Periodic Solutions of Periodic Delay Lotka Volterra Equations and Systems

Transcription

1 Joural of ahacal Aalyss ad Applcaos 255, 2628 Ž 2 do:6aa27248, avalabl ol a hp:wwwdalbraryco o Prodc Soluos of Prodc Dlay LokaVolrra Equaos ad Syss Yogku L Dpar of ahacs, Yua Ursy, Kug, Yua 659, Popl s Rpublc of Cha ad Yag Kuag Dpar of ahacs, Arzoa Sa Ursy, Tp, Arzoa Subd by al L Sh Rcvd Ocobr 2, 999 By usg h couao hor of cocdc dgr hory, suffc ad ralsc codos ar obad for h xsc of posv prodc soluos for boh prodc LokaVolrra quaos ad syss wh dsrbud or sadpd dlays Our rsuls subsaally xd ad prov xsg rsuls 2 Acadc Prss Ky Words: LokaVolrra quao; posv prodc soluo; dsrbud dlay; sa-dpd dlay; Frdhol appg INTRODUCTION A hallark of obsrvd populao dss h fld s hr oscllaory bhavor A a purpos of odlg populao racos s o udrsad wha causs such flucuaos Idd, h vry frs LokaVolrra sys s h rsul of such a ffor Thr ar hr ypcal approachs for odlg such bhavor: Ž roducg or spcs o h odl, ad cosdr h hghr dsoal syss Žlk prdaorpry racos ; Ž assug ha h pr capa growh fuco s dpd ad prodc ; ad Ž akg o accou Ths work was copld whl hs auhor was vsg a h Dpar of ahacs, Arzoa Sa Uvrsy X $35 Copyrgh 2 by Acadc Prss All rghs of rproduco ay for rsrvd 26

2 PERIODIC DELAY LOTKAVOLTERRA EQUATIONS 26 h dlay ffc h populao dyacs, 4 Grally spakg, approach Ž s rahr arfcal, whl Ž ad Ž phasz oly o aspc of raly Alhough all of h ar good chass of grag prodc soluos Žad hrfor offr so xplaaos for h of obsrvd oscllaory bhavor populao dss, dos o gv us ay sgh o whch s h ral grag or doag forc bhd h oscllaory bhavor f oly o such chas s cosdrd Naurally, or ralsc ad rsg odls of sgl or ulpl spcs growh should ak o accou boh h sasoaly of h chagg vro ad h ffcs of dlays Exsg rsuls o h xsc of prodc soluos prodc syss Ž populao odls, parcular of fall o o of hs hr cagors: Ž h rsuls of h applcaos of h coraco prcpl or h flucuao prcpl, whch sablsh boh h xsc ad aracvy of h prodc soluos prodc quaos wh dlay 7, p 8 ; Ž 2 h obsrvao ha h prodc soluo xss wh hr s o dlay ad hs prodc soluo ras so wh h dlay s a ulpl of h prod of h prodc quao 3, 3 ; Ž 3 h rsuls of h applcao of or s asypoc fxd-po hor, 2 Whl hs hods of allow h vsgaor o addrss h sably ssus of h prodc soluos, h codos for xsc ar of ucssary, urous, dous, srg, ad dffcul o sasfy Spcfcally, all of h abov hods ar ll sud o probls wh sa-dpd dlay quaos By ployg h powrful ad ffcv cocdc dgr hod, w foud ha h xsc of prodc soluos prodc odls wh or whou sa-dpd dlay rqurs oly a s of aural ad asly vrfabl codos Ths codos ar radly sasfd ay ralsc populao odls Such a approach was frs adopd by L 8 for a spcfc prdaorpry odl, whr g, Ž x r Ž xk Ž, pž x x Ths srogly suggss ha sasoal ffcs o populao odls dd of lad o sychroous soluos I addo, w ay coclud ha wh boh sasoaly ad dlay ar prs ad dsrv cosdrao, h sasoaly s of h grag forc for h of obsrvd oscllaory bhavor populao dss ovad by h laboraory work of h group ld by albach 4, ad Frda ad Wu sudd h xsc of a prodc soluo of a sgl spcs populao growh odl, ad hy obad TEORE A Assu ha h dlay fucoal quao K KŽ Ž

