 0 j Z Z t Z T 1 1 T e e ( uj (s)ds)e−iλj t dt = uj (t)e−iλj t dt, lim lim f rac1T e T →∞ T 0 iλj T →∞ 0 0 for j = 1, 2, . . . , n. It follows that Z t Z t ∧[ uj (s)ds − m( uj (s)ds)] = ∧(uj (t)), 0

j = 1, 2, . . . , n;

0

hence Z

t

Z t u(s)ds − m( u(s)ds) ∈ V1 ⊂ X

0

0

Rt

Rt Note that 0 u(s)ds − m( 0 u(s)ds) is the primitive of u(t) in X, we have u(t) ∈ Im(L), that is V1 ⊂ Im(L). Therefore, Im(L) = V1 . Furthermore, one can easily show that Im(L) is closed in Z and dim ker(L) = n = codim Im(L); therefore, L is a Fredholm mapping of index zero. The proof is complete.



Lemma 3.3. Let N : X → Z, N y = (Gy1 , Gy2 , . . . , Gyn )T , where Gyi = ri (t) − Fi (t, exp{yi (t)}) − fi (t, exp{yi (t − τii (t))}) Z 0 n X − µii (t, s) exp{yi (t + s)}ds + Gij (t, exp{yj (t)}) −σii

+

n X

j=1,j6=i n X

Z

0

gij (t, exp{yj (t − τij (t)}) +

j=1,i6=j

j=1,j6=i

µij (t, s) exp{yj (t + s)}ds, −σij

for j = 1, 2, . . . , n. Set P : X → Z, Q : Z → Z,

P y = (m(y1 ), m(y2 ), . . . , m(yn ))T , Qz = (m[z1 ], m[z2 ], . . . , m[zn ])T .

Then N is L-compact on Ω, where Ω is an open bounded subset of X.

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Proof. Obviously, P and Q are continuous projectors such that Im P = ker(L),

Im(L) = ker(Q).

It is clear that (I − Q)V2 = {(0, 0, . . . , 0)}, (I − Q)V1 = V1 . Hence Im(I − Q) = V1 = Im(L). Then in view of Im(L) = ker(Q) = Im(I − Q),

Im(P ) = ker(L),

we obtain that the inverse KP : Im(L) → ker(P ) ∩ dom(L) of LP exists and is given by Z t Z t KP (z) = z(s)ds − m[ z(s)ds]. 0

0

Thus, QN y = (m[Gy1 ], m[Gy2 ], . . . , m[Gyn ])T , KP (I − Q)N y = (f (y1 ) − Q(f (y1 )), f (y2 ) − Q(f (y2 )), . . . , f (yn ) − Q(f (yn )))T , where

t

Z

(Gyi − m[Gyi ])ds,

f (yi ) =

i = 1, 2, . . . , n.

0

Clearly, QN and (I −Q)N 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 {z k (t)} ⊂ Im(L) = V1 and z k (t) = (z1k (t), z2k (t), . . . , znk (t))T uniformly converges to z(t) = (z 1 (t), z 2 (t), . . . , z n (t))T ; that is, zik → z i , as k → ∞, i = Rt 1, 2, . . . , n. Because of 0 z k (s)ds ∈ Z, k = 1, 2, . . . , n, there exists σi (0 < σi < i ) Rt such that Ki (Hi , σi , φi ) ⊂ Ki ( 0 zik (s)ds, 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.

It is easy to see that li is the length of the inclusion interval of Ki (Hi , σi , φi ) and Ki (Hi , i , φi ), i = 1, 2, . . . , n. Hence, for any t ∈ / [0, li ], there exists ξt ∈ Rt k Ki (Hi , σi , φi ) ⊂ Ki ( 0 zi (s)ds, σi , φi ) such that t + ξt ∈ [0, li ], i = 1, 2, . . . , n. Hence, by the definition of almost periodic function we have Z t k z k (s)dsk 0 Z t = max sup | zik (s)ds| 1≤i≤n t∈R

0 t

Z

zik (s)ds|

≤ max sup | 1≤i≤n t∈[0,li ]

Z −

zik (s)ds

Z +

0

1≤i≤n t∈[0,l / i]

0

(3.3)

t+ξt

zik (s)ds|

0

Z ≤ 2 max sup | 1≤i≤n t∈[0,li ]

