LEt R, Z and N + denote the sets of real numbers,

Transcription

1 AENG nternational ournal of Applied Mathematics 48: AM_48 9 Dynamics of Almost Periodic Mutualism Model with Time Delays Yaqin Li Lijun Xu and Tianwei Zhang Abstract By using some new analytical techniques modified inequalities and Mawhin s continuous theorem of coincidence degree theory some new sufficient conditions for the existence of positive almost periodic solutions to a mutualism model with bounded and unbounded delays are obtained Further the boundedness and global asymptotic stability have been studied To the best of the author s knowledge so far the result of this paper is completely new The work of this paper extends and improves some results in recent years Finally some examples are given to illustrate the main results in this paper ndex Terms Almost periodicity; Coincidence degree; Mutualism model; Bounded and unbounded delays NTRODUCTON LEt R Z and N + denote the sets of real numbers integers and positive integers respectively Related to a continuous function f we use the following notations: f l inf fs f sup fs f M sup fs 1 T f lim fs ds T T Mutualism or cooperation is found in many types of communities For example some species of Acacia require the ant Pseudomyrmex in order to survive see 1 blue-green algae can grow and reproduce in the absence of zooplankton grazers but growth and reproduction are enhanced by the presence of the zooplankton see n 3 Gopalsamy had proposed the following model to describe the mutualism mechanism: dn 1t dn t r 1 tn 1 t r tn t K1t+α 1tN t 1+N t N 1 t Kt+α tn 1t 1+N 1t N t 11 where r i denotes the intrinsic growth rate of species N i i 1 The carrying capacity of species N i is K i in the absence of other species while with the help of the other species the carrying capacity becomes K i + α i N 3 i /1 + N 3 i i 1 The above mutualism can be classified as facultative obligate or a combination of both Naturally more realistic and interesting models of population interactions should take into account both the seasonality of the changing environment and the effects of time delay n system 11 the mutualistic or cooperative effects are not realized instantaneously but take place with time delays Manuscript received September 19 17; revised February Yaqin Li is with the Department of Mathematics Kunming University Yunnan 6514 China yqlikm@163com Lijun Xu is with the School of Mathematics and Computer Science Panzhihua University Panzhihua Sichuan 617 China pzhxlj@16com Tianwei Zhang is with the City College Kunming University of Science and Technology Kunming 6551 China zhang@kmusteducn Correspondence author: Tianwei Zhang zhang@kmusteducn Therefore by way of Mawhin s continuous theorem of coincidence degree theory Li 4 investigated the existence of positive periodic solutions for a periodic mutualism model with bounded delays: dn 1t dn t r 1 tn 1 t N 1 t ν 1 r tn t N t ν K 1t+α 1tN t µ t 1+N t µ t K t+α tn 1t µ 1t 1+N 1t µ 1t 1 where r i K i α i CR α i > K i µ i ν i CR r i K i α i µ i ν i are functions of period ω > i 1 Next Li and Xu 5 studied the following model of two species mutualism with unbounded delays: dn 1t dn t r 1 tn 1 t N 1 t ν 1 t r tn t N t ν t H 1t+β 1t H t+β t snt s ds snt s ds 1sN1t s ds 1sN1t s ds 13 Under the assumption that r i H i β i and ν i are continuous periodic functions with common period ω a i > K i i C and i s ds 1 i 1 they showed that system 13 admits at least one positive ω-periodic solution by way of Mawhin s continuous theorem of coincidence degree theory n real world phenomenon the environment varies due to the factors such as seasonal effects of weather food