Almost periodic solution of an impulsive multispecies logarithmic population model

Transcription

1 Wang and Zhang Advances in Difference Equaions 215) 215:96 DOI /s y R E S E A R C H Open Access Almos periodic soluion of an impulsive mulispecies logarihmic populaion model Li Wang * and Hui Zhang * Correspondence: mazhilaoma@163.com Deparmen of Applied Mahemaics, Norhwesern Polyechnical Universiy, Xi an, Shaanxi 7172, P.R. China Absrac In his paper, some easily verifiable condiions are derived for he exisence of almos periodic soluion of an impulsive mulispecies logarihmic populaion model in erms of he coninuous heorem of coincidence degree heory, which is rarely applied o sudying he exisence of almos periodic soluion of an impulsive differenial equaion. Our resuls generalize previous resuls by Alzabu, Samov and Sermulu. Besides, our echnique used in his paper can be applied o sudy he exisence of almos periodic soluion of an impulsive differenial equaion wih linear impulsive perurbaions. Keywords: coincidence degree heory; almos periodiciy; impulsive mulispecies logarihmic populaion model 1 Inroducion Many evoluionary processes in naure are characerized by he fac ha heir saes are subjec o sudden changes a cerain momens and herefore can be described by an impulsive sysem. Basic heory of impulsive differenial equaions can be found in he monographs [13]. Gopalsamy [4] proposed he following single species logarihmic populaion model: N )=N) a)b) ln N)c) ln N τ) )). By uilizing he coninuous heorem of coincidence degree heory, he exisence of periodic soluion and almos periodic soluion of his model is invesigaed in papers [5] and [6], respecively. In paper [7], Alzabu and Abdeljawad consider he exisence of periodic soluion of he impulsive differenial equaion N )=N )a)b) ln N)c) ln N τ))), k, N k + )=N k) 1+γ k e δ k, k =1,2,... By employing he conracion mapping principle and Gronwall-Bellman inequaliy, he exisence and exponenial sabiliy of posiive almos periodic soluion of his model are obained in paper [8]. Using he same mehod as [8], Yang and Li [9] dealwihheex- isence of almos periodic soluion of an impulsive wo-species logarihmic populaion model wih ime-varying delay. By means of he Cauchy marix, Wang [1] gives sufficien 215 Wang and Zhang; licensee Springer. This is an Open Access aricle disribued under he erms of he Creaive Commons Aribuion License hp://creaivecommons.org/licenses/by/4.), which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly credied.

2 Wangand ZhangAdvances in Difference Equaions 215) 215:96 Page 2 of 11 condiions for he exisence and exponenial sabiliy of periodic and almos periodic soluion for he delay impulsive logarihmic populaion: N )=N )a)b) ln N)c) ln N τ))), k, e N+ k ) = e 1+p k)n k )+c k, k =1,2,... Some scholars are ineresed in impulsive differenial equaions or he exisence of almos) periodic soluion. Relaed resuls can be found in he lieraure [1114]. By using fixed poin heory and consrucing a suiable Lyapunov funcion, Chen [15] sudies he exisence, uniqueness and global araciviy of posiive periodic soluion and almos periodic soluion of he delay mulispecies logarihmic populaion model [ N i )=N i) r i ) a ij ) ln N j ) b ij ) ln N j τij ) ) ] c ij ) k ij s) ln N j s) ds, i =1,2,...,n. In his paper, we consider he impulsive mulispecies logarihmic populaion model [ N i )=N i) r i ) a ij ) ln N j ) b ij ) ln N j τij ) ) ] c ij ) k ij s) ln N j s) ds, k, 1.1) N i + k ) =1+dik )N i k ), i =1,2,...,n, k =1,2,..., where r i ), a ij ), b ij ), c ij ), τ ij ) are nonnegaive almos periodic funcions in sense of Bohr, k ij s) and k ij s) ds <+, {d ik } is an almos periodic sequence, { k } is an equipoenially almos periodic sequence, r = sup R,i,,...,n τ i,j ), i, j =1,...,n. By uilizing he coninuous heorem of coincidence degree heory, we obain some sufficien condiions for he exisence of almos periodic soluion of Eq. 1.1). Our resuls generalize previous resuls obained in [6] and are easier o verify han he condiions obained in paper [15]. The coninuous heorem of coincidence degree heory has been exensively used o sudy he exisence of periodic soluion of a differenial equaion, regardless of he equaionbeingwihimpulseorwihou. Usinghisheorem,papers [6] and[16] invesigae he exisence of almos periodic soluion of a populaion model wihou impulse. To our bes knowledge, he coninuous heorem has no been used o prove he exisence of almos periodic soluion of an impulsive mulispecies logarihmic populaion model. Besides, our echnique used in his paper can be applied o sudy he exisence of almos periodic soluion of an impulsive differenial equaion wih linear impulsive perurbaions. The remaining par of his paper is organized as follows. We presen some preliminaries in he nex secion. In Secion 3, by employing he coninuous heorem of coincidence degree heory, we esablish a crierion for he exisence of almos periodic soluion of sysem 1.1).

