Non-Linear Effort dynamics for Harvesting in a Predator- Prey System
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1 ounl of ul Siene Reeh ISS e ISS 5-9 Online Vol.5 o on-line ffo ni fo Hveing in eo- e Se Reenu Rni Suni Gh Deen of hei Inin Iniue of Tehnolog Rooee Rooee Uhn Ini -il of he oeoning uho: eenuii@gil.o Ab In hi e non-line hveing of e i oniee in e-eo e. The eo i oniee o be of oifie Lelie- Gowe e. The effo i en ni vible. The e e of he e e eeine n he nil behvio of he e fo i ll e e i iue une ein oniion. ee oniion fo globl bili of he e i nlze he oiive ineio euilibiu oin. ueil iulion e ie ou o eloe he ni of he e fo he uible hoie of ee. Kewo: oifie Lelie-Gowe eion online Hveing bili ueil Siulion.. Inouion The eologil non-line oel of ineing oulion hve been uie eenivel b n uho.in een e he nee fo oe foo/ eoue i gowing lge le whih h eule in ove eloiion of evel biologil eoue. ven hee i globl onen o oe he eoe fo eloiion lge le. Theefoe hee i nee fo uinble eveloen oli in viou hee of hun ivi o oe eoe. In iul bio-eonoi oeling i onene wih ienifi ngeen of he eloiion of enewble eoue lie fiheie n foe. Hene hveing of eoe h been of inee o eonoi n eologi. A hveing oli efe o he ngeen of biologil eoue b eill onolling he eio ineni n e of hveing. The i objeive hee i o iize ouivi wihou eleing o iving he o o einion. In een e n wo on oil ngeen of enewble eoue e one. An eellen inouion o oil ngeen of enewble eoue i given b Cl [3] n lo oliie ele o bionoi eloiion of enewble eoue e iue b hi. Hveing h ong i on he ni of biologil eoue. The evei of he i een on he nue of he ileenion of hveing eg. Bill hee e of hveing egie e eoe in lieue i onn hveing ii ooionl hveing h n iii nonline hveing H / Holling e-ii. n eehe hve nlze heil oel uing non-line hveing b onieing iffeen e of gowh eening uon he eie n hei ineion [5 6 ] e. Zhu n Ln [6] uie Lelie- Gowe oel wih onn hveing in e. Though hee e nueou wo on eo- e e inooing he hveing of he eie. Lelie Gowe eo e oel n oifie Lelie Gowe eo-e une he uion of he nonline hveing on e eie h been uie b Gu n Bnejee e l. [5]. Bu in hi e we onie nil eion oel of oifie Lelie Gowe eo- e inooing he non-line hveing on e whee effo i en ni vible. The een e el wih ni eion oel in he e of eeo e fihe e while he oel we u hee i eeill be on oifie veion of he Lelie Gowe hee whee onl he e eie i ubjee o non- line hveing. The in i of hi e i o eine he effe of hveing on uh in of e - eo e wih he effo ni whih n give he be oible benefi hough hveing o he oie while evening einion of he eie.. The heil oel Le n e he eniie of e eo oulion n he hveing effo ie. The Holling e- II funionl eone n non-line hveing i oniee fo logiill gowing e eie n eo i ue o be oifie Lelie- Gowe e. The heil oel fo he ni of he e govene b he following e of iffeenil euion: 8
2 ounl of ul Siene Reeh ISS e ISS 5-9 Online Vol.5 o f. g h > > >. The onn n e he ee uing onl oiive vlue. The ee n eeen inini gowh e of e oulion gowh e fo he eo envionenl ing i en-oune e of eo o e iu e of he euion of eo oulion ie n o e uni n iffne ee eeivel. 3. Unifo Bounene Theoe: All he oluion of he e. whih in he egion R3 e unifol boune. oof: le u onie funion uh h Inouing oiive onn n ewie he bove euion we ge Le u > Then we hve: Solving bove ineuli we ge All he oluion of he e. whih in he egion R3 e onfine in he egion: Thi ove he eul. 4.iene of uilibiu oin Fo he oel. hee ei i non-negive euilibiu oin whih e given below:. i ivil euilibiu oin.. i he il euilibiu oin on -i. 3. / i he il oin on -i in he ben of e n hveing. 4. i boun euilibiu oin on -lne in he bene of hveing effo. The ; 4 4 li } > ; R R. 3