Z ≤ 2 max | 1≤i≤n

zik (s)ds

+ max sup |

0

t+ξt

t

Z

0

0

t

zik (s)ds|

Z + max sup | 1≤i≤n t∈[0,l / i]

li

zik (s)ds| + max i . 1≤i≤n

0

t

zik (s)ds

Z − 0

t+ξt

zik (s)ds|

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Rt From this inequality, we can conclude that 0 z(s)ds is continuous, where z(t) = (z1 (t), z2 (t), . . . , zn (t))T ∈ Im(L). Consequently, KP and KP (I − Q)N y are continRt uous. From (3.3), we also have 0 z(s)ds and KP (I − Q)N y are uniformly bounded in Ω. Further, it is not difficult to verify that QN (Ω) is bounded and KP (I − Q)N y is equicontinuous in Ω. By the Arzela-Ascoli theorm, we have immediately conclude that KP (I − Q)N (Ω) is compact. Thus N is L-compact on Ω. The proof is complete.  By (1.2), Fi (t, x), fi (t, x), Gij (t, x) and gij (t, x) can be represented expansion into power-series at αi , βi , γij and δij of x in form of ∂Fi |(t,αi ) x + o(x2 ), i = 1, 2, . . . , n; ∂x ∂fi fi (t, x) = fi (t, βi ) + |(t,βi ) x + o(x2 ), i = 1, 2, . . . , n; ∂x ∂Gij Gij (t, x) = Gij (t, γij ) + |(t,γij ) x + o(x2 ), i 6= j, i, j = 1, 2, . . . , n; ∂x ∂gij gij (t, x) = gij (t, δij ) + |(t,δij ) x + o(x2 ), i 6= j, i, j = 1, 2, . . . , n; ∂x Fi (t, x) = Fi (t, αi ) +

respectively, where o(x2 ) is a higher-order infinitely small quantity of x2 . By (1.3), we can conclude that Fi (t, αi ) = 0, fi (t, βi ) = 0, Gij (t, γij ) = 0, gij (t, δij ) = 0. For convenience, we denote ∂Fi |(t,αi ) := bii (t), ∂x ∂Gij |(t,γij ) := bij (t), ∂x

∂fi |(t,βi ) := cii (t), ∂x ∂gij |(t,δij ) := cij (t), ∂x

for i 6= j, i, j = 1, 2, . . . , n. By (1.3), bij (t) > 0, cij (t) > 0 for i, j = 1, 2, . . . , n. Theorem 3.4. Assume the following conditions hold: (H1) ei := m[r Pni (t)] > 0, m[bij (t)] > 0, m[cij (t))] > 0, i, j = 1, 2, . . . , n; (H2) dii > i6=j,j=1 dij , where dii = m[bii (t)] + m[cii (t)] + 1, dij = m[bij (t)] + m[cij (t)] + 1, i 6= j, i, j = 1, 2, . . . , n. 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 (1.1), 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 Lemma 3.2 and Lemma 3.3, respectively. Then we can obtain that L is a Fredholm mapping of index zero 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 = λN y,

λ ∈ (0, 1),

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we obtain n h X y˙ i (t) = λ ri (t) − bii (t)eyi (t) − cii (t)eyi (t−τii (t)) + bij (t)eyj (t) j=1,j6=i n X

+

+

+

cij (t)eyj (t−τij (t)) −

Z

0

µii (t, s)eyj (t+s) ds

−σii

j=1,i6=j n Z 0 X

(3.4) µij (t, s)e

yj (t+s)

ds − o(e

2yi (t)

2yi (t−τii (t))

) − o(e

j=1 −σij n X


i o(e2yj (t) ) + o(e2yj (t−τij (t)) ) ,

)

i = 1, 2, . . . , n.

j=1,j6=i

Assume that y(t) = (y1 (t), y2 (t), . . . , yn (t))T ∈ X is a solution of (3.4) for some λ ∈ (0, 1). Denote θ¯ = max sup yi (t),

θ = min inf yi (t). 1≤i≤n t∈R

1≤i≤n t∈R

On the both sides of (3.4), integrating from 0 to T and applying the mean value theorem of integral calculus, we have h


