M 2 STABILITY ANALYSIS AND OPTIMAL CONTROL ON THE MODEL HARVESTING OF PREY PREDATOR WITH FUNGSIONAL RESPONSE TYPE III
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1 Poeeding of Intentionl Confeene On Reseh, Implementtion nd dution Of Mthemtis nd Sienes 5, Yogt Stte Univesit, 7-9 M 5 M STBILITY NLYSIS ND OPTIML CONTROL ON TH MODL HRVSTING OF PRY PRDTOR WITH FUNGSIONL RSPONS TYP III Mohmmd Rif i, Subhn, Setidjo Wino 3 Mhsisw S Juusn Mtemti, ITS Sub ifi@mtemti.its..id Dosen Juusn Mtemti, ITS Sub Subhn@its..id 3 Dosen Juusn Mtemti, ITS Sub Setidjowino@its..id bstt osstem is the eipol eltions between living things nd thei envionment. Reltionships between ognisms o individuls will not be septed fom the poess of eting nd being eten. On eologil siene event lled the food hin. In the food hin, thee is the tem pe nd peo, whee both e intedependent of eh othe. The behvio o hteistis of the peo-pe n be modeled mthemtill nd histoill hs been developed b esehes. Peo-pe sstem bsi model in genel, fist intodued b Lot-Volte, lte developed b Leslie nd ontinued b Holling-Tnne. On the model of Holling-Tnne developed esponse funtion on the funtion lled peo Holling. In this eseh, the stbilt nlsis nd optiml hvesting ontol e done on peo pe sstem with Holling esponse funtions III. Consideing the eosstem, espeill in the se thee is bite to et, intetion so tht the ppopite hvest sttegies e needed fo mimum ommeil benefit while mintining the sustinbilit of the speies. In this se, the optimum hvesting mount obtined using the Piniple of Mimum Pontgin. Kewods : Model of Pe peo, Funtion esponse Holling tpe III, Hvesting, The piniple of Pontgin s mimum.. INTRODUCTION ve living thing equies othe living things to eh othe life suppot eithe dietl o indietl. Smbioti eltionship ous between podues, onsumes nd deomposes. In eosstems, thee is eltionship between ognisms nd the envionment e quite omple nd mutull influene eh othe. The eltionship between the elements of biologil nd nonbiologil led to the eologil sstem lled eosstems. While the stud of intetions between ognisms nd thei envionment, nd the othe is lled eolog. Siene olog len bout living things tht n sustin life b ognizing the eltionship between living things nd innimte objets in the neighbohood. With mthemtil modeling, sientists n lso pedit the stbilit of the intetion between the two speies. Stbilit in question is the totl popultion of speies tht is not depleted o etint so tht intetion emins. In the eltionship intetion pttens this involves the biogeohemil les, ie flow of eneg nd food hin. While the food hin is n event et nd be eten within n eosstem with ptiul sequene. In the food hin is the tem pe nd peos. Pe is M-
2 M Rif i, et l/ Stbilit nlsis nd... ISBN n ognism tht is eten, usull nown s the pe. While the peo is n ognism tht tes, usull nown s peo. Dnmi eltionship between pe nd peo is ve inteesting to len nd beome one of the topis e often disussed mong esehes. Ove the pst few es, mn esehes hve studied the peo pe models, but the nowledge bout the effets of hvest on peo pe popultion is still limited, whees eologil sstems e often ve distubed b humn eploittion tivities. Fo emple, due to pid dvnes in tehnolog s well s signifint inese in the humn popultion, the numbe of fish in the wold hs been edued dstill. This ous due to the eessive hvesting food esoues in the se, so n esult in the etintion of the speies. Given in the mine eosstem e intetion-eten mel, so tht the ppopite hvest sttegies e needed to obtin the mimum ommeil benefit to eep the pesevtion of the speies. Mn studies hve been ied out ginst peos pe. mong them is the eseh done b Tpn Kum K (5 who nlze the dnmi behvio of the model peo-pe with Holling esponse funtion of tpe-ii nd ssess the hvesting effot. Hung, et l (6 disuss model peo pe with Holling esponse funtion of tpe-iii ombines the potetion of pe. The hve nlzed the model nd disuss some signifint qulittive esults fom biologil viewpoint. Tpsi