Meng Fan *, Ke Wang, Daqing Jiang. Abstract

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1 Mathematical iosciences 6 (999) 47±6 wwwelseviercom/locate/mbs Eistence and global attractivity of positive periodic solutions of periodic n-species Lotka± Volterra competition systems with several deviating arguments Meng Fan *, Ke Wang, Daqing Jiang Department of Mathematics, Northeast Normal University, hangchun 324, People's Republic of hina Received 29 September 998; received in revised form 5 February 999; accepted 3 March 999 Abstract In this paper, we study the eistence and global attractivity of positive periodic solutions of periodic n-species Lotka±Volterra competition systems y using the method of coincidence degree and Lyapunov functional, a set of easily veri able su cient conditions are derived for the eistence of at least one strictly positive (componentwise) periodic solution of periodic n-species Lotka±Volterra competition systems with several deviating arguments and the eistence of a unique globally asymptotically stable periodic solution with strictly positive components of periodic n-species Lotka±Volterra competition system with several delays Some new results are obtained As an application, we also eamine some special cases of the system we considered, which have been studied etensively in the literature Some known results are improved and generalized Ó 999 Elsevier Science Inc All rights reserved AMS: 34K5; 92D25; 3425; 34D2 Keywords: Positive periodic solutions; Global attractivity; Lotka±Volterra competition system; Deviating arguments; oincidence degree; Lyapunov functional * orresponding author si@nenueducn Supported by National Natural Sciences Foundation of People's Republic of hina No: /99/$ ± see front matter Ó 999 Elsevier Science Inc All rights reserved PII: S ( 9 9 ) X

2 48 M Fan et al / Mathematical iosciences 6 (999) 47±6 Introduction In recent years, the application of theories of functional di erential equations in mathematical ecology has developed rapidly Various mathematical models have been proposed in the study of population dynamics, ecology and epidemic Some of them are described as nonautonomous delay di erential equations Many people are doing research on the dynamics of population with delays, which is useful for the control of the population of mankind, animals and the environment One of the famous models for dynamics of population is the Lotka±Volterra competition system Owing to its theoretical and practical signi cance, the Lotka±Volterra systems have been studied etensively [± 3,5,6,8±22] To consider periodic environmental factors, it is reasonable to study the Lotka±Volterra system with periodic coe cients A very basic and important ecological problem associated with the study of multispecies population interaction in a periodic environment is the eistence and global attractivity of periodic solutions Such questions arise also in many other situations In this paper, we investigate the following periodic n-species Lotka±Volterra competition system with several deviating arguments " # _y i t ˆ y i t r i t Xn a ij t y j t s ij t ; i ˆ ; 2; ; n; : where a ij 2 R; ; ; r i ; s ij 2 R; R ; are -periodic functions with r i :ˆ Z r i t dt > ; R i :ˆ Z jr i t j dt; a ij :ˆ Z a ij t dt P ; i; j ˆ ; 2; ; n: :2 In mathematical ecology, Eq () denotes a model of the dynamics of an n- species system in which each individual competes with all others of the system for a common resource and the intra-species and inter-species competition involves deviating arguments s ij The assumption of periodicity of the parameters r i ; a ij ; s ij is a way of incorporating the periodicity of the environment (eg seasonal e ects of weather condition, food supplies, temperature, mating habits etc) The growth functions r i are not necessarily positive, since the environment uctuates randomly; in bad conditions, r i may be negative The eistence and attractivity of periodic solutions of some special cases of Eq () has been studied etensively in the literature [±3,5,8±,5,6,8,9,2,22] Shibata and Saito [7] eamined a two-species delay Lotka±Volterra competition system It has been shown that the time delays in a

