A Predator Prey Model with Discrete Time Delay Considering Different Growth Function of Prey

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1 Avancs n Al Mahmacal Boscncs. ISSN Volm, Nmb,. -6 Innaonal Rsach Pblcaon Hos h:// A Pao P Mol wh Dsc Tm Dla Consng Dffn Gowh Fncon of P M.A. Ha Damn of Al Mahmacs Unvs of Wsn Onao, Lonon, Onao, N6A 5B7 Canaa E-mal: mha7@wo.ca, amnl@ahoo.com Absac Th namcs of a ao - nacon mol s s h consng ffn gowh fncons of an nclng a sc m la o mol h m lags bwn h ca of h an s convson o vabl bomass. Lna sabl analss vals ha n h absnc of la fo h monoonc gowh fncon of, h cosnc lbm s a cn, b f h gowh fncon of s logsc, hn h cosnc lbm s locall asmocall sabl f < an os no s f. I s shown ha f, oc solon ass n cas of h monoonc gowh fncon of as Hof bfcaon occs who an conon. In h cas of h logsc gowh of whn, h oc solon s ossbl hogh Hof bfcaon n can conons. wos: Dlas, Bomass, Sabl, Poc Solon, Hof bfcaon. Inocon In h al wol, h bosh s an moan zon fo bologcal acvs ha a manl sonsbl fo h changs n colog an nvonmn an h gowh a of ffn scs manl n on colog, cang caac of nvonmn c. As a consnc gowh a of h cs s an moan ma fo h ao nacon mol. Th co-snc of nacng bologcal scs has bn of ga ns n h as fw cas an has bn s nsvl sng mahmacal mols b sval sachs [,,,, 5, 6]. A ao- mol who la was s b Dbb [6] consng ffn gowh of ao. Followng [6], ffn gowh fncons of olaon a cons h. In man sng ao- ODE mols [6,,, 5], h m la fo h

2 M.A. Ha convson of bomass fom o ao olaon w gno. Th al s ha n h ao aon, h la s ofn cas b h convson of consm bomass n o h ao bomass, ma b n h fom of bo sz gowh o ocon. Fan an Wolkowcz [8] s a ao- mol n h chmosa wh sc m la. Followng [6, 7, 8, ], a ao mol s oos nclng a sc m la o mol h m lags bwn h ca of h an s convson o vabl bomass, consng ffn gowh fncons of. Moov, h m s ncl n ao aon whch accons fo aos hos nac wh a m b bfo gvng ocon o gowh m ns la.. f w assm a consan ah a δ fo hos aos ha svv n gsaon o ha mans h obabl of svvng bwn h m lags fo convng bomass. In olaon namcs, a fnconal sons of h ao o h ns fs o h chang n h ns of aach n m ao as h ns changs. Fo smlc, Hollng I fom.. h, of fnconal sons s cons fo boh cass. Th man os of hs, s s o analz h namcs of h ao nacng olaon mol o ffn gowh fncon, nclng sc m la fo h ca of h an s convson o bomass an h m. Analss a shown fo monoonc gowh fncon of n scon - an logsc gowh fncon of n scon -. Mol fomlaon Th oos mol s g h h Sbjc o h followng nal conons: θ φ θ, θ [,, φ θ φ θ, θ [,, φ H no h ns of olaon, s h ns of h ao olaon, g no h gowh fncon of olaon an h no h fnconal sons of h ao on, an s h ah a of h ao olaon. Assm ha h gowh a of ao ns onl on h olaon. Two gowh fncons fo h olaon a, g,

3 A Pao P Mol wh Dsc Tm Dla g,,, wh s h cang caac of h nvonmn. Scon- Consng h monoonc gowh fncon of an Hollng I, fnconal sons fo ao, h Mol bcoms. Posv of h solon I s moan o show osv fo h ssm. as h sn - ao olaons. Bologcall, osv mls ha h olaon svvs. Fo oof hs, followng Zh an Zo [9], w hav h followng hom:, b an solon o ssm. wh h nal conons. Thn w hav h followng:, fo. φ C an, Thom.. L θ, φ θ [,] R Poof: To ov fo [,, fom h fs aon n., follows ha, Now o ov ha on [,, sos ha h ss sch ha, an fo [,. Thn, [8]. Fom h scon aon of., w hav,

4 M.A. Ha a conacon, hfo,, [. Elba an Sabl Analss Th Mol. has wo lbm ons:, E s a val lbm whch s bologcall mannglss an E,,, s a cosnc lbm whch s bologcall manng fll. W a onl ns h n analzng h bologcall manngfl cosnc lbm. Th lnazaon of. abo an lbm, gvn b. Th assoca chaacsc aon s gvn b Now w fn F. A h lbm on E,,,. bcoms F F. If, hn. bcoms