3 Wangand ZhangAdvances in Difference Equaions 215) 215:96 Page 3 of 11 2 Preliminaries In his secion, some lemmas and definiions, which are of imporance in proving our main resul in Secion 3,willbepresened. Definiion 2.1 [17]) ϕ ) CR, R n ) is said o be almos periodic in sense of Bohr if ε >, here exiss a relaively dense se Tϕ, ε) suchhaifτ Tϕ, ε), hen ϕ) ϕ + τ) < ε for all R.DenoebyapR, R n ) all such funcions. Definiion 2.2 [3]) Le ϕ ) =ϕ 1 ), ϕ 2 ),...,ϕ n )) be a piecewise coninuous funcion wih firs kind disconinuiies a he poins of a fixed sequence { k }.Wecallϕ almos periodic if: 1) { k } is equipoenially almos periodic, ha is, ε >, here exiss a relaively dense se of ε-almos periodic common for any sequences { j k }, j k = k+j k ; 2) ε >, δ >such ha if he poins, belong o he same inerval of coninuiy and < δ,hen ϕ )ϕ ) < ε; 3) ε >, here exiss a relaively dense se Tϕ, ε) such ha if τ Tϕ, ε), hen ϕ)ϕ + τ) < ε for all R which saisfy he condiion i > ε, i =,±1, ±2,... Denoe by APR, R n ) all such funcions. Now, we inroduce some basic noaions. Suppose f apr, R n )orf APR, R n ), we use f o denoe he se of Fourier exponens of f, modf ) o denoe he module of f, mf ) o denoe he limi mean of f.wesupposehad ik and τ ij ) insysem1.1) saisfyhe following condiions: A1) < k < 1 + d jk), < k <τ ij ) 1 + d jk) are posiive almos periodic funcions, inf R < k < 1 + d jk)>, j =1,...,n; A2) k ij s) ln < k <s 1 + d jk) ds are almos periodic funcions, i, j =1,...,n. Thereino, he definiion of < k < 1 + d jk)j =1,2,...,n) is as follows: 1,, 1 ], 1 + d jk )= 1 + d j1 ) 1 + d jk ), k, k+1 ], k =1,2,3,... < k < Remark There exis a grea deal of funcions saisfying assumpions A1) and A2). For insance, le { k } be an arbirary equipoenially almos periodic sequence, τ ij ) =τ >, d j,k = d j,k+t,1+d j,k )>,k =1,2,...and1+d j1 )1 + d j2 )...1+d jt )=1,hen < k < 1 + d jk) is a posiive almos periodic funcion wih disconinuous poins k,andinf R < k < 1 + d jk )>. < k <τ ij ) 1 + d jk) is also a posiive almos periodic funcion wih disconinuous poins k + τ, i, j =1,...,n. Besides, we suppose ha k ij )saisfy k ij s) ds = + k ij u) du <+ and, [, sup i Z i 1 k ij )= +1],, oherwise, hen k ij s) ln < k <s 1 + d jk) ds, i, j =1,2,...,n. Thus, assumpion A2) holds.