0 ≤ λ m ri (t) − m bii (t) eyi (ξii ) − m cii (t) eyi (ηii −τii (ηii )) n X

+

n X

m bij (t) eyj (ξij ) + m cij (t) eyj (ηij −τij (ηij ))

j=1,j6=i 0

Z


m µii (t, s) e

−

j=1,i6=j n Z 0 X yj (ζii +s)


m µij (t, s) eyj (ζij +s) ds

ds +

−σii

j=1

−σij n X



− m o(e2yi (t) ) − m o(e2yi (t−τii (t)) ) +

m o(e2yj (t) )

(3.5)




j=1,j6=i n X

+


i
m o(e2yj (t−τij (t)) ) + m |y˙ i (t)| ,

i = 1, 2, . . . , n,

j=1,j6=i

where ξij ∈ [0, T ], ηij ∈ [0, T ], ζij ∈ [0, T ], i, j = 1, 2, . . . , n. In the light of (3.5), we have h


λ m bii (t) eyi (ξii ) + m cii (t) eyi (ηii −τii (ηii )) + m o(e2yi (t) ) Z 0

i + m µii (t, s) eyj (ζii +s) ds + m o(e2yi (t−τii (t)) ) −σii n n h X X


≤ λ m ri (t) + m bij (t) eyj (ξij ) + m cij (t) eyj (ηij −τij (ηij )) j=1,j6=i

+

n Z X j=1

+

j=1,i6=j

0


m µij (t, s) eyj (ζij +s) ds +

−σij

n X j=1,j6=i

m o(e2yj (t−τij (t)) )

n X


m o(e2yj (t) )

j=1,j6=i


i


+ m |y˙ i (t)| ,

i = 1, 2, . . . , n.

(3.6)

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On both sides of (3.6), taking the supremum with respect to ξij , ηij , ζij and letting T → +∞, we obtain n       X         m bii (t) + m cii (t) + 1 eθ ≤ m ri (t) + m bij (t) + m cij (t) + 1 eθ ; j=1,j6=i

that is, dii eθ −

n X

dij eθ ≤ ei ,

i = 1, 2, . . . , n,

j=1,j6=i

which imply θ ≤ ln B,

(3.7)

where B = max1≤i≤n {Bi }, Bi =

dii −

e Pn i

j=1,j6=i

dij

.

On both sides of (3.6), taking the infimum with respect to ξij , ηij , ζij and letting T → +∞, we obtain (3.8) θ ≤ ln B. On the other hand, according to (3.4), we derive h λri (t) − y˙ i (t) ≤ λ bii (t)eyi (t) + cii (t)eyi (t−τii (t)) Z 0 i + µii (t, s)eyi (t+s) ds + o(e2yi (t) ) + o(e2yi (t−τii ) ) ,

(3.9)

−σii

for i = 1, 2, . . . , n. Integrating on both sides of (3.9), from 0 to T , and using the mean value theorem, we obtain h



λm ri (t) < m y˙ i (t) + λ m bii (t) eyi (ξi ) + m cii (t) eyi (ηi −τii (ηi )) Z 0

(3.10) + m µii (t, s) eyi (ζi +s) ds + m o(e2yi (t) ) −σii
i + m o(e2yi (t−τii (t)) ) , i = 1, 2, . . . , n, where ξi ∈ [0, T ], ηi ∈ [0, T ], ζi ∈ [0, T ], for i = 1, 2, . . . , n. On both sides of (3.10), take the supremum and infimum with respect to ξi , ηi , ζi , respectively, and let T → +∞, then for i = 1, 2, . . . , n, we have         m ri (t) < m bii (t) + m cii (t) + 1 eθ ,         m ri (t) < m bii (t) + m cii (t) + 1 eθ , namely,   m ri (t) ei    e >  , = dii m bii (t) + m cii (t) + 1   m ri (t) ei θ    e >  , = d m bii (t) + m cii (t) + 1 ii θ

which imply that θ > ln C,

θ > ln C,

(3.11)

10

K. ZHAO, J. LIU

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 where C = min1≤i≤n ei /dii . Combing with (3.7), (3.8) and (3.11), we derive that for all t ∈ R, i = 1, 2, . . . , n, ln C < θ ≤ yi (t) ≤ θ < ln B + 1.

