Optimal and Extreme Economics with polynomial model

Transcription

1 Dhbe Brch of Nol Reerch Techologcl Uver «MISS» Dere of oer Scece Ol d Eree Ecooc wh olol odel Mhdf Y Dhbe 3

2 I he lovg eor of re d broher Profeor Docor of hcl h. Sc. M. Y Prefce The oogrh devoed o vego o-clled Eree Ecooc Model corced b hor d reer redcg o he lgebrc rereeo N d correodg wh ree of ber rooed d veged Polol Model for of crve hgh degree P N A Y Z A Y Z ele whch be coderg odel ecooc he for of d d d Af d d M P v P P d da : e d Af A A A A. d Edor Dr. Sl Afr

3 3 oe Abrc.. 3. Rereeo of cole obec b Polol odel eree rege..4. The corco of ol Ecooc Model The ol odelg rodco cl of reglro rodco of obb-dogl O he eergec heor of olo growh Polol Model of Develoe of oe he Wor odo b d wh og Selee - odfco ehod of he ere eghbor Eree Ecooc d ecooc r Model of erco of core wh dffere ecooc level of develoe Model of olo rblece d crohe eree ecooc.5. ool eere..54 The oclo 56 Aed. Polol odel foro ecr.. 57 Aed. Algebrc rereeo of ree ber of oe dfferel odel. The Model of Tree Nber...58 Referece...7

4 4 Abrc For corced Eree Ecooc odel d reer redcg o he lgebrc rereeo N d correodg wh ree of ber rooed d veged Polol Model for of crve hgh degree P N A Y Z A Y Z ele whch be coderg odel ecooc he for of d d d Af d d M P v P P d da : e d Af A A A A d d o be rodcg ecer d ffce crer of rg ecooc cr roble d d A f d where = e of he cl e = e of lbor deered N dd hre of he ol coe G gog o cl vee e of coo A=A echologcl level P=P rce fco B=B fco of bl of lbor deered f e d B fco of brh re of ower lbor F fco of deh re N=N olo of olo roble wh regrd o e-ge ce drbo.

5 5 e word: Polol odel cl lbor fco odel ecooc ecooc cr ol coe crohe r odel ree ol ree le ber ree.. Rereeo of cole obec b Polol odel eree rege Theore. e obec or roce oere o h he e e h codo he e decrbed b olol. e Proof. e he fco E... G G E chrcere he e of obec or roce or bce o e d oe of he oeror relg he chge of e of he obec or roce geerl d dreco. The he e of dreco chge of e of obec geerl. I rl o e h he fored where d. We e he followg rcle whch h or rccl erreo: A e or obec oere o h whole w eree.e. he be oe ee he fre. Bed o h rcle we ge: M wherem...: gve ber. e. Eqo decrbe he be erforce of he e or obec d eqvle o eqo of e или N or Y Y

6 6 oreover we hve... c c c c c c. The Proof: The ece. We hll rodce dego Z Y Y. e he codo e lce he = Z 3 e' how vld.e. M Z 3 e Z... he deco of he eqo 3 he hvg eered dego Z fro 3 we hve he followg e:... Z Z 3 A Z... he deco 3 e o ee Z hece deer of e 3 eql o ero

7 7. Rell we l ehod of hecl dco d 3. I c be ed h =34. e how vld vld decoog deer o + elee of le receved:

8 8 A w how d M where.e. e. Fro eqo 3 we hve: Z и Z. Hece Z т.е. fro here for Z Z Z M. Ad h eql he reched. Th we hve M Z. The ffcec: The eqo le e lce. e' rove vld 3. e' dege M Z. I e o ee h fro codo д д he e of he eqo follow d fro here or. To l eql o fro o he we hve. d he o of of fco M ''. e' clcle he vle of fco. I e o ee h

9 9 Z Z Z = Z Z.e. Z Z. Ad hece M Z ;. I e o ee h Z Z d Z Z fro here Z.e. e lce 3. Th Z. The le roved. Th he eqo 3 o ee d he ree of ber rore b h eqo rereeed o fg. lo o ree. e. The ree of ber o he rore eqo d 3 le gve. The here rforo whch rle he olo 3 o he olo 3.e.