supplies mating habits harvesting So it is usual to assume the periodicity of parameters in the systems However if the various constituent components of the temporally nonuniform environment is with incommensurable nonintegral multiples see Example 11 periods then one has to consider the environment to be almost periodic 6-1 since there is no a priori reason to expect the existence of periodic solutions Hence if we consider the effects of the environmental factors almost periodicity is sometimes more realistic and more general than periodicity Example 1 Let us consider the following simple population model: Ṅt Nt sin t sin 3t Nt 14 π n Eq 14 sin t is -periodic function and sin 3t is 3π 3 -periodic function which imply that E- q 14 is with incommensurable periods Then there is no Advance online publication: 8 May 18

2 AENG nternational ournal of Applied Mathematics 48: AM_48 9 a priori reason to expect the existence of positive periodic solutions of Eq 14 Thus it is significant to study the existence of positive almost periodic solutions of Eq 14 So the aim of this paper is to investigate the following almost periodic mutualism model with bounded and unbounded delays: dn 1t dn t K 1t+α 1tN t µ t 1+N t µ t + H1t+β1t snt s ds snt s ds r 1 tn 1 t N 1 t ν 1 t r tn t K t+α tn 1t µ 1t 1+N 1t µ 1t + Ht+βt 1sN1t s ds 1sN1t s ds N t ν t 15 where r i K i H i α i β i µ i and ν i are continuous nonnegative almost periodic functions i C and i s ds 1 i 1 Obviously systems are special cases of system 15 t is well known that Mawhin s continuation theorem of coincidence degree theory is an important method to investigate the existence of positive periodic solutions of some kinds of non-linear ecosystems see 13-6 However it is difficult to be used to investigate the existence of positive almost periodic solutions of non-linear ecosystems Therefore to the best of the author s knowledge so far there are scarcely any papers concerning with the existence of positive almost periodic solutions of system 15 by using Mawhin s continuation theorem Motivated by the above reason our purpose of this paper is to establish some new sufficient conditions on the existence of positive almost periodic solutions for system 15 by using Mawhin s continuous theorem The paper is organized as follows n Section we give some basic definitions and necessary lemmas which will be used in later sections n Section 3 we obtain some new sufficient conditions for the existence of at least one positive almost periodic solution of system 15 by way of Mawhin s continuous theorem Some illustrative examples are given in Section 4 PRELMNARES Definition x CR R n is called almost periodic if for any ɛ > it is possible to find a real number l lɛ > for any interval with length lɛ there exists a number τ τɛ in this interval such that xt + τ xt < ɛ t R where is arbitrary norm of R n τ is called to the ɛ-almost period of x T x ɛ denotes the set of ɛ-almost periods for x and lɛ is called to the length of the inclusion interval for T x ɛ The collection of those functions is denoted by AP R R n Let AP R : AP R R Lemma f x AP R then x is bounded and uniformly continuous on R Lemma 7 8 f x AP R then t xs ds AP R if and only if t xs ds is bounded on R Next we present and prove several useful lemmas which will be used in later section Lemma 3 1 Assume that x AP R C 1 R with ẋ CR For arbitrary interval a b with b a ω > let a b and s b : ẋs } then ones have xt x + ẋs ds t b Lemma 4 1 f x AP R then for arbitrary interval a b with b a ω > there exist a b a and b + such that x x and x xs s Lemma 5 1 f x AP R then for arbitrary interval a b with b a ω > there exist a b a and b + such that x x and x xs s Lemma 6 1 f x AP R then for n N + there exists α n R such that xα n x 1 n x where x sup xs Lemma 7 8 Assume that x AP R and x > then for t R there exists a positive constant T independent of t such that 1 T t+t t xs ds x 3 x T T Lemma 8 4 Let fx y a 1 a 1 b e y c 1e x a a b 1 + e x c e y and Ω x y T R : x + y < M} where M a i b i and c i are constants a i b i i 1 and M > max lna i /c i lnb i /c i i 1 } Then degfx y Ω T } Lemma 9 Let fx y a 1 + b 1 a e y c 1e x a + b a 1 + e x c e y and Ω x y T R : x + y < M} where M a i b i and c i are constants b i a i i 1 and M > max lna i /c i lnb i /c i i 1 } Then degfx y Ω T } Proof: Set Ψx y ι a 1 + b 1 a ιe y c 1e x a + b a 1 + ιe x c e y ι 1 For x y ι R 1 it is then easy to see that a 1 + b 1 a ιe y c 1e x < b 1 c 1 e x < as x M a + b a 1 + ιe x c e y < b c e y < as y M a 1 + b 1 a ιe y c 1e x a 1 c 1 e x > as x M Advance online publication: 8 May 18

3 AENG nternational ournal of Applied Mathematics 48: AM_48 9 a + b a 1 + ιe x c e y a c e y > as y M Hence Ψx y ι for x y ι Ω 1 t follows from the property of invariance under a homotopy that degfx y Ω T } degψx y Ω T } The proof is complete 1 MAN RESULTS The method to be used in this paper involves the applications of the continuation theorem of coincidence degree This requires us to introduce a few concepts and results from Gaines and Mawhin 9 Let X and Y be real Banach spaces L : DomL X Y be a linear mapping and N : X Y be a continuous mapping The mapping L is called a Fredholm mapping of index zero if ml is closed in Y and dimkerl codimml < + f L is a Fredholm mapping of index zero and there exist continuous projectors P : X X and Q : Y Y such that mp KerL KerQ ml m Q t follows that L DomL KerP : P X ml is invertible and its inverse is denoted by K P f Ω is an open bounded subset of X the mapping N will be called L-compact on Ω if QN Ω is bounded and K P QN : Ω X is compact Since mq is isomorphic to KerL there exists an isomorphism : mq KerL Lemma 1 9 Let Ω X be an open bounded set L be a Fredholm mapping of index zero and N be L-compact on Ω f all the following conditions hold: a Lx λnx x Ω DomL λ 1; b QNx x Ω KerL; c degqn Ω KerL } where : mq KerL is an isomorphism Then Lx Nx has a solution on Ω DomL For f AP R we denote by 1 Λf ϖ R : lim T T T } fse iϖs ds the set of Fourier exponents of f Now we are in the position to present and prove our result on the existence of at least one positive almost periodic solution for system 15 Let ν : maxν M 1 ν M } maxsup Theorem 1 Assume that ν 1 s sup ν s} F 1 inf K i s α i s i 1 F inf H i s β i s i 1 F 3 Φ i > where Φ i : r i s K i s + H i s i 1 Then system 15 admits at least one positive almost periodic solution Proof: Under the invariant transformation N 1 N T e u e v T system 15 reduces to dut K r 1 t 1t+α 1te vt µ t 1+e vt µ t + H1t+β1t sevt s ds e ut ν1t sevt s ds : F 1 t dvt K r t t+α te ut µ 1 t 1+e ut µ 1 t + Ht+βt 1seut s ds e vt νt 1seut s ds : F t 3 t is easy to see that if system 3 has one almost periodic solution u v T then N 1 N T e u e v T is a positive almost periodic solution of system 15 Therefore to completes the proof it suffices to show that system 3 has one almost periodic solution Take X Y V 1 V where V 1 z u v T AP R R : ϖ Λu Λv ϖ θ } V z u v T k 1 k T k 1 k R } where θ is a given positive constant Define the norm } z max sup us sup vs z X Y then X and Y are Banach spaces with the norm Set L : DomL X Y Lz Lu v T u v T where DomL z u v T X : u v C 1 R u v CR} and ut F1 t N : X Y Nz N vt F t With these notations system 3 can be written in the form Lz Nz z X t is not difficult to verify that KerL V ml V 1 is closed in Y and dim KerL codim ml Therefore L is a Fredholm mapping of index zero see Lemma 3 in 31 Now define two projectors P : X X and Q : Y Y as u mu u P z P Qz z X Y v mv v Then P and Q are continuous projectors such that mp KerL and ml KerQ m Q Furthermore through an easy computation we find that the inverse K P : ml KerP DomL of L P has the form t u K P z K P us ds m t us ds v t vs ds m t vs ds Then QN : X Y and K P QN : X X read u mf1 QNz QN v mf Advance online publication: 8 May 18