(3.12)

Denote M = max{| ln C|, | ln B + 1|}. Clearly, M is independent of λ. Take Ω = {y = (y1 , y2 , . . . , yn )T ∈ X : kyk < M }. It is clear that Ω satisfies the requirement (a) in Lemma 2.5. When y ∈ ∂Ω ∩ ker(L), y = (y1 , y2 , . . . , yn )T is a constant vector in Rn with kyk = M . Then QN y = (m[G1 ], m[G2 ], . . . , m[Gn ])T , y ∈ X where yi

Z

yi

0

µii (t, s)eyi ds

Gi = ri (t) − Fi (t, e ) − fi (t, e ) − −σii

+

+

n X

n X

Gij (t, eyj ) +

j=1,j6=i Z 0 n X j=1,j6=i

gij (t, eyj )

j=1,i6=j

µij (t, s)eyj ds,

i = 1, 2, . . . , n,

−σij

and m[Gi ] = m[ri (t)] − (m[bii (t)] + m[cii (t)] + 1)eyi +

n X

(m[bij (t)] + m[cij (t)] + 1)eyj

j=1,j6=i

= ei − dii eyi +

n X

dij eyj ,

i = 1, 2, . . . , n.

j=1,j6=i

There exist positive integers l, k ∈ {1, 2, . . . , n} such that xl = y = min1≤i≤n {yi } and xk = y = max1≤i≤n {yi }. If QN y = (0, 0, . . . , 0)T , then we have dll ey = dll eyl = el +

n X

dlj eyj ≥ el ,

j=1,j6=l

dll ey = dll eyk = ek +

n X j=1,j6=k

n X

dkj eyj ≤ ek +

dkj ey ,

j=1,j6=k

namely, el > C, dll e Pn k ey ≤ ≤ B, dll − j=1,j6=k dkj ey ≥

which imply that ln C < y ≤ yi ≤ y ≤ ln B, i = 1, 2, . . . , n. Thus, y = (y1 , y2 , . . ., yn )T ∈ Ω, this contradicts the fact that y ∈ ∂Ω ∩ ker(L). Therefore, QN y 6= (0, 0, . . . , 0)T , which implies that the requirement (b) in Lemma 2.5 is satisfied. When y ∈ ∂Ω ∩ ker(L), y = (y1 , y2 , . . . , yn )T is a constant vector in Rn with kyk = M . Then QN y = (m[G1 ], m[G2 ], . . . , m[Gn ])T 6= (0, 0, . . . , 0)T ,

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EXISTENCE OF POSITIVE ALMOST PERIODIC SOLUTIONS

11

which implies that the requirement (b) in Lemma 2.5 is satisfied. If necessary, we can let M be greater such that y T QN y < 0, for any y ∈ ∂Ω ∩ ker(L). Furthermore, take the isomorphism J : Im(Q) → ker(L), Jz ≡ z and let Φ(γ; y) = −γy + (1 − γ)JQN y, then for any y ∈ ∂Ω ∩ ker(L), y T Φ(γ; y) < 0, we have deg{JQN, Ω ∩ ker(L), 0} = deg{−y, Ω ∩ ker(L), 0} = 6 0. So, the requirement (c) in Lemma 2.5 is satisfied. Hence, (3.1) 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. Examples Consider the following two species Lotka-Volterra type cooperative system with time-varying delays and distributed delays:  x(t) ˙ = x(t) r1 (t) − F1 (t, x(t)) − f1 (t, x(t − τ11 (t))) Z 0 − µ11 (t, s)x(t + s)ds + G12 (t, y(t)) −σ11 0  + g12 (t, y(t − τ12 (t))) + µ12 (t, s)y(t + s)ds , −σ12  y(t) ˙ = y(t) r2 (t) − F2 (t, y(t)) − f2 (t, y(t − τ22 (t))) Z 0 − µ22 (t, s)y(t + s)ds + G21 (t, x(t))

Z

(4.1)

