Multi-Team Prey-Predator Model with Delay

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Osborn Webb
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1 h S L o 5-5 l-tm y-o ol h Dly Shn ly n F ly hms Dpmn Fly of Sn Kng Khl Unsy h 9 S hms Dpmn Fly of Sn l-h Unsy ss 75 gyp hms Dpmn Fly of Sn nso Unsy nso 55 gyp ml's shhly7@yhooom mohfhy@mnsg R Jn ; Rs y ; p Jn s Th mn gol n hs pp s o onn h nsgons of h mpon sysm s [] y onsng ly ml-m py-po mol In h sn of ly sy h onons of h xsn n sly pops of h qlm pons Fo h fll gnl mol h ly onons n hh h n no hng n sly Usng h s m ly s fon pm s fon h Hopf fon os hn h ly psss hogh l l Rsls f y omp smlon Kyos Dly po-py mol Hopf fon Inoon hml Sns Ls n Innonl S l Sns lshng Co In h nl ol ho sps os no xs lon lso omps h h oh sps fo sp foo o p y oh sps Thfo s mo ologl sgnfn o ons h ff of nng sps hn sy h ynml hos of ologl mols Dly-ffnl qons xh mh mo ompl ynms hn ony ffnl qons sn m ly ol s sl qlm o om nsl n s h poplons o fl Tm lys h n nopo no ologl mols y mny hos fo xmpl [] [] [5] [7] [8] [9] [] In n ys ly py-po sysm s on of h mos mpon fls of ns s [] [] [] [] [] In hs pp foml ml-m py-po mol h m ly no h gson of h po Th sls sho h f h gson ly s smll nogh h ss ll kp sl n h long n f h gson ly of po s lg nogh h ss ll polly fl n h long m y H h lol sly of h pos qlm of hs mol s nsg Th onons n hh h pos qlm s lolly sympolly sl on Un h sm onons nmly h h sm pms h sly of h pos qlm of po-py mol ol hng o h noon of gson m ly fo po oo h h on of m ly h pos qlm of h mol sjs o Hopf fon Th o hol sls l y nml smlons h h hlp of ynml sof TL Th pp s ogn s follos In Son h mol s l Son s o o h nsgon of h onons fo lol sly In Son lnon of h ly mol s n h onons fo h nono-hopf fon slsh hs h mn sls of hs pp W sho h h ns of ly sls h sysm n ss h on of po osllons W ons n xmpl o lls h n xp O onlng mks psn n son 5

2 Shn ly n F ly l-tm y-o ol h Dly hml mol In hs Son popos sysm onsss of o ms of pys h nss n sply n h on m of po h nsy Th ssmpons of hs mol s follo In h sn of ny pon h m of pys gos logslly; hs s n Th ff of h pon s o h py goh y m popoonl o h py n po poplons; hs s h ms Th ms of pys hlp h oh gns h po h h m s xs In h sn of ny py fo ssnn h po's h sls n ns y h s h m 5 Th py's onon o h po goh s ; h s popoonl o h ll py s ll s h s of h po poplon s on h o ssson ly s [] h s h follong py po mol Tm ly s no o n h o mol In hs pp o m s o nsg ho h m ly ffs h ynms of sysm Fom h o ssmpons h m ly hh s h m q fo h gson of po h mol qons om h h offns n pos onsns n I s l h h m n h py qons mns h h pys hlp h oh g n fogng n n ly nng gns pon o h hs hlp os only n h psn of po hs s psn y h m n h pys qons Th mol ho m ly In hs Son ll sy h sysm ho ly In pl ll fos on h xsn of ql n h lol sly Ths nfomon ll l n h nx son h sy h ff of h ly pms on h sly of h sy ss Th ql of h sysm

3 Shn ly n F ly l-tm y-o ol h Dly n h follong o no solons hh mns ll ms oxs n h xsn onons sn sly nlyss s on h Jon mx; J fo h ql sho h h ogn s nsl qlm pon hh hs o pos gnls Smlly hs o pos gnls So s nsl qlm pon y h sm y hs o pos gnls So s nsl qlm pon lso h qlm pon hs on pos gnl So ll of hm nsl qlm pons Th qlm solon s lolly sympolly sl n h onon Th qlm solon 5 s lolly sympolly sl f Th fs nnl qlm solon s lolly sympolly sl f Th son nnl qlm solon To nsg h sly of h pos sy ss ons h ln sysm of Th Jon mx s gn y

4 8 Shn ly n F ly l-tm y-o ol h Dly Th hs qon of h ln sysm s gn y h / / 5 Thn ong o h Roh-H ll oos of h ng l ps f hn h pos qlm pon s lnly sympolly sl n h follong onons 7 - Th mol h m ly In hs Son fos on nsg h sly of ql n Hopf fon of h pos qlm of h sysm To sy h sly of h sy ss l s fn Thn h ln sysm of h sysm s gn y 8 Thn xpss sysm 8 n mx fom s follos 9 h n ms gn y n

