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Grazir1g a CoirnrlOil Pasture ir1 a. l~a11dorrl E11 virour11on t'1 '2
llodnoy n(1ctrd . J)ppart.uwuf, of A!?,Ticnltur(l lhlivPrsity of <.~HvPnsland 4072
Allstralia e ... Inctil: r.1H'nrd'n1umHbox.uq.o7..au
Siinou l{atn IJt~partnlNlt of Ecouon1ks
UnivPrsHy of (~twenslnud 4072 Australia
January 10~ lOH7
1 PnpPr pn)s<'lliPd nt flu' .u·'it Anmml Confnrmtc·t' of t.lw Aw•tru.lian Agtknlt;tm~t Hnd HPsnmc(• Et·ownuks Sod<'f'V. Bmtnllwad.- .lauuatv 20 .. 25 :tmn.
i!\Vn would lilw f,o thank Ti;n Pnt<'<·ll~ Jynthi ViJ·~wn. Gali, Anthou~t llloc:,t:h, Culio Brmvll and ,Jolm Mntt for diHt•u~-;siuns un vnriutm t:t.'{JW(•f:H uf thi~ t)nJWl'
A ttl<whasf'k <h·namh· ntt\<lPl of nplilmtl ~toddng of n t'OJiliHOH ntng<~ is prt"' ~wnt ''<1. ThP 1nndPl aHPmpfH I o PaptnrP tlw prPdowiun.nt ptnrwtty dp;ht~~ fpafun·s of 'lunhilP snh!1ish•nt'P pahloruliHTH iu nrid aud sNni·ttrid tHUJ{datuh.;. Grazjnp; dp~bts m·p Ulnth'l1c><l Oil fftp Oll(\ hand vin l:hP itt<lirP<'t. dfcjd: or off~ tnk<* nu avnilahlP Jm"tnn• awl on thP othPt lmnd via tPBOlU'<'l' BlH\ritly, (nkhP mvrlap) hPf WPPH t·nnt.twtiug lH1tds. UtH'Pttai.nt)1 h~ itu·m·poratNl via Pnvf ..
ronnwutal tmis<•. The nnaJ~mis .is t'mlfdli:t Prtl. with n\tlfP dynatnks lwlng in <lis(•quilihthttu ..
1 ltttrodla.ct:.iotl lJn ~tndm~'f;k t;ntth·nl tlwory lm,'i l>r~Cl\ npplk~d. in ugrit'Ultitll'al n.nd tt~s<mr<~(~ P<'onmuh·s by a uurnhN· nf nnthm-s. Pindy·ek {:1984) givt~H n aut'VtW or ttppli-. cat'lnus to r<~m'Wttbl<> rPHotH~<·<~ ancl .ltx~rt.~l<'t (19trl) stttVt~yfs nppHc~~.ttiot-.,.q t;o Agri<~ntt.m·('. Atlplh·ations hwnlving <'Otnmnu. ptopt~rty lHW(\ t.ypit~ally bci~H t<}.; stdd.Pd t:o Hfm'"t(\twwnbl<~ t't~ouret>s nud luW(1 .irwadnhly itlvolv<~l post;nlatitlg l'f'tt:aiuty to situpllly tlw mmlysis1•
Stmlh~\.!i of optimal stocking lmVt' t~Nwrally umd<•· us«~ of eiU1(W <Uscrotn Hum s1:ot~lm ... .:;t.k nptimn.l ('Ollt.r(:)l tlwoty (P.g. P<'tting-s ( 1994.)) or stodul..qtk d)rua.rnh' prngTttrtunhll\ (\Vang (l mH and lfln2)) and Passmm·(~ ttnd Drown (1902),. an (!)\('Ppt,inn is UPatd (109tw,) Wh() llHt'd no \.'t'>Uta·ol t:o stnc.ly opt.itu:.\1 stncking:.z. 'TJw~w st ndit~H havP igttot'<'<l t.h(• ptohll\m of }H7<)lHWty dght;s~ itt parfkulm· the grfwing of <'ot1U1lon pastnrP!-i whkh hnR h<l('ll tJw f<wns of tlw stntk lit t\tnt,m'P:~. dynmuk rnmh•ls UwL in<·m·por:ltP t\ommnn JWO(Wtty sneh ns Ch,mhont nud \Vnn t H)Hfi) hav~~ lw<•n mrn·p ori<1ntnJ<•<l tnwanls, Ushriri(1S.
thn•stty and utin1ng prnhht•n~. ln this J>nJWr WP ut tPmpl t u l'<•nwd~· this HitHatinn hy dPVPlopin~ n. stoduu; ..
f'h- ctifi'pn~ntin.l wmw of thP ~tlt'Zittp> of a <'nmmnu pmd:urn ftmn tlw lto twt .. S}Wf.•t\vP. ~rlw :;olntinu nl Htm·lnJ..'1Lit• diifc•rPntin.l garu(ls was first, p!'OpOS<'d by P:ri(~dnmn {1972). Bnt it Wf,:m't uutil th(1- dPVPlnprrwnt or t.lw ttidutiqll(l of Da;vis aud \ht.nli~·n t hnt tlw r-~olutiou of Sll<'h r)rohlNtni lw<·mn<' f\WtlJ•t;kally tract:abh~.
l)P~pi.t;p t:ht' llll£)U1't.llll('P nf holh f~RllW fht~)t'jf and \ltn•ertaint~y iu rf\'iOl\l\~H (1eoumnit·~ stcwhn.'1tk diff't-tPntiPJ gauws luwP rnn'l.\· hPNII\pJ)li(l(l tn ~~(\i:.iOHl'<'<' ptnh]Pms. Cl<'lUllont n.nd \\·an '108!i) tU'<' a notnhh~ (~Xt~ept:iou·t.
'Tlw f Pehnknl diUit•nlt i<'s asstwh~t Pd with Sf'llving systNU.B of puttial dif .. ferPntial ttquut.mn.s luwP k·d tn I h(\ mutlynis of st;oelmst.it• diffc!t<mtin.l gun.ttts lwiug l'(1St l'i<'f'Pcl tn fbrmul pl:nnf.e;; of tlw PXist.PU<'(• of Hnlttf.ions and to Jlttnt<•dt•nl sin.tldatinns.
A fW('OUd ttpprmwh would ht' to t•om;tnwt· a sohttinH nn<l tlwu v~r.;:y t.hat, it is iu fuvt n solution. This apr)tnadt hn.s hPNt J:nt<.lly usl~l itt Uv~ Ut(1t·attlrt'~
1I'hP lit<)raturP nn tlw t1XiHt'PlW·<' nf Hnhitint.IS hwnlves t.wn hn.<iiJ~ appr<1adH~s. 'the (Jlr.!mnov nr wc~al\ .. fWHSt'~ nJutt·nl approadt and the.' (Wf.'UJH~f;irm nwusur\\
u"''"ig;;i· (~~~;~l;;;;.:-~"~d \~~·~l-'{l OR5) ~Otlli"t" tN'hniqm•F~}HWP n.l!icJ bt~n. ttsti(J. bwlndlng dlst·t~1t.~ thl:ll' d(lt.etntitlislic optbttnl
<~<mtrol Vltt.nln (1092) nild t('<'t1JlflY fuzzy cor1trol' tht><>ry vn.n l.\tx)tt'n nnd KhtY\H\ (lUCJu) !•For nu .. ov~J'Vit'W of flu~ static Ht(it:)t m:q f!('(i l3•J:~rd ( l995h) -1 n should Jmw(l~,'t'l~ ht' noti?'f1 HIM t ht*J'(t i~ n dJ.Wt'lu})hlt\ lif;~rnhtr~ (.Jtl couth)UOUs·fiUW
pritu•iph' ~t.gr•nf: prohlNus Omt t•oinld 1,(~ htt(ltpf(ltr:cl f\..4 gu:nws, nltbough. UtJt bi: the ll('US~ ·1 llH'til\ hN'f>,
~\PJH~O(~c!h. Bot.h of tlu*l<' ttpproadt<"S ltn.v(~ bt'lc!n l\ .. ~!<1 t;c) study ganics in whid1 noitw follows n \Vit\m~t· pto<~(}Ss and ganws in Wllidl nols" fnUows a. Poi~;aou (ju1np) pr<wt·~s.
Tlw adwlntaJj~(l of lmth lltrl};t~ apt)ron.Pbus iH that~ tlu~y clo not; toquirn t.ho SO]\lttOtl of a sys1.t1JU nf partial dlfl't'trnt,ia.} (X}t.Utt.)OHS. rJ'hUS lUCH~lSitlJ~ tlm t.r:wtahility of stm·hm;tic diffetou.tial g:.mws. t;hn <lisad~nt;age ls that~ a t~OJ1 .. <'n't<\ t>r<H~(~lnn~ for Hw t~nnstnwl.inn nt' an eqnlllhdnxu t<'lnainl{ tUl<'l<~a.r.