4 AENG nternational ournal of Applied Mathematics 48: AM_48 9 fut Qfut K P QNz z ml fvt Qfvt where fx is defined by fxt t Nxs QNxs ds Then N is L-compact on Ω see Lemma 33 in 31 n order to apply Lemma 1 we need to search for an appropriate open-bounded subset Ω Corresponding to the operator equation Lz λz λ 1 we have ut λr 1 t vt λr t K 1t+α 1te vt µ t 1+e vt µ t e ut ν1t sevt s ds 31 + H1t+β1t sevt s ds K t+α te ut µ 1 t 1+e ut µ 1 t + Ht+βt 1seut s ds a 1seut s ds e vt νt Suppose that u v T DomL X is a solution of system 31 for some λ 1 where DomL z u v T X : u v C 1 R u v CR} By Lemma 6 there exist two sequences T n : n N + } and P n : n N + } such that ut n u 1n u u sup us n N + 3 vp n v 1n v v sup vs n N + 33 From Φ i > i 1 and Lemma 7 for a R there exists a constant ω ν + independent of a such that 1 a+t ri r i s ds T a 3 r i 1 a+t Φi Φ i s ds T 3 Φ i T ω i 1 For n N + we consider T n ω T n and P n ω P n where ω is defined as that in By Lemma 4 there exist T n ω T n T n ω and T n + such that u u and u us s 34 ntegrating the first equation of system 31 from to leads to K1 s + α 1 se vs µs r 1 s 1 + e vs µs + H 1s + β 1 s ze vs z dz ze vs z dz e us ν1s ds which yields that α1 s K 1 s r 1 s 1 + e + β 1 s H 1 s vs µs ze vs z dz +e us ν1s α 1 s + β 1 s ds which implies from F 1 -F that +ν r 1 se us ν1s ds r 1 sα 1 s + β 1 s ds r 1 se us ν1s ds By the integral mean value theorem and there exists s + ν s ν 1 s such that r 1 4 eus ν1s ν ν r 1 eus ν1s 1 r 1 se us ν1s ds +ν e us ν1s 1 ν +ν r 1 s ds 1 r 1 sα 1 s + β 1 s ds 3 r 1 αm 1 + β1 M which implies from 34 that u ln6α1 M + β1 M 35 Let s T n : us } t follows from system 31 that dut us ds ds K1 s + α 1 se vs µs λr 1 s 1 + e vs µs + H 1s + β 1 s ze vs z dz e us ν1s ds ze vs z dz λr 1 s α 1 s + β 1 s α 1s K 1 s 1 + e vs µs β 1 s H 1 s ze vs z dz eus ν1s ds λr 1 sα 1 s + β 1 s ds Tn r 1 sα 1 s + β 1 s ds T n ω r1 M α1 M + β1 M ω 36 By Lemma 3 it follows from that ut u + us ds 1 ln6α1 M + β1 M + r1 M α1 M + β1 M ω : ρ 1 t u n T n which implies that ut n ρ 1 n view of 3 letting n + in the above inequality leads to u lim ut n ρ 1 37 n + Advance online publication: 8 May 18

5 AENG nternational ournal of Applied Mathematics 48: AM_48 9 Similar to the argument as that in 37 we can obtain that v ln6α M + β M + r M α M + β M ω : ρ 38 Taking π max } ω 4rM 1 e ρ1 ν 1 + e ρ1 4rM e ρ ν 1 + e ρ Φ For n Z by Lemma 5 we can conclude that there exist n π n π + π n π and n π + π + such that u u and u us s 39 ntegrating the first equation of system 31 from to leads to K1 s + α 1 se vs µs r 1 s 1 + e vs µs + H 1s + β 1 s ze vs z dz ze vs z dz which yields from that e us ν1s ds 1 r 1 s K 1 s + H 1 s ds 1 + e ρ K1 s + α 1 se vs µs r 1 s 1 + e vs µs + H 1s + β 1 s ze vs z dz ds ze vs z dz r 1 se us ν1s ds +ν r 1 se us ν1s ds + +ν +ν r 1 se us ν1s ds + r M 1 e ρ1 ν which implies from F 3 and that + e ρ e ρ r 1 se us ν1s ds r 1 s K 1 s + H 1 s ds r 1 se us ν1s ds + rm 1 e ρ1 ν +ν r 1 se us ν1s ds + rm 1 e ρ1 ν +ν π Φ r 1 se us ν1s 1 ds + 31 ρ +ν 4 + 4e n view of 31 by the integral mean value theorem and 39 there exists s 1 + ν s 1 ν 1 s 1 such that Φ 1 eus1 ν1s1 r ρ 1 s ds 4 + 4e +ν r M 1 e u ν r M 1 