3 M Fan et al / Mathematical iosciences 6 (999) 47±6 49 two-species Lotka±Volterra competition system can lead to chaotic behaviour To our knowledge, few papers have been published on the eistence and global attractivity of periodic solutions of Eq () The main purpose of this paper is to derive a set of `easily veri able' su cient conditions for the eistence and global attractivity of positive periodic solution of Eq () The present paper is organized as follows In Section 2, we study the eistence of periodic solutions with strictly positive components of Eq () by using the continuation theorem of coincidence degree theory proposed by Gaines and Mawhin [7] In Section 3, we investigate the global attractivity of strictly positive periodic solutions of n-species Lotka±Volterra competition system with several delays, which is a special case of Eq (), by using the method of Lyapunov functional We de ne a suitable Lyapunov functional and obtain a set of easily veri able su cient conditions In Section 3, we also eamine the uniform persistence of Eq () and analyze the e ect of the time delays on the uniform persistence In Section 4, as an application, we study some special cases of Eq (), which have been studied etensively in the literature The eamples show that our general easily veri able su cient conditions di er from the known results and improve and generalize some known results 2 Eistence of positive periodic solutions In this section, we study the eistence of strictly positive (componentwise) periodic solutions of Eq () For the reader's convenience, we shall rst summarize below a few concepts and results from Ref [7] that will be basic for this section Let X ; Z be normed vector spaces, L : Dom L X! Z be a linear mapping, and N : X! Z be a continuous mapping The mapping L will be called a Fredholm mapping of inde zero if dim Ker L ˆ codim Im L < and Im L is closed in Z If L is a Fredholm mapping of inde zero there eist continuous projectors P : X! X and Q : Z! Z such that Im P ˆ Ker L; Ker Q ˆ Im L ˆ Im I Q It follows that LjDom L \ Ker P : I P X! Im L is invertible We denote the inverse of that map by K P If X is an open bounded subset of X, the mapping N will be called L-compact on X if QN X is bounded and K P I Q N : X! X is compact Since Im Q is isomorphic to Ker L, there eist isomorphisms J : Im Q! Ker L In the proof of our eistence theorem below, we will use the continuation theorem of Gaines and Mawhin ([7], p 4) Lemma 2 (ontinuation theorem) Let L be a Fredholm mapping of inde zero and let N be L-compact on X Suppose (a) For each k 2 ;, every solution of L ˆ kn is such that 62 ox;

4 5 M Fan et al / Mathematical iosciences 6 (999) 47±6 (b) QN 6ˆ for each 2 ox \ Ker L and degfjqn; X \ Ker L; g 6ˆ : Then the equation L ˆ N has at least one solution lying in Dom L \ X Lemma 22 The domain R n ˆ f y ; ; y n T j y i > ; i ˆ ; 2; ; ng is positive invariant with respect to Eq () Proof Since 8 < y i t ˆ y i ep : Z t r i s Xn! 9 = a ij s y j s s ij s ds ; ; i ˆ ; 2; ; n; 2: the assertion of the lemma follows immediately for all t 2 ; since considering the biological signi cance of Eq (), we specify y i >, that is y ; ; y n T 2 R n The proof is complete Theorem 2 Suppose that a ii > and n r i > X j6ˆi a ij r j a jj epf r j R j g; i ˆ ; 2; ; n hold Then Eq () has at least one positive periodic solution of period say y t ˆ y t ; ; y n t T and there eists a positive constant such that ky k 6, where ky k ˆ P n ma t2 ;Š jy i t j 2 =2 Proof y Lemma 4 in Ref [] and the assumptions in Theorem 2, it is not di cult to show that the system of algebraic equations X n a ij ep u j ˆ ri ; i ˆ ; 2; ; n 2:2 has a unique solution u ; u 2 ; ; u n T 2 R n Making the change of variable y i t ˆ epf i t g; i ˆ ; 2; ; n 2:3 then Eq () is reformulated as Let _ i t ˆ r i t Xn a ij t ep j t s ij t ; i ˆ ; 2; ; n: 2:4 X ˆ Z ˆ f t ˆ t ; 2 t ; ; n t T 2 R; R n ; t ˆ t g; 2:5

5 M Fan et al / Mathematical iosciences 6 (999) 47±6 5! kk ˆ Xn 2 =2 ma j i t j for any 2 X or Z : 2:6 t2 ;Š Then X and Z are both anach spaces when they are endowed with the norm k k Let r t Xn a j t ep j t s j t N ˆ n A r n t Xn a nj t ep j t s nj t L ˆ _; P ˆ Z t dt; 2 X ; Qz ˆ Obviously, Ker L ˆ f j 2 X ; ˆ h; h 2 R n g; 8 9 < Z = Im L ˆ z j z 2 Z; z t dt ˆ : ; Z A for any 2 X ; z t dt; z 2 Z: 2:7 2:8 and dim Ker L ˆ n ˆ codim Im L: 2:9 Since Im L is closed in Z, L is a Fredholm mapping of inde zero It is easy to show that P and Q are continuous projectors such that Im P ˆ Ker L; Ker Q ˆ Im L ˆ Im I Q : 2: Furthermore, the generalized inverse (to L) K P : Im L! Ker P \ Dom L is given by K P z ˆ Z t Thus QN : X! A! z s ds Z Z Z Z t z s ds dt: " # r t Xn a j t ep j t s j t dt " # r n t Xn a nj t ep j t s nj t dt A 2:

6 52 M Fan et al / Mathematical iosciences 6 (999) 47±6 K P I Q N : X! A! Z t Z " # r s Xn a j s ep j s s j s ds " # r n s Xn a nj s ep j s s nj s ds A Z Z t Z Z t " # r s Xn a j s ep j s s j s ds dt " # r n s Xn a nj s ep j s s nj s ds dt A Z " # t r 2 s Xn a j s ep j s s j s ds : Z " # t r 2 n s a nj s ep j s s nj s ds A 2:2 learly, QN and K P I Q N are continuous Using the Arzela±Ascoli theorem, it is not di cult to show that K P I Q N X is compact for any open bounded set X X Moreover, QN X is boundedthus, N is L-compact on X with any open bounded set X X The isomorphism J of Im Q onto Ker L can be the identity mapping, since Im Q ˆ Ker L Now we reach the position to search for an appropriate open, bounded subset X for the application of the continuation theorem orresponding to the operator equation L ˆ kn; k 2 ;, we have " # _ i t ˆ k r i t Xn a ij t ep j t s ij t ; k 2 ; ; i ˆ ; 2; ; n: 2:3 Assume that ˆ t 2 X is a solution of Eq (23) for a certain k 2 ; : Integrating Eq (23) over the interval ; Š, we obtain

7 Z that is, X n " r i t Z Xn a ij t ep j t s ij t # dt ˆ ; i ˆ ; 2; ; n; a ij t ep j t s ij t dt ˆ r i ; i ˆ ; 2; ; n: 2:4 It follows from Eqs (23) and (24) that Z Z j_ i t j dt ˆ k r i t Xn a ij t ep j t s ij t dt that is, Z M Fan et al / Mathematical iosciences 6 (999) 47±6 53 < Z jr i t j dt Xn Z ˆ r i R i ; i ˆ ; 2; ; n; a ij t ep j t s ij t dt j_ i t j dt < r i R i ; i ˆ ; 2; ; n: 2:5 Since t 2 X, there eist n i 2 ; Š such that i n i ˆ min i t ; i ˆ ; 2; ; n: 2:6 t2 ;Š From Eqs (24) and (26), we see that Z a ii epf i n i g 6 a ii t epf i t s ii t g dt < r i ; i ˆ ; 2; ; n; 2:7 that is, ( ) r i i n i < ln ; i ˆ ; 2; ; n: 2:8 a ii From Eqs (25) and (28), one obtains Z ( ) r i i t 6 i n i j_ i j dt < ln r i R i ; i ˆ ; 2; ; n: On the other hand, there also eist g i 2 ; Š such that a ii 2:9

8 54 M Fan et al / Mathematical iosciences 6 (999) 47±6 i g i ˆ ma i t ; i ˆ ; 2; ; n: 2:2 t2 ;Š From Eqs (24) and (22) it follows that r i ˆ Xn Z Z 6 Xn ˆ Xn a ij t ep j t s ij t a ij t ep j g j dt dt a ij ep j g j ; i ˆ ; 2; ; n; 2:2 then from Eq (29), one obtains a ii epf i g i n g P r i X j6ˆi n P r i X j6ˆi a ij ep j g j r n j a ij ep r j a jj o R j ; 2:22 which imply 8 r i P n o 9 n >< a ij r j =a jj ep r j R j >= j6ˆi i g i P ln >: a ii >; :ˆ i; i ˆ ; 2; ; n: From Eqs (25) and (223), we have i t P i g i Z 2:23 j_ i j dt > i r i R i ; i ˆ ; 2; ; n: 2:24 Eqs (29) and (224) imply ( ( ) ) r i ma j i t j < ma ln r t2 ;Š i R i a ii ; i r i R i :ˆ H i : 2:25 learly, H i are independent of k Set H ˆ Pn H i 2 =2, where is taken su ciently large such that the unique solution of Eq (22) satis es k u ; u 2 ; ; u n T k <, then kk < H Let X :ˆ f ˆ ; ; n T 2 X jkk < Hg It is clear that X veri es the requirement (a) in Lemma 2 When 2 ox \ Ker L ˆ ox \ R n ; is a constant vector in R n with kk ˆ H Then