~rlw Girsmwv approadl invoiV($ ~nthstituting tlm o:tigitl&l \V.it~llt~t rnca .. sur<- for a JlWasttrP ·w·hi<'h p~mtanh-<1B a solution of U1P uud<trlylnr; s:w~t.<~lll of :;t.och(~l:ltk dHfPwntia.t equat.inns t~nd thNl solviug tlw SO•<!<t.lle<l lsruJ.ns equatiOJl (a Haxnil1.nnlan) to lind 1:1 :;nlntion to Uu' wuuP.
Tlw <wt·upati<m nwastU'P apprmwh invnlvc~s (lf'Linbt~); fltl '~atJgtlWnfir~l :-;tt\h~ spm·(~" to irwhtd<> HtrHtPp\it 1H n.c.; p;:u·t nf tlw H(,ut<' sp;wtt. Tlw rPHult iH that;~ <Jy .. Wltuit- pro~~t·mln.uluP, prohlPnt <·an bP transfortn<~l i.ut;n a lhwm· progt·alun1Iug ptnhlNn. Pnfo.rttnmtdy~ this tuvthml js uot· snUkiPntly dt'VPlnpPtl to a st;ag<1
wlJ<'t'P appli<~.utinn.~ at<' lH>hsihlP. ThP sta.-uh1rd tJ.·patmPHl oi t.ltiH apprmwh i~'> Botkar a.nd Ghosh {1UH2).
Gh~(.'fl t lw pl'PvttlPnt'<l of p;auw th<'<JtPt k prohlPm::r i11 l"(~SNn'P<~ t•c.t:nlOJ..tlies and Hw huport.rUH'P nf \lll\'N'taint·y iu nm.uy rt'~KHU't'<' P<.,ot;omi(' J>roh1tmlB. tlw d(~\>rPloprueu.t of o·a.d~ahlP f~(lPlUJlqnns fc)r solviH~~ st()elw.st;i(• <llflb·ent;ial gan.ttl.'l would lH~ a sigrufkaltt dt!V<•lopnwnt in irltproviug tJw apt>lkation uud tlu.' popuhu·ity nf strwlu\stie diff(~r< 1f~tial gatuf• fonuu'lations of nat;uraltt>Smtre<1
a.llmmtion prohlNnH. In thiH p:if)f'l\ W(• Jn'<'SPnt: au applit!aMnn of a Girsa.nov approu.eh t;o th(~
solnti.nn of a stodluu~tk dif:f<•tPutial p,auw of gtazing a t·ommoJ1 pastlttP trhP al)Jll'Otwh t\};NlluvnlVl>S ~1p}>l.Vill~ tllf' IviartJngal.<• approaeh nr Cox and Huang awlKarat.i'-a,..;. LPhn<~~l<y awl Shl'PV<' tlutt i~ cnmtumilJ· llSPd f n soh~<· atodHL."1tk t~nnt.rnl ptf)l·,t{~tus 1u fhmw·<\ t.o tlw study nf a stonhu.stk diff<>n~ntial garun.
Tlw apprmwh tbPl'PfotP diff(·l·s frmu tl1<' Wi<' of tlw GirsanrN tJ•ansfhrru h1 tht- lil·<'ratnr<• on. sl .. odm.HtJe dilfPl'Prttial gatu.<1~.
An 6h~jPPtive of Hw pa]wr L~ tn anub'zn tlw t.ra(lPnfl' n~latimmhip hPfWt•PH
Uv<•stm~k ntnnhc'J'$ and thP nfft;a.k(- r,1.U.P in. a eouunnu ptnpPtty H<1tting. 'rhiH JS dmw hy t~Ol.lSttUdiJlf~ H· llJ.OC}d hasPd OU a. {)OS)J}\·(~ tJWOt~' nf (•l.'OUOUlle
lwhavimu· {a tlul()ry of Uw ratimml vastm·aUat) aud tt nonuu.th'<~ t.lwory of institutiort.'i (tlw Uwo:ry or mN1muism d(:.Sign). 11lis apptrm,d.t diffPtS ll'OI,IJ, a Nmv .. lJu;tit~ufional :tppr<>Udl in tlvLt instituUons su<~h fl$ gra.zh'1g tights at<1
vi<-Wt•d from a twnua.t,iv(• tf~tlwr tlmn a pol'dt;iv(~ pet'SIH~<·tJwli. This dhJPdiw" mnst f()rnair.. an itttPnd£><1 objP<•t;ivP t)(•nrli.ng tJw d<)V(llop ..
wmdd llkt» t.u tlnu:•k Vijnyn GnU {ut· elnJ:it}'il)J!.; tlw tl(t.l.tltt\ of lh~ disthlrUon btlt\v•vn tlH;o xww iulltitlltionnl N'OU~Jnlit•f1. atw th<l tltt:>or~· otm(~(:hnnlstnc.k"'dgn.
2
n:wnt of a t.lnl(>ry of uwdJu.nism d<'Bign fnr dynan~k v;atm~s t)r T<!SO\tnl<l c.x: .. ploitatiott.
2 Tl1e Model Consid(1t tlw sit.uation of ('oruntnn proJwrt~· s~rmdng in n. suhsist.t~rw<l pastnrn.l st~d.nr. A ('Orttmon sitmtt1on in ln.tgt1 parts of Afrkn. and Asia~ \\"p r<~~trkl t:ht~ u.ualysis tn n two play<·r p;anw.
1'lw nh.lt'cti\'(l fmt.t•tiona.l of <•<w.lt pht~~\'l' ls dnu:m't t1l'il',v<l h:~' djs<·mmttl(l PXpt1t•t.r!ll utility~ whPn•1w 11t.ility in <~ndt twrind ls r<'J>l'<~l'nt<~l by a C<.>hh· Dnu~lus flmt•tion in lot~ fonu.
nNP' .ft E IJ."' Ia itw 111 ·l b l"A "t !<- ilt II ;;,} Plnr<~
wll! 1l'P u, is thP nfftnlw ru.t~.•. !h 1lu• HlZt> of <'(Wh lwnl/lhH'1<.1 .• j thtt dh·wnnnt. taJP~h. ·n1us pastoral hnmu•lmlds f:u·r• a t.t·a<h·off lwtw<-PH tlw h.-t·dsiz<· nnd n(ft.akP t'<'<JUh'(lcl fnr nnbsistt'nt'<• (~utviYtil).
'llr·· dytmntit·s of th{~ sysnln1. nrP l'P}lf< 1~Pn1t 1d p,<,ru•t(~lly hy tlw 1ollm\•if11~ sysWm of Ito diUPr<)tHln-1 ('quatinns.
(
d;r ). (. f (.r ~ u, t) ) ( rr(:rl NW {•r .. 11) cor (;t\::) ) ( dwo ) du .· ,. ··. !ll. (J'.1J .. ~;.uJ). ·.dtt cor( .• r.uJ rr{u) cm·(u • .c) .• ilw, •.• d: · !l'l (;t\ :rJ •. z. 'Uz') · ·. em• f;r.::) t•a~t (JI.:.) rr(::) dw~
\~r·lwr<' .r is t.}JP awulahlt· pnstm·p lUPasm·pd in hv<'~tm·k uuitfir'. Tlw eontrol ptnhlPlll is of tlw lmlJrPt't' tyyw wlw.r<· fJJc (•ont;roi vu.rhthl<~
(u1 .. "l.ll) ot offt(~kP ratPs ha\·p n.n iuclirPd 1mp:wt nu tlw Rtat.P wu:iahl<• a· (pns.. 1 ur(') viu t ht\ l otal t•bmlW* in sttwk tmmlH1l'S ( dy. d::.). ( )[t.nlw rat.PH at'<l dJnrl<'tl
sinmlt.HnPcmsl~~. so th<' ganw is of thP !\u~h t~~tw7 • lt. lH itupntt~ant to notp in dwnsln~ spudfi£- furW( i(1IU~l fm·n.m fC1t ,q 1 (;t,1/t Z, 'Ut)
and !J't,(J',11~ z~ u~) tlmt tlw go,nw sttlWhlrt• of t}w prohk•m is :nnt puml)¥ !•<~(l·· nouth· hut i$" also lndtWll(l v.ia. tl1P degrtlt~ of ''nklw Hlmtinlf that ot·<~m:s 1 )(l... tw<~t'l1 tlw two (ot IUOr<') ht)rds+ .•. _, ___________ _
6c.g. i:lry sheep l'<JUhra.Jent., ttoplcul. livestock units~ n)iitttnl Utlit ~t~t\ts ur SOliW t>qHivaletlt lr!(1i).$~ttC
1'l''h~ d~Jfinition of lmslt <'quUihrium in dyuumir r;~tn<lfl diftr):'f; ronwwhat front t'h<1 swtk lit<'tnhU'P. 'l~h~ <llstint>tioul~ ho\\'(•\'Pt' !l(JUlc\\·hnt t~h11irul. thc:- h:it,t>tt:!;h•d r•.n«:l<'r .iHr(lrr~ttt'<i tu t,h(llit~tat Hrt.,, f<W a disC'ussiutt of indired c·~mttQl ptolMms~ f'~('ll,.t:Jfndwt:;,, ()