4 Wangand ZhangAdvances in Difference Equaions 215) 215:96 Page 4 of 11 Before sudying he exisence of sricly posiive AP soluions of sysem 1.1), we firsly consider he following equaion: [ y i )=y i) r i ) a ij ) ln 1 + d jk ) < k < b ij ) ln < k <τ ij ) 1 + d jk ) c ij ) k ij s) ln 1 + d jk ) ds < k <s b ij ) ln y j τij ) ) a ij ) ln y j ) c ij ) k ij s) ln y j s) ds ], 2.1) where i =1,2,...,n. The soluions of Eqs.1.1) and2.1) saisfy he following relaion. Lemma 2.3 Suppose ha A1) is saisfied, he following resuls hold: 1) If N i ) APR, R) is a posiive soluion of Eq.1.1), hen y i )= < k < 1 + d ik) 1 N i ) is a posiive ap soluion of Eq.2.1), i =1,2,...,n. 2) If y i ) apr, R) is a posiive soluion of Eq.2.1), hen N i )= < k < 1 + d ik)y i ) is a posiive AP soluion of Eq.1.1), i =1,2,...,n. Proof Since < k < 1 + d ik) APR, R) andinf R < k < 1 + d ik) >,i = 1,...,n, from [3] weknow < k < 1 + d ik) 1 APR, R). Therefore, if y i ) apr, R), hen N i ) = < k < 1+d ik)y i ) APR, R); if N i ) APR, R), combining N i k + )=1+d ik)n i k ), hen y i )= < k < 1 + d ik) 1 N i ) apr, R). Similar as [18], he res of he proof of Lemma 2.3 can be obained easily, we omi i here. InorderoinvesigaeEq.2.1), we ake y i )=e x i),eq.2.1) can be ranslaed o where x i )= r i) a ij )x j ) b ij )x j τij ) ) c ij ) k ij s)x j s) ds, 2.2) r i )=r i ) a ij ) ln 1 + d jk ) < k < b ij ) ln 1 + d jk ) < k <τ ij ) c ij ) k ij s) ln 1 + d jk ) ds, i =1,2,...,n. < k <s Obviously, if Eq. 2.2) hasap soluion, hen Eq. 2.1) has sricly posiive ap soluion. I follows from Lemma 2.3 ha Eq. 1.1) has sricly posiive AP soluion. Consequenly, we mainly sudy he exisence of ap soluion of Eq. 2.2). To do so, we firsly summarize a few conceps. Le X and Z be real Banach spaces, L : dom L X Z be a linear mapping, N : X Z be a coninuous mapping. L is called a Fredholm mapping of index zero if dim Ker L =

5 Wangand ZhangAdvances in Difference Equaions 215) 215:96 Page 5 of 11 codim Im L < and Im L is close in Z.IfL is a Fredholm mapping of index zero, here are coninuous projecs P : X X, Q : Z Z such ha Im P = Ker L, Im L = Ker Q = ImI Q). I follows ha L dom L Ker P :I P)X Im L is inverible. We denoe he inverse of ha map by K p.if is an open subse of X, he mapping N will be called L-compac on if QN ) is bounded and K p I Q)N : X is compac. Since Im Q is isomorphic o Ker L, here exiss an isomorphism J : Im Q Ker L. The following resul is proved in [19]. Lemma 2.4 Coninuous heorem) Le Xbeanopenboundedse, le L be a Fredholm mapping of index zero and N be L-compac on. Assume 1) For each λ, 1), every soluion x of Lx = λnx is such ha x / ; 2) For each x Ker L, QNx ; 3) degjqn, Ker L,). Then Lx = Nx has a leas one soluion in dom L. 3 Main resuls In his secion, by means of he coninuous heorem of coincidence degree heory, we invesigae he exisence of sricly posiive AP soluion of Eq. 1.1). To do so, we ake X 1 = { x =x 1,...,x n ) ap R, R n) : modx i ) modf), λ xi, α 1 > λ > α } {}, { Z 1 = z =z 1,...,z n ) AP R, R n), z i )areap funcions wih disconinuous poins k, modz i ) modf), λ zi, α 1 > λ > α, aλ j, z i ) } <+ {}, Z 2 = X 2 = { h =h 1, h 2,...,h n ) R n}, where α and α 1 are given posiive consans, and F is a given almos periodic funcion in sense of Bohr. Define X = X 1 X 2, Z = Z 1 Z 2 wih he norm φ = max 1 i n sup R φ i ), φ =φ 1, φ 2,...,φ n ) X or Z. Lemma 3.1 X and Z are Banach spaces equipped wih he norm. Proof Firsly, we can easily obain ha X is a Banach space, hence we only need o prove ha Z is a Banach space. If {z k =z 1k,...,z nk )} k Z 1 converges o z =z 1,...,z n )uniformly, i follows from he properies of AP funcions ha z APR, R n ). Similar as [16], he fac modz i ) modf), λ zi, α 1 > λ > α can be obained easily. Now, we asser ha aλ j, z i ) <+, i =1,...,n.Ifno, M >, m >andī such ha m aλj, zī) > M. 3.1) Since {z k } is a Cauchy sequence, for 1/2 m, K such ha for any m, n > K, aλ, zī, m )aλ, zī, n ) = lim 1 T zī, T + 2T m )zī, n ) ) e iλ d < 1, 2m λ R. 3.2) T

