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Journal of Software Engineering and Applications, 20 12, 5, 26-29 doi:10.4236/jsea.2012.512b006 Published Online December 2012 (http://www.SciRP.org/journal/jsea) Copyright © 2012 Sci R es. JSEA Periodic Solution of n-Species Gilpin-Ayala Competition System with Impulsive Perturbations Kaihua Wang, Zhanji Gui* Scho ol of Mathematics and S tatistics, Hainan N ormal University, Haiko u, Hainan, China. Email: *zhanjigui@sohu.com Received 2012 ABSTRACT The principle aim of this paper is to explore the existence of periodic solution of n-Species Gilpin-Ayala competition system with impulsive perturbations. Sufficient and realistic conditions are obtained by using Mawhin's continuation theorem of the coincidence degree. Further, some numerical simulations show that our model can occur in many forms of complexities includin g p e r iodic oscilla tion and chaotic s trange attractor . Keywords: Periodic Solution; Impulsive Perturba tions; Ma whin’s Continuation The orem 1. Introduction The dynamics of Ayala-Gilpin competitive system, which was first introduced by Ayala et al. [1], has been widely studied by many authors [2-6]. However, the corresponding problems with periodic coefficients and impulsive perturbations were studied far less often [7]. In this paper, we will study the following impulsive Gil- pin-Ayala s ystem: 1, ()() () () () 1() , () () ()() ()(),, i i ii nj iij j ji ij i ikik ikkikk zt rtzt zt zt t Kt Kt ztztztpztt t θ α = ≠ +− − = ⋅ −− ∆= − == ∑ (1.1) where )(tzi represents the density of the thi species at time t ; )(tr i deno tes the intr insi c gr owth r ate of the thi species; )(tKi means the environment carrying capacity of species i in the absence of competition; )(t ij α ( ji ≠ ) measures the amount of competition be- tween the species i x and j x ; i θ is a positive con- stant and provide a nonlinear measure of intra-specific interference; i k p are constants. In system (1.1), we give two hypotheses as follows. (H1) )(),( tKtrii and )(t ij α ( jinji ≠= ,,,1, ) are all nonnegative T - periodic functions defined on R . (H2) 01 >+ i k p and there exists a positive inte- ger q such that Ttt kqk+= +, i k iqk pp = + . 2. Existence of Positive Solutions To prove our result s, we need the noti on of t he Ma whi n’s continuation theorem formulated in [8]. Lemma 1 ([8]) Let X and Y be two Banach spaces. Consider an operator equation NxLx λ = where :L Dom YXL → is a Fredholm operator of index zero and ]1,0[∈ λ is a parameter, then there exist two projectors XXP →: and YYQ →: such t hat =PIm Ker L and =LIm Ker Q . Assume that YN →Ω: is L - compact on Ω , where Ω is open bounded in X . Fur- thermore, assume that a) for each )1,0(∈ λ , Ω∂∈x Dom L, NxLx λ ≠ ; b) for each Ω∂∈x Ker L , 0≠QNx ; c) { Ω,degJQN Ker } 00, ≠L , where →QJ Im: Ker L is an isomorphism and deg{*} represents the Brouwer degree. Then the equatio n NxLx = has at least one solution in Ω Dom L . For the sake of convenience, we shall make some preparation. Let RI ⊂ . Denote by ),( n RIPC the space of functions n RItx →:)( which are conti- nuous at It∈ ,k tt≠ , and are left continuous for *Corresponding author. Periodi c Solution of