Ds (7 disusses the hvesting of pe peo fish in the infeted e toins. Lv Yunfei, et l ( disusses the hvesting of Phtoplnton-Zooplnton models. Tpn Kum K ( investigted the stbilit nlsis nd optiml ontol model peo pe sstem with n ltentive to peo feeding. Chbot, et l ( emines the globl nd bifution nlsis of peo-pe sstem ombines time potetion, b using esponse funtion Holling tpe-ii. Fom thei esults tht the eistene of potetion hs n impotnt effet on the eistene of peo nd pe popultions. Lusin Ptm (3 investigted the optiml ontol on pe bioeonomi models Holling II peo esponse funtion with time del. Bsed on these studies, it is in this ppe desibes n investigtion of the stbilit nlsis nd optiml ontol peo pe model with Holling III esponse funtion. Stbilit nlsis in question is to now the stbilit of vious equilibium point, nd used to detemine the dnmis of intetions between speies of popultion. While the optiml ontol ole is to obtin the optiml hvesting effot so s to obtin the mimum benefit fom n eonomi inome.. BSIC TORY. Holling Response Funtion Response funtions desibe the te of peion nd vilbilit of food (pe. In genel, the esponse funtion is developed oding to the tpe of peo nd pe vilbilit. If thee is peo pe sstems suh s: d p( f (, d f (, Then f (, the esponse funtion, nd sttes pedsin pmetes. Some tpes of esponse funtions tht hve been developed e s follows : Holling Tpe I M-
3 Poeeding of Intentionl Confeene On Reseh, Implementtion nd dution Of Mthemtis nd Sienes 5, Yogt Stte Univesit, 7-9 M 5 ( Model Holling tpe I, hve the ssumption tht the level of peion ous linel with inesing densit of pe, until it ehes the mimum peion te. F H ( t ( t Holling Tpe II Model Holling tpe II desibes the eltionship between peo pe b ssuming the eistene of time hndling (pe on pe tht is the time it tes peo to pe, subdue nd spend pe in units of time. F H ( t ( t Holling Tpe III Model Holling tpe III illusttes the gowth te of peos. Holling models lso desibe the uent deline in the level of peion on pe densit is low. Besides, thee is tenden peos powl in othe popultions when the popultion begn to deese eten. F H ( t, n n. Dinmi Sstem with Response Holling III Hee is n peos pe dnmi sstem with Holling tpe III funtion nd the pesene of dditionl ftos in the diet of peos nd hvesting effot in both speies d ( d ( ( whee (t is size popultion of pe t t, (t is size popultion of peo, intinsi gowth te peo deth te, ing pit, is the te of peion pe, is onvesion te of peos, nd eh one is in pe ptue powe oeffiient nd peos, is effot hvesting..3 Stbilit Sistem Definition. Given fist-ode diffeentil eqution d f (. point * lled equilibium point if it meets f ( *. d( t Theoem. Sstem ( t smptotill stble if nd onl if ll the eigenvlues of, is ( hve negtive el pts nd is denoted b Re( (. i. Optiml Contol Poblem In genel, the fomultion of the optiml ontol poblem is s follows : n i M-3
4 M Rif i, et l/ Stbilit nlsis nd... ISBN To desibe pemslhn mens modeling the poblem into mthemtil models in the fom of vible iumstnes. b Detemine the objetive funtion Detemine the bound onditions nd phsil onstints on the stte of nd o ontol..3 Pontgin Mimum Piniple The following e the steps in solving optiml ontol poblems with the piniple of Pontgin. Build hmilton funtion smbolized b H is : H ( ( t, u( t, ( t, t V ( ( t, u( t, t ( t f ( ( t, u( t, t ' H Solve H the eqution to whih ontol is to mimize H to u is : in ode u to obtin the sttion ondition u * ( t 3 B u * ( t whih hs been geneted in step, will get the new Hmilton optiml * funtion H 4 Solve n eqution, with n sum of stte vibel H ( t * H ( t dn o-stte eqution * 5 Substitution of the esults obtined in step 4 into the eqution u * ( t in step to obtin optiml ontol. 