Existence of Periodic Solution for a Non-Autonomous Stage-Structured Predator-Prey System with Impulsive Effects

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1 Appled ahemacs 55-6 do:.6/am.. Pblshed Onlne arch (hp:// Exsence o Perodc Solon or a Non-Aonomos Sage-Srcred Predaor-Prey Sysem wh Implsve Eecs Absrac eng W Zolang Xong Ypng Deng Deparmen o ahemacs Nanchang Unversy Nanchang Chna E-mal: xong6@6.com Receved November ; revsed Janary 5 ; acceped Janary 8 In hs paper we sded a non-aonomos predaor-prey sysem where he prey dspersal n a wo-pach envronmen. Wh he help o a connaon heorem based on concdence degree heory we esablsh scen condons or he exsence o posve perodc solons. Fnally we gve nmercal analyss o show he eecveness o or heorecal resls. Keywords: Perodc Solon Concdence Degree heory Sage-Srcred Implsve. Inrodcon In recen years non-aonomos predaor-prey sysems have been wdely sded [-6]. here has been a growng neres n he sdy o mahemacal models o poplaons dspersng among paches n he nare world [7-9]. In he classcal predaor-prey models s sally assmed ha each ndvdal predaor adms he same ably o eed on prey. However s deren or some speces whose ndvdals have a le hsory ha aes hem hrogh wo sages mmare and mare where mmare predaors are rased by her parens so many models wh me delays and sage srcre or boh prey and predaor were nvesgaed and rch dynamcs have been observed [6-]. In hs paper we are consdered he eecs o prey dson n wo paches and maraon delay or predaor on he dynamcs o an mplsve predaor-prey model. We dscss he derenal eaon: (See.) Where we sppose ha he sysem s composed o wo x and paches conneced by dson. x represen he denses o prey speces n pach I and II a me y and y represen he denses o he mmare and mare predaor a me n pach II respecvely. x x can dse beween pach I and II whle he predaor speces s conned o pach II. repre- sens a a s he nrnsc consan me o mary. d x x x y d x x ( ) r s ds c e x y d y y r( s) ds y x x a r x x x a r x y c x y y c e x y x x x x y y y y (.) r growh rae; s he carryng capacy; a d s he dspersal rae o prey speces; cs a con- s he capre rae o mare predaor. verson ecency. d s he deah rae o he mma- s he rae o nra-specc re predaor. Copyrgh ScRes.

2 56. F. WU E A. compeon. and represen he annal brh plse o x y a Z. We mae he ollowng assmpons or or model: ) a r d d c and r are connos posve perodc ncons; ) and are consans and here exss a posve neger sch ha.. Prelmnares Denoe by PC J RJ R he se o ncons : J R whch are pecewse connos n and have pons o dsconny. e PC J R denoe he se o ncons wh dervave PC J R. We dene he Banach space o perodc ncons PC PC R wh sp : and PC wh PC PC max ( ) we wll consdered he PC PC PC wh he norm We dene: d ( ) PC PC. PC PC mn max [ ]. [ ]. Exsence o Posve Perodc Solons In hs secon we sdy he exsence o posve perodc solons o sysem (.). Beore sang or resl on posve perodc solons o sysem (.) we need he ollowng lemma: emma. ([]). e X be an open bonded se. e be a Fredholm mappng o ndex zero and N be compac on. Assme ) or each x s any solon o x Nx sch ha x ; ) or each QNx or each x Ker ; ) deg JQN Ker. hen he eaon x Nx has a leas one solon n Dom. heorem. I he sysem (.) sases ln (H) a d ln (H) a d ln a d e ln mm r (H) ce c e hen he sysem (.) has a leas one perodc posve solon. Proo. e x e x e y e y e hen d d e d rsds a r e d e a r e e c e d e c e e rsds c e e e ln ln ln (.) One can easly see ha sysem (.) has one hen perodc solon e e e e x x y y s a posve perodc solon o sysem(.). hs n wha ollows or goal s o show ha sysem (.) has a leas one perodc solon. Here we rewre and e wh. Dom PC PC PC : N PC PC PC Z Copyrgh ScRes.

