Journl of Phsis: Confrn Sris PAPR OPN ACCSS Sbili nd Oiml rvsing of Modifid Lsli-Gowr Prdor-Pr Modl To i his ril: S Toh nd M I Azis 08 J. Phs.: Conf. Sr. 979 0069 Viw h ril onlin for uds nd nhnmns. This onn ws downlodd from IP ddrss 48.5..8 on /07/08 7:6
Th nd Inrnionl Confrn on Sin ICOS) IOP Conf. Sris: Journl of Phsis: Conf. Sris 4567890 979 08) 0069 doi :0.088/74-6596/979//0069 Sbili nd Oiml rvsing of Modifid Lsli-Gowr Prdor-Pr Modl S Toh nd M I Azis Drmn of Mhmis snuddin Univrsi Tmlnr 9045 Mssr Indonsi -mil: smsuddin@hoo.om Absr. This r sudis modifid of dnmis of Lsli-Gowr rdor-r oulion modl. Th modl is sd s ssm of firs ordr diffrnil uions. Th modl onsiss of on rdor nd on r. Th olling II s rdion funion is onsidrd in his modl. Th rdor nd r oulions r ssumd o b bnfiil nd hn h wo oulions r hrvsd wih onsn ffors. isn nd sbili of h inrior uilibrium oin r nlsd. Linrizion mhod is usd o g h linrizd modl nd h ignvlu is usd o jusif h sbili of h inrior uilibrium oin. From h nlss w show h undr rin ondiion h inrior uilibrium oin iss nd is loll smoill sbl. For h modl wih onsn ffors of hrvsing os funion rvnu funion nd rofi funion r onsidrd. Th sbl inrior uilibrium oin is hn rld o h mimum rofi roblm s wll s n rsn vlu of rvnus roblm. W show h hr iss rin vlu of h ffors h mimizs h rofi funion nd n rsn vlu of rvnus whil h inrior uilibrium oin rmins sbl. This mns h h oulions n liv in oisn for long im nd lso mimiz h bnfi vn hough h oulions r hrvsd wih onsn ffors.. Inroduion Mhmil modling ssoid wih h dnmis of rdor nd r oulions hs bom gr rsrh hm in h fild of mhmil olog nd in h fild of fishris mngmn. Thr r som fnomn in h fishris inluds oulion s rdor nd h ohr oulion s r. Th dnmis of h rdor nd r oulions dos no inlud onl wo oulions bu som fnomn show h involvmn of mor hn wo oulions in h nvironmn. Sin h oulion for ml fish givs bnfi hn h oulion is onsidrd s so. Th oulion is hn mngd o giv mor bnfis nd h oulion rmins susinbl. Thr r som rrdor modls wih hrvsing. Th rdor-r fishr modl wih sliv hrvsing for r s [-4] sliv hrvsing for rdor s [5-8] nd h wo oulions r hrvsd s [9]. Th dnmis of rdor nd r oulions vi Lo-Volrr modl hv bn nsivl onsidrd b mn uhors. In Lo-Volrr modl h rdor jus dnds on h siz of r oulion. This modl dos no onsidr h siuion whn h siz of r drss hn h rdor will s ohr r. Lsli modld h ff of his fnomn. Lsli-Gowr rdor-r modl is nohr roh o h dnmis of rdor-r. This modl is sruurll diffrn from h lssil Lo-Volrr modl. Som rsrhrs hv onsidrd nd modifid h Lsli-Gowr Conn from his wor m b usd undr h rms of h Criv Commons Aribuion.0 lin. An furhr disribuion of his wor mus minin ribuion o h uhors) nd h il of h wor journl iion nd DOI. Publishd undr lin b Ld
