Stability Analysis of Wangersky-Cunningham Model with Constant Effort of Harvesting

Transcription

1 Vol. No. 4-5 Jana 6 abl Analss of Wangsk-Cnnngham Mol wh Consan Effo of Hasng amsn Toaha Absak In hs pap w ons anoh pao-p mol wh m la whh s all h Wangsk-Cnnngham mol. In hs mol h a of hang of h pao poplaon pns on h nmbs of p an pao psn a som pos m. Th mol s hn mpo b onsng a onsan ffo of hasng no h gowh a of h p an pao poplaons. Th mho s n hs analss s lnazaon h mol aon h qlbm pon an hn nspng h gnals o mn h sabl. W fon ha h ss a pos qlbm pon fo h mol wh an who hasng. Th m la an n nsabl an Hopf bfaon an also o. om plos of ajos of h p an pao poplaons a also gn. aa n: Wangsk-Cnnngham mol m la onsan ffo hasng Hopf bfaon.. Inoon Th nams laonshp bwn paos an h p has long bn an wll onn o b on of h omnan hms n boh olog an mahmaal olog o s nsal sn an mpoan Bman 99. Ths poblm ma appa smpl mahmaall a fs sgh n fa ofn hallngng an ompla. A ah haas bhao of h pao-p nams s h ofn obs osllao phnomnon of h poplaon szs. om pao-p mols whh a ons n Ba an ang 998 po h sam kn of mhansms of pong osllao solons. Anoh ommon mhansm of osllao solon s o no m las n h mols whh s a mo als appoah o h nsanng of h pao-p nams. A smpl an naal wa o o osllaon s o nopoa a sngl s la no pao qaons Ba an ang 998. Loka-Vola p pao-p mol wh Mhals-Mnn p fnonal spons an hb p of pao-p mol as wll known as Hollng-Tann pao-p mol ha bn lop b nopoang h m la n h pao qaon. L an Wang 4 ha ons a non aonomos pao-p ffson ssm wh Hollng III fnonal spons an a onnos m la an h sl show ha h ssm s pssn n an ffson a ff an h pos po solon s globall asmpoall sabl. L an ang ha s h sffn onons fo h sn of h pos po solons n po la Gas-p pao-p ssms an h sls na ha whn boh sasonal an m la a psn h sasonal s ofn gnang Jsan Mamaka Faklas Mamaka an Ilm ngahan Alam Unsas Hasann mal: samsn@ahoo.om

2 amsn Toaha 4 fo fo h ofn obs flaon n poplaon nss nlng h nhnl osllao pao-p nams. A mol wh m la n a pao-p ssm has bn ons n an Ms an h sls show ha h ssm s pmann n som appopa onons an sffn onons a sablsh fo h global asmpo sabl of h pos qlbm pon of h ssm. Gol an ang 4 ha fomla a gnal an obs pao-p mol wh sag s wh onsan maaon m la an pfom a ssma mahmaal an ompaonal s. Th sls na ha f h jnl ah a s nonzo hn fo small an lag als of maaon m la h poplaon nams aks h smpl fom of a globall aa sa sa. In a pao-p mol a 3 has ons an sss h sl hasng of fshs b nopoang a m la n h hasng m an h sls show ha h m la an as a sabl qlbm pon o bom nsabl an n a swhng of sabls. In hs pap w ons anoh pao-p mol wh m la whh s all h Wangsk-Cnnngham mol. In hs mol h a of hang of h pao poplaon pns on h nmbs of p an pao psn a som pos m. W n ons h onsan ffo of hasng no h mol. In h analss h sffn onon fo h sn of h pos qlbm pon an hasng ll a ons an also h m la magn s s.. Wangsk-Cnnngham Mol mol In h s of pao p mol Vola n ang 993 has nsga h a b F F wh an a h sz of p an pao spl an all paams an fnons a nonnga. Fo smla naons Wangsk an Cnnngham n ang 993 also s h mol fo pao p mol wh m la. Th paam s h a of nas of h p poplaon s h ah a of h pao poplaon s h offn of ff of paon on s h offn of ff paon on s a pos m la an psns h ang apa of h p poplaon whn h s no naon bwn p an pao. Th m la s a onsan bas on h assmpon ha h hang a of h pao poplaon pns on h nmbs of p an of pao psn a a an pos m. Whn w p n mol h mol s o a pao p mol

3 amsn Toaha 4 whh has bn analz n h pos hap s also Toaha al. 7. Ths mol has an qlbm pon whh s globall asmpoall sabl whn. Man an an ha analz h Wangsk-Cnnngham mol wh h p s has wh onsan qoa. Th ons mol s H wh H s a pos onsan o wll known as onsan qoa of hasng. Th pos qlbm pon of h mol s gn b H an. Th show ha h m la an n nsabl an osllaons a Hopf bfaons an swhng of sabl os. Th analss s also al whn H. 3. Wangsk-Cnnngham Mol wh Consan Effo of Hasng W ons h pao an p poplaons n mol wh h wo poplaons a sbj o onsan ffo of hasng. Th mol wh hasng s as follows Fo analss w assm ha E E. E E E. H w assm ha h wo poplaons a has b applng h sam ffo. Fo ampl n fshng h fshs s h sam n an boa. Th pos onsan E s h ffo of hasng. Thn h mol boms Th mol an b wn as E E. 3