10 Y 4... N N N Y... N N. The roof. e be he olo. Trforo 3 he for of N N N... ; 3... N e... N he olo. A llg b we hve N rforo we hve d hece N N. Here. Bed o h N 4 Forl 4 forl for growg ree d c be led vro feld of rl cece. Trforo 3 wh = N N N where rfor olo N o olo N d he 4 e follow: N

11 Th for cole obec decrbed b dfferel oeror of eqo or g he vecor E whch deered fro he rereeo redced o decro b fe ber of eleer obec ch olol 4 for.e. crve of hgher degree ch N Pobl ellcl ed for d roeco or for 3 Y Z Y N where Z = + Y d ll he olo of he eqo for = re rereeed. Noe h N P olol of degree - d re he cl of o degeere crve hgher degree whch ver well ed o roec foro. Rer. The bc rel of he gve wor co he decro of cole obec of Ecooc e:... ; o M b e of vecor E whch defed fro rereeo * N N d redced o decro b e of fl ber of eleer obec of e olol be relve : A A N M o r

12 N.e. crve hgh degree P N A * * * Y Z A Y Z. The corco of Eree Ecooc Model I ow [ -3] h cve eero of cefc ehod o rcce of oder ecoo boh here of fcre of he dr d here of grclrl rodco bece roe fere of or e. I eecll how b codero of oe qeo whch he deco coeced wh creo of he rc cefcll rove ehod roble of ecoo d ecooc develoe. The deco of hee ho qeo oble who rco of oder ehod of hecl cece. reo of he cefc devce for reerch d forecg codo of ecooc reorce ll over he world oe of he or e roble. Develoe of ehod of qlve reerch d hece he qve forec of e of ecooc develoe rll ded ll-rod dg of reer of be ecooc ce d he core hoe or oher vle of reer hroogeo d ocl fcor. Th eere wh rel e re rher eeve log d freqel dble herefore here ece of develoe of vro or of hecl odel. B e of hecl odel here w oble qlve d eerel dg of coeqece of hoe or oher led co ochg fcog of ecooc e drec eere wh whch re dble. The roble of dg of o vle of reorce e of he c l d lbor e qeo d o re ecoo d o hvg decded oble o d effecve cv of he be ecoo. Eecll he roble co forecg e of he cl gree he be fcre before core d rclr before he core of he IS he codo of ecoo of core of he IS ow low ge of ecooc develoe. Hge ecooc receo hold eo lbor re. Forecg of reorce he eeve

13 3 bblogrh devoed o hecl qeo. Sce. Mr' wor d lo Mw N.G.' wor Zg V.B. M N.A. PerovA.A. heroveov D.S. Srov N.I. Sh. herbov A.V. Alrev N.S. d of oe oher ce de vro ec hecl odelg of ecooc e d forecg of her codo. Oe of he fr hecl odel of l e he odel of he cl vew of fcre obb-dogl d ower - odel Mlh. I rclr odel of he cl vew of h or h rodco fco o codered qeo of oo of fcre o reer of fcre d cceed odel Mlh h growh re rooroll ber d her o e o cco fcor of ge d ce. I vew of e e - ge d e - ge - l drbo wor Volerr Jeffre Vebb Aleeev Svrehev Moeev d oher re devoed o develoe of odel of dc of olo. To oe of he gfc heoe of cece of l e beg he heoeologcl heor of growh of he olo of he world S.P. whch wh good ccrc he olo of he Erh drg rher log e erreed growh herbolc growh owg o qre-lw deedece of growh re o ber. I hee d oher wor be of corco of he devce qlve d qve vego of olo ber re corored. Develoe of odel vew of ge d l drbo of olo d he roble coeced o roeco of rre bologcl d re codered wor. I hee wor qeo of correce of odel of bologcl olo re codered vew of ge rcre d l drbo. Soe de fro he ecfed wor were ed for he decro of codo of e of lbor.e. ower wh he frewor of ower odel of ower. For wde cl e ge-ce drbo odel decrbed b he egrl-dfferel eqo rl dervve qeo of odelg of e of he cl d ower rher geerl ce re ded. Oe of chevee of or wor oo of drl odel d coeced wh he of ecooc e ch «he cl - rodco». The roe of he gve boo co develoe of odel d ehod of reerch of ecooc e vew of e ge drbo of ower d reerch of roble oo rodco ower wh he frewor of odel wh eree roere. I he followg qeo re codered. The olo of roble of oo of rodco d ecooc e defed o reer e of reorce. orco he be odel of fcre lo re coeced wh he ecooc e.