Th nd Inrnionl Confrn on Sin ICOS) IOP Conf. Sris: Journl of Phsis: Conf. Sris 4567890 979 08) 0069 doi :0.088/74-6596/979//0069 modl wih vrious ss s [0-]. Mos of rdor-r fishris modls wih hrvsing r rld o h onomi roblms inluding mimum rofi nd ol disound n rvnu s [ 6 9]. Bsd on h wors bov w rsn modifid Lsli-Gowr rdor-r modl wih onsn ffors of hrvsing for h wo oulions. W nls h sbili of h inrior uilibrium oin nd no onl o drmin h riil vlu of h ffors h mimiz h rofi bu lso o minin h sbili of h uilibrium oin. Th oiml hrvsing oli of rsn vlu of rvnu is solvd using Ponrgin s miml rinil.. Lsli-Gowr rdor-r modl W onsidr rdor-r fishr in n nvironmn whih inluds wo oulions rdor nd r. If w l ) dnos h siz of r nd l ) b h siz of rdor im hn h growh r of rdor-r oulions bsd on h Lsli-Gowr modl is givn b following ssm d r b ) d d ) s. d Th funion ) in modl ) is rdion funionl rsons whih msurs h lvl of rdor onsumion o h r. Firsl Lsli onsidrd h funion ) is roorionl o h siz of r. Th mning of ll rmrs in modl ) n b sn for ml in [6 ]. B onsidring h olling II funionl rsons of h rdor o h r h modl is hn modifid nd boms d r b d ) d s. d W onsidr h rdor nd r oulions r onomill vlubl hus hs wo oulions r hrvsd wih onsn ffors. Modl ) is hn ndd nd boms d r b d ) d s. d In h modl ) rmrs nd s h hbili offiin for h r nd h rdor oulions rsivl. Prmrs nd s h hrvsing ffors sisfing h ondiions 0 i i m for i nd som vlus of i m. For h nlsis w l r r nd s s. From h modl ) w hv fiv non ngiv uilibrium oins nml T 0 0 0 s r T 0 T 0 T nd T4 4 4 whr b / A A 4A A A s / A A 4A A 4 A s 4 4
Th nd Inrnionl Confrn on Sin ICOS) IOP Conf. Sris: Journl of Phsis: Conf. Sris 4567890 979 08) 0069 doi :0.088/74-6596/979//0069 A b 0 A s r b nd A s r. In ordr for h uilibrium oins T nd T4 b is in h firs udrn w ssum h A 4A A 0. Cs. A s r 0 h uilibrium oin T boms n inrior uilibrium oin whil h uilibrium oin T 4 will no b onsidrd bus 4 0. Cs. A s r b 0 nd 0 A A h uilibrium oins 4A T nd T 4 bom inrior uilibrium oin. In his s h uilibrium oin T m b sbl or no sbl whil h uilibrium oin T 4 is n unsbl sddl oin. For nlsis w us h Jobin mri o g h linr modl nd b vluing h Jobin mri h uilibrium oin T w hv r b J T. s 4) Th olnomil hrrisi 4) ssoid wih h Jobin mri J T is givn b f ) d I J i. f d d d d d d whr d d T r b 4 d 0 4 d 0 nd 4 s. Rfrring o h Rouh-urwiz ririon [] h inrior uilibrium oin T d d4 nd d d4 dd 0 r sisfid. is loll smoill sbl whn h ondiions 0 ml. For modl ) w s h rmr vlus s r b 0. 00 s 9. 5. 5 nd 7 in rori unis. Thn w hv uilibrium oins T 0 0 0 T 0 0 T 00 0 T 68.89 5468. nd T 4.8 80.66. For h uilibrium oin T w g ignvlus from h Jobin mri s 0.00.50i whih mns h uilibrium oin T is unsbl sirl. For h uilibrium oin T 4 w g h ignvlus from h ssoid Jobin mri s 0. 544 nd. 84 whih mns h h uilibrium oin T 4 is unsbl sddl oin s figur. I is s o h from h hs orri h h osiiv is nd osiiv is r sbl mnifold whil h uilibrium oin T 4 is unsbl sddl oin nd uilibrium oin T is unsbl sirl. In his siuion hr is sbl limi l in h firs udrn.