e u which implies that u ln Further we obtain from system 31 that nπ +π n π nπ +π n π 4r M eρ 311 us ds λr 1 s K 1 s + α 1 se vs µs 1 + e vs µs + H 1s + β 1 s ze vs z dz e us ν1s ds ze vs z dz r1 M K M 1 + α1 M e ρ + H1 M + β1 M e ρ + e ρ1 π : Θ 1 31 t follows from that nπ +π ut u us ds n π ln 4r1 M 1 + Θ 1 eρ : ρ 3 t n π n π + π 313 Obviously ρ 3 is a constant independent of n So it follows from 313 that u inf us inf 3} ρ 3 n Z 314 Similar to the argument as that in 314 we can obtain that v ln 4r1 M 1 + rm K M + α M e ρ1 eρ +H M + β M e ρ1 + e ρ π : ρ 4 Set C ρ 1 + ρ + ρ 3 + ρ 4 + C + 1 where C is taken sufficiently large such that C > max ln mr iα i + β i ln mr } ik i + H i i1 mr i mr i Clearly C is independent of λ 1 Consider the algebraic equations QNz for z u v T R as follows: mr 1 α 1 + β 1 mr1α1+β1 mr1k1+h1 1+e mr v 1 e u mr α + β mrα+β mrk+h 1+e mr u e v Similar to the arguments as that in and we can easily obtain that ρ 3 u ρ 1 ρ 4 v ρ Then z X u + v < C Let Ω z X : z X < C} then Ω satisfies conditions a and b of Lemma 1 Finally we will show that condition c of Lemma 1 is satisfied By Lemma 8 we have deg QN Ω KerL deg QN Ω KerL where deg is the Brouwer degree and is the identity mapping since mq KerL Obviously all the conditions of Lemma 1 are satisfied Therefore system 3 has one almost periodic solution that is system 15 has at least one positive almost periodic solution This completes the proof Advance online publication: 8 May 18

6 AENG nternational ournal of Applied Mathematics 48: AM_48 9 Remark 1 By Theorem 1 it is easy to obtain the existence of at least one positive almost periodic solution of Eq 14 in Example 1 although there is no a priori reason to expect the existence of positive periodic solutions of Eq 14 Assume that all the coefficients of system 15 are ω- periodic functions we have Corollary 1 Assume that F 1 -F 3 hold Suppose further that all the coefficients of system 15 are ω-periodic functions Then system 15 admits at least one positive ω- periodic solution Theorem Assume that F 3 holds Suppose further that F 4 inf K i s α i s i 1 F 5 inf H i s β i s i 1 Then system 15 admits at least one positive almost periodic solution Proof: Proceeding as in the proof of Theorem 1 we see that 34 holds ntegrating the first equation of system 31 from to leads to K1 s + α 1 se vs µs r 1 s 1 + e vs µs + H 1s + β 1 s ze vs z dz e us ν1s ds ze vs z dz which yields from F 4 -F 5 that r 1 se us ν1s ds K1 s α 1 s r 1 s 1 + e vs µs H 1 s β 1 s + ze vs z dz + α 1s + β 1 s r 1 s K 1 s α 1 s +H 1 s β 1 s + α 1 s + β 1 s r 1 sk 1 s + H 1 s Similar to the argument as that in we can obtain that u ln6k M 1 + H M 1 + r M 1 K M 1 + H M 1 ω : ρ and v ln6k M + H M + r M K M + H M ω : ρ 316 By a parallel arguments as that in there exist ρ 3 and ρ 4 such that u ρ 3 v ρ Set C ρ 1 + ρ + ρ 3 + ρ 4 + C + 1 where C is taken sufficiently large such that C > max ln mr iα i + β i ln mr } ik i + H i i1 mr i mr i Clearly C is independent of λ 1 Consider the algebraic equations QNz for z u v T R as follows: mr 1 α 1 + β 1 + mr1k1+h1 mr1α1+β1 1+e mr v 1 e u mr α + β + mrk+h mrα+β 1+e mr u e v Let Ω z X : z X < C } From the proof in Theorem 1 it is easy to see that Ω satisfies conditions a and b of Lemma 1 Further by Lemma 9 condition c of Lemma 1 is also satisfied Obviously all the conditions of Lemma 1 are satisfied Therefore system 3 has one almost periodic solution that is system 15 has at least one positive almost periodic solution This completes the proof From Theorem and Corollary 1 we also have the following corollary: Corollary Assume that F 3 -F 5 hold Suppose further that all the coefficients of system 15 