14 4 Reerch of he coeced roble coeced o decrbg codo of e of he cl d lbor reorce d o he b of led fcol of e of lbor egrl fro ber of lbor. A bo of he receved eqo for fcol of lbor ce whe ber of h olo deed o ge d l fcor held. A hecl bo l he hecl odel coeced fro e of he cl d lbor. Mhecl odelg of dc of ower h log eogh hor. Oe of he fr wor h dreco hold co odel Mlh bo eoel growh of ber of he h olo whch h erved b bc o o creo of hecl odel. I rl o e he followg ge of hecl odel logc odel whch h fored b for lo of rerble wor Volerr o d o o. I hee d beqe wor he bg eo gve he develoe of roble of corco d bl of do odel. Th he bc hecl devce of odelg hee wor re he oler eqo of dc of ower d olo. Alo ecer o ecf. Mr' ecooc wor whch h crefll ded codo of ecoo wh he hel of ee dgr. Model Mlh he logc odel d oe oher odel hve eerel cowledgee dg dc of olo. A he e e o eogh eo fr w gve odelg of dc of ower vew of ge rcre cl of he dfferel eqo. For ecooc e vew of ge rcre d l drbo or wor here re oe de o qeo of corco of hecl odel vew of e ge drbo. I he offered boo he geerl ehod he deco of he rore hecl roble wh he hel ecl eered fcol offered d roved. We hll rooe or odel Ecooc whch codered or wor [4-4].. Prodco odel. I ow h ecoo h fcre d drbo of erl beef. We hll dege hrogh Y q de rodco or he ol coe cle of he cor d fco of he cl-n lbor- d rodcv of echcl rogre - A: Y = A f

15 5 where Yf.re o egve fco.ccordg o he gve lw he q de rodco grow wh growh of e of he cl wor d ere of he crre level of echcl rogre d hee fcor re rr fcor of growh of ecoo. A Y = Y + - Y he r of he receved coe deged hrogh I = Y d refer o e he vee d oher r deged hrogh d refer o coo. Bede he cer r of he ggrege rof of he cor whch hold go o e rche G. To ggrege rof ecer o dd e of re eor - Nх. Th we hve he followg eqo of blce of drbo of erl beef: у = С+I+G+N. I ecer o oe h he e of coo deed o he vlble coe h = -T. Here Тe of e whch go o ocl ecr e oor d e of ocl rce elderl ec. For corco of he eqo of hecl ecoo we hll e dvge of odelg fcre d he blce eqo. e codo e lce:. All ecooc fco d reer deed fro e of reer r e where -e he r-rel re of ere e- re of eerl echge ge of lbor -l vrble =... R he R- of rego.. Re of fcre re defed b re of drbo: dy d d d Y. Model of he cl. e =. e of he cl re of rodco ed b worer oer reorce vle of reer eql + = +. The = + - e g of he cl for ervl of reer. Hece = I d fro here wh he cco we receve he eqo of he cl d f d 3 wh here deged d d r dr d d de d D d e