Th nd Inrnionl Confrn on Sin ICOS) IOP Conf. Sris: Journl of Phsis: Conf. Sris 4567890 979 08) 0069 doi :0.088/74-6596/979//0069 Figur. Plo of rjoris round h inrior uilibrium oins. ml. For modl ) w s h rmr vlus s r b 0. 00 s 8 0 nd 5 in rori unis. Th uilibrium oins of h modl r T 0 0 0 T 0. T 000 0 T 94.76 4906.0 nd T 4.4 7.. In his s h uilibrium oin T 4 dos no is in h firs udrn. Th uilibrium oin T is smoill sbl wih h ignvlus 7.6 4.568i s figur. Figur. Plo of rjoris round h uilibrium oin T. Bionomi uilibrium d d Th biologil uilibrium is found b solving h ssm 0 nd 0. Th onomi d d uilibrium is obind whn h rods from h sl of hrvsd biomss uls o h oss usd in hrvsing iviis. W ssum h TR Y) whr Y ) N is h ild of hrvsing is h ri of h biomss is hrvsing ffor is h hbili offiin nd N is biomss. W lso ssum h ol os of hrvsing is roorionl h hrvsing ffor TC. Th rofi funion is dfind s TR TC. Th rofi funion of hrvsing for h modifid Lsli-Gowr. Th bionomi rdor-r modl is * * * * uilibrium ) s [4] is found b solving simulnousl h uions r b 0 0 s nd ) 0. 4
Th nd Inrnionl Confrn on Sin ICOS) IOP Conf. Sris: Journl of Phsis: Conf. Sris 4567890 979 08) 0069 doi :0.088/74-6596/979//0069 W hn rl h sbl inrior uilibrium oin T o h mimum rofi roblm. Th uilibrium oin T is n inrior uilibrium if h ondiions 0 i i m for i nd A s r 0 h is r s 5) r sisfid. Th rofi funion h uilibrium oin T is wrin in h form ). Th roblm is drmining ir of ffors whih sisfis ondiion 5) nd for i h mimizs h rofi funion ). W lso nd o minin 0 i i m h inrior uilibrium oin T is lws smoill sbl. ml. For h roblm of mimum rofi funion w s h rmr vlus s r b 0.00 s 8 5 nd in rori unis. T 0 5 5 nd 5 in rori unis. Thn w hv h uilibrium oin T whr 466.6667 66.6667 500 66.6667.6667 0. 66.6667 7.970.840 0.60 0.060 / 4666.6667 66.6667 7.970.840 0.60 0.060 /. 500 In ordr for uilibrium oin T boms n inrior oin h hrvsing ffors nd mus sisf h ondiions A 0 whr A 0 5 0 nd 0 i ) i m for i. W u m 5 nd m 5. In h ohr words h uilibrium oin T boms n inrior oin whn D whr D : 0 5 0 5 0 5 0. Th rofi funion ssoid wih h uilibrium oin T is givn b ). Afr subsiuing h vlus of nd nd hn simlifing w g ) 466.6667 666.6667 666.6667 40 5 666.6667 5000 7.970.840 0.60 0.060 / ) 4666.6667 66.6667 500 7.970.840 0.60 0.060 / 5. * * W hv sionr oin.469 4.906 * *.469 4.906 in D. Th onl riil oin sisfis h ondiions nd lso mimizs h rofi wih h vlu * * of ) 8767. 56. B ling h vlu of hrvsing ffors 5
6 4567890 Th nd Inrnionl Confrn on Sin ICOS) IOP Conf. Sris: Journl of Phsis: Conf. Sris 979 08) 0069 doi :0.088/74-6596/979//0069 4.906.469 * * w g h uilibrium oin 96.64449.077 T. Th olnomil hrrisi of h Jobin mri h uilibrium oin T is givn b 4.7.80804 ) f whih hs h ignvlus 696 6. nd 7869 5.. Undr his siuion wih h vlus of ffors h lvl of 469. * nd 906 4. * hn h rdor nd h r oulions will susin for long riod of im nd lso mimiz h rofi funion. 4. Oiml hrvsing oli Our objiv in his roblm is o mimiz h n rsn vlu of rvnus whih is givn b d J ) ) 0. 6) Th smbol ss h insnnous r of disoun. W nd o mimiz J subj o h onsrin uion ) b using Ponrgin s mimum rinil [5]. Th onrol vribls ) nd ) r subj o h onsrins m ) 0 i i for i. Th milonin uion is. s b r 7) W s 0 nd 0 s h nssr ondiions for h onrol vribls nd o b oiml. From h milonin funion 7) w hv 0 nd 0. Thn w g nd. From h milonin uion w lso hv b r nd. s From h Ponrgin s mimum rinil nd w g 0 b r 8) 0. s 9) B subsiuing nd ino h uions 8) nd 9) w g nd. Agin fr subsiuing 5 nd