5 A Pao P Mol wh Dsc Tm Dla 5 Wh ± ± Whn, h a no al oos an wo l magna oos. Thfo, s a cn. Now w can amn whh Hof bfcaon wll occ o no. Cons, G,.5 G G ± ± ± G G / ± ± R ± Th ansvsal conon os no sasf. Thfo, f Hof bfcaon os no hol. In cas of osv la,.. h chaacsc aon fo h lnalz aon aon h on E, s gvn b P Q.6 Wh P Q If, L, b a l magna oo of.6. Now sbsng n E..6 F cos sn cos sn

6 6 M.A. Ha sn cos cos sn Now saang h al an magna as w oban h ssm of anscnnal aons R sn cos.7 cos sn S.8 Sang an ang.7 an.8 w g, B, as,,., as. Thfo w hav a osv sch ha aon.6 has l magna oos. Elmnang sn fom.7 an.8, w g cos Thn cosonng o s gvn b accos Hof- bfcaon W wll now show ha R No ha all oos of.6 n connosl on s Bsnbg & cook, 99, an as ncas, a oo of.6 ma n h gh half lan onl b cossng h magna as s. Ba & ang. Ths as ncass, oos of.6 ma coss h magna as onl hogh a a of non zo l magna oos. To s f h s an sabl swch as cosss, w ak h

7 A Pao P Mol wh Dsc Tm Dla 7 hl of som sls b Cook an Van n Dssch n Thom of []. W fs look fo l magna oos of, of.6. Eaon.6 mls Q P an hs mns a s of ossbl vals of. O am s o mn h con of moon of as s va. Tha s, w mn R R sgn sgn Θ Now, ffnang.6 wh sc o, w g [ ] Thfo R sgn Θ R sgn R sgn 6 R sgn 6 sgn sgn

8 8 M.A. Ha sgn sgn W hav R, Thfo, h ansvsal conon hols an hnc Hof-bfcaon occs a,.so as ncass.., a oc solon wll occ whch s h cas of Hof-bfcaon. Hnc f, h s a a of l magna oos an s sn cn. Whn ncass o..,, h s anoh a of l magna oos. Scon- Now w cons h logsc gowh fncon of h P an Hollng I, fnconal sons fo ao on olaon. Th mol bcoms. Posv of solons Posv of h ssm. can b asl ov as lk as w ov fo ssm.. Elba an Sabl Analss Th Mol. has wo lbm ons:, E s a val lbm whch s bologcall mannglss an δ E,, s a cosnc lbm whch wol b bologcall manngfl ff δ.. < δ ln. Fo snc, hs lbm on w n o b assm

9 A Pao P Mol wh Dsc Tm Dla 9. W a onl ns h n analzng h bologcall manngfl cosnc lbm. Th lnazaon of. abo an lbm, gvn b. Th assoca chaacsc aon s gvn b Now w fn F. A h lbm on δ E,,,. bcoms F F. If, hn. bcoms

10 M.A. Ha W cons ± an H h al a of s ngav, hfo h sabl ns on anoh gn val. Thom.: If, hn E s locall asmocall sabl f < an E os no ss f. In cas of osv la.., h chaacsc aon fo h lnalz aon aon h on E s gvn b P Q.5 Wh H P Q <

11 A Pao P Mol wh Dsc Tm Dla If, L, b a l magna oo of.5. Now sbsng n E..5 F sn cos sn cos cos cos sn sn sn cos Now saang h al an magna as w oban h ssm of anscnnal aons sn cos.6 cos sn.7 Sang an ang.6 an.7 w g,

12 M.A. Ha ± Snc < < < δ.8 Ths sson.8 s alwas as all h aams h a osv; hfo w hav a osv sch ha aon.5 has a l magna oo. W can also fn h val of cosonng o n a smla fashon ha w hav ala scss n scon. Hof- bfcaon W wll now show ha R Ths wll sgnf ha h ss a las on gnval wh osv al a fo

13 A Pao P Mol wh Dsc Tm Dla. Also, h conons fo Hof bfcaon [] a hn sasf lng h oc solon. To s f h s an sabl swch as cosss, w ak h hl of som sls b Cook an Van n Dssch n Thom of []. W fs look fo l magna oos of of.5. Eaon.5 mls Q P an hs mns a s of ossbl vals of. O am s o mn h con of moon of as s va. Tha s, w mn R R sgn sgn Θ Now, ffnang.5 wh sc o, w g [ ] Thfo R sgn Θ R sgn R sgn R sgn sgn