3 262 LI AND KUANG Ž has a pos, -prodc, couously dffrabl soluo K Th h quao x x x xž Ž, Ž 2 has a pos -prodc soluo, whr Ž, Ž, Ž, ad Ž ar couously dffrabl -prodc fucos I s asy o s ha codo Ž of Thor A s dffcul o vrfy Our a purpos hs papr s o oba vrfabl suffc codos of h xsc of posv prodc soluos of so prodc dlayd populao odls by usg h couao hor of cocdc dgr hory Spcfcally, w shall cosdr h followg prodc LokaVolrra quaos ad syss W wll cosdr frs h followg quaos wh dsrbud dlays: x x r a x b xž d Ž, Ž c xž d Ž,, Ž 3 Ž x x r a x b xž d c xž d Ž Ž 4 Nx, w wll cosdr h prodc LokaVolrra quao wh sa-dpd dlays, Ž x x r a x b x, x Ž c x Ž, xž Ž 5 r, Eqs Ž 3 Ž 5, r, a, b, c CŽ R, R Ž,,;,, ar ogav -prodc fucos I addo, w assu ha r, a, ad CŽ R, R Ž,,;,, ar ogav -prodc fucos wh rspc o hr frs argu Ž, ad Ž, Ž,,;,, ar couous -prod wh rspc o hr frs argus ad odcrasg wh rspc o hr scod

4 PERIODIC DELAY LOTKAVOLTERRA EQUATIONS 263 argus ad Ž,,;,, ar posv cosas, ad Ž,,;,, ar odcrasg fucos W wll also sudy h prodc LokaVolrra cooprav sys wh sa-dpd dlays, x x c a x a x Ž b x, x Ž,, x Ž, Ž 6 whr, 2,,, ad h prodc LokaVolrra cooprav sys wh dsrbud dlays, x x c a x a x b x Ž d Ž Ž 7, 2,, I sys Ž 6 ad sys Ž 7, c, a, b Ž,, 2,, Ž ar ogav -prodc fucos I sys 6, C R, R Ž, 2,, ar -prodc wh rspc o hr frs argus I sys Ž 7, Ž, 2,, ar posv cosas, ad Ž, 2,, ar odcrasg fucos sasfyg Ž Ž,, 2,, Our rsuls graly prov ad gralz h corrspodg rsuls of Frda ad Wu ad Tag ad Kuag 2 For h cologcal rlvac of Eqs Ž 3 Ž 5, sys Ž 6 ad sys Ž 7, w rfr o Kuag 7 ad h rfrcs cd hr For so ohr rsuls cocrg h sablsh of h xsc of prodc soluos of fucoal dffral quaos va cocdc dgr hory, w rfr h radr o al ad awh 6 ad awh 9 For h work cocrg h xsc of prodc soluos of sa-dpd dlay quaos, w rfr h radr o Sh ad Kuag ad al ad Lul 5 To ak us of h couao hor of cocdc dgr hory, w d o roduc a fw oaos L X, Y b ral Baach spacs, l L: Do L X Y b a Frdhol appg of dx zro, ad l P: X X, Q: Y Y b couous