Th nd Inrnionl Confrn on Sin ICOS) IOP Conf. Sris: Journl of Phsis: Conf. Sris 4567890 979 08) 0069 doi :0.088/74-6596/979//0069 / s s 4s ino h uions s nd w g h vlu of onrol vribls nd. Th vlus of nd mimiz h rsn vlu of rvnus J. ml 4. For h roblm of mimizing rsn vlu of h n rvnus w s h rmr vlus s r b 0. 00 s 8 0 5 nd 0. 9 in rori unis. T 5 0 0 5 nd 0. 005 in rori unis. Thn w hv h uilibrium oin T whr 466.6667 50.0000 500 66.6667 7.970 0.9.840 0.60 0.0540 /.6667 0.000 66.6667 Th djoin vribls r.640 nd 8. 806 uilibrium r 9. 464 9. 909 J 0. 005 4666.6667 50.0000 500 7.970 0.9.840 0.60 0.0540 /. 0.005 0.005 5. 6 5 nd. T W g h ffors 940.9754 9.097. Th ignvlus rls o h nd 0. 07696. Th djoin vribls r 0. 005 4. 9989 nd. Thrfor w g h mimum vlu of rsn vlu of h n rvnus 5 0.005 7 4.0677970 d 8.54590 0. 5. Conlusions Th modifid Lsli-Gowr rdor-r oulion modl wih onsn ffors of hrvsing m hs wo inrior uilibrium oins nml T nd T4 4 4. Whn h uilibrium oin T iss in h firs udrn i m b sbl or unsbl. Whn h uilibrium T4 iss in h firs udrn i is lws unsbl sddl oin. Whn h uilibrium oins T nd T4 is in h firs udrn whr h uilibrium oin T is unsbl sirl hr iss sbl limi l in h firs udrn. Wih h rsriion of hrvsing ffors 0 i i m hr iss rin ondiion suh h h inrior uilibrium oin T rmins sbl nd lso givs mimum rofi. Th rdor nd h r oulions n liv in oisn lhough h wo oulions r hrvsd wih onsn ffors. B using Ponrgin s mimum rinil w found h hr iss rin vlu of hrvsing ffors nd whih is ssoid wih h sbl uilibrium oin T h mimiz h n rsn vlu of rvnus. Anowldgmns This rsrh is suord b Minisr of Rsrh nd ighr duion of Rubli Indonsi vi LPM snuddin Univrsi grn omiion of Profssorshi Rsrh Shm r 07. This finnil suor is highl rid. 7
Th nd Inrnionl Confrn on Sin ICOS) IOP Conf. Sris: Journl of Phsis: Conf. Sris 4567890 979 08) 0069 doi :0.088/74-6596/979//0069 Rfrns [] ou F Jing M nd Mng X Y 0 A dnmi modl for fishr rsour wih rsrv r nd ion Journl of Alid Mhmis h://d.doi.org/0.55/0/79479 [] Chrbor K Chrbor M nd Kr T K 0 Bifurion nd onrol of bioonomi modl of r-rdor ssm wih im dl Nonlinr Anlsis: brid Ssms 5 6-65 [] Gu R P Bnrj M nd Chndr P 0 Bifurion nlsis nd onrol of Lsli Gowr rdor r modl wih Mihlis Mnn r-hrvsing Journl Diffrnil uions nd Dnmil Ssms 9-66 [4] Kr T K 00 A Dnmi rion modl of r-rdor ssm wih sg-sruur for rdor Modrn Alid Sin 4 5 8-95 [5] Lsmi G M V Gunsrn M nd Vij S 04 Bifurion nlsis of rdor modl wih hrvsd rdor Inrnionl Journl of nginring Rsrh nd Dvlomn 0 6 4-5 [6] Chrbor K Chrbor M nd Kr T K 0 Oiml onrol of hrvs nd bifurion of r-rdor modl wih sg sruur Alid Mhmis nd Comuion 7 8778-879 [7] Qu Y nd Wi J 00 Bifurion nlsis in rdor-r ssm wih sg sruur nd hrvsing Journl of Frnlin Insiu 47 7 096- [8] Chrbor K Jn S nd Kr T K 0 Globl dnmis nd bifurion in sg sruurd r-rdor fishr modl wih hrvsing Alid Mhmis nd Comuion 8 5 97-990 [9] Toh S Kusum J Khruddin nd Mwrdi 04 Sbili nlsis nd oiml hrvsing oli of r-rdor modl wih sg sruur for rdor Alid Mhmil Sins 8 59 79-794 [0] Ji C Jing D nd Shi N 009 Anlsis of rdor-r modl wih modifid Lsli-Gowr nd olling- II shms wih sohsi rurbion Journl of Mhmil Anlsis nd Aliions 59 48-498 [] ou F Wng X nd Chvz C C 0 Dnmis of sg-sruurl Lsli-Gowr rdorr modl Mhmil Problms in nginring doi:0.55/0/494 [] Yu Q 06 Dnmis of modifid Lsli-Gowr rdor-r modl wih olling- II shms nd r rfug Sringrlus. 5: 46. doi:0.86/s40064-06-087-7 [] Ksh L 005 Mhmil modls in biolog Phildlhi: SIAM) [4] Clr C W 985 Bioonomi modlling nd fishris mngmn Nw Yor: John Wil & Sons) [5] Grss D Culins J P Fihingr G Trglr G nd Bhrns D A 00 Oiml onrol of nonlinr rosss. Brlin idlbrg: Sringr-Vrlg) 8