−σ22

Z

0

+ g21 (t, x(t − τ21 (t))) +

 µ21 (t, s)x(t + s)ds ,

−σ21

√ √ √
√ where r1 (t) = 2+sin 2t+sin 3t, F1 (t, x) = 2+sin 3t+sin 5t sin x, f1 (t, x) = √ √
√ √ 2 + cos 3t + cos 5t sin x, G12 (t, x) = 14 (2 + cos 2t + cos 3t) sin 2x, g12 (t, x) = √ √ √ √ √ 2+sin 2t+sin 3t sin 3x, τ11 (t) = esin 2t+sin 5t , τ12 (t) = esin t+cos 2t , r2 (t) = 2 − 6 √ √ √ √
√ sin 2t − sin 3t, F2 (t, y) = 2 + sin 3t − sin 5t sin y, f2 (t, y) = 2 + cos 3t − √ √ √ √ √
3t 3t sin 4y, g21 (t, y) = 2−sin 2t+sin sin 5y, cos 5t sin y, G21 (t, y) = 2−cos 2t+cos 8 10 √ √ √ sin 2t+cos 5t sin t−cos 2t τ22 (t) = e , τ21 (t) = e , µij (t, s) are positive almost periodic functions on R × [−σij , 0] and continuous with respect to t ∈ R and integrable with respect to s ∈ [−σij , 0], where σij are nonnegative constants, moreover R0 µ (t, s)ds = 1, i, j = 1, 2, . . . , n. Obviously, αi = βi = γi = 2π. By a simple −σij ij calculation, we have √ √ √ √ b11 (t) = 2 + sin 3t + sin 5t, c11 (t) = 2 + cos 3t + cos 5t, √ √ √ √ 2 + sin 2t + sin 3t 2 + cos 2t + cos 3t , c12 (t) = , b12 (t) = √2 √ √ 2 √ b22 (t) = 2 + sin 3t − sin 5t, c22 (t) = 2 + cos 3t − cos 5t, √ √ √ √ 2 − cos 2t + cos 3t 2 − sin 2t + sin 3t b21 (t) = , c21 (t) = , 2 2 e1 = m[r1 (t)] = 2, d11 = m[b11 (t)] + m[c11 (t)] + 1 = 5, d12 = m[b12 (t)] + m[c12 (t)] + 1 = 3,

e2 = m[r1 (t)] = 2,

12

K. ZHAO, J. LIU

d22 = m[b11 (t)] + m[c22 (t)] + 1 = 5,

EJDE-2013/157

d21 = m[b21 (t)] + m[c21 (t)] + 1 = 3,

then, in the matrix d11 = 5 > 3 = d12 , d22 = 5 > 3 = d21 and M = max{| ln C|, | ln B + 1|} = 1, Ω = {y = (y1 , y2 , . . . , yn )T ∈ X : kyk < 1}. Therefore, all conditions of Theorem 3.4 are satisfied, and system (4.1) has at least one positive almost periodic solution. Acknowledgments. This work are supported by grant 11161025 from the National Natural Sciences Foundation of China, grant 2011FZ058 from the Yunnan Province natural scientific research fund, and grant 2011Z001 from the Yunnan Province education department scientific research fund. References [1] T. Cheon; Evolutionary stability of ecological hierarchy, Phys. Rev. Lett. 4 (2003) 258-105. [2] L. Chen, H. Zhao; Global stability of almost periodic solution of shunting inhibitory celluar networks with variable coefficients, Chaos, Solitons & Fractals 35 (2008) 351-357. [3] Z.J. Du, Y.S. Lv; Permanence and almost periodic solution of a Lotka-Volterra model with mutual interference and time delays, Applied Mathematical Modelling 37 (2013) 1054-1068. [4] K. Ezzinbi, M. A. Hachimi; Existence of positive almost periodic solutions of functional via Hilberts projective metric, Nonlinear Anal. 26 (6) (1996) 1169-1176. [5] A. Fink; Almost Periodic Differential Equitions,in: Lecture Notes in Mathematics, vol.377, Springer, Berlin, 1974. [6] P.Y. Gao; Hamiltonian structure and first integrals for the Lotka-Volterra systems, Phys. Lett. A. 273 (2000) 85-96. [7] K. Geisshirt, E. Praestgaard, S. Toxvaerd; Oscillating chemical reactions and phase separation simulated by molecular dynamics, J. Chem. Phys. 107 (1997) 9406-9412. [8] R. Gaines, J. Mawhin; Coincidence Degree and Nonlinear Differetial Equitions, Springer Verlag, Berlin, 1977. [9] Z.Y. Hou; On permanence of Lotka-Volterra systems with delays and variable intrinsic growth rates, Nonlinear Analysis: Real World Applications 14 (2013) 960-975. [10] A.M. Huang, P.X. Weng; Traveling wavefronts for a Lotka-Volterra system of type-k with delays , Nonlinear Analysis: Real World Applications 14 (2013) 1114-1129 . [11] A.J. Lotka; Elements of Physical Biology, William and Wilkins, Baltimore, 1925. [12] Y.K. Li, Y. Kuang; Periodic solutions of periodic delay Lotka-Volterra equations and systems, J. Math. Anal. Appl. 255 (2001) 260-280. [13] Y.K. Li; Positive periodic solutions of discrete Lotka-Voterra competition systems with state dependent delays, Appl. Math. Comput. 190 (2007) 526-531. [14] Y.K. Li; Positive periodic solutions of periodic neutral Lotka-Volterra system with distributed delays, Chaos, Solitons & Fractals 37 (2008) 288-298. [15] Y.K. Li, K.H. Zhao, Y. Ye; Multiple positive periodic solutions of n species delay competition systems with harvesting terms, Nonlinear Analysis: Real World Applications 12 (2011) 10131022. [16] Y.K. Li, X.L. Fan; Existence and globally exponential stability of almost periodic solution for Cohen-Grossberg BAM neural networks with variable coefficients, Appl. Math. Model. 33 (2009) 2114-2120. [17] Y.K. Li, K.H. Zhao; Positive almost periodic solutions of non-autonomous delay competitive systems with weak Allee effect, Electronic Journal of Differential Equations 100 (2009) 1-11. [18] Y.G. Liu, B.B. Liu, S.H. Ling; The almost periodic solution of Lotka-Volterra recurrent neural networks with delays, Neurocomputing 74 (2011) 1062-1068. [19] Z. Li, F.D. Chen, M.X. He; Almost periodic solutions of a discrete Lotka-Volterra competition system with delays, Nonlinear Analysis: Real World Applications 12 (2011) 2344-2355. [20] X. Meng, L. Chen; Almost periodic solution of non-autonomous Lotka-Voltera predatorprey dispersal system with delays, Journal of Theoretical Biology 243(2006) 562-574.