ll ls HSHlltll<~l tlmt: tl:w tbP1 <'lU'l.'(~nt Bltlf:(\ of t;hc- Sf;at<' Vfitiubl<' '"}:~\~)tUt<''' is t•(mtp1< 1h~ly nhHN'Vi\hl<\ 1'flw prohl<~'m l}t UtPt<.fort• 'ktmwrt ns a ft~f(,u·kfwill!tt; !Jl'IJtu' or n !/lUlU' of tu•tfn·t al»m''f'lltt/um 71lrt]lf'.tl bzltnt:tr' .iif1ntttr]i(~.r;;~. It: shmild b(' nc>t(l(l t:lmJ, hnUt Frir•chnan o.l\d navis mul Vm.·niyn ('Onsid('tNl t.lw (~Sf$(1 Wh<W(' tlw statt~ is partially nhs<\l'Vnbl{•. P;wth\1 ohsm·vahility t',f t.h<l Rt;ntn vnrh\blc is· l\Unlognus to f1latk (Hl7urn t•mw<\pt nf nwt<'ttlflnlmWlB Ulnt m·c· <tonu:norlly Ulitxl in fish<'th'S prohl<'ntH. Both t("SI1Hl't'(• C'<'.l'lftOtuist;s ;md N'ologi~ts lmv~ t:<•udPd t.o truat Uw prohh.'nt of rmrth~lly ohf«'l"VPd stntt~ vnk'J~thlc'S hy tlslng a disnt'f(l tinn• frmu<'Wnt k. Rt·rkt.l~r SJW~~kitlp~ thiH ls not ~~ c~ort<'<'t nr rignrntt.'{ ttppt'<>ndl to tlw pJ:nhlt1U of' nbs(•rvnhilit)• of t1w b)YstNH. ':rtw Hllpl'mtt1l Ufi<~d hy both [t'tH'<lnuut awl Davis nntl Yarrtiya mailttnlnR tJu' e<mthnxouH finw frnrm.\-wntk hnt nssnmt~ rmls 1mnial nhRt•tWLl'ltli.ty nf tlu~ systNu. Jn. Ul.is piqwr Wt' it!;ttt1tP thi:-~ mnrtt t•utnplt•x .t·tt~(·lJy 'L\).r;nmi.np~ t.hat po..-;t(lrnUsts know tl1tt t'lll'I'PUt hystNu :,fat·<·. ~1.1htk nssnmptiott ~wcrms plnnsihh' f(u trmlitk>w\1 pns1 nralistG wlm ntm~£' witlt tlwir lwnls nnd llPtd hv<'Silh~k mm'P intNto;;iV'Ply tlmu tlwir Attstwtliau nr ~t>rf'h Am<>tkun ecumh'rtmris. nHhongh it nmy nnt l.w nppttHH'int·p tbr t•nntirmmm shwking stt·at.(•~\i<~ woHt fr<•cpwnt'!:r m~<~l hy Australian u.nd NortlJ .Anwrit•;.m ).m~tuu~U~.~t~.
ThP mmh•t l'P})t'(~(·tll'~ a t•nnt hlHOili\ f ilUP annlngtlP of l1(~trillA'H rnork•l of optinml stlH~ldnt~ ih thP Hahd. with tlw ncldrlfi'f(•,~tm;(t ()f <·ommon pt~arwtty. Cnntmnn prnpPl't\' i·· ~T·ru1')0tftlPtl hy using MswArthnr and '!v1ay's: tnodPl nf nklw ovPrlnp. tlw c}(·~~rc•P of nidlt' nvPdnpt(j}Hl!S('ntinp., the d~W'<• nl' <10ntptlt.i .. Hon for pt.h'1ttU'('. ln JWtwral.. Jtht(•n n hlWdt•s tlw p()pUlf.:t,ions nf tlt<' lJ, S{W<'ins pnss<~~ tlw fnUowin.A dytHttuic>H~
\\lWl'H u,, at:<· t·owrwt it ion rw uh:Jw O\'<'dap parutnt,tVI.'H. "rhi.H tnndc~l is n t>Pt-sinu of tlw l.ntka*YdJlntta lllmlPl.
1t di(fPt'!-\ front flw Lotk}t .. Voitm'l'H. l)lOdtll in nw intetpi'(lf;at.iun of Uw pa .. ranwtPtH .. TlJ('Rt~ Htta no lPtlg;t,lr lmterprpt.c~l asplwuonl()tlnl.ngi(•fll (~onRf,~tnts h•tt as int:r~~Tals nl' tt'Hmln't' utlliz:at~irm~
'"'N:iootuun (1072)~ J>. 80. '~.'he tert.t1 'l\lutk4vinn game is llU,dbut;(l(i by Fri~dm~tn tc::. Fl(tmlJ'•l~· Nut(l tlJ(\ term 1\f1wkovinn g~mt(l J)rNlf&tes th~ tc'.rrn ~lllrkuv WltJ"W that hM b<'en pur.mluriZ(Id by l~td<•nb(,rg muJ Th·'*' (lOYl}. CtHI<'rett~lr ill:;, fli'{SUJIJNl t;)u:i.f.; t;ht' drlCt and diffusion tr>l*lnr.; dfi)H'lld uniJ Oil th<- rHtrNH stntt'- of th~:~ sy~tt>m, Also noh' tlll\1> A Mr.dcc:winu gruur tlUfm·n from t\ Mnrkm,· gtmic lJ:t t ht}.t Uw stntt~ dyiu\tllkl1 follow n stoclnL.~tir dlff~r(lnthd eq•mtion
t.~"J j"/.(;r)f3(:r)d;r
Wh(\l't· J\(~t·) tl'\ a om,,<UntPnsimml l'PSutltt'<' <'Onthunutl. tl.Hd !J.(a:) t•ht- );'(\, smu:eP ntili:.mtirm (dNIHHy} furwt.lon \1f t.lH~ i.oU1 srwdt\<?i l't·'Spet~t;ivPly ;i.,i;h S.[Wdf'.Ru.
Not<• tlli~ sy~tPlll of dUft•rt'tt(.ial Pqnntinnslwemrws a. syst~IH1 of no rliffm> ~'nt'ial c•qnat ions if otw :·wts.
wlwn· ··*tf) iB a \\"httP uois{• tN·w nnd J;., tlw UWflU int;thi~H' swn\v1J.t rnt<' of Uw 1 t h spP<'tt~.
By ittlTmht\'ing a ('n.padt:v wnu into this sysl.<•tn of P<JUatinn:; one• nbt:nins a s~·sl'< 1lll 'Jl t•tmph~il ~hwha."itk lnAiHtk P<}ttatJous with nklw ovnriaJ).
(·. f.'·.l' )·· du
th
( r, .. i·t.cc. •.· ) dw1 tlw'l
T\t.if4 lW ,.l<il ('tnul )hW!':i {(tahu·t~ii nf hnth Porrinp,}t opt:imal st;mikJng ruodt'l nml !\.lnfs uivhP tW<•rlap wmhA iu stwh a \Vay u.s tn ali ow t.lw irw<HjJOJ:nt.lou of t·nmmon prnpt~rty i.ntn n (~tn,.hmd.h·) dlffpt;md:inl gam<· nf optin:ml Rto<~khtA·
Tlw ltn apprmwh providPH for a rum·c' gmmral versinn of t;lw st;ntc~ aucl fTausitinn apprmwlt tn rnrlgt' d)"nmnh~s in tJmt it dews uot .iWt'(~sm•ily ussun1<• st.JttiunarHy and tw nlJnwing st<w.k lJ.umhPrs tc• hP suhjPd tnl'fmdnrn fhwf.u., atinus talw:; ittto <'nnsid<•l'l~tions fhwtnations in awtilahi(~ ddnkinp; wat(~r t~hat. ar<'. in mlditlon tn lht<'f·unf·ions ht :wnilabhl forapp. c•omn1mt hl tuany ndd i~n<l ~wmi~m:id \'( 1p;imts.
tn m•dt1t' to HolvP tlw ahnvc~ Bt:rwbu~tk ditf<·t~nntinl gtww of g't'nzing sol1l(!
usc• of tuW\Sllr<1 t.lul(>r.v is tnquinl<l. In pm·t.ktilar, t.lw usual tC1('ltniqtm ofsolv"' ing. tlw H:nnilton ... ltW(Jbi., Bc~thmm <!<JUatian ,is not muployt~t Inst<!ad usn b!l :rrmdf) of tlw Gh:sannv nwu.sUl'(' t;:rlttlafol'l:tl t<'<lhrtiqlw {S(1(! Apptmdi:x:)~ Two aflfll'Oltt'h('}i tu Rolving i:ihwlmst!t" {'Ollttnl prnhlems hfl\'(1 lwcm attgp,~: .. ~t;<~d hr
t\w JlH\UWn:HI.tkniJ~ it mn> \)('of intl1J'<•st. tu :Uoh.• tl1M l'(;r) .is in fntt a (h'i:i(')l44i Inurtiun. Stl{• Adornhm ( ws:.n for t~ dis~usaimt ()f Otr!'f}~$ f\llj('tivntt nnd OJ)«!rUtQr
tht.~ot~· n$:. nppli(.>(t lo Rt udml:ltk diffrrNtt.itlt ('<jUntious.
t;hr- litorutur(\ t;lmt rn~lkl\ \.4q(\ of Uw C~lrsanov nlcn..•nu:o tnn.asfotm tcdltuqn<~. Th!' fitst npprondt knmvn as llw uu,.t.hotl n£ t>fwis aml Va.w~lyu. (1973} .. in .. volvt~ t:ranstm·miug tlw rn:uhlPm int.o n, Hn.tnlltnn.imJ. 'rh(' S(l(~oml approudt nttrihut:vd tnn Km'tttrus. t.dtrwzk): nnd Shtc\vc~ flrtd Oox f\ml Huang[12]is <1{muuott\y u~.<lo(l in Unandal (l(~mtonlit·s. th.is apprm\(:h inv61V(:S tram~f(ntnhtg Mw statv t~qm1.tions into lvhutingr\IJ~ t,lmtPby nllnwing tJl(\ solution nf n stm·hnst it' (•cmtrol £>l'flhlNn nsinp; standawl La.grangiau tf:'dinlq•ws. Both npproadws uvoid tlw Xt,qttjl'<'nmnt of Holving .(\. syst('rn of pattial difl(1f'mthtl £1qttatinus (syst:Nn of Hlttuilt.ott···:Jm~)hi~O!'lltwm <•qnnt,ions).