5 264 LI AND KUANG procors such ha I P Kr L, Kr Q I L ad X Kr L Kr P, Y I L I Q Do by LP h rsrco of L o Do L Kr P, do by K P:ILKr P Do L h vrs of L P, ad do by J: IQ Kr L a soorphs of I Q oo Kr L For covc, w also c blow h couao hor 2, p 4 LEA A L X b a op boudd s ad l N: X Yba couous opraor whch s L-copac o Ž, QN: Y ad K Ž I Q N: Y ar copac Assu P Ž for ach Ž,, x Do L, Lx Nx; Ž for ach x Kr L, QNx, ad dg JQN, Kr L,4 Th Lx Nx has a las o soluo Do L I hs papr, w shall us h oao u u d ad u ax už,, whr u s a couous -prodc fuco 2 EXISTENCE OF POSITIVE PERIODIC SOLUTIONS IN EQUATIONS Th obcv of hs sco s o drv suffc codos for h xsc of posv prodc soluos Eqs Ž 3 Ž 5 To us h couao hor of cocdc dgr hory, w ak X Y y Ž CŽ R, R : y y Ž4 Wh h or, X ad Y ar Baach spacs S L : Do L X, Ly y, whr Do L y Ž C Ž R, R 4 Df wo procors P ad Q as Py Qy y d, y X Clarly, Kr L R, IL y X : y Ž d 4 s closd X ad d Kr L cod I L c, L s a Frdhol appg of dx Furhror, hrough a asy copuao, w fd ha h vrs K P of LP has h for K P :ILDo L Kr P, u KP Ž y yž s ds yž s ds du,, W ar ow a poso o sa ad prov our frs rsul

6 PERIODIC DELAY LOTKAVOLTERRA EQUATIONS 265 TEORE 2 Assu ha hr xs pos cosas C ad C such 2 ha d Ž, C ad d Ž, C 2 Ž Ž I addo, suppos ha a d Ž, for, Ž ad a b d Ž, d Ž c d Ž, d Ž Th Eq Ž 3 has a las o pos -prodc soluo Proof L y Ž l x; Ž h Eq Ž 3 bcos yž y Ž y r a b d Ž, y c d, 2 Ž I s asy o s ha f Eq Ž 2 has a -prodc soluo y* Ž, h x* Ž xpžy* Ž s a posv -prodc soluo of Eq Ž 3 So, o copl h proof, suffcs o show ha Eq Ž 2 has a -prodc soluo W df N: X X as yž y Ž Ny r a b d Ž, y c dž Ž

7 266 LI AND KUANG Noc ha QN: X X aks h for yž y Ž QNŽ y r a d b d Ž, y c dž d Ž By so copuao, w ca show ha K Ž I Q P N: X X aks h for KP Ž I Q NŽ y s yž s yž s s Ž s rž s až s b Ž s d Ž s, ds s yž s cž s dž s ds sž s r a yž y b dž, d d Ž y c dž d d Ž ž /½ yž r a d 2 y b dž, Ž y Ž 5 c d d Th grao for of h rs of boh QN ad K Ž I Q P N ply ha hy ar couously dffrabl wh rspc o ad ha hy ap boudd couous fucos o boudd couous fucos By h AscolArzla hor, w s ha QNŽ, K Ž I Q N P ar rlavly copac for ay op boudd s X Thrfor, N s L-copac o for ay op boudd s X Corrspodg o h

8 PERIODIC DELAY LOTKAVOLTERRA EQUATIONS 267 opraor quao Ly N,,, w hav y yž y Ž y r a b d Ž, y c d, 22 Ž Suppos ha y Ž X s a soluo of Eq Ž 22 for so Ž, Choos, such ha y ax y, Th s clar ha y Fro hs ad Eq Ž 22, w oba y yž Ž rž až b Ž d Ž, yž Ž c Ž d Ž,, whch pls ha Thus y Ž rž až b Ž d Ž, y rž až bž dž, Ž df ½ 5, Ž c, w hav ax r a b d, A I follows fro Eq 22 ad 23 ha yž l A Ž 23 df 2 2 Ž y r aa AC AC A Ž 24