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[21] Y. Muroya; Persistence and global stability in Lotka-Volterra delay differential systems, Appl. Math. Lett. 17 (2004) 795-800. [22] X. Meng, L. Chen; Periodic solution and almost periodic solution for nonnautonomous LotkaVoltera dispersal system with infinite delay, J. Math. Anal. Appl. 339 (2008) 125-145. [23] Z. Teng, Y. Yu; Some new results of nonautonomous Lotka-Volterra competitive systems with delays, J. Math. Anal. Appl. 241 (2000) 254-275. [24] X.H. Tang, D.M. Cao, X.F. Zou; Global attractivity of positive periodic solution to periodic Lotka-Volterra competition systems with pure delay , Journal of Differential Equations 228 (2006) 580-610. [25] V. Volterra; Variazioni efluttuazioni del numero dindividui in specie danimali conviventi, Mem. Acad. Lincei 2 (1926) 31-113. [26] Y. Yu, M. Cai; Existence and exponential stability of almost-periodic solutions for higherorder Hopfield neural networks, Math. Comput. Model 47 (2008) 943-951. [27] K.H. Zhao, Y. Ye; Four positive periodic solutions to a periodic Lotka-Volterra predatoryprey system with harvesting terms, Nonlinear Analysis: Real World Applications 11 (2010) 2448-2455. [28] K.H. Zhao, Y.K. Li; Four positive periodic solutions to two species parasitical system with harvesting terms, Computers and Mathematics with Applications 59 (2010) 2703-2710. [29] X.H. Zhao, J.G. Luo; Classification and dynamics of stably dissipative Lotka-Volterra systems , International Journal of Non-Linear Mechanics 45 (2010) 603-607. [30] K.H. Zhao; Existence of Almost-Periodic Solutions for Lotka-Volterra Cooperative Systems with Time Delay, Journal of Applied Mathematics 2012 (2012). Kaihong Zhao Department of Applied Mathematics, Kunming University of Science and Technology, Kunming, Yunnan 650093, China E-mail address: [email protected] Juqing Liu Department of Mathematics, Yuxi Normal University, Yuxi, Yunnan 653100, China E-mail address: [email protected]