are ω-periodic functions Then system 15 admits at least one ω-positive periodic solution Now we give some assumptions: F 6 inf K 1 s α 1 s and inf K s α s F 7 inf K 1 s α 1 s and inf K s α s F 8 inf H 1 s β 1 s and inf H s β s F 9 inf H 1 s β 1 s and inf H s β s From the proof of Theorems 1- we can easily show that Theorem 3 Assume that F -F 3 hold suppose further that F 6 or F 7 is satisfied Then system 15 admits at least one positive almost periodic solution Theorem 4 Assume that F 1 and F 3 hold suppose further that F 8 or F 9 is satisfied Then system 15 admits at least one positive almost periodic solution Theorem 5 Assume that F 3 and F 6 hold suppose further that F 8 or F 9 is satisfied Then system 15 admits at least one positive almost periodic solution Theorem 6 Assume that F 3 and F 7 hold suppose further that F 8 or F 9 is satisfied Then system 15 admits at least one positive almost periodic solution From Theorems 3-6 we can easily show that Corollary 3 Assume that all the coefficients of system 15 are ω-periodic functions and F -F 3 hold suppose further that F 6 or F 7 is satisfied Then system 15 admits at least one positive ω-periodic solution Corollary 4 Assume that all the coefficients of system 15 are ω-periodic functions and F 1 and F 3 hold suppose further that F 8 or F 9 is satisfied Then system 15 admits at least one positive ω-periodic solution Corollary 5 Assume that all the coefficients of system 15 are ω-periodic functions and F 3 and F 6 hold suppose Advance online publication: 8 May 18

7 AENG nternational ournal of Applied Mathematics 48: AM_48 9 further that F 8 or F 9 is satisfied Then system 15 admits at least one positive ω-periodic solution Corollary 6 Assume that all the coefficients of system 15 are ω-periodic functions and F 3 and F 7 hold suppose further that F 8 or F 9 is satisfied Then system 15 admits at least one positive ω-periodic solution n 4 Li obtained that Corollary 7 4 Assume that F 1 -F 3 hold Suppose further that all the coefficients of system 1 are ω-periodic functions Then system 1 admits at least one positive ω- periodic solution Remark Obviously the works in this paper extend and improve the result in 4 V BOUNDEDNESS Theorem 7 n system 15 assume that S 1 H M i β M i ν M i µ i t µ i for all t R i 1 Then every solution of system 15 satisfies lim sup t N i t N i K M i + α M i i 1 t R Proof: From system 15 it leads dn 1t r1 M N 1 t K1 M + α1 M N 1 t r M N t K M + α M N t dn t By the comparison theorem of differential equations we have lim sup N i t Ki M + αi M i 1 t This completes the proof V GLOBAL ASYMPTOTC STABLTY Let k i : sup K i s α i s i 1 Theorem 8 Assume that S 1 holds then system 15 is globally asymptotically stable Proof: From Theorem 1 we know that system 15 has at least one positive almost periodic solution N 1 N T Suppose that N 1 N T is another solution of system 15 Let x 1 x T ln N 1 ln N T and x 1 x T ln N 1 ln N T then system 15 is transformed into Define dx 1t dx t d x 1t d x t r 1 t r t r 1 t r t K 1t+α 1tN t µ 1+N t µ N 1 t K t+α tn 1t µ 1 1+N 1t µ 1 N t K 1t+α 1t N t µ 1+ N t µ N 1 t K t+α t N 1t µ 1 1+ N 1t µ 1 N t V t V t + V 1 t + V t 51 where V t x 1 t x 1 t + x t x t V 1 t r M 1 k 1 t t µ N s N s ds t V t r M k N 1 s N 1 s ds t µ 1 By calculating the upper right derivative of V along system 51 it follows that D + V t sgnx 1 t x 1 tx 1t x 1t +sgnx t x tx t x t r l 1 N 1 t N 1 t r l N t N t +r M 1 k 1 N t µ N t µ +r M k N 1 t µ 1 N 1 t µ 1 5 Further by calculating the upper right derivative of V 1 V and V 3 along system 51 it follows that D + V 1 t r M 1 k 1 N t N t r M 1 k 1 N t µ N t µ 53 D + V t r M k N 1 t N 1 t r M k N 1 t