14 M.A. Ha 6 sgn 6 sgn Thfo, h ansvsal conon wll hol an hnc Hof-bfcaon wll occ a,... R,, f an onl f.9 an. Rsl an scsson I s boal wll known ha as hso as wll as cn conons can nflnc olaon namcs an sch nacons has mova h nocon of m las n olaon gowh mols. In mos of h naal ssms, olaon of on scs os no son nsananosl o changs n h nvonmn o h nacons wh oh scs of olaons whn h commn. I s blv ha h m las hav a sablzng ffc n h mols of olaon namcs an ofn m las a sonsbl fo h olaon oscllaons n consan nvonmn. Dsc m la has abl o al h namcal bhavo of a mol ssm sgnfcanl. In hs a, a mahmacal mol has bn oos an analz o s h namcs of a ao- ssm o h m lags fo h convson of bomass an consng ffn gowh fncons of. Th mol has bn analz n wo cass: fs whn gowh fncon of olaon s monoonc an scon whn gowh fncon of olaon s logsc. In hs

15 A Pao P Mol wh Dsc Tm Dla 5 a, w hav also ma an am o nsan h ffc of gsaon la on namcal bhavo of a ao ssm. Gsaon la s h m nval bwn h momns whn an nval s kll an whn h cosonng bomass s a o h ao olaon. As h gowh a of ao scs soll ns on h amon of bomass a n ao s olaon ns o h kllng, h snc of gsaon la n ao s gowh affc h abnanc of aos, as h a som ossbls of ao s ah ng hs gsaon o bfo gong o ocon o gowh. Lna sabl analss vals h fac ha fo h monoonc gowh a of, n h absnc of la h cosnc lbm s a cn. B fo h logsc gowh fncon of, s locall asmocall sabl f < an os no s f. Bologcall mls ha fo mananng cosnc bwn h ao- nacng olaons, balanc gowh a of an h cang caac of an nvonmn s a ccal ma. Oscllaon n olaon ns s naal an commonl obsv n mos of h ao bas cossms. W obsv ha fo h monoonc gowh a of, n absnc of la, Hof bfcaon s no ossbl b n cas of osv la Hof bfcaon s ossbl who an conon an h s a oc solon whch s h cas of Hof bfcaon. Bologcall mls ha gsaon la s ccal fo a ao- nacng ssm. In cas of h logsc gowh of, Hof bfcaon s ossbl n h conons an. Rfncs [] Albch, F., an Gazk, H., Haa, A., an Wa, N., 97, Th namcs of wo nacng olaons, J. Mah. Anal. Al. 6, [] Cshng, J.M.,98, Two scs comon n a oc nvonmn, J. Mah. Bol.,. 85. [] Db, B., Das, B., an Hssan, J.,, A ao- nacon mol wh slf an coss ffson, Ecol. Mol.,, [] Fman, H. I., an Walman, P., 98, Pssnc n mols of h nacng ao- olaons, Mah. Bosc., 68,.. [5] Goh, B. S., 976, Global sabl n wo scs nacons, Mah. Bosc.,,. 8. [6] Db. A., 7, P-Pao Mol wh a Rsv Aa, Nonlna Analss: Mollng an Conol,, [7] Banoaha, M., an Banj, S., 6, A sag-sc ao mol wh sc m la, Al Mahmacs an Comaon 8,

16 6 M.A. Ha [8] Fan, G., an Wolkowcz, G.S..,, A Pao-P Mol n h Chmosa wh Tm Dla, Innaonal Jonal of Dffnal Eaons, Acl ID 87969, ags,o:.55// [9] Zh, H.,an Zo, X.,8, Imac of las n cll nfcon an vs ocon on HIV- namcs, J. Mahmacal Mcn & Bolog 5, 99-. [] Cook,., an van n Dssch, P., 986, On zos of som anscnnal aons, Fnkcalaj Ekvacoj 9, [] Hal, J.., 997, Tho of Fnconal Dffnal Eaons, Sng, Nw Yok. [] Collngs, J. B., 995, Bfcaon an sabl analss of a ma nn m ao- nacon mol ncooang a fg, Bll. Mah. Bol., 57, [] Thm, H. R., an Zhao, X.Q.,, A non-local la an ffsv ao- mol, Nonlna Analss: Ral Wol Alcaons, 5-6. [] Hs, S.B., 978, on global sabl of ao- ssms, Mah. Bosc., 9,., [5] Hason, G.W., 979, Global sabl of ao- nacons, J. Mah. Bol., 8,

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