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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() ,() ()()() ()(),,ii iinjiijj jiijiikik ikkikkzt rtztztzt tKt Ktztztztpztt tθα= ≠+− −= ⋅−−∆= − ==∑ (1.1) where )(tzi represents the density of the thispecies at time t; )(tri 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; )(tijα(ji ≠) measures the amount of competition be-tween the species ix and jx; iθ is a positive con-stant and provide a nonlinear measure of intra-specific interference; ikp are constants. In system (1.1), we give two hypotheses as follows. (H1))(),( tKtriiand )(tijα (jinji ≠= ,,,1, ) are all nonnegative T- periodic functions defined on R. (H2)01 >+ikp and there exists a positive inte-ger q such that Tttkqk+=+, ikiqkpp =+. 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 NxLxλ= 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 =LImKerQ. Assume that YN →Ω: is L- compact onΩ, where Ω is open bounded inX. Fur-thermore, assume that a) for each )1,0(∈λ, Ω∂∈xDom L, NxLxλ≠; b) for each Ω∂∈xKerL, 0≠QNx; c) {Ω,degJQNKer}00, ≠L, where →QJ Im: KerL is an isomorphism and deg{*} represents the Brouwer degree. Then the equatio n NxLx = has at least one solution in ΩDomL. For the sake of convenience, we shall make some preparation. LetRI ⊂. Denote by ),(nRIPC the space of functions nRItx →:)( which are conti-nuous atIt∈,ktt≠ , 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 Ittk∈= . Let)}({min0tuuTtL≤≤=,)}({max0tuuTtM≤≤=, ∫=TdttuTu0)(1,∫=TdttvtuTuv 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) ∑∑≠==>++nijjCijiLjiqkikjTerKTrp,111)1ln(α, where 1111ln 1ln(1)ln2ln(1 )|ln(1 )|.qjMkjkjjjjqqjjkkkkpKrTC rTppθ== =+ ++= ++ +++∑∑∑ Then system (1.1) has at least one positive T- periodic solution. Proof. Let )()( txiietz = (ni ,,1 =) (2.1) then the s ystem (1.1) becomes ()()1,()() 1()() ()( )ln(1)jiiixtxt ni iijkj jiijiikk keextrttt tK tKtxtpt tθθα= ≠=−− ≠∆=+ =∑ (2.2) In order to use Lemma 1, we set Tntxtxx))(,),((1=, )}()(|),({ txTtxRRPCxX n=+∈=, nqRXY ×=, then it is standard to show that both X and Y are Banach space when they are endowed with the norm |)(|sup|||| ],0[txx Ttc∈= and 2/122121)||||||(||||),,,(||qcqccxccx +++=. Set:DomLLX Y⊂→ as ))(,),(),(())((1qtxtxtxtLx ∆∆= , where Dom{|'()(,)}nLxXxtPC R R=∈∈, =+∈=∑∫=0)(|),,,(Im101qiiTqcdttyYccyL  and KerKer nLR=. At the same time, we denote YXN →: as )))((,)),(()),(,(())((11 qqtxtxtxtftNx ΦΦ=, where ()()1, 1 (,)() 1()()()jiiixtxt ni ijj jijinftxeert tKtKtθθα= ≠×= −−∑, ( )Tnkkkkpptx )1ln(,),1ln())((1++=Φ , where ni ,,1=, qk ,,2,1 =. Define two projectors P and Q as LXP ker:→,∫=TdttxTPx 0)(1;YYQ →:,+= ∑∫=0,,0,)(1),,,(101qkkTqcdssyTccyQ. It can be easily proved that L is a Fredholm operator of index zero, thatP, 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 NxLxλ=,)1,0(∈λ (2.3) Suppose that Tntxtxtx))(,),(()( 1= is a periodic solution of (2.3) for certain)1,0(∈λ. By integrating (2.3) over the interval],0[ T, we get ()01()01,()ln(1)()() ()()iiijqTxtiiikkixtnTi ijj jijrtrTpe dtKter ttdtKtθθα== ≠=− +++∑∫∑∫ (2.4) From (2.3), (2.4), we can obtain iqkikiTiApTrdttx ≡++≤∑∫=10)1ln(2|)(|  (2.5) Since )],,0([)(RTPCtxi∈, there exist },,,{],0[,21+++∈qiitttT ηξ, such that )(inf)(],0[txxiTtii∈=ξ, )(sup)(],0[ txx iTtii ∈=η, It follows fro m (2.4) that ( )()01()1()ln(1)ii iiiiTx xtiiMiiqiikkrtrTee dtK KtrT pθξ θθ=≤≤+ +∫∑ which implies Periodi c Solution of n-Species Gilpin-Ayala Competition System with Impulsive Perturbations Copyright © 2012 SciRes. JSEA 28 iiMiqkikiiiBKpTrx≡+++≤∑=θξln)1ln(11ln)(1 Thus we get 011( )()|( )||ln(1)|| ln(1) |qTii iiikkqiiik ikxt xxtdtpBAp Cξ==≤+++≤+++ ≡∑∫∑ (2.6) In particular, we have iii Cx ≤)(η. On the other hand, from (2. 4 ), we have ()1()1,ln(1)1ii ijjqxiiikLkinxi ijLj jijrTrTp eKr TeKθηηα== ≠≤−+++∑∑. The n we get ()1 1,1ln(1 )jii iqnCxiikii ijLLkj jiijrT eprTr TeKKθηα== ≠≥+ +−∑∑ Because of (H3) we have 1 1,ln() ln()()1ln ln(1 )jLiiiiiqnCikiiijLkjjijiiK rTxprTr TeKDηθαθ== ≠−≥++ +−≡∑∑ Thus we get 011( )()|( )||ln(1)|| ln(1) |qTii iiikkqiiik ikxtxxt dtpDAp Eη==≥−− +≥ −−+≡∑∫∑(2.7) From (2.6 ) and (2.7) , it follows that |}||,max{||)(| iiiiECFtx =≤ Obviously, iF (ni ,,1=) are independent ofλ. Thus, there exists a constant0>F, such that { }Fxx n≤||,|,|max 1. LetFFFr n+++>1, }||:||{ rxXxc<∈=Ω, then it is clear that Ω sat-isfies condition (a) of Lemma 1 and N is L- compact on Ω. when( )1, ,KerTnnxx xLR=∈∂Ω=∂Ω, x is a constant vector in nR with xr=. Thus 0QNx ≠. Let : ImJQ→KerL, ( ,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 () ()iixt zt=. In (1.1), we take2n=, kTtk=, ,sin6.05)(1ttr+= ,cos4.04)(2ttr −= ,sin3.02)(1ttK += ttK sin1.02)(2+=, 5.11=θ, 6.12=θ, tt cos1.08.0)(12 +=α, tt sin2.09.0)(21 +=α. Obviously, ),(1tr ( ),2tr ),(1tK ),(2tK ,12α ,21α satisfy (H1). ,3.01=kp 2.02=kp. If 2π=T, then system (1.1) under the conditions (H5 ) ha s a uni q ue π2-periodic so lution (In Figures 1-3, we take TTxx ]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 )(1tx evolved in system (1) with 2π=T. Figure 2. Time-series of )(2txevolved in system (1) with 2π=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 takeTTxx ]5.0,5.0[)]0(),0([21=. Figure 3. Phase portrait of periodic solutions of system (1) with 2π=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. REFERENCES [1] Ayala F J, Gilpin M E and Eherenfeld J G “Competition between species: Theoretical models and experimental tests,” The or e t. Popul. Bi ol . 4 (1973) 331-56. [2] Gilpin M E and Ayala F J “Global models of growth and competition,” Proc. Natl. Acad. Sci. USA. 70 (1973) 3590-93. 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