3 . F. WU E A. 57 ln ln N ln and C C Ker : R. C C Where Q s dened by d a g d b QZ. hd c j d d Frhermore K :Im KerPDom s gven by KPZ hs P d a sdsd a g d b g sdsd b. hd c hsdsd c j d d jsdsd d d ln d ln ln d KP I QN d sdsd ln sdsd ln sdsd ln s dsd sd sd ln sd ln sd ln In order o apply he emma. we also need o nd an approprae open and bonded sbse. Corrsponng o he operaor eaon N here we can ge ln ln ln Sppose (.). By negrang over (.) s a perodc solon o ad ln () r e d e d a d ln r e e d e d d ln ce e d r sds c e e d () e d r sds c e e d (.) Accordng o (.) and (.) we have Copyrgh ScRes.

4 58. F. WU E A. d a d d r e d e d a dln (.) d a dln (.5) d dln (.6) d e d (.7) Scnce PC sch ha mn max. e v max hen v PC ) or b v and hen a r e a r e ; ) or b v and a r e a r e. hen Dnoe amax a a pmn r r max hen v Dv a pe (.8) v ln Inegrang (.8) over we ge ln hereore v a p e d. aln e d e d (.9) aln ln p p ln d aln ln (.) p a d ln Accordng o he orh eaon o (.) we have rsds e d c e e d De o r e d c e d r c e e d () (.) (.) e d e d (.) From (.) and (.) we have r e d c e (.) c e ln r Accordng o (.7) and (.) we ge d e d r c e e d d r r c e c e ln Accordng o he hrd eaon o (.) we have ce d dln Do o and we have ln c e e d d c e ln dln (.5) Copyrgh ScRes.

5 . F. WU E A. 59 ln d c e ln (.6) dln d ln From he rs eaon o (.) we have r e d r e d ad ln So a dln ln r ln d ln adln r ad ln m From he second eaon o (.) we have r e d a dln e a d ln e ln r () ( ) () d ln[ ( )] ln[ ( )] a d e ln r (.7) a d ln[ ( )] m (.8) From (.) we have ( ) ( ) ( ) () e e d e d r m () ce e d r m ce ln e d r m r ce c e ln m Accordng o he hrd eaon o (.) we have mm r ce c e e d d e ln Smlarly we have ln mm r ce c e e ln d ln d ln e ln d hs we have ( ) mm r ce c e d ln m (.9) (.) sp max m m m m D Denoe max D D D D D where D may be aen scenly large sch ha each solon o Eaons (.) adre de ln a d re e de ln r sds ce c e e d e ln r sds c e e e (.) Copyrgh ScRes.

6 6. F. WU E A. sases D hen. : Dom X as he orm Denoe ad re ln a d re ln d e ln c e e e rsds de e de r sds ce c e e Where s a parameer. Wh he mappng we have or Ker. So we now ha. Obvosly he algebrac Eaon (.) has a ne solon. adre ln a d re ln (.) d e ln r sds c e e e From he concdence degree heory we can oban deg JQN Ker deg ( ) Ker.. Nmercal Analyss In hs paper we have ocsed on he dynamcs complexy o a sage-srcred sysem wh dson and mplsve eecs. By sng he mehod o concdence degree we oban he scen condon or he exsence o a leas one posve perodc solon. In hs secon we gve he nmercal resls. x x [.6cos.5 x ] ( cos [ x x ] x x [5..sn. x ] (.5sn ) xy (.sn )[ x x ] y. sn x y.8. sn( e xy.y.5cos y y.75cosy.8 e xy (. sn x x x x y y y y (.) Nmercal analyss ndcaes ha he complex dynamc behavor o sysem (.) depends on he vales o mplsve perrbaons n model (.). Or heorecal resls are conrmed by nmercal smlaons. we can see ha he dynamc behavor o he sysem (.) has obvosly vared as he mplse vale changng. e... s easly proved ha he sysem (.) sases all he condons o heorem. ha mean he sysem (.) has a leas one posve perodc solon (Fgre ). As mplses ncrease he perodc oscllaon o sysem (.) wll be desroyed (Fgre ). 5. Acnowledgmens hs wor s sppored by Naral Scence Fondaon o Jangx Provnce. (No. 9GZS) 6. Conclsons here s mch prevos wor repored on non-aonomos sage-srcred sysem or dsve sysem. hs movaes s o sdy a non-aonomos sage-srcred predaor-prey sysem wh mplsve eecs. As poned o n Secon we bl sysem (.). In Secon we gve some prelmnares. In Secon by sng he mehod o concdence degree we oban he scen condon or he exsence o a leas one posve perodc solon. In Secon we gve he nmercal smlaons on he dynamc behavors o he sysem hrogh wo examples. B we dd no dscss he global sably o he perodc solons perodc solon o sysem (.). We Copyrgh ScRes.