n-Species Gilpin-Ayala Competition System with Impulsive Perturbations Copyright © 2012 SciRes. JSEA 27 Itt k∈= . Let )}({min 0 tuu Tt L ≤≤ = , )}({max 0 tuu Tt M ≤≤ = , ∫ = T dttu T u 0 )( 1 , ∫ =Tdttvtu T uv 0)()( 1 , where )(tu , )(tv are T periodi c functions. Theorem 1. Suppose (H1) and (H2) hold, furthermore, the following co nditions are satisfied. (H1) ∑∑ ≠== >++ n ijj C iji L j i q k i k j Ter K Trp ,11 1 )1ln( α , where 1 11 1 ln 1ln(1)ln 2 ln(1 )|ln(1 )|. qjM kj k j jj j qq jj kk kk pK rT C rT pp θ = = = + ++ = + + +++ ∑ ∑∑ Then system (1.1) has at least one positive T - periodic solution. Proof. Let )( )( tx ii etz = ( ni ,,1 = ) (2.1) then the s ystem (1.1) becomes () () 1, ()() 1() () () ( )ln(1) j ii i xt xt n i iijk j ji ij i ikk k ee xtrttt t K tKt xtpt t θ θ α = ≠ =−− ≠ ∆=+ = ∑ (2.2) In order to use Lemma 1, we set T n txtxx))(,),(( 1 = , )}()(|),({ txTtxRRPCxX n=+∈= , nq RXY ×= , then it is standard to show that both X and Y are Banach space when they are endowed with the norm |)(|sup|||| ],0[txx Tt c∈ = and 2/122 1 2 1 )||||||(||||),,,(|| qcq ccxccx +++= . Set :DomLLX Y⊂→ as ))(,),(),(())(( 1q txtxtxtLx ∆∆= , where Dom{|'()(,)} n LxXxtPC R R =∈∈ , =+∈= ∑ ∫ = 0)(|),,,(Im 1 0 1 q ii T q cdttyYccyL and Ker Ker n LR= . At the same time, we denote YXN →: as )))((,)),(()),( ,(())(( 11 qq txtxtxtftNx ΦΦ = , where () () 1, 1 (,) () 1()() () j ii i xt xt n i ij j jij in ftx ee rt tKt Kt θ θ α = ≠× = −− ∑, ( ) T n kkkk pptx )1ln(,),1ln())(( 1 ++=Φ , where ni ,,1= , qk ,,2,1 = . Define two projectors P and Q as LXP ker:→ , ∫ =Tdttx T Px 0)( 1 ; YYQ →: , += ∑ ∫= 0,,0,)( 1 ),,,( 1 0 1 q kk T qcdssy T ccyQ . It can be easily proved that L is a Fredholm operator of index zero, that P , Q are projectors, and that N is L - compact on Ω for any given open and bound subset Ω in X . Now we are in a position to search for an appro- priate open bounded subset Ω for the application of Lemma 1 correspo nding to operator equation NxLx λ = , )1,0(∈ λ (2.3) Suppose that T ntxtxtx))(,),(()( 1= is a periodic solution of (2.3) for certain )1,0(∈ λ . By integrating (2.3) over the interval ],0[ T , we get () 0 1 () 0 1, () ln(1)() () ()() ii i j qTxt ii ik ki xt nT i ij j jij rt rTpe dt Kt e r ttdt Kt θ θ α = = ≠ =− ++ + ∑∫ ∑∫ (2.4) From (2.3), (2.4), we can obtain i q k i ki T iApTrdttx ≡++≤ ∑ ∫ =1 0)1ln(2|)(| (2.5) Since )],,0([)( RTPCtx i∈, there exist },,,{],0[, 21+++ ∈ qii tttT ηξ , such that )(inf)( ],0[ txx i Tt ii∈ = ξ , )(sup)(],0[ txx i Tt ii ∈ = η , It follows fro m (2.4) that ( )() 0 1 () 1 () ln(1) ii iii i T x xt i i M ii qi ik k rt rTee dt K Kt rT p θξ θ θ = ≤ ≤+ + ∫ ∑ which implies Periodi c Solution of n-Species Gilpin-Ayala Competition System with Impulsive Perturbations Copyright © 2012 SciRes. JSEA 28 i i M i q k i k i ii B Kp Tr x≡ + ++ ≤ ∑ = θ ξ ln)1ln( 1 1ln )( 1 Thus we get 01 1 ( )()|( )||ln(1)| | ln(1) | q Ti i iiik k qi iik i k xt xxtdtp BAp C ξ = = ≤+++ ≤+++ ≡ ∑ ∫ ∑ (2.6) In particular, we have iii Cx ≤)( η . On the other hand, from (2. 