16 For ele f d d f followg d: r d f 6 h we hll receve he clcl eqo of he cl: h we hve he eqo of he cl he 3. Model of bor. Wor of hoe eole who hve devoed he o wor hoe re q flflled worer of hor. Now odelg ecoo for defo of reer of wor he odel ed: where re of growh of he olo eqo. lerl h wh he frewor of he gve odel or fcor he erdo ge floor ol re o e o cco. I h coeco we hll e h wor defed fcol ower: R N dd 4 here oel fco of worg N=N - ber of worg o R ge d < < he oe of e ; - ccordgl he l d l ge of he worg here of fcre. A how or wor fco N=N he deco of he followg roble: N F N N / N R 5 Here f r dr. d N N d N /. F B - ccordg o fco of deh re d brh re of he lbor olo S border of re R; R = R R ый rego. Poel fco of worg he deco coeced o 5 roble [4] - [4]. I he eoed wor how h he fco of wor deered wh he d hel 4 fe he eqo d where re of growh of he olo re he deco of he followg o-clled eqo of rvvl re: d d D

17 B e d F d 7 Here e - fco of rvvl re fcor of brh re F - orl re coeffce < <. The eqo 6 h oe l erl roo d ccog ber cole - coeced roo =. For he l roo e lce > f h = f h= < f h < d dt dg dn d MP MP where =-. 3 d di d d MP =. Fro here vew of he eqo 3 4 d vle of reer МРК - we hve: da в d 8 where. Th he eqo for he cl wor d rodcv of echcl rogre hve he followg d: e c where h he oel of lbor. Hece 4. Model of echolog rodcv level he Mere of he crre echologcl level. A dy da df f d f A A d d d d 7 d d d di dg dn d d d dr d d d d d h wh he cco d d d d di d d da A f d A f d A MP d d d fro 7 we hve: d f d f d МРК в MP 6 d d

18 d f / d d / d I= Y = - Y 9 d в / d d d 8 / where he deco 6. To he eqo 9 ecer o dd lo he eqo: dt dg dn d 3 d d d d. The rgh r of he eqo re re of growh of e T G N. The re ecer for defg fro codo of o of oe ecooc crero or fro codo of o co fcol d o o: 3. I e 9 he followg dego cceed: d v D d r e. I order h f MP d d d. MP oe d I e h d d h f MP h MP Y f MP I G N d blce of ecoo 3. I r o ol de o choce of re chge of e re of he e rche d re eor d re of hre of he coe gog o cl vee. If MP h occr cree or redco of he ol coe deedg o r of fco. Deedg o e of vle of reer = r e he e of he eqo 9 - wll be rfored or e of he ordr dfferel eqo or e of he eqo rvedvdl dervve. For ele f we hll receve e of he ordr dfferel eqo of order: d d Af d d N dd R

19 da d d d A A A A 9 MP 3 MP dn dt dg d 3 d d d d. 5. Proere of Prodco. Uder rodco we hll derd e of elee bc d rrod foro d lbor reorce rel of o fcog of whch " he cl d lbor " wll be rfored o fl rodc or ol coe. The rforo of whch crre o h ro refer o b rodco fco d deged hrogh Y=Af where he -e of he cl fed cl fcol of lbor reorce whch deed fro oel of lbor reorce edco ervcebl floor d ge d lo ber of he worer A echologcl lbel. A he oe ll over he world dgh hree odelg of rodco.. Prodco obb-dogl Y Y where Y he ol coe he rore cl d lbor reorce. b. Prodco wh elcll of relcee ES Solow: Y Y [ ]. c. The co rooro rodco P eoev Y=Y {/ / } d.µ-rodco - : =A / >> < <or rodcg f f Y we hve