9 268 LI AND KUANG Igrag 22 ovr,, w oba Fro hs, w hav y r d b dž d Ž y c Ž, d Ž yž a d yž y Ž a d c Ž, d r d, whch pls ha hr xs a posv cosa A ad a po, 3 such ha yž A 3 Ž 25 Fro 24 ad 25, w oba By hs ad 23, w oba Do whch y yž y d yž y d A A y ax l A, A 2 A 3 4 A ax l A, A A A, ž / ž / ½ 4 Ž A l r a b d Ž, d 5 Ž c d, d, ad ak y Ž X : y A 4; h s clar ha sasfs codo Ž La A Wh y R, y s a cosa wh y A

10 PERIODIC DELAY LOTKAVOLTERRA EQUATIONS 269 c, y y QNy r a b d Ž, d Ž y c d Ž d Ž Furhror, ak J I: IQ Kr L, x x ad by a sraghforward copuao, w s ha dgjqn, Kr L, Th cocluso ow follows fro La A Ths copls h proof Slar o h proof of Thor 2, o ca prov h followg TEORE 22 Suppos ha a b for, Th Eq Ž 4 has a pos -prodc soluo Proof Th proof s rly slar o h proof of Thor 2 ad s od Rark Fro h proof of Thor 2, o ca s ha Eq Ž 3, v f so of h s ad s or all of h ar, h cocluso of Thor 2 ras ru Th followg s our rsul abou h xsc of a posv prodc soluo of sa-dpd dlay LokaVolrra Eq Ž 5 TEORE 23 Suppos ha a b for, Th Eq Ž 5 has a pos -prodc soluo Proof Slar o h proof of Thor 2, o copl h proof, suffcs o show ha h quao y r yž a b xp y yž, Ž Ž c xp y Ž, yž

11 27 LI AND KUANG has a -prodc soluo To hs d, l Ny r yž a b xp y yž, yž c xp yž Ž,, ad L, P, Q, X ar h sa as hos h proof of Thor 2 Corrspodg o h oprao quao Ly y,,, w hav ½ yž yž y r a b xp yž Ž, yž Ž 5 c xp y Ž, Ž 26 Suppos ha y Ž X s a soluo of Eq Ž 26 for a cra Ž, Choos, such ha y ax y Th y, By hs ad Eq Ž 26, w hav Thrfor, ha s, yž yž Ž r a b xp y, yž c Ž xp yž Ž, yž rž až bž, or yž r a b ½ 5 df ax r a b B,, y l B 27

12 PERIODIC DELAY LOTKAVOLTERRA EQUATIONS 27 Fro Eq 26 ad 27, w oba df 2 y r ab B b B c B Ž 28 Igrag Eq 26 ovr,, w hav c, yž yž ½ Ž 5 a d c xp y, d ½ 5 yž b xp yž Ž, d r d ½ 5 yž yž Ž a d c xp y Ž, d r d, whch pls ha hr xs a posv cosa B ad a po *, 3 such ha yž * B 3 Ž 29 Fro 28 ad 29, w hav y yž * y d Fro hs ad 27 follows ha * 3 2 yž * y d B B 4 y ax l B, B 2 B 3 Th rs of h proof s slar o h proof of Thor 2 ad s hus od Idaly, fro Thor 2 or Thor 22, w hav COROLLARY 2 Suppos ha bž c Ž Th h quao x x a b x c x has a pos -prodc soluo, whr a, b, c, ar couous -prodc fucos