\\~(1 Nuplo~· t.ll<' ~.lmtinp;ah• t P<1t.niqn(~ of Cox ~tnd Huang tn Uw prolllNu nf soh·ing 1~ stndut.stu· dHft't<'nt.lnl EVtUW of t•nmmou ptop<>tt,v IU';tzing.
Tn flu~ lmst of nnr lmnw1( 1<\fl,\• llus ;tptn·mwb lm~~ not prl'Viously ht'<m ;~pplw(l tn flu• Ntml~· nf !'tm;hn,~tit diffPtc*nHnl ~~n.rm~ m rr'Sotn•c'<* t'l'ntlcmlic proh· lt1!Uh.
3 SolutiJJt"l C'un~hh·r tlw fnl11\WIHJ~. ~tm·lmsth dUf'ttt<•nlial ,t',HltlP
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Iu cn:tlc>r to solt,·c· thiH N;anw WP ntfPtllt~t to transfortu tlH~ tltift tt!ttUH ln tJlwh a wa.}· f'lmJ tlwstafl•t'qnation~hN·m.tw l\Jartit1g:~ft~. this t;henaUowsorw tn a{,ply stm1<1m·(l 'l;IHl~taus~nn it'\·lmiquPs ·t.o d<'t<trnrim• thr• t~.)tinmJ ('nnf.rol ttth~<i of t•iult playttt'll2. Cox anclllnatl14l· lusPtting tlw~w npthwtl ('Ontrol t•nlPs b<n·k imu tlw !'Jhtl'(' P<iHHHon u.nrl tlw ohjN·tiV(• nuw1icU'J AlV(.~ u solution tn Uw J.,: • .nw.
llN~~\llRt' tln\ first. stat<' <'<ltu~ti<)t\ (t»\l'Jhtt'<l dynntttit\'1) is lndt~p(mdont~ of the th0 ('<)nt,rnl vnrinhlP (olf't',n.lw tntc1) dw f(1fmltmxt. Lngrn.nghm l'lf!<~lllOt~ eont.f~hl n. ptt~tnt<' Ponstrnint (tlw shndnw pdef' nf })a.'ltUt<' is ~('l:(')). 'Pnst;ur<' is tlwtnfot<~ ltrUl<'r'Valntl<l and OlW would <\XlWt~l. OVt'fJ;tn.zit.lg tn l'<'HUlt .• \Vb<'t.lWl' Or jlQ1~ fl)ifl m·<~m·s dPtwnds nn.llw ~lynmult'H or t.lH' (•cmtto\ltlfl syRt'NU~ Tlms~ although lmstun'tlyumuit·s tlm'!R unt eunst.rnin tlw tlw hPrdors opt.lmal offU~k(~ d(~i~ision it (hws huV(' 1tt1 hnpad on livt~'itoPk dynmnks nnd 'tk<• vt•J:~m,. ,Th(1 optim.nl st~wkinv~ gnmr~ bPiup; lllnyt)<llwtwt>t'U :twnlt~1·s is tlmrnfot<~ i'ft1.hPr indin\<·t~ and Rt1llWWlmt H\ll)l.\t.•.
lu tlH' !\fnrt.lntwl<• ttpprm:u~h n~t'<l hPr<· ·w<l tlwtPfm:-t~ onl~t Ut't:~l ttl t~ormrdPt' twn Htat<l fi(JHHt.iuu~ in tlw .follnwhlf.\·
l:xomulinp, tlw drift t(1t'tu\\<H l\iVt>R.
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,}' .r As:mrninp; tlu• vru:mrwc· · <'tl\tnt"mtt<·t• mntdx is nouHitw,nlm·· u.ud :tt.H.l.Y thPJ'(\..
Hlt<' lw iJwN·tctd. tlw < Hrhanm· u·an~fonu {in Vt1t't or JnrmJ~
( rl.t·l· )· d:
'\Vt~ nnw irltrndttl'P tlu~ f(•llnwiug risldt~Ss variuhlPs t<1pn~Pnting lwrd <Jy .. umnit·H.
\VP thri::l curry out tlw ibllnwint~ t'lltttl!~t.' of varltthlos {~t;~ ~, f/> apply tl.w hn dmh1 tnlP
7
·w l ,,.(!)) 1.' 1i} rl• t .£.' {:,<try H ! f'IW(U. II}) dli:dJ G(11y) + ·! ron(11· .::) ) lld'.J
.t!{t) '<.~{0) r l~'Jd~t r ·(' l (ft\7} ,, ~t·m·(~;,.:::.))' dth t·l1 (:rrrz) + }.~l'm 1 (JJ.~l)· .. dt1'~
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E\'ahta.tiHJ.~; tht-st· in f'lw lnuit a~ I .~ "X' aml uuting that u · 1ln(:tttt and ... :.;ntu HJllJlu~ that tlw LJLS i~ .:lt-t'o {iLs lnnp, u .. ~ JJ{'Xj. nnd t("X·) .U'P
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.# d,lil !, [f (···· l (ttJJ} ~- ·~ ••ot'ttb .~J)· rlth i r (·.· ·.l (rr!J) 4 ~~ t'al'(U. z)). rill'"' uH1} iu !I Ju . .11 •. ./11 . n 1,# •
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Tht>tt UlHPrt'iU~, I! rUt<i t•\·atuatin~t; af t nand lakint~ \'XtiPf't~~t'inns of both siclt~~ w«· nhtaiu tlJ«· f~iHO\\ luv .•
.um.~ J..(uJ, .x) '" If [irx· U1 (u, 1q )(, ttdtll~~] A [u (O}
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\Vt' n1pn1~Pnt lwnh'rH pl.'<1ft'n·;wns for liv(~HtmU\ (Rerv.h't'R) awl uwat/trmtw~t (otft;nkt!) usiug a Cnhh~Ooup,ln.f.l utility fmlPt.inn in log fot1.11;
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Thin is Jl(lC<!RSt·u:v for tlw <'onstrulnl tn ·tw n !vlmtingnhl, Cmwc\Ulng tho (~xrH•d,atinn nmlint<1gnttinn OJ.>tWnto.t·s on lmU1 sldos. glv(~,
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\V(\ still rw•?d l.o Ptin1inatc\ tlw Lav.rmwJan J>arantPtPJ'H in ordm~ to obtain (\. < lm.;t'{) lhnn t-iohtt ion.
Appl,vhtp, tlw ~·out inunm1 t huP VPn1hJH of BayP'H t lwnrPtn[2fi},
wlwr" in oul t·nxP " u~ :- ~ alhrw~; ns 1 o dtmwp t hl· PXlwd.ntiotl luwk to tlw orif~innl UlPUHitl'P, thiH gl\·p~ {frmu tlw t•on:;l·.rnhtt)~
1 \ ·l /,',)i' lJ 1 ... · 1 >; l ··{/)};. · mtl'y(/)dl!}•l!. I'll • t t
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'l'u:-~Nting ut awl u~ intn tllP~P r<'ntTnughJJl~ nnd Pvahmtiog lh(: :intvp;rals and Jiually niuq llifyhJg yiPl< h~.
aj . JJ(l}lt·mt
tfhi~-t l'PHHlt dliP,W'HtH tlmt lwnh't·~ bmw ofl'tuhJ.~ rk'l'hdous ou L11P <'IUTPUf: tntnl lwrd HlZ<' tlm.l tlw,v wonld t1XJ>Pt·t wnltiplit•<l n g1·owth flwtnr fh1' thP lwrcl in tho. absPIWt1 Clf Pll\'il;nutrwntal nohw mljH!'Itvrl hy Uw d.iHi.'tHIUf rnf,(l, ThiH PXJ,l(lf'f:t><t
10
Htn(·k h~v<'l is an ovr~resUumtt\ nf nd.llallivnst;od<: lJ.lllUh<ll'.S aa it; fails li<> tt1.kc iuto rWl'nttllf. tb(-' dhni.ni!1hing nnturtll lU('J'Htl.,t.;(• or Uw lwtd 1\S it; apprmu~hes t;lu~ earrylng eajml'ity audlt also failH tn takP int;o n<'<'OUllt ntty ptopnr:t,y .rights PX(.('l'ttnili UPS.