µ 1 N 1 t µ 1 54 Together with 5-54 it follows that D + V t r l 1 r M k N 1 t N 1 t r l r M 1 k 1 N t N t t T Therefore V is non-increasing ntegrating of the last inequality from T to t leads to that is V t + r l 1 r M k +r l r M 1 k 1 < + t T + + which implies that t T t T N 1 s N 1 s ds N s N s ds V N 1 s N 1 s ds < + N s N s ds < + lim N 1s N 1 s lim N s N s s + s + This completes the proof Theorem 9 Assume that F 1 -F 3 and S 1 hold Then the almost periodic solution of system 15 is globally asymptotic stable Advance online publication: 8 May 18

8 AENG nternational ournal of Applied Mathematics 48: AM_48 9 V SOME EXAMPLES AND SMULATONS Example Consider the following almost periodic system: N 1 t where dn 1t dn t K 1t+α 1tN t 1 1+N t 1 + H1t+β1t e s N t s ds snt s ds N 1 t sin 3t N t K t+α tn 1t 1+N 1t + Ht+βt e s N 1t s ds 1sN1t s ds N t 5 K1 t K t H1 t H t α1 t α t β1 t β t sin 1t cos 1t sin t cos t sin 1t cos 1t sin t + 5 cos t By a easy computation it is not difficult to verify that F 1 - F 3 in Theorem 1 are satisfied By Theorem 1 system 61 admits at least one positive almost periodic solution Example 3 n system 61 let K1 t sin t H1 t sin t K t cos t H t cos t α1 t sin t α t cos t β1 t sin t + 5 β t cos t + 5 By a easy computation it is not difficult to verify that F 1 - F 3 in Corollary 1 are satisfied By Corollary 1 system 61 admits at least one positive π-periodic solution Example 4 n system 61 let K1 t 15 + sin 1t K t 5 + cos 1t H1 t H t α1 t cos 5t α t sin β1 t 5t β t By a easy computation it is not difficult to verify that F 3 - F 5 in Theorem 3 and S 1 in Theorem 8 are satisfied By Theorem 3 and Theorem 8 system 61 admits at least one positive almost periodic solution see Figure 41 which is globally asymptotic stable title yslice:6m;zslice:15m FontName Times Roman FontSize ; New Fig 1 State variables N 1 and N of Example Remark 3 n Example 4 sin 1t is 1π 1 -periodic function and cos 5t is 5π 5 -periodic function So system 61 is with incommensurable periods Through all the coefficients of system 61 are periodic functions the positive periodic solutions of system 61 could not possibly exist However by Theorem 3 the positive almost periodic solutions of system 61 exactly exist see Figure 1 Example 5 n system 61 let K1 t sin 5t + cos 1t K t cos 1t H1 t sin t H t cos t α1 t sin 5t + cos 1t α t cos 1t β1 t sin t + 5 β t cos t + 5 n Example 4 sin 5t + cos 1t is almost periodic function which is not periodic function Similar to the argument as that in Example it is easy to obtain that system 61 admits at least one positive almost periodic solution Example 6 n system 61 let K1 t 15 + sin t K t 5 + cos t H1 t 5 + sin t H t 5 + cos t α1 t sin t β1 t α t cos t β t sin t cos t N 1 N By a easy computation it is not difficult to verify that F 3 - F 5 in Corollary 5 are satisfied By Corollary 5 system 61 admits at least one positive π-periodic solution Remark 4 t is obvious that the results in papers 453 couldn t obtain the existence of positive almost periodic solutions for the models in Example and Examples 4-5 Therefore our works extend the results in papers 453 ACKNOWLEDGEMENTS This work is supported by the Scientific Research Fund of Yunnan Provincial Education Department 14Y388 and the Research Program of Kunming University in China XL151 Advance online publication: 8 May 18