6 Wangand ZhangAdvances in Difference Equaions 215) 215:96 Page 6 of 11 Since z k converges o z uniformly, for λ l zī and 1 2, n l l > K, s.., aλl, zī) < aλl, zī,nl ) 1 +, l =1,2,...,m. 3.3) 2l Since n 1, n 2,...,n m > K,from3.2) aλj, zī,nj ) < aλj, zī,n1 ) 1 +, j =1,2,...,m. 3.4) 2m Hence, combining 3.1)-3.4) M < < m aλ j, zī) m < aλ j, zī,nj ) + 1 m < aλ 2 j j, zī,nj ) +1 m aλj, zī,n1 ) m < aλj, m zī,n1 ) +3<+. The arbirariness of M leads o a conradicion, so he asserion holds. Therefore, Z 1 is a Banach space. Similarly, we can obain ha Z is a Banach space. The proof is complee. Lemma 3.2 Le dx1 L : X Z, Lx 1, x 2,...,x n )= d, dx ) 2 d,...,dx n, d hen L is a Fredholm mapping of index zero. Proof Obviously, Ker L = X 2.WeproveIm L = Z 1. Firsly, for any ϕ 1 + ϕ 2 = ϕ Im L Z, ϕ 1 Z 1, ϕ 2 Z 2,since ϕs) ds apr, Rn ), ha is, ϕ 1s) ds + ϕ 2s) ds apr, R n ). From [2] weknow ϕ 1s) ds apr, R n ), hen ϕ 2 =.Henceϕ Z 1, Im L Z 1.Secondly, for any ϕ =ϕ 1, ϕ 2,...,ϕ n ) Z 1, wihou loss of generaliy, we suppose ϕ.since ϕs) ds apr, Rn ), furhermore ϕ i s) dsm ϕ i s) ds) = ϕ i, i =1,2,...,n, hen ϕ is) ds m ϕ is) ds) is he primiive of ϕ i in X, ϕ Im L. Therefore, Z 1 Im L. To sum up, Z 1 = Im L. Besides, one can easily show ha Im L is closed in Z dim Ker L = n = codim Im L. Therefore, L is a Fredholm mapping of index zero. Remark If f apr, R n )and λ f, λ > α >,henf has an ap primiive funcion. I does no hold for an AP funcion. From [2] weknowhaiff APR, R n ), λ f, α 1 > λ > α >, i=1 aλ i, f ) <+, henf has an AP primiive funcion. Tha is he reason why in Lemma 3.1 we ake Z 1 like ha.

7 Wangand ZhangAdvances in Difference Equaions 215) 215:96 Page 7 of 11 Se N : X Z, Nx 1, x 2,...,x n )=Nx 1, Nx 2,...,Nx n ), Q : Z Z, Qz 1, z 2,...,z n )= mz 1 ), mz 2 ),...,mz n ) ), P : X X, Px 1, x 2,...,x n )= mx 1 ), mx 2 ),...,mx n ) ), where Nx i )= r i ) a ij )x j ) b ij )x j τij ) ) c ij ) k ij s)x j s) ds, hen we have he following. Lemma 3.3 N is L-compac on is an open, bounded subse of X). Proof Firsly, i is easy o show ha P and Q are coninuous projecors such ha Im P = Ker L, Im L = ImI Q)=Ker Q, where I is an ideniy mapping. Hence L dom L Ker P :I P)X Im L is inverible. We denoe he inverse of ha map by K p. K p : Im L Ker P Dom L has he form Thus, K p z = K p z 1, z 2,...,z n ) = z 1 s) ds m ) z 1 s) ds,..., )) z n s) ds m z n s) ds. QNx =QNx 1,...,QNx n ), K p I Q)Nx = f x 1 ) ) Qf x 1 ) ),...,f x n ) ) Qf x n ) )), where f x i ) ) = QNx i )=m r i ) Nxi s)qnx i s) ) ds, a ij )x j ) b ij )x j τij ) ) ) c ij ) k ij s)x j s) ds. Obviously, QN and I Q)N are coninuous. In fac, K p is also coninuous. For any z =z 1,...,z n ) Z 1 and for any 1 > ε >,lelε) denoe he inclusion inerval of TF, ε). Suppose z =z 1, z 2,...,z n ) Z 1 = Im L,hen zs) ds apr, Rn ). Since z i s) ds \{} = z i s) dsm z i s) ds) = z i, i =1,2,...,n,