3 ounl of ul Siene Reeh ISS e ISS 5-9 Online Vol.5 o euilibiu level n e he oluion of he following euion: Solving we ge oin ; 3. ei fo >. ^ ^ 5. i boun euilibiu oin on - lne in he bene of hveing effo. Thee euilibiu 4 level e he oluion of he following euion: Solving we ge he vlue : L n ; whee L L ^ ^ 4 L > L i oiive if n L 6. i he uniue ineio oin of he following euion: 5 ; L L n L e oiive fo n i oiive fo. L 6. Lol Sbili of Viou uilibiu oin Hee we iu he lol bili oniion fo feible euilibiu oin of he e. be uon he n lineizion ehniue n hen uing he well-nown Rouh- Huwiz ieion o eeine he nue of eigenvlue of he obin i evlue he euilibiu oin[]. The obin i of he e. n oin i given b: f f f f g h n g g h L g h h
4 ounl of ul Siene Reeh ISS e ISS 5-9 Online Vol.5 o.3 5 h g f 6.. Sbili of oin ; The obin i evlue he euilibiu oin i given b : The eigen vlue of evlue euilibiu oin e > n 3. Hene he oigin i le oin wih unble nifol in ieion n ble nifol in well in -ieion. 6.. Sbili of oin ; The obin i evlue he euilibiu oin i given b: The eigen vlue of evlue euilibiu oin e > n. The euilibiu oin i le oin > wih unble nifol in - ieion n ble nifol in -ieion n lo in -ieion if Sbili of oin / ; The obin i evlue he euilibiu oin / i given b: The eigen vlue of evlue euilibiu oin e 3 n. If hen he euilibiu oin / i loll oill ble. Ohewie i i le oin wih n unble nifol in -ieion n ble nifol in well in ieion Sbili of oin : The obin i evlue he euilibiu oin i given b:
5 ounl of ul Siene Reeh ISS e ISS 5-9 Online Vol.5 o The heii euion oie o he i 3 iel he eigen vlue follow: i If n hen he euilibiu oin 3 i loll oill ble. I i le oin wih n unble nifol in -ieion fo Fo hee i hne of bifuion Sbili of euilibiu oin The obin i evlue he euilibiu oin i given b: Coeoning o euilibiu oin 4 one of he eigen vlue i > n he ohe wo n be obine he eigen vlue of he following i: If i T 4 he he oin 4 i le oin. ii If T 4 he we ge i of igin oo whih how h hee i e of hof bifuion Sbili of Ineio uilibiu oin Theoe: The uniue oiive ineio euilibiu i oill loll ble ovie oniion 6. i ifie. The obin i evlue he euilibiu oin i given b: 4 4 > ^ ^ 4 ^ ^ 4 5 5
6 ounl of ul Siene Reeh ISS e ISS 5-9 Online Vol.5 o The heii euion of he bove obin i bou he euilibiu oin 5 i given b: 3 A A A3 whee A 33 A A Uing Rouh- Huwiz Ciei he oniion fo lol bili of oin 5 e A> A> n AA-A3> oe h A> if 6. Alo A> n AA-A3> fo he oniion 6. Thu he ineio euilibiu oin 5 i loll oill ble fo he uffiien oniion Globl Sbili Theoe: The ineio euilibiu oin i globll oill ble. oof: Conie Lonov funion V uh h: V log log log A V i zeo he euilibiu oin i.e.. n oiive fo ll ohe vlue of V Thu ineio euilibiu oin i globll ble fo he bove oniion 7.. 3
7 ounl of ul Siene Reeh ISS e ISS 5-9 Online Vol.5 o ueil Siulion In hi eion nueil iulion e ie ou fo iffeen hoie of ee o inveige he ni behvio of he e. We he nueil inveigion eeing ll he ee fie ee. Le ; A B e 6 5 eo 4 C ffo Tie Tie Tie Figue : Tie eie nli of e oulion eo oulion n ffo fo iffeen vlue of o wih iniil level A B Y X Y X.8 8 Figue : he oi of e oulion eo oulion n ffo fo iffeen iniil level oeoning A.. n B..8 In he figue- ig A B n C give long e behvio of jeoie of e- eo oulion n effo w.. ie. whih how h fo he iniil oniion 4 35 in he ineio fo he iffeen vlue of o n eeing ohe ee fie ll he