Control of a Lake Network Invasion: Bioeconomics Approach

Transcription

1 of a Lak Nwork Invaon: Boconomc Approach Gra Lak Invaon by Aln Spc Al Poapov, Mark Lw and Davd Fnnoff Cnr for Mahmacal Bology, Unvry of Albra. Dparmn of Economc and Fnanc, Unvry of yomng Zbra Mul Ruy crayfh Sa Lampry Economc Impac: Avrag Co o Zbra Mul by Plan yp a of 995: ydrolcrc facl $83,. Fol ful gnrang facl $45,. Drnkng war ramn facl $4,. Nuclar powr plan $8,. Damag o Rcraon, Chang n Ecoym Bach covrd by hll mll, clard war, l por fh Cruy boa Zbra mul prad 988 3

2 Ruy Crayfh n Norh Amrca ow do nvadr prad? o h Gra lak varou way manly by hp n balla war. hn h lak ym naurally o land lak and bwn hm manly hrough fhng and boang qupmn. Prvnon qupmn wahng Analy ung opmal conrol hory rarchr from 5 unvr: D. Lodg and Gary Lambr (U Nor Dam), M. Lw (U Albra),. MacIaac (U ndor),. Shogrn and D. Fnnoff (U yomng), Bran Lung (McGll) 5 yar proc Collaborav proc bwn bolog conom and mahmacan hp:// Clark C.. Mahmacal Boconomc. h opmal managmn of rnwabl rourc. 99. an Koon G. C. and Bul E. h Economc of Naur,. Man da: Modl an cologcal ym a a dynamcal ym Includ human acvy and co/bnf Drmn h opmal harvng/managmn va opmal conrol hory Invadr dynamc + co/bnf opmzaon (ngrav boconomc modl) Prvnon/ Epn Populaon lvl Inrocon ranporaon Dpral Opmzaon Lo/chang Co/Bnf Gnral nvaon modl wh conrol Modl nclud dynamc of h nvadr n h lak u, pobl conrol mnmzaon of co (or mamzng bnf ( u, h ) + w ( u, c, c ) F [ u, h, c ] mn w, Mnmz co or mamz bnf

3 Macrocopc modl for nvaon prad Invaon dcrbd n rm of proporon of nfcd lak pn I /N. Invadr propagul ar ranpord from lak o lak by boa (nny A ), probably of urvval A, ncra n numbr of nfcd lak N p rng (N N I )N I A A, Ap (, A A A N Invadr Prvnon ffor a nfcd and unnfcd lak: and (ffor/lak/m). Probably of propagul capng ramn a nfcd lak a, and a unnfcd lak b. ahng ffcncy a, and b. Aum ffc of wo uccv prvnon ramn ar ndpndn: a( + )a( )a( ) a( ) p( ) b( ) p( ) Dynamc quaon for proporon of lak nvadd: A nfcon no ra conrol wh a( ) b( ) rcon n ra o conrol ffor Ap( k p( nonlnar dynamc ranmon ) p( f (, Co Dcounng and opmaly Invaon co: g ($/lak/m) dcra n bnf or ncra n co Prvnon co: w a nvadd lak oal nvaon co/lak: C w a unnvadd lak gp( + w ( ) ( ) ( )( ( )) p + w p nvaon lo conrol co Co funconal Dcounng funcon oal co rng m nrval : Dynamcal quaon for proporon of lak nvadd p( (opmzaon conran [ (, ( ] ρ( ρ( p( ) Opmal conrol problm: A Infcon ra wh no conrol a( ) b( ) Ap( k C(,, mnmz by choong ( and( ) Dcounng Co dpndng on a of ym p and conrol and Rcon n ra o conrol ffor p( Nonlnar dynamc ranmon ) p( f (, Goal: mamz (amlonan) Dynamcal quaon for hadow prc µ( wh rmnal conon Dynamcal quaon for proporon of lak nvadd p( wh nal conon Opmaly (ma n, ) conon a any Mamum Prncpl (, µ ) C(, + Currn valu amlonan r µ Dcounng p Ra of chang of hadow prc Ra of chang of proporon lak nvadd f (,,, >, > Growh dynamc Co funcon Chang wh of p µ prc g(, µ ), <,, <,. p() p Inal proporon of nvadd lak f (,, Nonlnar dynamc Growh dynamc µ ( ) Fn m horzon w p k µ w Opmaly conon ( or, A p < ( ) ( ) p k A p p or, µ < hr yp of conrol. ( µ,,, p < p S. Rcpn conrol ( µ,,, p > 3. No conrol, µ > µ S ( p S 3

4 Non-ovrlappng conrol rgon rmnal m pcf opmal racory -conrol -conrol No conrol g( ( µ,,, µ ), f ( ( µ,,, g(, ( µ, µ ), f (, ( µ,, g(,, µ ), f (,, p Fnh hr a m, pp prc Proporon nfcd lak Rcpn conrol mall: No conrol nrmda:donor, hn No conrol larg: Donor, hn Rcpn hn No conrol h currn valu amlonan mamzd by,, ( µ,,, ( µ, Sar hr a m hn hr no dcounng (r), oluon can b calculad analycally from f g rmnal m p p p Proporon nfcd lak For any gvn, hr and opmal racory wo dffrn pha plan rprnaon (p-µ plan, conrol-p plan) Proporon nfcd lak Effc of h dcoun ra Proporon nfcd lak prc proporon nfcd pha plan prc Rcpn conrol prc Rcpn conrol Sold ln: No dcounng (oluon calculad analycally) Dahd ln: h dcounng (oluon mu b calculad numrcally) co proporon nfcd pha plan co Rcpn conrol co Rcpn conrol Proporon nfcd lak Proporon nfcd lak Oucom wh and whou dcounng hou dcounng lvl Proporon nfcd lak No conrol opmal h dcounng lvl Proporon nfcd lak ffcncy kk k vard. hck old (, hck dahd (, hn old p(, hn dahd unconrolld p(, A, p.3, g.5, r, 5,. 4