4 ), we have () 1 () 1, ln(1) 1 ii i jj qx ii ik L ki nx i ij L j jij rT rTp e K r Te K θη η α = = ≠ ≤−++ + ∑ ∑ . The n we get () 1 1, 1 ln(1 ) j ii i qnC xi ikii ij LL kj ji ij rT eprTr Te KK θη α == ≠ ≥+ +− ∑∑ Because of (H3) we have 1 1, ln() ln() () 1 ln ln(1 ) j L ii ii i qnC i kiiij L kjjij i i K rT x prTr Te KD ηθ α θ == ≠ − ≥+ + +− ≡ ∑∑ Thus we get 01 1 ( )()|( )||ln(1)| | ln(1) | q Ti i iiik k qi iik i k xtxxt dtp DAp E η = = ≥−− + ≥ −−+≡ ∑ ∫ ∑ (2.7) From (2.6 ) and (2.7) , it follows that |}||,max{||)(| iiiiECFtx =≤ Obviously, i F ( ni ,,1= ) are independent of λ . Thus, there exists a constant 0>F , such that { } Fxx n≤||,|,|max 1 . Let FFFr n+++> 1 , }||:||{ rxXx c <∈=Ω , then it is clear that Ω sat- isfies condition (a) of Lemma 1 and N is L - compact on Ω . when ( ) 1 , ,Ker Tn n xx xLR=∈∂Ω=∂Ω , x is a constant vector in n R with xr= . Thus 0QNx ≠ . Let : ImJQ→ Ker L , ( ,0,,0)dd→ . A direct computatio n gives deg{,ker,0}0JQN LΩ≠ . By now we have proved that Ω satisfies all the re- quirements in Mawhin’s continuation theorem. Hence, (2.1) has at least one T - periodic solution. By of (2.1), we derive that (1.1) has at least one positive T - periodic solution. The proof is complete. 3. An Illustrative Example To e asy to call functio ns, let () () ii xt zt=. In (1.1), we take 2n= , kTtk= , ,sin6.05)( 1ttr+= ,cos4.04)( 2ttr −= ,sin3.02)( 1ttK += ttK sin1.02)( 2 += , 5.1 1= θ , 6.1 2 = θ , tt cos1.08.0)( 12 += α , tt sin2.09.0)( 21 += α . Obviously, ),( 1tr ( ) , 2 tr ),( 1tK ),( 2 tK , 12 α , 21 α satisfy (H1). ,3.0 1= k p 2.0 2 = k p . If 2 π =T , then system (1.1) under the conditions (H5 ) ha s a uni q ue π 2 -periodic so lution (In Figures 1-3, we take TT xx ]5.0,5 .0[)]0(),0([ 21 = ). Because of the influence of the period pulses, the influence of pulse is obvious. Figure 1. Time-series of )( 1 tx evolved in system (1) with 2 π =T . Figure 2. Time-series of )( 2 tx evolved in system (1) with 2 π =T . Periodi c Solution of n-Species Gilpin-Ayala Competition System with Impulsive Perturbations Copyright © 2012 SciRes. JSEA 29 But if 2=T , then (H2 ) is not satisfied. Periodic os- cillation of system (1.1) under the conditions (H5) will be destroyed by impulsive effect. Numeric results show that system (1.1) under the conditions (H5) has gui cha- otic strange attractor (see Figure 4) [9]. In Figure 4, we take TT xx ]5.0,5.0[)]0(),0([ 21 = . Figure 3. Phase portrait of periodic solutions of system (1) with 2 π =T . Fig ure 4. P ha se p ortrait of ch a ot ic st range a tt ract or of sys- tem (1) with 2=T . 4. Acknowledgement s This work is supported jointly by the Natural Sciences Foundation of China under Grant No. 60963025, Natural Scienc es Fo undat ion of Ha ina n Pro vince und er Gr ant No. 110007 and the Start-up fund of Hainan Nor mal Uni ver- sity under Project No. 00203020201. 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[7] Wang Q, Ding M M, Wang Z J and Zhang H Y “Exis- tence and attractivity of a periodic solution for an N-species Gilpin-Ayala impulsive competition system” Nonlinear Ana: Real World Applications 11 (2010) 2675-85 [8] Gaines R E and Mawhin J L “Coincidence degree and nonlinear diferential equations ” Springer-Verlag, Berlin, 1977. [9] Zhang J and Gui Z J “Periodic solutions of nonautonom- ous cellular neural networks with impulses and delays,” Nonlinear Analysis: Real World Applications. 10 (2009) 1891-1903. |