9 M Fan et al / Mathematical iosciences 6 (999) 47±6 55 Z " # r t Xn a j t ep j dt QN ˆ Z " # r n t a nj t ep j dt A r Xn a j ep j r n Xn a nj ep j 6ˆ : A 2:26 Furthermore, in view of the assumption in Theorem 2, it is easy to prove that degfjqn; X \ Ker L; g ( ( )) ˆ sign n det a ij nn ep Xn 6ˆ : 2:27 u j y now we have proved that X veri es all the requirements in Lemma 2 Hence, Eq (24) has at least one -periodic solution t in X Set y i t ˆ ep i t ; then by Eq (23) we know that y t ˆ y t ; ; y n t T is a positive -periodic solution of Eq () The boundedness of t implies the eistence of positive constant in Theorem 2 The proof of Theorem 2 is complete Remark 2 Theorem 2 is also valid for both the advanced type and the mied type Lotka±Volterra competition system From Theorem 2, we can see that the deviating arguments s ij ; i; j ˆ ; 2; ; n have no e ect on the eistence of positive periodic solution of Eq () Remark 22 Li [5] discussed the eistence of positive periodic solutions of periodic N-species competition system with time delays, which is a special case of Eq () Theorem 2 improves and etends the results in Ref [5] 3 Global asymptotic stability of positive periodic solutions We shall assume, henceforth, that s ij ; i; j ˆ ; ; n are non-negative constants That is, in this section we will con ne ourself to periodic n-species Lotka±Volterra competition system with several delays, which is a special case of Eq ()

10 56 M Fan et al / Mathematical iosciences 6 (999) 47±6 De nition 3 Let y t ˆ y t ; ; y n t T be a strictly positive (componentwise) periodic solution of Eq () and we say y t is globally asymptotically stable (or attractive) if any other solution y t ˆ y t ; ; y n t T of Eq () together with the initial condition y i s ˆ / i s P ; s 2 s i ; Š; / i > ; i ˆ ; ; n 3: where s i ˆ ma 6 j 6 n s ji and / i s is continuous on s i ; Š, has the property X n lim jy i t y t! i t j ˆ : 3:2 It is immediate that if y t is globally asymptotically stable then y t is in fact unique Lemma 3 ([4]) Let f be a nonnegative function de ned on ; such that f is integrable on ; and is uniformly continuous on ; Then lim t! f t ˆ : Theorem 3 Assume that the conditions in Theorem 2 hold Furthermore, suppose that s ii ; i ˆ ; 2; ; n and min a jj t > X n ma a ij t ; j ˆ ; 2; ; n: 3:3 t2 ;Š t2 ;Š i6ˆj Then system () has a unique periodic solution with strictly positive components, say y t ˆ y t ; ; y n t T, which is globally asymptotically stable and there eists a positive constant > such that ky k 6 Proof y Theorem 2, there eists a strictly positive (componentwise) periodic solution y t ˆ y t ; ; y n t T of Eq () and there is a positive constant > such that ky k 6 To complete the proof of Theorem 3, we only need to show that y t is globally asymptotically stable onsider a Lyapunov functional v t de ned by 2 3 v t ˆ Xn 6 4jln y i t ln y i t j X n Z t a ij u s ij jy j u y j u j du7 5 ; t s ij Obviously t P : j6ˆi 3:4

11 v 6 Xn jln y i ln y i j v t P Xn n X j6ˆi! ma a ij t sup j/ j t y j t js ij < ; 3:5 t2 ;Š t2 s ij;š jln y i t ln y i t j; t P : 3:6 A direct calculation of the right derivative D v of v t leads to D v t 6 a ii t jy i t y i t j X n a ij t s ij jy j t y j t j A where ˆ Xn 6 Xn a jj t jy j t y j t j X n a ij t s ij jy j t y j t j A a jj t jy j t y j t j X n ma a ij t jy j t y j t j t2 ;Š A 6 l Xn l ˆ min min a jj t 6 j 6 n@ X n t2 ;Š i6ˆj jy j t y j t j; t P ; 3:7 i6ˆj ma a ij t t2 ;Š A > : Integrating from to t on both sides of Eq (37) produces then v t l Xn X n Z t M Fan et al / Mathematical iosciences 6 (999) 47±6 57 Z t jy j s y j s j ds 6 v < ; t P ; 3:8 jy i s y i s j ds 6 v =l < ; t P ; 3:9 and hence P n jy i t y i t j 2 L ; : y Eqs (35)±(38), one obtains