In Hw ahsNw(~ of di .. !·WOlU:tl~iug lwrdPtH would t.ll<'H'fm'<' ('Ull t.oo tuauy sf;<wk. Dis(~cmnl hlp.; R<•rvti£i t.o UlnelJorut<• t.lw hPrdvrs op1 huisw by n~tlneing tlHAr off .. t nlw rn.tPB .. Hi1~b dit-wnuuJ, ral;v.s do~w t.n mw . ( 1) f l'nd t.o n.C.J.just tlw. ofrt;ak(~~~r rat.v downwardH hy V<\r;v littlP. J,.m\· dis<>ouut J'NJ,('S (•Jm;p to zero (0) tm.1d t.o a<Um~t. I h(l dhwmmt rat<~ downwnrds hy n P.tPat d<~~tl. lu Uw la.ttPt en .• ~(\ t~h.<~ <. JUnkP r~th• wnn1d lw sruall.
HPc~nnH<·lwrdPt'H vahw t'lli'J.'ttut ut.iJH.v highur by clh-wonntinR; tlw,\~ w.ill t<~nd tn t•nlll:ltock in onlt~r to tht-rPhy s~aiu addltinrtn111tllit);· AI· lnwPl' dis<·cmut rat.ps Clwy vahw tlw Hi hstuJ of h<'twfit.H i.,ht.ain<'<l ftot.u lwldiug UveHtnd\ tuor<• hip,hly.
ttf> <h~IPl'UUllP flu• PXHd iHIJHtd of thiH lwh:wimtt• Ott pastnr<' W<' lWPcl
to mml~7t' tlw «'tllltrolJpfl syHtl.'tn h.\r hl:-it'rtin:{ tlw optinml .(•nnt.rolH lnto tlw systPtu of stcH'!taHfk dHfPrNllhtJ *'<JHHt.ionh. ·rhis lflqntt(~H unnwrknl snlutinn oftlw c'IIUirnllPd h~'HiNu nf<Wkt.t•utiali'<Jltatinns. ThP n•"ittlf.s nf fl1is awd;vr;is nt'P l H'P!i<'U t 1 '< 1 in liPct i nus n.
5 SustaiJlability· ru,d stability
\V'hpflwr or unt. om• fhvours Ptu•lo:-nu;tt nr t·ouunon pro}wr.ty IlTJtu n pnli<·y p(•rspo(•fiw• woul.tl n'quirP Bnllw hOJt of um·n1ai.ivP 5tah1HW1lt. ccm<'t1rn.iup~ Un• lH'rlonunw·p nf i•owwou ]H'OPPl'ty iu HtP f:tn of c•twirorurwutalon<·t•d,ainty.
T.Jw }>lTJhlPHI is how tloPs nrw dPlitu• smtw ('1'it Pda of sttst.nhmhUH)· in n stot•luiHHt· HPt;tint~· Hnsf.u.irmh!P ~~iPlcl ~·ow•t~ptn (P.g. ntnxiHllllU HttshurmhlP yiPld) nn· h:tsPd on sf.<,.tdy~~oif.nt.<! <lR~nutpthHlH whi<'h lm\~t· a c]ifl{•n'nt int{'rprP· t.afinu inn Hlodmst.k sr'ttiug. Binnmnir t>quiHlma alsn lm\'<' lJttlP uwaniHf~ in this HPt.t i.ng ns ~qmrt frow Uw fhd. Hmt Wto HHP utility nttrlnnt }Wnfit !urH'ttnns~ I']H'Y' at'P an f~SH<'ntially a sf nf.i<• c-titi<'riH.
A :tnunhttt.· of npprom·lws lll'(• possilik• Hndr a,t.; mmlyHiug tlw nw;m <•>tit iJUJJ'
h'olH a srt.-;t niunblP rPgion, jtJ our r~tHP. n ual'ural dH>kP ~~ ~·~:~ n.s (.r, 11·· z} f.~ .n:!. AltPrwtliv,{v .. onf1 1uighl~ (jgwwinp; uoiHP) amd,vzt~ tlw stnbilit~r of (.t~, y, z)
m.'naud Utrf rwigin. IuHtahiHt.y 'wnnld l.ltPU hnph" sust.uina.hil.H.y and st.ahHity ,~·nuld i mpl,v · tlmt tlm syHf.<'U.l iK tnumHf.ahJnblc>. Tlw irnpm<t of r•twlosure tuny i.llPU lw mudy~Ptl h;v tiPfifi.n~. H 1 aud auJ.v~ltw. t.lw dynamics nf tJ.tP eont;rnll<lfl ~Wnft>m of sfodm!'tk f.Jir!t•t·Pnl'ial P<JHaliotlH Pit1U'1' lHIUH'dcaUy m; hy t•nr':t~~·.hJ~ uuf u Ntnh.iH I y aualy~is.
CHlwl' r•tHt•du I hat might lw HStl(l iudtHlP nul .(t(•I.Prlniunt.inll nf wlwtlwr
11
Uw dliptl(• futwt~inn of tlw ('nrn'Hpmuling llalniltnn<hwohl .. l)r•Uuttttl <!<JlU\f;ion is n L~'llj)lUHW fmwHnn ('))' t\(){; {S(~{' Siljalt {1!178) fot a h'<~a.t.nwnt, or t.hia t.~ar:w.).
r:l'h<~st~ £'xamplwt all mnoHlJ.t, h:l ('l'it('tia n£' UtP su~tn.inahility (in U.1<>. seus(•) of Uw stability nf thv syHl.t1Ul. hnt nnt UtP n~lli!'lll•<• (Holling) nf f1lw sysf.Pitt.
As mw <'lf tlw authors (BPard) hn» atptlPd PlsPwhotv {u}~ thfl Holl.i.ng ros\Ht'tll't' of n. siodmHt k dymuuit.· s)!'itPtn. is h<'!11; m•nlyr.~lt'l from t.lw l)(lt'SJWctlvn of HH\at\ PXit tiUW front a dPJiUPd r<;Ji;lOil (hm;in nf n.tttn\'t.lnn). \\!p Will not mldrPss Ott' isstw of llolliuv.; .n~HiPW'P in. tltis lHllWl'.
In tlw ahs<'lH'P nr mt npptlllH'i;t{t' <'1 it (Ida of Rttsi'aiuahHity. U\llJl(•rh·at t\tUll .. ysis is lil\<·1,\~ to shP<l smnP lif~ht on how· au'itl'tiunhlP t.lw Ry;;tprn is. 'Ib t.his nud w<~ n.nal)w.<~u thP t•onttnll<'tl dttt.PrtuiHi:iHt· awl Alot'hn.stic· sy~;t<lttJ!-t~ in nt:dr•r to ,gniu sonu.• insip,ht hll n Ow hu}.uwt t 1l diff<'rPnt prnJwrty rights tPP.iHws nn thn (l.vnatnh·::; 'lf t loth 1 m.st m·p aml lwtds,
6 Nultleri<!al A11alysis !'httHPl'kal aual)·ulH or t lw l'OHI rolh><l :-;.y~d.PIH wns t>atric~l oat iu nrdN· tn cln· h'rttrhw what hupad tlw opt hnal snntP~ic~ of tlw h\:n lwnh'.l'!'i is lik({\· tn ltaVP ou thP clytmmks of a com..tuun pruptnly J>aHtnn,w. Jn onlPr t.o fhdlif.at<' tlw analysis of syHtt~nmtit• rttlntiou!'hip!-1 WP initiall~· l'PHtrh·t tlw analysis t'.o f]Pf'PnltiniRt·it· cl)ma.ntks.