13 272 LI AND KUANG Rark 2 Coparg Thor A wh Corollary 2, w s ha h rqur of Corollary 2 s uch asr o vrfy ad ha w d o assu ha a, b, c, ar dffrabl owvr, h assupo of b Ž c Ž s srogr ha wha s rqurd by Frda ad Wu, sc hs codo fac surs ha Ž dfs a coracg appg o so subs of X ad hus has a fxd po whch corrspods o a posv prodc soluo 3 EXISTENCE OF POSITIVE PERIODIC SOLUTIONS IN SYSTES I hs sco, w sa ad prov our rsul abou h xsc of prodc soluos for syss Ž 6 ad Ž 7 W shall aga us h couao hor of cocdc dgr hory W l X Y Žu Ž, u Ž,,u T CŽ R, R : u Ž 2 Ž 4 Ž T u,, 2,, ad u, u,,u u Ž 2 Wh hs or, X ad Y ar Baach spacs L L : Do L X b T T Ž 2 Ž 2 L u, u,,u u, u,,u, whr Do L Žu Ž, u Ž,,u T C Ž R, R 4 2 W df h wo procors P ad Q as u d u u u P Q, X u u u u d Clarly, Kr L R, I L Ž u, u,,u T X : u Ž 2 d,, 2,, 4 s closd X ad d Kr L cod I L c, L s a Frdhol appg of dx Furhror, hrough a asy copuao, w ca fd ha h vrs of LP has h for K P:ILDo L Kr P, such ha u Ž s ds u d d u K P u už s ds u d d

14 PERIODIC DELAY LOTKAVOLTERRA EQUATIONS 273 TEORE 3 Assu ha h sys of quaos y y c 2 a a b ž /,,2,, 3 Ž T has a uqu soluo y, y,, y R I addo, suppos ha a ax a b ax b,,,,,2,, Th sys Ž 6 has a las o pos -prodc soluo Proof Slar o h proof of Thor 2, o copl h proof, suffcs o show ha h sys of dffral quaos u Ž u Ž u c a a b u Ž Ž, u Ž,,u Ž,,2,, 32 has a -prodc soluo, whr c, a, b h sa as hos sys Ž 6 L N: X X ad u N u ad,,, 2,,, ar už u Ž 2 u Ž Ž, už,,už b c a a u Ž u Ž c a a u Ž Ž, už,,už b

15 274 LI AND KUANG Noc ha QN: X X aks h for už u Ž 2 c a d a d u QN u u Ž u Ž c a d a d u Ž Ž, už,,už b d u Ž Ž, už,,už b d ad ha K Ž I Q N: X X aks h for P už s u Ž s c Ž s a Ž s a Ž s 2 x u Ž s Ž s, už s,,užs b Ž s ds x u Ž s u Ž s c Ž s a Ž s a Ž s u Ž s Ž s, už s,,užs b Ž s ds už u Ž c a a dd 2 u Ž u Ž c a a dd

16 PERIODIC DELAY LOTKAVOLTERRA EQUATIONS 275 u Ž Ž, už,,už b dd u Ž Ž, už,,už b dd u Ž u Ž ž /½c a d a d už u Ž ž /½c a d a d5 2 ž / u Ž Ž, už,,už b d 2 u Ž Ž, už,,už b d 2 ž / W ca s ha N s L-copac o for ay op boudd s X Corrspodg o h opraor quao Lx Nx, Ž,, w hav for, 2,,, u c a u Ž u Ž u Ž Ž, u Ž,,u Ž a b Ž 33 Suppos ha Žu Ž, u Ž,,u T X s a soluo of sys Ž 33 2 for so, Choos,,,, 2,,, such ha u ax u ad u u Ž,,2,,,,

17 276 LI AND KUANG Th, s clar ha u ad u,,2,, Fro hs ad sys 33, w oba ha for, 2,,, už už Ž Ž I Ž c a a b, Ž 34 už Ž, už,,už ad c a u Ž a u Ž b Ž 35 už Ž, už,,už I follows fro 34 ha už už cž 2 až až Whch pls ha or už Ž, už,,už b už už cž 2 až až u Ž b už už cž 2 až až u Ž b

18 PERIODIC DELAY LOTKAVOLTERRA EQUATIONS 277 Tha s, or Thrfor, ž / už 2 a až bž c u Ž c Ž Ž Ž Ž 2 a a b c,,2,, c už l,,2,, 36 I follows fro 35 ha c, c a u Ž Ž Ž,,2,, ž /, ž / 5 c c už l l,,2,, Ž 37 ½ a a By 36 ad 37, w hav c ½ ž / 5 c df u ax l, l,, a,2,, Now, l r s suffcly larg such ha h Ž T Ž uqu soluo y, y,, y of sys 3 sasfs y, y,, y T 2 2 L Ž u, u,,u T X : Ž u, u,,u T 4; h s 2 2