20 Y < < I how wor [] fro ES rodco follow obb- Dogl rodco d he co rooro rodco P d fro µ-rodco or ll rodco. I hll o h ll e of rodco.e. he rodco fco hold f o followg codo:. f [ ] [ ].e. erce vrble oohl re chg d rel of cv of fcre - he ol coe rher oohl vre chge q of ed reorce. I rl forecg lrge e for ele ecoo of cor.. F = f =.e. bece hogh of oe drl reorce f of fcre oble. f 3. A К> d > e h growh of ed q of fed cl d growh of ber of he worer rel growh of he ol coe. f f 4 I.e. codo of re ecooc growh of fcre who echcl rogre grow of eee ol of oe drl reorce rel decreeredco of effcec g. 5. f f or f f d. 6. Th codo rovde h roorol growh he q of ed reorce occr roorol growh de rodc or ol coe. We hll oce h ll bove eoed fco b o hee codo. 6. The bc reer of rodco. The bc reer of fcre re: = he e of he cl he oe of e.e. = ; = fcol of lbor reorce = d/d lg or of relcee f f reorce E E d / d / f f. elc relcee of re coeffce of elc of e o f f reorce re ed cl rodcv. We hll oce h for ele how how fed cl eee of lbor er

21 of d o he corr for he reervo of he ol coe forer level f =f с c be releed cree. Preer defe eed of chge of lg or of relcee of reorce. Fcor of elc of e o reorce E E how o how of erce wll chge fcre of he ol coe chge of eee of he rore reorce of fcre o oe erce. I ecer o oe h f o eer coce of vle =/ rodco we hve: =F where Y=Y/Y F. =f d F > F <. 7. Model of he bc reorce. Followg or wor we hll wre geerl ecooc odel for defo of e of he bc drl reorce = = d coo = : d d N " f O. f Here he hre of he ol coe Y=Af gog o roce of fcre re d ge of he worer - oel fco of he worer defed he deco of he followg roble A B o f A. B. f. re gve fco N=N he ber of he worer of ge he oe of e : N N F N N N N B N d F N B re gve fco of he rge d F. - e he dehfco of he worer d B. fco of her brh re N - l ber of he worer. Solvg roble we fd fco = d N=N d he we hll defe fcol of lbor reorce =. A ow d of fcre Y=Af fro roble we hll defe dc

22 of he e of fed cl.e. e of he cl = < < d coo = oe of e. e Defo. Ecooc e coeced wh fcre Y=A f we hll e e cog fro he followg elee: I where = = = re he deco of e. 8. Geerl odel rodco: µ-rodco. A red bove fro ES rodco obb-dogl rodco follow rclr d co rooro rodco. I or wor b w offered followg odelg rodcoa= f=y: Y Y. We hll oce h hgher reled odelg rodco ecl ce of he gve fcre re. Fro rodco fco follow fco ES whch hve oed ore geerl h obb-dogl fco d co rooro fco. Th fro rodco fco of re follow ll ow rodco fco. We hll clcle reer of fcre. El o ee h er d herefore.e. he or of relcee fco of. The eg of elc relcee re defed follow:. oeffce of elc of e o reorce re defed ccordgl der he forl:

23 3 E E d / / / / f f f f E E E E. We hll coder offered fcre. A F =f ecer o fd l A Y l El o ee h Y A l l l d herefore l l A.e. К f Y F К. A F eqvocll defe fco f we hve receved he ee h he fco obb-dogl d ecl ce or fco. Slrl d we hve l Y A Y or } { Y F d herefore } { Y Y Y f.e. hve receved fco wh co rooro. We hll defe lg eg of ecooc reer =

24 E E 4. E / [ ] E [ ] Ol odel of rodco. El o ee h red rodco fco obb-dogl ES fro co rooro b oe reer o oed.e. he codo of he rore rodco co be roved. The µ- fco offered o reer oed. El o ee h dy d Y Y d Y d d Sbg = * he forl 4 we hll receve:.e. e lce Y Y 3 Model rodco 5 we hll e be odel rodco d rore ecooc e coeced wh fcre 3 be ecooc e. Fro 3 we hve