Fi~.l shnwH pashH't• Htul lwt·d dynnusks of n c•ouuunt1 pasttu:P wit;lt ;t
5ot;r~ J.>astuni shminr~ nth' 'lwtwt~'n lwnlPrs (pttn~ •··ouunon pt·r•Iwtty) HHd nn httl'iusk pasintP p,rnwth tntP of 2:i\ l'OJt(~Spnucliltgto u.n lUuumJ p~l..<;t'Utf~.
l!PnlPrs tlh·wmmt tlw 11ttnn' at a ntiP whi<'h Pn.nt•spcmds ~tpproxirmtt<~l~· to t•ttll tatP Pqaal {or Hlighll~· lwlnw} flw iuttinslt· g,towth rah, nf tlw lwrd U ~ c t~{. ). this ~<~n·t'S t.o uminfaiu tlist·cm:ut ralPH ln .a .t;PJnth·Pty r!~alisth• rdnHnnship to l~l'nwth rah•s. ~o that tlw lwnl rlnPH nnt t•ollnpsP fhtrmgh nvPn·nlling. Pi,~HtP ·1 slmws thP lwr<l nnd J>astHrP dywunks untlPr ('OlHll\OH propPt1.v~ t·ompal'iHJ~ this wHh l~.iguw 2 •• mw !iPPB Hmt
nlthongh PtH'}Osltrt' lPacls tn a higltPl' hfPady-Htni<'}>H.~t.tU'<'l~·\~et\ th(• varjn, .. Uon iu hoth pastur<1 and. lwrd hiomnss has h't'rt~~uwd. Tlms cmdnstU'(' <~xpmwA )wrd(~t:s fn gtPatnr ri')l\$ thatt ('OUUUOU J>l'Ol>Pt'fS. 'J'hiH i'<'SHlt. iR ('OUSiRfPhf. With and in nwt Pnulh·nu; oue of Uw <rxplanatimls fm· cnunu.on propmt.v thaf; ha:~ lwt>ll nf tlw <'Pl)h•r nf tlw dc1bal.t• in nw statk Htt1tnltU:P{4l.
ltth!rr~ting iB nJso t:lw (•asp illustru.t<'d hs l'•'igtU't• 3~ in whidt tho intdu .. Sic- i!;JYlWf.h rH{(• of l>~t."if.,)lf(l )ta~ lwou itwnimwd h'ClUI. 2.7 to :J lu Otdot· i.wlnt~(1
~~lt:Pat"N Huc·t uatinl.ls ht past.u.rP hlrnua~s. in ord(l.t' t.n ll'prr•Rf+nt n. disllfJUilihrhuu 10A:·~lmplf, fit~t ordrl' Euk~;: ~~ursion sdH'm~ WJ~~.> liSNl., whit·h nlthuUJ~h not fht? Ulo'it.
fi(.'('lll'f!.lt' of J.("t·hnlqw~ lmtt tlw \'htliP of ·>.;irupJii.;~Us.
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Pn\·irnnnwuL ltrulN· tht.~£~ Prit'mu~tl:nw<~~ t·onmmn prnpPtt~"' t>row~ ntort• su&tainal)h• tlmn t•t:wlt,~UI'f~.
Etwlo~UU:'('. lPmi~ tu tlv• t·olht.pst> of thPlwrd hl sud1 a situt\tion bP(•;:un;t\ H tPsults in hnth P,l:(!<\h•r ymstnrP awllwrd var.mhiHtY~ t;lmt-~. im~rnasiup; tlw risk nf t hP s)·:ttt•m t·dllapsuu;. Fl~~llfP 4 pcn1 rnys. t lw latt('l' <·as<•.
lt if-\ huporh.mt tn nott· that this tnllUPrkal mml~fsis is{unn~ tJtP hup:wt nf un1hr· on tlw !-tySt<•m dy·umui<•s. Xnisc' l'(\U l<·ad tn tll(* {btma.llnu (U' (\(•SttlWtlOtl
of Umit <'J'<*lPs cmd ntlwr· nttra<'tnrs. Ht£'l'<•Ii)l'<' il umy luwP b(1th <t stuhiliziuv~ t~todmstit· :rr~lotmneP} nt d•,ssta.hili'l.iu;~ {t•.g. noiH(' huhwPd t•huo~) inflmm<'<' driJwwlmp~ on tlw Htt.P of tlw JH•ttttrhu.t. inn. lu m:dt•r tn a.rudyZ<' tJwsP t•ffet•ts n tnntwt:kul nuubrsis oft lw eoutrolh)(l t-iyntt-tlt nfstm~hastk cllffflll!Utlal t1quations l8 ll(l('(),tif1a}'~~.
Figm·<• 5 sl1ows tlw rt~sults of snd1 au ~uuil>•sis for th<) t•asP of t•mtuuon J>rop!•tt).
<hu· of thP int«•tt~St'iup. lPntnn~ of n(!dinp. uolsfl to tlw sy*'t<•.ru iH that t.hP cl~·uamh·~ nf lh£· two lwnb tnay dh'('l'~~· as a. rPsult nf <~ttvirnmm,nt;nl intltww·<~ o!tlltt> hPrd. Hnnw lu\H1Pts nnw Jw hu~kh•r t.lt.an <lthtlrs.
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h• the ~todm!-iti~· ~Ptt!tu~ Pudo~unt i~ PVPU h~~ dt~sixnl)lt' fhatl iu flw (}tl,.
tc)nuiuist k ~N tu:w, n:., FW,m·p CJ ill~.tHt ratM"''. Etwlnsun~ }p;:ul.s tn thP rnllaJ>St of tlw bPnl aml tn far tuN'P t~rratk pastnr,. •lywuuit'H. Pdval,c• prtllWtt~~ dons uot apfw<n· t'u lhi.uclh~ '1'ihk YPt.V Wt·•ll nt nU.
7 Policy ltni>li<!a.tiox:ts Tlw rmlh·y imp!kalion."i h('(IUJ ·dP(U'. ;.\f tPntpl~ at ()nfhn~<.'<(l NH'1oz;ntlt in UJany dP\'r'lnplrw, <::un.uttl,(t!-l lm.:;(• tw<'U Jm~t·if.n'd lu f.('l'WH of tlw h'~Jp,pdy nf tlw <'OHl· xuon~. CmlHidPrnhlr- tlu\m·r-tknl. nnd <1TJ.J})iti.<·al ttvhknrt•lu1..1.l t•.aU<I(l lht• u:ago<ly nf flw .('C>nuuons Into qtw.stion. 'rh<• b.wk t)f hioet~nnomk rnml<•ls of Huutuou ptotwtty grm~iug nnd tim ntilizntinn of statie unt.l dPtk~l'lni:nistk ttwtho(t'i hnH nwnnt that,, rmly J>nttit'nlur cu:iJWds ()f tlw Nwlmnu·c• vro.rsns c~c).ntU.tnn l,ll'Ol_." <•rty dPhat<~ hav<~ tw<1fl di~t·t.lHH<'<l. aud't as l'Pnt. <Hs~c;ip:.\t.Jn.n. St'f\ti:t~l·itJg or fi(~a.lH ('('nnmuh~. ln t·onHt.rlwthx~\ n. d,v.ruuuk Htodw"~tk Juml<*l of <~umulmlJ>lOJWrty Ju·aziuf~, WP lnh'P bhown thn.t t·mumon prnpntty has u~ unuJlwr nf n<h·ant~tlJt~~ ovrjr privntt\ protwrty wlwn mw h1kPH risk lttt.o at'<'«mnJ. ln a rnwlnn1 PUYirnJJ"
tn<~tlt, ~twh JJR nn ndd m: ~t1mi··add tangnln.ttd thnt tmw lm in <liR<!<J,UUihtiutn {n(>nst;nt;imUl.t'Y dist;rillnt;lmtR) W<' ftnfl t.bnt prhJllt<' ptopurt.y is not, n sna1;a.ht:"" n.bl<• hmt;H~udcm chw t.n it •.s pnor ahiUt;,v tn lumdh' tmvh~mtttU}tltai liuntuat.iona.
ll is tlw 1·iak prnpPl'th~~ of c•mumou 1n·opm•ty Bystm:us t.hnt mc.plain t;lwit t•olmst.utlss ~t-'1. n nwm1s nt on~unizing gruzing tight:~ in llUU~Y ~wifll.lncl ~m:nl .. at'Hl r<tgious nf t.lu~ world.
8 Conclllsiotl ln this parwr., W<' haNP t-tnployPd n 1\.tart.irmal<\ apptmltih t:n snbtP a Htoeh~l.Htk dif.f<~nmtiul J!,n.tut• nl grm'lll~\ n t•onmmu pnRlJll'P in a random nuvirmuuont•. NunH\tif•fd mmlyHI~ nf tlw ronttnUPd dPI.('ttnini~tk H,YRtmu wn .. 11 ttwn naml to ntml~~rt' tlw imtmd nf Ptwh>snn· on thv Pc·olopJcnl n.nd £•nltnrrd sustaiwihilit..v nf: a t•cmuunn ptotwrt~· sushsisten<·t• paHtr>ral ~WRt:om. ~"('hP rf'HttltH arn (l(msistnnt. ·with t.hP Btat it• lit PrtttlnP nn <·mmuon pxopttrt~· and thP d~'lHLIUk Jit.Pt'ntun• em nptimal stm~kinP. uwlt'r snl<• owuPt'Hhip. At' t.lw !Ultn<' t.inw howPvr-r .. Uwy J\tr .. tlwx t•all htf.n q1wst inn. nw aj:iptnprint·tmPss nf tlw Htrup~Pd~, C>f Uw t•onmtnns'u US U t \wor<•t.knl htL~i.l'l fr>r pnlicy dPt'iSlOtlH. \Vhil:-;t PUJ'IC>SlU'(I J<~adH t.n hit!,lWl' lU<'Im J>nstnn• hicmmss~ lwc·nmw tlw trns:wdy of t.}w (•muntmts tlw<'ll'~V involvP~ n larg,•ly stn:tk smal~~h~. it ignon':i vnriatinns in ps.t.~o~tutf• blotuf.\SH. AH a uwml:-; of nmtm~~inp; risk £•nnuunn Jll'C>}Wl'f'\· Jms advautngN; ov<~r t'ttdmntl'(1 in wrrus of lnwot varinhllit y of Ht(H'k rmmht~n .. t~ucl pnHtltrc~ hirm1m1~.