19 278 LI AND KUANG clar ha sasfs codo Ž La A Obsrv ha wh Ž u T, u 2,,u R, u, u 2,,u T bcos a cosa vcor wh u c, u u ž / c 2 a a b u QN u u u c 2 a ž a b/ Furhror, l J I: IQ Kr L, Ž u T,,u x,, x T Bya sraghforward copuao, w fd dgjqn, Kr L, Accordg o La A, h proof s copl By Thor 3, w hav COROLLARY 3 Assu ha h sys of quaos y y c 2 a a b ž /,,2,,, Ž T has a uqu soluo y, y,, y R I addo, suppos ha a ax a b ax b,,,, Th h sys x x c a x a x b x Ž,,2,,,,2,, has a las o pos -prodc soluo, whr c, a, b,,,, 2,,, ar -prodc fucos

20 PERIODIC DELAY LOTKAVOLTERRA EQUATIONS 279 Fally, slar o h proof of Thor 3, w ca prov TEORE 32 Assu ha h sys of quaos y y c 2 a a b ž /,,2,,, Ž T has a uqu soluo y, y,, y R I addo, suppos ha a ax a b ax b,, -,,,2,, Th sys Ž 7 has a las o pos -prodc soluo Rark 3 I sys Ž 7, wh so of h or all of h ar, h cocluso of Thor 32 s sll ru REFERENCES I Frda ad J Wu, Prodc soluos of sgl-spcs odls wh prodc dlay, SIA J ah Aal 23 Ž 992, R E Gas ad J L awh, Cocdc Dgr ad Nolar Dffral Equaos, Sprgr-Vrlag, Brl, K Gopalsay, R S Kulovc, ad G Ladas, Evroal prodcy ad dlays a food-ld populao odl, J ah Aal Appl 47 Ž 99, U albach, Lf abl daa ad populao dyacs of h rofr Brachoous calycflorus pallas as flucd by prodcally oscllag praur, Effcs of Tpraur o Ecohrc Orgass, pp 27228, Sprgr-Vrlag, dlbrg, J K al ad S Lul, Iroduco o Fucoal Dffral Equaos, Sprgr-Vrlag, Nw York, J K al ad J L awh, Cocdc dgr ad prodc soluos of ural quaos, J Dffral Equaos 5 Ž 974, Y Kuag, Dlay Dffral Equaos wh Applcaos Populao Dyacs, Acadc Prss, Boso, Y L, Prodc soluos of a prodc dlay prdaor-pry sys, Proc Ar ah Soc 27 Ž 999, J L awh, Prodc soluos of so vcor rardd fucoal dffral quao, J ah Aal Appl 45 Ž 974, R ay, Sably ad Coplxy odl Ecosyss, Prco Uv Prss, Prco, NJ, 974 L Sh ad Y Kuag, Prodc soluos of dlay dffral quaos of hrshold-yp dlays, Oscllao ad Dyacs Dlay Equaos ŽJ Graf ad J al, Eds, pp 5376, Coporary ahacs, Vol 29, A ah Soc, Provdc, 992

21 28 LI AND KUANG 2 B R Tag ad Y Kuag, Exsc, uquss ad asypoc sably of prodc soluos of prodc fucoal dffral syss, Tohoku ah J 49 Ž 997, B G Zhag ad K Gopalsay, Global aracvy ad oscllaos a prodc dlay-logsc quaos, J ah Aal Appl 5 Ž 99, T Zhao, Y Kuag, ad L Sh, Global xsc of prodc soluos a class of dlayd Gaus-yp prdaor-pry syss, Nolar Aal 28 Ž 997,