5 Shn ly n F ly l-tm y-o ol h Dly 9 Th hs qon of h sysm 9 s gn y xp I h s h / / I s knon h h qlm s sympolly sl f ll oos of osponng hs q h ng l ps W ons q h h s q n ssm h ll h oos of q h ng l ps Ths s qln o h ssmpon 7 Dno h gnls of h hs q h n pn on h ly Sn h qlm of h OD mol s sl follos h hn y onny f s sffnly smll sll h n s sll sl If fo n l so h s ply mgny oo of q hn h sy s loss s sly n nlly oms nsl hn oms pos In oh os f sh n os no xs h s f h hs q os no h ply mgny oos fo ll ly hn h qlm s lys sl W shll sho h hs n s fo h hs q Clly s oo of q f n only f Spng h l n mgny ps h os sn sn os ng p h sqs of oh h qons on 5

6 Shn ly n F ly l-tm y-o ol h Dly L Thn q 5 oms h Lmm Th s ls on pos oo of h f Th ls on pos oo of h f n Rmk h fnon s h W os h h xs on o o xm pons of h f h smn oo f n o oos of h pos Th l l of ly fo hh h pos qlm pon s sl f s gn y os n n m T 7 Thom ssm h n hn Hopf fon os po h s pos o ng oof O ssmpon gs o pos oos of h fnon h W ons h sgn of h follong nssly onon R 8 y onny h l p of oms pos hn n h qlm oms nsl oo Hopf fon os hn psss hogh h l ls Dffnng hs q h sp o g Ths gs

7 Shn ly n F ly l-tm y-o ol h Dly sgn sgn R R R sgn sgn sgn sgn sgn If s h pos o ng hn h nssly onon hols n hn Hopf fon os Ths ll l fom h follong nml xmpl ml smlon

8 Shn ly n F ly l-tm y-o ol h Dly Fg Lf fgs Th s of ly = Th solons n shos ongn o h qlm solon s m nss Rgh fgs Th s of ly = 5 Th solon n h qlm pon loss s sly n sl Hopf fon os h s fmly of po solon fs fom qlm s m nss Fg po y pplyng TL xmpl L Fo h s of pm ls h nq pos sl qlm s gn y 59 Ssng hs pm ls n q on h Solng q 9 g on pos l of h s = 889; on pos l of n 9 Th xs l l of ly = sh h h pos qlm fs o po solons hn ls n = 55;77; 8799 W h o ss of ly n o h h sm nl ls kn sh h In h fom s p s n h Fgs f h ly s slghly sho hn h l on = 585 h solon og o h qlm solon In h l s f h ly s slghly long hn h l ly os Hopf fon hh osll s shon n h Fgs 5 Conlsons ny nmls l n gop Dffn gops sh on h hn hs gops my oop omp h h oh o fom po-py sysm In hs ok psn n mol fo popy mol h ly qlm solons h lol sly s ologl lon of o mol s o oopng ms of glls n s k y on po Fo h fll gnl mol h ly onons n hh h n no hng n sly Usng h s m ly s fon pm s fon h Hopf fon os hn h ly psss hogh l l Rsls f y omp smlon knolgmns W hnk h fs fo y sfl sggsons hh hlp s o mpo onsly h psnon of o sls Rsh plly sppo y h Dnshp of Snf Rsh Kng Khl Unsy KS KKU-SCI--5 n KKU-SCI--5

9 Shn ly n F ly l-tm y-o ol h Dly Rfns [] S ly fons n po-py mol h ffson n mmoy Innonl Jonl of fon n Chos [] S sng K Cook Vlly Tnsm Dsss Spng ln 99 [] J Cshng Ingo-Dffnl qons n Dly ols n oplon Dynms Spng Hlg 977 [] F ly To-py on-po mol Chos Solons & Fls [5] K Goplsmy Sly n Osllons n Dly Dffnl qons of oplon Dynms Kl Doh 99 [] J Ksk D K Sklly K H ss hol on of nfon sk ongs of h nonl my of Sns of h US [7] Y Kng Dly Dffnl qons h pplons n oplon Dynms m ss Yok 99 [8] Donl Tm Dlys n ologl ols Spng Hlg 978 [9] G Spn R Dynml Sysms Sly n Chs Fnons Longmn UK 989 [] S Roy n J Chophyy Dss-sl pon my l o py xnon hml hos n h ppl Sns [] Y Xo n L Chn olng n nlyss of po-py mol h ss n h py hml osn [] Jnh Zhng n G n Sly of h-sps symoss mol h lys onln Dynms [] X Chngjn Tng Xnh Lo oxn n H Xof fon nlyss n ly Lok Vol po py mol h o lys onln Dynms 9-8

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