7 . F. WU E A. 6 Fgre. Dynamc behavor o he sysem (.) wh nal vales τ =.and mplsve perrbaons θ =. θ =. φ =.. Fgre. Dynamc behavor o he sysem (.) wh nal vales τ =.and mplsve perrbaons θ =. θ =. φ =.. Copyrgh ScRes.

8 6. F. WU E A. leave hese aspecs or re research. 7. Reerences [] Y. Naaa Y. roya Permanence or Nonaonomos oa-volerra Cooperave Sysems wh Delays Nonlnear Anal Vol. No. pp do:.6/j.nonrwa.9.. []. V. on Srvval o hree Speces n a Nonaonomos oa-volerra Sysem Jornal o ahemacal Analyss and Applcaons Vol. 6 No. Febrary pp do:.6/j.jmaa [] S. H. Chen J. H. Zhang and. Yong Exsence o Posve Perodc Solon or Nonaonomos Predaor-Prey Sysem wh Dson and me Delay Jornal o Compaonal and Appled ahemacs Vol. 59 No. Ocober pp do:.6/s77-7()5-5 [] Z. H. X. B. Ch and. S. Chen Global Aracvy o Nonaonomos Sage-Srcred Poplaon odels wh Dspersal and Harves Jornal o Compaonal and Appled ahemacs Vol. 66 No. Aprl pp. -5. do:.6/j.cam..8. [5] Z. D. eng and. S. Chen Unorm Perssence and Exsence o Srcly Posve Solons n Nonaonomos oa-volerra Compeve Sysems wh Delays Compers & ahemacs wh Applcaons Vol. 7 No. 7 Aprl 999 pp do:.6/s898-(99)87-5 [6] J. Y. Wang Q. S. and Z. S. Feng A Nonaonomos Predaor-Prey Sysem wh Sage Srcre and Doble me Delays Jornal o Compaonal and Appled ahemacs Vol. No. Ags 9 pp do:.6/j.cam.8.. [7] X. X. and. H. Hang Permanence and Perodc Solons or a Dsve Rao-Dependen Predaor-Prey Sysem Appled ahemacal odellng Vol. Febrary 9 pp do:.6/j.apm.7.. [8] Z. X. H G. K. Gao and W. B. a Dynamcs o a hree-speces Rao-Dependen Dsve odel Nonlnear Anal Vol. 7 November pp [9] C. J. X X. H. ang and. X. ao Sably and Bbrcaon Analyss o a Delayed Predaor-Prey odel o Prey Dspersal n wo-pach Envronmens Appled ahemacs and Compaon Vol. 6 Jly pp do:.6/j.amc... [] S. Q.. S. Chen and Z. J. Exncon and Permanence n Nonaonomos Compeve Sysem wh Sage Srcre Jornal o ahemacal Analyss and Applcaons Vol. 7 No. Ocober pp do:.6/s-7x()9- [] Z. and F. D. Chen Exncon n Perodc Compeve Sage-Srcred oa-volerra odel wh he Eecs o oxc Sbsances Jornal o Compaonal and Appled ahemacs Vol. No. Sepember 9 pp. -5. do:.6/j.cam.9.. [] X. W. Jang Q. Song and. Y. Hao Dynamcs Behavors o a Delayed Sage-Srcred Predaor-Prey odel wh Implsve Eec Appled ahemacs and Compaon Vol. 5 No. Febrary pp. -9. do:.6/j.amc.9.. [] R. E. Ganes and J.. awhn Concdence Degree and Nonlnear Derenal Eaons Sprnger Berln 997. Copyrgh ScRes.