4 43 amsn Toaha 4 wh E E an E. W assm ha E. Ths assmpon s ma o gaan h nns gowh of p poplaon s ga han h ffo of hasng so ha h poplaon an nas n sz. Th qlbm pons of mol 4 a E E an E. In o o ha a pos qlbm pon w assm ha ha s E. Un hs assmpon h qlbm pon E os n h fs qaan. Fo mol 4 s o mol. Consqnl h qlbm pon E of mol 4 s also globall asmpoall sabl whn E whn. In o o nsan h loall asmpoall sabl of h qlbm pon E n h mol wh m la w analz h assoa lnazaon mol wh pbaon. L an. Thn sbs no mol 4 o g. Af smplfng an nglng h po ms w ha h lnaz mol. Analzng h loal sabl of h qlbm pon E n h mol wh m la s qaln o analzng h sabl of h zo qlbm pon n h lnaz mol. Fom h lnaz mol w ha h haas qaon f 5

5 amsn Toaha 44 wh. Fo h haas qaon boms an 6. 4 whh has h oos 7 W ha an. Un h assmpons E an hn an a boh pos nmbs. Fh h gnals of h haas qaon 6 ha nga al pas. Thom. L an E. Thn h qlbm pon asmpoall sabl. E of mol 4 s oof. Fom h onon E w g h qlbm pon E n h pos qaan. n hn h gnals of h haas qaon 6 ha nga al pas. Thn w onl ha h qlbm pon E s asmpoall sabl. ha Now fo f s a oo of h haas qaon 5 hn w os sn os sn. paang h al an magna pas w ha Eqalnl qang boh ss gs sn os os sn. sn os os sn. 4 sn sn sn os sn os os. os 8 Ang boh qaons an gopng b pows of w oban h followng foh g polnomal

6 amsn Toaha 45 fom whh w oban Fom qaon follows ha f an hn h qaon 9 os no ha an al solons. To fn h nssa an sffn onons fo h nonsn of m la n nsabl w s h followng hom. Thom. a 3 A s of nssa an sffn onons fo an qlbm pon o b asmpoall sabl fo all s. Th al pas of all h oos of a nga.. Fo all al an wh. Thom 3. If onon an Thom 4.7 a sasf hn h qlbm pon asmpoall sabl fo all. E s loall oof. Fom h assmpons E an also w ha ha an a boh pos. Thn h gnals of h haas qaon 5 ha nga al pas. Fom onon an Thom w onl ha h qlbm pon E s loall asmpoall sabl fo all. L s ons mol 3 wh paams an. 3 E Fo E. Th qlbm pon of h mol s h Jaoban ma of h mol assoa wh h qlbm pon E has gnals Ths mans ha h qlbm pon of h mol who m la s sabl. W an f ha h onons a sasf ha s. 764 an In hs as h s no m la ha ns nsabl. om ajos of wh aos m las a gn n Fgs an.

7 amsn Toaha 46 Fg. Tajo of wh an.. Fg. Tajo of wh an.. Fg wh m la. spall ns o h qlbm pon. Fgs wh m la. shows ha h qlbm pon s sll sabl n f w nas h al of m la. Th ajo of sll ns o h qlbm pon as m gos on b h s no mo a an pan of h bhao of h ajo. Fom w an f ha h s a nq pos solon of f W an now fn h al of shows ha h ajo of. k b sbsng sn os os sn. no qaon 8 an solng fo. W g Mlpl h fs qaon wh an h son qaon wh an sba o g. os Agan mlpl h fs qaon wh an h son qaon wh an a hm o g 3. sn

8 47 amsn Toaha Thn w g k k 3 wh an an wh k. Agan f 4 an 4 hol hn h a wo pos solons of. bsng no qaon 8 an solng fo w oban k k 5 wh an an wh k. Dffnang qaon 5 wh sp o w oban hfo Fom qaon 5 w ha hn w oban Ths sgn sgn sgn sgn

9 amsn Toaha 48 sgn Fom qaon 9 w know ha hn w oban sgn 4. sgn Thom 4. If onon s sasf hn h qlbm pon sgn. 6 E s asmpoall sabl fo an nsabl fo. Fh as nass hogh h qlbm pon bfas no small ampl po solons wh as k. k oof. Fo an n h assmpons E an hn an a boh pos. I mans ha h qlbm pon E s asmpoall sabl. Hn followng h poof of Thom n ang 993 pag 66 h qlbm pon E mans sabl fo. W now ha o show ha. Ths wll sgnf ha h ss a las on gnal wh pos al pa fo. Moo h onons of Hopf bfaon a hn sasf lng h q po solon s Hal 977 Bo 983 an Nafh an Balahanan 995. Fom 6 an follows ha hn w g sgn. sgn sgn 4 Thfo h anssal onon s sasf an hn h Hopf bfaon os a. Ths ompls h poof. Now w ons agan mol 3 wh paams an. 3 E E. Th qlbm pon of h mol s Fo h Jaoban ma of h mol assoa wh h qlbm pon E E has