jeoie onvege o i ineio euilibiu oin whih en h ineio euilibiu i oill loll ble ovie.5. Alo we ee h he vlue of o will inee hen evenull hveing effo will eee n in eule e oulion inee well eo oulion lo. Bu fo >.5 effo will en o zeo. Figue-A eeen he lne jeoie of eie n effo wih he iffeen iniil level whih eeen h he ineio oin i oill globl ble fo..5 fo he eiene of ineio euilibiu oin. The figue-b eeen he lne jeoie of iffeen bioe wih he iffeen iniil level he ineio whih onvege o he oin on he boun lne i.e. - lne oeoning o.8 fo he oniion >.5 eeing ohe ee fie.. Theefoe fo he oniion >.5 i how h fo he eve iniil oin on he - - e onvege o he oin on - lne oill whih en h hee i on hveing of he e oulion fo hi oniion i i no ofible o fiheen o o hveing. 5. Conluion Thi e i onene wih he u of nonline hveing n he onevion of eologil eoue fo wo-ienionl e eo e nil e. We hve oniee e eo oel wih Holling e-ii funionl eone n nonline hveing of e. The hveing effo i en ni vible. The oniion fo eiene n lol oi bili of viou euilibiu oin hve been eine. I i eblihe h he oeiene of e n eo oulion een uon he oe hveing egie 4
8 ounl of ul Siene Reeh ISS e ISS 5-9 Online Vol.5 o n hene we n voi he i of einion o ove eloiion of he eie. B nlil n nueil eul we hve eine h fo fie vlue of ie e uni n ohe ee he vlue of o i ineing hen hveing level will o eee. fe ie we will fin level of o whee hveing effo will en o zeo. So fo he o-eiene of e- eo oulion long wih effo ni hee i eiion on vlue of o o h hee n o-ei ll he eie wih oil level of hveing effo. Refeene Aziz-Aloui.A. Dhe Oie. Bounene n globl bili fo eo e oel wih oifie Lelie- Gowe n Holling e-ii hee Al. h. Le Aziz-Aloui. A. Su of Lelie- Gowe- e i-ohi oulion Cho Solion n Fl Cl C.W. heil Bio-eonoi: he Oil ngeen of Renewble Reoue Wile ew Yo 976. Gh S. Singh B.. The ni of foo web oniing of wo e n hveing eo ho Solion n fl Gu R. Chn. Bifuion nli of oifie Lelie- Gowe eo- e oel wih iheli- enen e e hveing. h Anl Al Kihn S. V. Sinivu. D.. n Kln B. Conevion of n eoe hough oil ion Bullein of heil Biolog Lenzini. Rebz. on-onn eo hveing on io- eenen eo- e oel Al. h. Si Li Y. Xio D. Bifuion of eo e e of Holling n Lelie e Cho Solion Fl u. D. heil Biolog vol. I Singe- Velg Belin 3. eo L. Diffeenil uion n Dnil Se Singe ew Yo 996. hn T. Chuhui K. S. A ni eion oel of wo- eie fihe wih ion onol inuen: A il heoei nli ologil oeling Sinivu.D.. Bio-eonoi of enewble eoue in eene of eo online Anl. RWA Xio D. enning L. Bifuion of io- eenen eo-e e wih onn e hveing SIA. Al. h Zhng. Chen F. Su Q. Wu T. Dni behvio of hveing Lelie Gowe eo e oel. Diee Dn.. So.. Zhu C.R. Ln K.Q. he oi Hof bifuion n lii le of Lelie Gowe eo e e wih hveing e. Diee Conin. Dn. S. Se. B Reenu Rni: She i uuing he hd IIT Rooee ine ul. He eeh e i heil oelling of Renewble Reoue. She h olee he.s. Hn fo njb Univei Chnigh uing he eion 8-. She h one he guion fo Guji College Fehgh Shib ffilie o unjbi Univei il uing 4-7. Suni Gh: She i woing ofeo IIT Rooee. He in e of inee e Bio- hei oue Aliion Flui Dni heil oelling of nvionenl n ologil Se. She h one he hd IIT Knu in 98. She h olee he.s. in Alie hei fo Univei of Rooee in 977 n B.S. in h hi n Chei fo eeu Univei in 975. She h Ceifie of ei Iniuion of nginee w: fo he e in ounl of Iniuion of nginee Ini She i we Chnello Gol el: fo euing highe in ll ogue oue in he univei 977 n lo A Khol Gol el: Fo euing Highe in.s She h n ubliion in iffeen e 5
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