3. Result nd Disussion 3. quilibium Point Peo Pe tintion quilibium Point quilibium point of etintion of pe nd peos is ondition when the pe nd peo popultions do not eist, nmel t the time. Suppose the equilibium point of etintion of pe nd peo is denoted S, S (, b the equilibium ( point is obtined when the pe nd peo popultions do not eist (etint is S (,. Peo tintion quilibium Point peo etintion equilibium point is ondition when thee e no peos, d d nmel when nd. Peo etintion equilibium point nd the similities obtined fom ( nd (. If this peo etintion equilibium point denoted b S, S (,, the peo etintion equilibium point S (, is ( (, with nd. Titi setimbng pe peo hidup besm quilibium point between the two speies oeist is ondition when the pe nd peos living togethe o both speies is not etint, nmel when nd. quilibium point between the two speies oeist is denoted b S (, S (,. Vlues nd obtined though the following onditions. M-4
5 Poeeding of Intentionl Confeene On Reseh, Implementtion nd dution Of Mthemtis nd Sienes 5, Yogt Stte Univesit, 7-9 M 5 d Of view of eqution (, then b using the tems obtined: ( d Of view of eqution ( using the onditions obtined: nd then p (4 then subtitusin eqution (4 into eqution (3 then beomes (3 p ( (5 p p So tht the equilibium point when the pe nd peos living togethe is S (, p ( (, p, with p nd p p 3. nlsis of stbilit of the sstem Dnmil sstems pespetive on equtions ( nd ( will be fomed Jobin mti s follows : ( J ( ( (6 Futhemoe, the stbilit of the sstem will be nlzed t eve point of equilibium of the model peo pe Holling tpe III with the ddition of ltentive hvesting food nd thei effots. Stbilit of quilibium Point Pe nd Peo tintion Bsed on Jobin mti in eqution (6, then we obtin the hteisti eqution Jobin mti with the etintion of pe peo equilibium point s S (, follows : det ( ( thus obtined tu ( M-5
6 M Rif i, et l/ Stbilit nlsis nd... ISBN M-6 In ode fo sstem is sid to be stble if its eigenvlues negtive vlue, so tht must be met when the onditions on the pevious setion eplined tht vlue. So tht in ses whee the pe nd peo etintion sstem is sid to be unstble. Stbilit of quilibium Point tintion Peos Bsed on Jobin mti in eqution (6, then we obtin the Jobin mti with etintion peo (, S equilibium point s follows : J ( (7 Then loo fo the hteisti eqution of eqution (7 ( ( det Thus obtined eigenvlues s follows: tu ( In ode fo sstem is sid to be stble if its eigenvlues negtive vlue, so tht must be met the onditions, whees in the pevious setion eplined tht vlue. So tht in ses whee n etint peo ws sid to be unstble sstem. Stbilit Of quilibium Point Peo Pe Coeist Bsed on Jobin mti in eqution (6, then the Jobin mti obtined b equilibium point peo pe oeist s, ( S follows : J ( ( ( (9 Then loo fo the hteisti eqution of eqution (9
7 Poeeding of Intentionl Confeene On Reseh, Implementtion nd dution Of Mthemtis nd Sienes 5, Yogt Stte Univesit, 7-9 M 5 M-7 ( ( ( det ( ( ( whee : ( ( ( ( ( ( 3 ( ( ( ( The point, ( S is stble, if the oots of the hteisti is negtive on the eln. So tht the equilibium point when the peo pe living togethe is, ( S sid to be stble if it stisfies the onditions nd 4. Conluding Rems Fom the disussion peo pe dnmi sstem with Holling III funtion in the pesene of ftos hvesting nd supplement food ltentive to the peos will be stble if the peo pe popultions living togethe. So tht the intetion between the two speies. Whees if one etint speies will ffet othe speies. This ould destbilize eosstems nd the pesevtion of ntue. Refeenes. Finizio N, nd Lnds G (998. Odin Diffensil qution With moden pplition. Clifoni : Wsdswoth Publishing Compn. Hung Y, Chen F, Zhong L (6. Stbilit nlsis of Pe-Peo Model With Holling Tpe III Response Funtion Inopoting Pe Refuge. Jounl of pplied Mthemtis nd Computtion 8( K, T.K., Ghosh, B. ( Sustinbilt nd Optiml Contol of n ploited Pe nd Po Sstem Though Povision of ltentive Food to Peo. lsevie. BioSstem. Pge Nidu, S.D (. Optiml Contol Stem. US : CRS pess LLC
8 M Rif i, et l/ Stbilit nlsis nd... ISBN Tpsi Ds, R,N. Muheje nd K.S Chudhui. (9. Hvesting of Pe-Peo Fishe in the pesene of Toiit. Jounl pplied Modelling 33 (9 8-9 M-8
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