12 58 M Fan et al / Mathematical iosciences 6 (999) 47±6 X n Therefore jln y i t ln y i t j 6 v t 6 v < ; t P : 3: jln y i t ln y i t j 6 v ; i ˆ ; 2; ; n; t P : 3: Furthermore min y i t epf v g 6 y i t 6 ma y i t t2 ;Š t2 ;Š ep fv g< ; i ˆ ; 2; ; n; t P : 3:2 The boundedness of y t and Eq (32) imply that y i t are bounded above and below by positive constants for t P Since y i t and yi t are bounded for t P with bounded derivatives (from the equations satis ed by them), it will follow that P n jy i t yi t j are uniformly continuous on ; y lemma 3 obtains X n lim jy i t y t! i t j ˆ : 3:3 Now the proof is complete orollary 3 Under the conditions in Theorem 3, system () is permanent Essentially, condition (33) in Theorem 3 indicates that the periodic solution is a global attractor if the undelayed intra-species competition dominates over the delayed inter-species competition This amounts to saying that individuals compete more with those of the same species rather than with those of the other species; in other words, intra-species competition is more signi cant that the inter-species competition Such assumptions and results are consistent with the results derived from the autonomous version of competition models Under the condition in Theorem 3, the system is permanent The delays in interspeci c competition are harmless for the permanence of system () On the other hand, as long as we restrict our consideration to the autonomous case, namely r i t ; a ij t ; i; j ˆ ; ; n are positive constants, then the unique globally asymptotically stable periodic solution will be given by positive equilibrium Remarks 3 We should point out that the results of Theorem 2 and Theorem 3 remain valid if some or all of the terms with discrete delays in Eq () are replaced by continuously distributed nite (or in nite) delays, which have been proved in our other papers

13 4 Applications M Fan et al / Mathematical iosciences 6 (999) 47±6 59 In order to illustrate some feature of our main results, in the following we will apply Theorems 2 and 3 to some one- or two-dimensional systems, which have been studied etensively in the literature Application onsider the delay logistic equation dy t y t s t ˆ r t y t ; 4: dt K t where r; K; s are continuous -periodic functions with R r t dt > ; K t > : Noting Eq (222) and applying Theorem 2 to Eq (4), we obtain the following theorem, which is an immediate corollary of Theorem 2 Theorem 4 System (4) has at least one strictly positive -periodic solution Zhang and Gopalsamy [23] study a special case of Eq (4) in which the period of the environmental parameters is related to the time delay in the selfregulating negative feed back mechanism It has been proved that if s t ˆ m; m is a positive integer, then there eists a positive -periodic solution of Eq (4) Obviously, Theorem 4 improves the results in Ref [23] It is interesting that for periodic logistic equation, the time delay has no e ect on the eistence of positive periodic solutions Application 2 onsider a two-species competition system in a periodic environment of the form dy t ˆ y t r t a t y t s t a 2 t y 2 t s 2 t Š; dt dy 2 t ˆ y 2 t r 2 t a 2 t y t s 2 t a 22 t y 2 t s 22 t Š; 4:2 dt where r i ; a ij ; s ij ; i; j ˆ ; 2 are continuous -periodic functions with R r i t dt > ; a ij t > : Eq (42) is di erent from the periodic two-species competition system considered in the literature, since r i t are not always positive Applying Theorems 2 and 3 to Eq (42), we have the following results Theorem 42 Suppose that r a 22 > r 2 a 2 epf r 2 R 2 g; r 2 a > r a 2 epf r R g 4:3 hold Then Eq (42) has at least one strictly positive (componentwise) -periodic solution Theorem 43 Suppose that s t ˆ s 22 t ˆ ; s 2 t ; s 2 t are non negative constants 4:5