8 Wangand ZhangAdvances in Difference Equaions 215) 215:96 Page 8 of 11 and modz i ) modf), hen mod z is) ds) modf). Therefore, here exiss < δ < ε such ha TF, δ) T z is) ds, ε). Le l be he inclusion inerval of TF, δ), hen l lε). For any / [, l], here exiss ξ TF, δ) T z is) ds, ε) suchha + ξ [, l]. Hence, by he definiion of almos periodic funcion, we have sup z i s) ds sup z i s) ds + sup +ξ z i s) ds z i s) ds R [,l] / [,l] +ξ + sup z i s) ds / [,l] 2 sup [,l] 2 l zi s) ds + sup z i s) ds + ε. / [,l] +ξ z i s) ds z i s) ds Hence, we can conclude ha K p is coninuous, and consequenly, K p I Q)N is also coninuous. In addiion, we also have K p I Q)N is uniformly bounded in, QN ) is bounded and K p I Q)N is equiconinuous in. SinceI Q)Nx Z 1 = Im L and Kp IQ)Nx = IQ)Nx, x, henmodk p I Q)Nx) =modi Q)Nx) modf). For any ε >, δ >suchhatf, δ) TK p I Q)Nx, ε), x,hencek p I Q)N is equialmos periodic in, According o [21], we can conclude ha K p I Q)N is compac, hus N is L-compac on. Noicing Lemmas ,forEq.1.1), we have he following resul. Theorem 3.4 If A1) and A2) are saisfied, m n a ij)+b ij )+c ij ) k ij s) ds)), i =1,2,...,n, hen Eq.1.1) has a leas one sricly posiive almos periodic soluion. Proof From he analysis above, we know ha in order o prove he exisence of sricly posiive AP soluion of Eq. 1.1), we only need o invesigae he exisence of ap soluion of Eq. 2.2). Define he isomorphism J : Im Q Ker L as an ideniy mapping. We search for an appropriae bounded open subse for he applicaion of Lemma 2.4. Corresponding o he operaor equaion Lx = λnx, λ, 1), we have x i )=λ r i ) a ij )x j ) b ij )x j τij ) ) ) c ij ) k ij s)x j s) ds, i =1,2,...,n. 3.5) If x X is a soluion of sysem 3.5), aking he limi mean for sysem 3.5), we obain m r i ) ) = m i =1,2,...,n, a ij )x j )+b ij )x j τij ) ) + c ij ) k ij s)x j s) ds) ),

9 Wangand ZhangAdvances in Difference Equaions 215) 215:96 Page 9 of 11 hen sup x i ) R inf x i) R m r i )) m n a ij)+b ij )+c ij ) k ij s) ds)), i =1,2,...,n, m r i )) m n a ij)+b ij )+c ij ) k ij s) ds)), i =1,2,...,n. Thus, here exiss a leas one R such ha xi ) m r i )) m n a ij)+b ij )+c ij ) k ij s) ds)) Since Lx Z 1, by a similar argumen as in Lemma 3.3 we can derive ha x i = sup xi ) xi ) + sup R R x i ) +2 +l x i s) ds Take τ [l,2l] TF, δ) T x i s) ds, ε), hen +1, i =1,2,...,n. 3.6) x i s) ds +1, i =1,2,...,n. 3.7) x + τ i s) ds x i s) ds = + τ x i s) ds ε 1, i =1,2,...,n. Inegraing sysem 3.5)overheinerval[, + τ], we obain λ + τ Consequenly, a ij s)x j s)+b ij s)x j s τij s) ) + c ij s) + τ λ r i s) ds +1. s k ij s u)x j u) du) ds +l x i s) ds + τ + λ x i s) ds λ + τ + τ r i s) ds a ij s)x j s)+b ij s)x j s τij s) ) + c ij s) s k ij s u)x j u) du) ds + τ 2 r i s) ds ) Combining 3.6)-3.8), we obain x i M 1, i =1,2,...,n,