A nuruhN' of PXt(1llHton!1 tn tlw Wot'l~ lH'tl!inntc~l lwr<l nrc~ intnnd<~L ~Pbmm hwlndt• Ulln1~*His of tuJ'HU rxit tinw-,: ll'tnll hu.Bht~ or att,rrwtinn. ExJNtdinp; t,}w ;:uml,vsls to itH·nrp<lrut·<j :;:Pmi·.:·nlhsistt•twt• <'(•nmmrht.S J•n.tlwt tlum prirt~ HltHhsiH .. H'tl('P ( 1t·(>i·lntlliP~, nnd t hP mmh·'~J~ of m ixPd H~*Ht Nus of prorwtty rights sndt tth
trn.nslnuunut Jt;tm~in~~· 'rhP lattt't' will t'Ntnir<\ Llw itltH>dttt'tJrm nf R<U4'10Wtlil~v mul nudf'iph~ P('illn~r.kn~ Jlr\tdtf'!-1 into tlw modPt
Gbnmlov's UlNl.}:lut~<\ u·anafonu t.Pdtniq'w t)x.plnits lmt,h tlw H1~don .. Nikodyul tb<"lOrPru fr()Ullnt~nsm'<• t.he<JtY aml t~lm Cftnuwou·'lvintt:iu.lhrmul~' (ttanslat~ion t.h<'onml) ftolu ptolmhility thr~oty t;n trnnsform cnw ptohnhnlty U!~aBnn~ int;o unoth<'t' l>)t wn.y nr fl Unmn· t.r;~tlSform or t.lw drift tm:nt ar a stoeha.,~.Jt;k di{f()r(1flt,i}\1 t•qnu.tion.
Giv<'U n nonmlfi<'lpaUng lJrowniml (\Vi(im~r) fhtwtionnl O(t,w) u·· 1 /(~yt) tlwn tlw following fttHt'timml known. ~~s tlw luulon .. Nikmlym dedvaUv<~ Jua.y lw dfllitwdl
Htwh t.hat di> · r(mu~.;Jl,u·· nt1d .Erl<W!twH L Tlw pro1mhilit:v nwasnw f) pos1wsst~ tllP following JU'<'~>(•rt.iPs~
i) /' lt~ ahsnlut.PlV t~onf'inuouH wilh rnstwt•t to P1".
H) Gh'Nl a \ViNwt·ptnc·<!!iS w, dt!Umd fot: fh<' uw~umrc• f> t:hr•th
Usilw; tlJls f'<'<·Jmiq'w tlw t'XistPtH'P of a un~al: H{'tuw nptimal t:ont,l·nl 'JUay lw shnwu12,
'tlw dmkP of/(J•, f) Hhould tH• xuudc~ ln f'nwh a Wil,\~ as shuplif.y tlw ptoh .. lt~ut. (•.g. lhwm·it<1 th<• dUftfrPntial <'qlmtint\H .
• ,-,,.,~ ... ,,-,,.""'• ., .• -..,~""'''''"'' .. ~""""~'"''•d .... ~ 11-l\'>·11"'"":~~ H' (.X. A.} ls A tl1NJ.'lUi'~bl~ spMc.> ntJ(l, j) rmd P positiVt+ mcmmtffi dclb1ed ox! (X. A) tb~xt
P itt ilh';olut~ly ron.tlnuuns wlt:b rt'»ipPd~ to P if for C:!Udl S('i. A i.H ,1 both PfA) .-:.- O· ~tld J1(A) ·~ n. !$(1<~ C~)lm (lUHO): p. l3l.
lJtl h(,• l111lH wMk i'!\""tlc:;-(1 f'(\(i:•ts t.u t.ttt <•plhnal r;oluU(,m: uf t.lw pt<Jblenl t•Xil'it'ing hl the s<mtre o[ Hdtwt~rtz tU~t ribuJhms. S(1• lU~b(iJ (l97l).
20
References Ill Adt'>tuio.n. G. (1983) 1 Stochnstu· .Sustrnu;) APndJ'lttiJ• ptOHR~ N(1
W Ym:k.
[2] Bnsat\ 'l\ mul OL'1(lPt\ (l,~t DtJna·mi'' N(mcN11U'tYtli11(; Gam.c• 1
tlwtn1J, A(~n ... d<~mh~ Pr<~Bs. Nnw Ybt:k~ l 082.
gl} I3Ntrd. R.~t {l90l>~\)t 011ftmal Stot'l.:in!J?.n n Buu·t·mu>rn.ie Morlrl of am.z,. it'I!J Ulif.h Phmt .. Hc,·M·tl()t(~ Cmn.z)t•fiUon1 l'¥npt•r ):1\:(lS(\ntt'<\ at. uu~ 39th rm ... 1mal ennfN~NH'<' nf tlw AnstrnUnn Agl·h~ult.m'.al·l~·mlnlnkH S<wit~t.y\ P<'t·tJh
Vdll"llittY· 24,~l'r.
HJ BPnl'<l, ltl\1. (lQOtth). :tl'hr I~<'O'twmJc ~l'ht•t•t1J of .l~rfl'liB'i1'f~ Ptuilmnli.sm, Di:-;e,ni~ion PHpt~r No. :~. Dil'it·ns:{icm 11apt'l'h in Ap;rk\lltm~al Ht•nnnmks. UulVI~rsH~' nl qiWPtl:1lnncL
[h}lkm·d. lt,~t (Hmrw)~ Rttl1'0Jl!J; SloJJilifJJ urullk."ltlil rwr~ of !Jwermwtn'it• Su.'ilr•·m.'l: A Nt~twm·A~ Tht'1'1twrlunn.mir·. w·t'bJWcftttr:~ Papl'l' pt'clfiC'llHll:1 at'. t~h(\ llmngnt}\l ConfPrPtH'P of t;lw Aust.rn.lhtH and NP\V i~<·almul H<·nlogif~al B(·onntnic.~s Sm'iN~\ ('~off:-; nm~bnnt\ N(WNU1Wl~.
[0] D('f\(l,q~ VJ~. {1071). f~;n:'iltlifH't' (>I Opfnn.al Hlut'luurNc Corrlml La:ws, SlAl\!1 ~l. Om1trn1, 9:!~* pp. 4·t0·472.
[71 B<'nllnhilh ~1. a.ml Jladrwr. H. (1002). 1'hc Jowl l~;rplmln.lwn of n, Prn .. tludit'l' A.'ls<•l: A Ornm·, Thfm'el'ic A1ltJr'Ofltll. Bt•t)t\Olnh• trtwm·y 2. pp. 15f,,tno.
(H) l3otkat·~ v.s. nnd Cllmsh\ M.l\ .. (1002)t Sftwlnu;/-w n.1jft·1'Nll7lll OartU*l5:
Oc(·mmlitm Aft1lUnt1't> lln.~nrl AtJptmu·h, .Tonr1ml of Optimization 'Them~y tmtl Apr;lkntion~o~. vol. Til. no.2. Pll· 350·m~5.
{H} Clarkl C.'.V., i'rfc)./hmrw.lif~d l1im·t•ortr1m·in.clt .lohtl \\~ilt~y ~~.rul !iOJ),'1. NPW
York 1970.
{lOl Cl(!mllmtL, S. H.nd \Vnn. l:tY. (1985), Dunarnit• Oomrrwn .l'ttJTJrrf]J l?.t'"' $(HL'I'l'Ni urul Btwitym:rntmJal Pmlilems~ :Jrmnml or ()pUmlzaHon trlwm;~t nn.d ApplkaHcmH, ~10:4. pp. 47 t ... un I
(ll\ Coltth D.L.~ ~~(!turu.f·c ~r/lt'M'1b Hirld1ihnmr. Btrl-1tnn lORn.
21
fl2} Ctlx, .1.('. n.nd. U\W.llgt c ... Jt\ (1989)\ OtJtimttl ('!MMUtntJtion aruJ. Porlfo .. lw Pnlit•iN, 1alwn A sse I PfitJ{J.,'i H}llmt• «t Dilrtu;ion PrruYJB#~ .lont\ud of gr·om)tnh·'rh{'<wy Ml~ aa.sa.
{la] 01wl~. f\.f.ll.A. ( 107:\)~ On lhc B;ri,~t~~r~e•t• of OJltimnl Poli!'i(w 1.11. StoclwsUc t•m;tml. SlA?vl .r. Cont.,~nll1!·1, pp. ns; .. not
[1~1.1 ll~wis~ 1\t.ll.A. nn<l Vf\rnlya, l1• (l07a), Dunamie J,t'OfJttmunirtJJ ·omuli~ fhms fur Pm·tinllu Ol>.qN"t•nblr) Sl.ocbashc $1J.'lh1t1u;~ StA'tvt ;J. Controlll:2) pp. 22(), 2Hl.
{Ui} H\llot t., n. (1070)t 71w f?.ri.91NU'(1 t>.f Vhltu· in Sl<whn.'li'it• J)tjfmrnf1Jd, Gnnws, SIA!vl ,J. Cohttol m\d Opthu~rnUon l·l!l .. pp. Sf>.,{),J.