14 6 M Fan et al / Mathematical iosciences 6 (999) 47±6 and r a 22 > r 2 a 2 epf r 2 R 2 g; min a t > ma a 2 t ; t2 ;Š t2 ;Š r 2 a > r a 2 epf r R g; min a 22 t > ma a 2 t t2 ;Š t2 ;Š 4:6 hold Then Eq (42) has a unique -periodic solution with strictly positive components, which is globally asymptotically stable Gopalsamy [8] also eamined a two-species periodic Lotka±Volterra system dy t ˆ y t r t a t y t a 2 t y 2 t Š; dt dy 2 t ˆ y 2 t r 2 t a 2 t y t a 22 t y 2 t Š; 4:7 dt where r i ; a ij 2 R; ; ; i; j ˆ ; 2 are -periodic functions It has been proved that if r l al 22 > ru 2 au 2 ; rl 2 al > ru au 2 ; a l > au 2 ; al 22 > au 2 : 4:8 Then Eq (47) has a unique positive periodic solution which is globally asymptotically stable, where r l i ˆ min r i t ; r u i ˆ ma r i t ; t2 ;Š t2 ;Š a l ij ˆ min a ij t ; t2 ;Š a u ij ˆ ma a ij t ; i; j ˆ ; 2: 4:9 t2 ;Š Obviously, our results for two-species competition system are di erent from the results obtained in Ref [8] For n-species competition system, our results are fresh and general Acknowledgements The authors are very grateful to the referees for their careful reading of the manuscript and a number of ecellent suggestions, which improve the representation of the present paper References [] Alvarez, A Lazer, An application of topological degree to the periodic competiting species model, J Austral Math Soc Ser 28 (986) 22 [2] S Ahmad, On the nonautonomous Volterra±Lotka competition equations, Proc Am Math Soc 7 (993) 99 [3] A attaaz, F Zanolin, oeistence states for periodic competitive Kolmogorov systems, J Math Anal Appl 29 (998) 79

15 M Fan et al / Mathematical iosciences 6 (999) 47±6 6 [4] I arbalat, Systemes d'equations di erentielle d'oscillations nonlineaires, Rev Roum Math Pures Appl 4 (959) 267 [5] JM ushing, Two species competition in a periodic environment, J Math iol (98) 385 [6] HI Freedman, P Waltman, Persistence in a model of three competitive populations, Math iosci 73 (985) 89 [7] RE Gaines, JL Mawhin, oincidence Degree and Nonlinear Di erential Equations, Springer, erlin, 977 [8] K Gopalsamy, Globally asymptotic stability in a periodic Lotka±Volterra system, J Austral Math Soc Ser 24 (982) 6 [9] K Gopalsamy, Globally asymptotic stability in a periodic Lotka±Volterra system, J Austral Math Soc Ser 29 (985) 66 [] K Gopalsamy, Stability and Oscillation in Delay Di erential Equations of Population Dynamics, Mathematic and its Applications 74, Kluwer Academic, Dordrecht, 992 [] I Gyori, G Ladas, Oscillation Theory of Delay Di erential Equations, Oford Science, Oford, 99 [2] JK Hale, AS Somolinos, ompetition for uctuating nutrient, J Math iol 8 (983) 225 [3] P Korman, Some new results on the periodic competition model, J Math Anal Appl 7 (992) 3 [4] Y Kuang, Delay Di erential Equations with Application in Population Dynamics, Academic Press, oston, 993 [5] YK Li, Periodic solutions of N-species competition system with time delays, J iomath 2 () (997) [6] RM May, WJ Leonard, Nonlinear aspects of competition between three species, SIAM J Appl Math 29 (975) 243 [7] A Shibata, N Saito, Time delays and chaos in two competition system, Math iosci 5 (98) 99 [8] HL Smith, Periodic solutions of periodic competitive and cooperative systems, SIAM J Math Anal 7 (986) 289 [9] P de Mottoni, A Schia no, ompetition systems with periodic coe cients: a geometric approach, J Math iol (98) 39 [2] HL Smith, Periodic competitive di erential equations and the discrete dynamics of a competitive map, J Di erent Eq 64 (986) 65 [2] A Trieo, Alrarez, A di erent consideration about the globally asymptotically stable solution of the periodic n-competing species problem, J Math Anal Appl 59 (99) 44 [22] F Zanolin, Permanence and positive periodic solutions for Kolmogorov competing species systems, Results Math 2 (992) 224 [23] G Zhang, K Gopalsamy, Global attractivity and oscillations in a periodic delay-logistic equation, J Math Anal Appl 5 (99) 274

Existence of Positive Periodic Solutions of Mutualism Systems with Several Delays 1

Advances in Dynamical Systems and Applications. ISSN 973-5321 Volume 1 Number 2 (26), pp. 29 217 c Research India Publications http://www.ripublication.com/adsa.htm Existence of Positive Periodic Solutions