10 Wangand ZhangAdvances in Difference Equaions 215) 215:96 Page 1 of 11 where 4 m r i )) M 1 = m n a ij)+b ij )+c ij ) k ij s) ds)) τ r i s) ds. Take = {x X, x M 1 +2}, heniisclearha verifies all he requiremens in Lemma 2.4,henceEq.2.2) has a leas one almos periodic soluion in and he proof is complee. Remark In his paper, we invesigae he exisence of almos periodic soluion of Eq. 1.1) in erms of he coninuous heorem of coincidence degree heory. The novely of his paper is no only he mehod bu also he resuls. In fac, if d ik =,c ij =,i = j =1,Eq.1.1) can be rewrien as N )=N) [ r)a) ln N)b) ln N τ) )]. Paper [6] obains he exisence of almos periodic soluion of he above equaion based on he assumpions ma)+ b)). Obviously, our resuls generalize previous resuls obained in paper [6]. Compeing ineress The auhors declare ha hey have no compeing ineress. Auhors conribuions All auhors conribued equally and significanly in wriing his aricle. All auhors read and approved he final manuscrip. Acknowledgemens The auhors would like o hank he Edior and he referees for heir consrucive suggesions on improving he presenaion of he paper. This work was suppored by he Fundamenal Research Funds for he Cenral Universiies 31214JC-Q187), he Naional Naural Science Foundaion of China No ). Received: 9 Ocober 214 Acceped: 29 January 215 References 1. Bainov, DD, Simeonov, PS: Impulsive Differenial Equaions: Periodic Soluion and Applicaions. Longman, London 1993) 2. Lakshmikanham, V, Bainov, DD, Simeonov, PS: Theory of Impulsive Differenial Equaions. World Scienific, Singapore 1989) 3. Samoilenko, AM, Peresyuk, NA: Impulsive Differenial Equaions. World Scienific, Singapore 1995) 4. Gopalsamy, K: Sabiliy and Oscillaion in Delay Differenial Equaions of Populaion Dynamics. Kluwer Academic, Boson 1992) 5. Li, YK: Araciviy of a posiive periodic soluion for all oher posiive soluion in a delay populaion model. Appl. Mah. 12, ) in Chinese) 6. Alzabu, JO, Samov, GT, Sermulu, E: Posiive almos periodic soluions for a delay logarihmic populaion model. Mah. Compu. Model. 53, ) 7. Alzabu, JO, Abdeljawad, T: Exisence and global araciviy of impulsive delay logarihmic model of populaion dynamics. Appl. Mah. Compu. 198, ) 8. Alzabu, JO, Samov, GT, Sermulu, E: On almos periodic soluions for an impulsive delay logarihmic populaion model. Mah. Compu. Model. 51, ) 9. Yang, BX, Li, JL: An almos periodic soluion for an impulsive wo-species logarihmic populaion model wih ime-varying delay. Mah. Compu. Model. 55, ) 1. Wang, Q, Zhang, HY, Wang, Y: Exisence and sabiliy of posiive almos periodic soluions and periodic soluions for a logarihmic populaion model. Nonlinear Anal. 72, ) 11. Chen, DZ, Dai, BX: Periodic soluion of second order impulsive delay differenial sysems via variaional mehod. Appl. Mah. Le. 38, ) 12. Li, Z, Han, MA, Chen, FD: Almos periodic soluions of a discree almos periodic logisic equaion wih delay. Appl. Mah. Compu. 232, ) 13. Li, XD, Song, SJ: Impulsive conrol for exisence, uniqueness and global sabiliy of periodic soluions of recurren neural neworks wih discree and coninuously disribued delays. IEEE Trans. Neural New. 24, )

11 Wangand ZhangAdvances in Difference Equaions 215) 215:96 Page 11 of Li, XD, Bohner, M: An impulsive delay differenial inequaliy and applicaions. Compu. Mah. Appl. 64, ) 15. Chen, FD: Periodic soluions and almos periodic soluions for a delay mulispecies logarihmic populaion model. Appl. Mah. Compu. 171, ) 16. Xie, Y, Li, X: Almos periodic soluions of single populaion model wih herediary effecs. Appl. Mah. Compu. 23, ) 17. Zhang, C: Almos Periodic Type Funcion and Ergodiciy. Science Press, Beijing 23) 18. He, M, Chen, F, Li, Z: Almos periodic soluion of an impulsive differenial equaion model of plankon allelopahy. Nonlinear Anal. 11, ) 19. Gaines, RE, Mawhin, JL: Coincidence Degree and Nonlinear Differenial Equaion. Springer, Berlin 1977) 2. Wang, L, Yu, M: Favard s heorem of piecewise coninuous almos periodic funcions and is applicaion. J. Mah. Anal. Appl. 413, ) 21. Corduneanu, C: Almos Periodic Funcions. Chelsea, New York 1989)