9 AENG nternational ournal of Applied Mathematics 48: AM_48 9 REFERENCES 1 DH anzen Coevolution of mutualism between ants and acacias in Central America Evolution KG Porter Enhancement of algal growth and productivity by grazing zooplankton Science K Gapalsamy Stability and Oscillations in Delay Equations of Population Dynamics Kluwer Academic Publishers YK Li On a periodic mutualism model ANZAM YK Li GT Xu Positive periodic solutions for an integrodifferential model of mutualism Appl Math Lett TW Zhang YK Li Y Ye On the existence and stability of a unique almost periodic solution of Schoener s competition model with puredelays and impulsive effects Communications in Nonlinear Science and Numerical Simulation T W Zhang X R Gan Existence and permanence of almost periodic solutions for Leslie-Gower predator-prey model with variable delays Elect Differ Equa T W Zhang X R Gan Almost periodic solutions for a discrete fishing model with feedback control and time delays Commun Nonlinear Sci Numer Simulat YQ Li TW Zhang Existence and multiplicity of positive almost periodic solutions for a non-autonomous SR epidemic model Bull Malays Math Sci Soc YQ Li TW Zhang Almost periodic dynamics in a nonautonomous bi-directional consumer-resource system with time delays Comp Appl Math TW Zhang Multiplicity of positive almost periodic solutions in a delayed Hassell-Varley type predator-prey model with harvesting on prey Math Meth Appl Sci TW Zhang Almost periodic oscillations in a generalized Mackey- Glass model of respiratory dynamics with several delays nt Biomath M Fan K Wang DQ iang Existence and global attractivity of positive periodic solutions of periodic n-species Lotka-Volterra competition systems with several deviating arguments Math Biosci YK Li Periodic solutions for delay Lotka-Volterra competition systems Math Anal Appl Y Chiba T Nose M to A to Estimating the user s state before exchanging utterances using intermediate acoustic features for spoken dialog systems AENG nternational ournal of Computer Science vol 43 no1 pp WL Al-Yaseen ZA Othman M Nazri Real-time intrusion detection system using multi-agent system AENG nternational ournal of Computer Science vol 43 no1 pp Z Chen Y Zhu Y Di S Feng Optimized self-adaptive fault tolerance strategy in simulation system based on virtualization technology AENG nternational ournal of Computer Science vol 4 no4 pp ZQ Zhang XW Zheng A periodic stage-structure mode Appl Math Lett HF Huo WT Li Positive periodic solutions of a class of delay differential system with feedback control Appl Math Comput FD Chen Positive periodic solutions of neutral Lotka-Volterra system with feedback control Appl Math Comput ZQ Zhang L Wang Global attractivity of a positive periodic solution for a nonautonomous stage structured population dynamics with time delay and diffusion Math Anal Appl C Zhao On a periodic predator-prey system with time delays Math Anal Appl K Wang YL Zhu Global attractivity of positive periodic solution for a Volterra model Appl Math Comput K Wang Existence and global asymptotic stability of positive periodic solution for a predator-prey system with mutual interference Nonlinear Anal: RWA YL Zhu K Wang Existence and global attractivity of positive periodic solutions for a predator-prey model with modified Leslie- Gower Holling-type schemes Math Anal Appl GD Zhang Y Shen BS Chen Positive periodic solutions in a nonselective harvesting predator-prey model with multiple delays Math Anal Appl AM Fink Almost Periodic Differential Equations Springer Berlin CY He Almost Periodic Differential Equations Higher Education Publishing House Beijing 199 Chinese 9 RE Gaines L Mawhin Coincidence Degree and Nonlinear Differential Equations Springer Berlin YK Li HT Zhang Existence of periodic solutions for a periodic mutualism model on time scales Math Anal Appl Y Xie XG Li Almost periodic solutions of single population model with hereditary effects Appl Math Comput Advance online publication: 8 May 18