[H1} l''rh~d.ulnth A. (Hl72)~ Stodwshc· Di11'(Jn'nfwl rlnmr.-;, ~Jnurual nf DUtl1r .. <mt.lal f;~qnatimu~ ll·~ pp. 70·' 1 mt
[17] lt\ulN•lwtl~· D. o.utl 'l'itnlt•, J .• <imtH' 1'lwm11, ~rrr PrPss~ Caruhrhlp,t·~ 1\iA. HWL
[181 Unrd, ~r. lr1/rurlnd7m1 lo 8/orlttv,tu· Dl,,lTf'1'(•ttlh.ll RrrwtHuuH, ~fnr•·t~l l)pl~kPr NPW Ynd\ 1!1HB.
[10] Gnhtt.~'·, :}.G. an<l .lohmtt.nn, D.1"'., {l~lls.). 'rlw H'orM of J'n,'l·iom.li;~Jm .. Hl•lhnVl'il l"'n'sH. Lrmdnn lUOO.
[20] OitHnnov, LV. (HlH0) 1 On ~tr'anB.{onwin!l ,A Cf"rlnw. Clos:.; ()! 8/ol'hm~lil' Ptm·eBsf's IJ1J Almultddy ( !onlhnwtt.'l' h"'nlud·li'Uficm af A1rasm'ui. Tlwm)~ nf ProhnhilHs mulltls Applh~aunnH 5:a. pp. 28n .:~01.
{21} Gh,it. A. (1078), Op/im.a.l Har'l'l'Hi'itl.{l m r·~Ufi~l·tiltW?l..'l ~tinw )tmtll 8toclws ... lie· amwth~ ~JnHwnulth·~d Biolit'it1UCt~ It pp. UJ ... 12a.
{221 llPI't~lPI\ G. (HUH). Dunatnir· Dn·i.~iflr>.'l 11nder JN.';k: At>t>lil'tt.ti()'n of .Tiel Sftwhosi1c• Coniml in Agf'irul11l1't\ AlilPdt<t1l .lmu·rml nrA~Tit•ult\tral E(•()~ nnnli<'S, Nnvt~tulwt\ pp. ll2f3 .. llH7.
[23} Ilnlling~ C\H. (l07a)t Rr.'1il·u~1wc• und 8ntbiUtr1 of E:colc1[/kctl S;lJ.qfnnH, Au .. mmt lh,viow o.f Eenlngy and SyHt.PUll\th~~.h Vt1l. 4. PI>· 1 .. 2~t
(24] :JohtiR(ltl, D.l~. 7'/u· Nalnrc" of Nommli,~tm,· A cmUJ'twn.ii1.'C' lc;ft1d11 of fWB .. lonll ·mi!Jt'ttlirntH i11 Brrulh1t~(·,~tfC't'n .4.CJin nml Norllul?'f1, 11/r·icn, 1JniV<1t:~ity of Chkng,o, llPtmrtuwuL of GPI1~taphy Ue~~~m~<·h Pntwr no.JlR. 1900.
[20} l\1\rl\tiZU.S, l. fUHl Sht<'V'<\ S. Brmm;.i(t'il motion mul Stm:Jw .. ,tic Calt~ithw, 2tld <~<HUon, Sprin!~<~.r ... VPrlng, Bm·lin, H)Ol.
[26) 1\on<.•zadd, Z.A. ~l'lw l~'mrwrnic.IJ o.f Pa,-;tornlimn: A OrxN~ StucltJ of Std,., Salwnw A.fricn, ll\·nuk Ca .. lit-~ u.t1cl C()., London :t 078.
[27} Vnu Koottlh\ O.C~ mul Kinyua, P. (1905) 1 l?ttnge J\1(tna.fJtlm.mJ O'n Prl1mte l~omds in }((!fl.1Jrt lo Profl•ct H1ildlt1t Po1wlat·iow~: A IJynami.r• Sittwlnt:lon A1mld 1nith l;lt.zzu Gont.rol~ Univnrsit;y of Bdiiish Col.umhln., D<!JltWtm<~nt; of At~l·ku.ltm:nl Et•(luomkH \Vorking PapHl' #05·1
[28] Li1
tl-1 ... (lOPii), Stnbilt17J rrnalJptis of rt noulwnw.f}mwns Mnrkmtian larul·· Bcapt mmld~ Ecnlq~kn.1 Ivloddlirtg 82, pp. 2·17~250.
[2D) Ludwit~' D. (l97Dh Optmwl.llm·t•f~Ht·in!J of n Raruloml;y Ji'luf'i'tta.UntJ Re·· saurt'l' l: Applicnlum u.{Pt,tlnr(Hdion Ah·lluulH, SJAtv1 J Appl. .Mat:h. :{7:1 pp. lGf). 18-t
{30} rvtanp;d, 1\l. Dn··zsim/, .mul Ooniml h» Uncer•lrlhl; /ll'!Wttr(~(' S;tpdctnH, A(~~\_ .. dPtnh1 Pn~Hs. NPw Ycn·k, HlH5.
{:UJ t'rll:L)'\ H.Jv1. (Hl7~l)\ On llu~ 1'hcm!J of N1:rlu· Otwrltq)1 tl'lworet.it'n.l Pnpulntion llio1op~t o, pp. 2H7~3~m.
{a2} Ps.msmnr<\ G. and Brown~ C'.G~ (l!J91). AnnltJ.'~Ur of BlLnfJ('l(J:rul D(~Jm .. datwn a,*>ing Stodwsfit· ~luntJ,Wil· Pto!J"'ttmmin!b At1Rtl'(ttian .Jonrnal of Agrkult.m·al gt;.ClJlQflUPf-h 35 (2), nng11st, pp. 131-157.
{H3] P(\rrings .. C. ( :tmH). Slrt·.-;BI shm k o:ml I hr Bu~t;lnitw.biltly ofllt•sourc•e U1 ~·· in .w:rni u:rid E-n•t1Jtotmtf!71IH,_ 'rlw Annals or Ht~p;iow~l Sdmll'(\ vol. 28; pp. Hl .. OH.
{3Jl PirHl;v\·k, H. (Hl8·l}, lhwt•r/(t'inlJJ ·in /he ':tlu'm7J (1/ JlnwwaiJlt~ Ilt:~1ontr.rt J,\1/n·A·t·IY, UnviPW of B<·onorunk StmliPH. pp. 28{J .. 30:3.
[!:~5] Hisht•t n.·\V. (1071 ). ~J/Ntk Boluti:mu; of a Pnrli<tl DiJfnn:mhcd Bqtw.Uon of J)yna.rnw Profltam:rning~ SlA11~J. Control Ot·11 pp. '51{) .. 528.
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!37] Siljak~ D. (HJ7H). lAn?lc--Bcale J>wwmic S;tpJtt?nm~ North--Holland~ N<!W Yotk.
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[m>} lbrolt L.A~, Lynn, J\.S. and Ck)(lft{\y, B.n. (Hl91), Long-1·1m, tU!T\':ru.s
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[·l0)1hrelli; tv!. (1977), Random E'rwtmnmJmt~~ ru1..Z Sloclw.slic Caleulu.s, ~rtw .. or('tka:l Populatinu Biology 12, pp. 140. .. 178.
[4l] Udthla, K. (1078), On B.ri.tlfcru'(~ of a, Nw;h Rqnilibtium Poiul in N .. Pent;an Notlzt•ro Suw Sllwhnsfil' DiJfr'tr•ulud Oam(~s, SJArvi .1. Oont;to1 and OpHnd~rtt.irm Hkll pp. H2 .. J. •. m.
H2l Vm·aiya, P. (Hl7G). N .. Pfa]Wl Siochm;fw DilTt·rcnfial Omucs~ SIA~f ,}. Control a.nd ( )pthuizat.iou 14::3~ pp. 5~8~5·.15.
[~.13} Vitt;ala, ~t {1992). Optnnal 1Iantf·slnlff of a J>lr;,n.f ... J!r•l'bi·omv· Sy.slrw: Dil'!w·n mul R(•indt•N' 1tt No'tlhl'Jtl. Fmlrwd, Ecnlngh·a}, l\;fnrl<:~lling vol. 00, Pl>· 23i{..255.
[4!1] \V<'tStohy, tL. \Vhlk*'r1 BH l'i(w~!\.1()irl t. ( H)89a)~ Opport·nni.sUc ttui.rUL[/C!"'
nu:nt for rrw,qr:l(lnd.'i not' at rftnililwimn., ,Jnurnnl of Hanv,o 11anagen.mnJ, 42 (4), July~ pp. 206~27-l.
[45] '\Vpstohy. }~..f.~ \Valk<~r~ B .. Noy .. fvlnir, L (1089h). Ron[J(~ A·1nna!J'mumf on t.lu! f,rt.<JL" <Jf ll Afmld 1ohirh dotr.'J ·not seek I o tl.~?tablish l.tn Ct}n·ililwimn~ Journal ()f Arid Envirotmwuts 17~ pp .. 235 .. 2SO.
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