10 amsn Toaha 49 gnals Ths mans ha h qlbm pon of h mol who m la s sabl. I s as o f ha h onon s sasf ha s Fh w ha.898 an.569. om ajos of wh aos m las a gn n Fgs 3 4 an 5. a b Fg 3. a Tajo of wh 3. an. 4 ; b Tajo of ao wh an. 4. a b Fg 4. a Tajo of wh 3. an. 8 ; b Tajo of ao wh an. 8.

11 amsn Toaha 5 Fg 5. Tajo of wh an Fgs 3a an 3b wh m la. 4 show ha h qlbm pon s sabl. Fgs 4a an 4b wh m la. 8 show ha h qlbm pon s nsabl. Th al al of m la o la magn s.569. Whn. 569 h qlbm pon s asmpoall sabl; whn.569 h qlbm pon loss s sabl; an whn.569 h qlbm pon boms nsabl an h s a bfang po solon s fg 5. Thom 5. L k b fn n qaon 5. If onons 4 a sasf hn h ss a pos ng m sh ha h a m swhs fom sabl o nsabl an o sabl. In oh wos whn h qlbm pon E s sabl an [ m whn h qlbm pon m m Thfo h a bfaons a h qlbm pon m E whn E s nsabl. k. k oof. n onons 4 a sasf hn o po h hom w n onl o f h anssal onons s Cshng 977 k an. Fom 6 an follows ha sgn sgn hfo k sgn 4 k.

12 amsn Toaha 5 Agan sgn hfo sgn 4 sgn k. Hn h anssal onons a sasf. Ths ompls h poof. Af smplfng w know ha whh w ha an fom. Fh w ha an hn. Consqnl s mpossbl o ha wo pos solons of an w f ha h s no an swh fom sabl o nsabl an o sabl fo mol Conlsons Th mol who m la has on pos qlbm pon an sabl. Th sam qlbm pon s fon fo h mol wh m la. Th sabl of h pos qlbm pon fo h mol wh m la s f fom h ff of m la whn onon s sasf. Whn onon hol h ss a la magn fo whh a Hopf bfaon o an whn h al of m la s ga han h la magn hn h qlbm pon boms nsabl. Whl whn onon 4 s sasf h m la an n nsabl an som Hopf bfaons o. W fon fn sabl nals fo h qlbm pon. Fo h mol wh m la an onsan ffo of hasng h sll ss a pos qlbm pon. Th pos qlbm pon ma o ma no b sabl. I s pn on h als of ffo of hasng an m la. Th m la an aff h sabl of h qlbm pon. Th s som onons fo h ffo an m la ha ass h sabl of h pos qlbm pon. fns [] Bman A.A. 99 Th ogns an olon of h pao-p ho Eolog 73: [] Bo H.W. 983 as po n loal bfaon ho In Bfaon Tho Mhans an hss. C.. B A. Aagnol an A. Lhnowz pp Doh: D. l blshng Compan. [3] Cshng J.M. 977 Ingoffnal qaons an la mols n poplaon nams. Hlbg: png-vlag.

13 amsn Toaha 5 [4] Gol.A. an ang Y. 4 A sag s pao-p mol an s pnn on maaon la an ah a J. Mah. Bol 49:88-. [5] Hal J Tho of fnonal ffnal qaons. Hlbg: png-vlag. [6] a T.. 3 l hasng n a p-pao fsh wh m la Mahmaal an Comp Mollng 38: [7] ang Y. 993 Dla ffnal qaons wh applaon n poplaon nams Nw Yok: Aam ss. [8] ang Y. an Ba E. 998 Global qala analss of a ao-pnn paop ssm J. Mah. Bol. 36: [9] L Y. an ang Y. o solons n po la Gas-p paop ssms ongs of Dnamal sms an Applaons 3: [] L Z. an Wang W. 4 ssn an po solon of a nonaonomos pao-p ffson ssm wh Hollng III fnonal spons an onnos la Ds an Connos Dnamal sm s B 43: [] Man A. an an. ao-p mols wh la an p hasng Jonal of Mah. Bol. 43: [] Nafh A.H. an Balahanan B. 995 Appl nonlna nams Nw Yok: John Wl & ons. [3] L. an Ms T. ao-p nams wh la whn p spsng n n-pahs nonmn Onln Jonal hp:// nagoa.a.jp/ppn/ -.pf [Ass on Jl 4] [4] Toaha. Hassan M.A. Ismal F. an Jn L.W. 7 abl analss of ppao poplaon mol wh hasng sbm o J. IIAM