5 Concluon- can dlay nvaon bu no op. Goal o dlay nvaon o a o ncra n bnf from a boconomc prpcv. Problm can b analyzd ung pha plan mhod. hr man rag for conrollng nvadr:, rcpn conrol, no conrol. Swchng occur bwn rag a h nvaon progr. Shor (.g., polcal) m horzon can yld no conrol a opmal. rag ar nv o dcounng. Dcounng rc arly nvmn n conrol and allow nvaon o progr quckly. C Modl non: radcaon A p ( h(,, p > p ( ) gp( + w ( p( + w( ( p( ) + whh nvaon lo conrol co radcaon co C lnar n h, bang-bang conrol: h or hh ma. h w h µ, wchng a µ w h n h pha plan Nw knd of oluon: compl radcaon If w radca nvadr by om momn, hn for > hr ar no lo and no co. Nw formulaon: fr rmnal m, fd nd a and hnc. Dffrn boundary conon C(, h) mn µ ( ) whh( ) µ ( ) h( ) ( ) wh, p ary of oluon: ochron vw Iochron of all nal a (µ) uch ha µ() rmnal valu: byond h conrol horzon A h coym rman and ll can brng bnf mu hav om valu (p ). hn ncary opmz co+rmnal valu. L a ym wh nvaon lvl p undr conrol ( proc bnf wh a ra (), hn w nd Compl radcaon racory Iochron wh appropra w 3, h.5, g 3, r and r ( k ) k h Bgnnng of opmal racory Bgnnng of ubopmal racory Eradcaon opmal Nw ffc: vral locally opmal oluon. Compl radcaon h opmum only for bg nough. r r ( ) + ( p( )) ma() ow o dfn (p())? No agrmn on h a prn. 5

6 Invaran rmnal valu L u dfn hrough nfn horzon problm. p()p. Dfn ( p ) ma () ( p ), alu prn co of mamum fuur bnf undr opmal managmn hn oluon of a fn m horzon opmal conrol problm wh rmnal co (p ) concd wh ( on (,) (( do no dpnd on ). Can b formulad n rm of mnmzng fuur co undr A oluon of an nfn-horzon problm (IP) nd a an nvaran of h dynamcal ym. horm. L h oluon of IP { (,p (} and unqu for ach p p() and h corrpondng nvaran nda. hn opmal conrol ( for fn-horzon problm wh h rmnal valu (p()) and h am p ( ( on (,). r ( p ) ma ( ) + ( p( )) ( p ) ( p, ) + ( p, ) r ( p, ) + ( p ( )) Ehr ( (, <<, p()p (), or a conradcon r ( p ) ma ( ) + ( p( )) ( p ) ( p, ) + ( p, ) r ( p, ) + ( p ( )) Eampl: no radcaon r. r.7 Proof: uppo p() p (), hn >, hn ( no opmal, hn ( no unqu <, hn ( no opmal ( (, << (opmaly prncpl) g < + r A w k g > + g.5;.5; 3. r A w k Eampl: wh radcaon No radcaon a h nd r. r.3 Compl radcaon Opmal racory Subopmal racory r. g h.5; h.5; w 3 Implcaon of rmnal valu for h problm wh plc paal dpndnc ( u, h ) + w ( u, c, c ), F, [ u, h, c ] + C mn Opmal conrol problm ym of N quaon; Infn-horzon problm only ady a ar mporan; a mall dcoun look for h b ady a; May b a condrabl mplfcaon: fr udy ady a hn choo a b way o hm N 6

7 Accounng for All ffc All ffc wh rnal flow All ffc populaon canno grow a low dny Canno b ngrad no h macrocopc modl Sngl lak dcrpon ( u ) + w, F( u) ρu( u a)( u) F No rnal flow, populaon go nc a mall u ak rnal flow, w< F mn, populaon ll go nc a mall u; Srong rnal flow, w> F mn, populaon grow from any u Eplc paal modl wh All ffc F w u ( u ) + ( ) w w Opmal nvaon oppng: fnd opmal paal conrol drbuon ha kp flow blow crcal a unnvadd lak can look for h opmal plac o op h nvaon B + u Eampl: Lnarly ordrd lak B B ( ) Numrcal oluon gv paal drbuon of conrol B B p( γ ) B B / (+(γ ) ) Concluon- Eradcaon of h nvadr can mak h problm of fndng opmal conrol mor complcad and gv nw rag; rmnal valu hrough nfn-horzon problm rc analy o ady a and racor ladng o hm a condrabl mplfcaon of analy pcally for hghdmnonal problm + mor ranparn managmn rcommndaon; All ffc allow o op nvaon whou radcaon; accounng for h rmnal valu lad o h naural problm of opmal nvaon oppng Acknowldgmn ISIS proc, NSF DEB NSERC Collaboraon Rarch Opporuny gran.. Rfrnc A.B. Poapov, M.A. Lw D.C. Fnoff. Opmal of Bologcal Invaon n Lak Nwork. ournal of Economc Dynamc and, 5 (ubmd). D.C. Fnoff, M.A. Lw A.B. Poapov. Opmal of Bologcal Invaon n Lak Nwork., 5 (n prparaon). A.B. Poapov, M.A. Lw. Opmal Spaal of Invaon wh All Effc., 5 (n prparaon). 7