25 5 * Y Y o r d for reorce we hve...@ The ol ecooc e. A rd * * * = * where =* =* he deco wh rodco fco f * * = * d coo * = - * we hll e he ol ecooc e. We hll oce h f o eer dego Z we hll receve he eqo of e + Y =Z whch h o he olo he whole ove ber > : /. The rore ecooc reer for o fcre 3 re rereeed : = - = * E * E E E. We fd lg eg of reer: A : E = E * E =. A E * der der * l. Fro receved rel follow h for fco obb-dogl

26 6 СЕS fro co rooro here re o be codo d ecer rodced oe reglro. 3. The ol odel rodco cl of reglro rodco of obb-dogl Occrrece of he heor of rodco fco cceed for crrg b 97 whe rcle of he Aerc ce of he ecoo of e h ered. Dogl P. Dogl d D. obb' hec D. obb «he Theor of fcre». I h rcle e w dere he ercl w o defe flece of he e cl d wor o o fcrg dr of USA. A ow odelg fcre obb- Dogl loo le: f A Y where f f where he d A heologcl level e of he cl e lbor reorce f ove co f f. Here reer chrcered degree e of reorce drg fcre. I ecer o oe h odelg fcre "rgd" fcre d doe o cce he l codo o oe reer. I h coeco here qeo o chge of re of chge of erce reer fco. For ele re of chge of degree of e of reorce he cl d lbor drg fcre he e of rgh le лини dvdl qre : M. A M we e e crvler le o here :... M. For fco = lo we hll e = = lo roble of o fco o e of M redced o he followg roble: f A M Z where : M. Hvg eered dego f A we hll receve

27 7 Z. Th he l roble co reece of reer M d degree of e of reorce drg fcre d he l codo of odelg fcre Z. where The ee. Te lce l l Z A f The roof. A e d A f l l d A f l l d d Th fro codo we hve l l.e. l l l l l l l d l l l e' rfor fco. I e o ee h

28 8 l l l l l l l l Af Af Z l l l l Fro here g o cco dego l l we hve e Z Z * * h w reqred o roved. We hll eer l Z Z he fro 3 we hve he eqo or d geerll for reorce we A ow [3-6] he eqo 4 we hve ccog ber of deco. Icldg he whole deco of d fro here owg defo e Y for ol odelg fcre обб - Dogl rereeed he for. e Z Z e e Oo of e of he cl der lw obb Dogl.The gve rgrh devoed o oo of roce of foro of he cl ccordg o lw obb Dogl d foro of lbor o экспоненциальному o he lw. A ow e of he cl fored ccordg o he lw: Af d d where f

29 9 he gve ove ber chrcere ecooc reer. For ele he e e degree of e of reorce. We hll e h where. Se chrcered ower d wh he frewor of he gve wor we hll e h where e re of growh of ower e. The ee. Th Model oed o ; lo he followg e e Af d d Rell h d d h g he logrh boh r we hll receve Af d d Af. We hll fd eree of fco d d Z o.e. d d d fro here d d herefore

30 3 Af d d Af. Th e Af d d h w reqred fcor. Ug e odel we hll co Af d d.th Model chrcered foro of he cl gree he lw of oo b obb-dogl d for ed ower der lw Mlh. A 3 oler odel for defo e of cl ccordg o odel 3 we e ehod of broe le Eeler.e. h f h... e The ol odel rodco cl of fcre of obb-dogl ce reorce. I he gve rgrh he odel oo of e of he cl cl of fcre obb - Dogl ce reorce offered. A ow he e of he cl ce reorce fored ccordg o he lw: П Af d d where f he gve ove ber chrcere ecooc reer he e - reorce e degree of e - reorce. We hll e h where > >.

31 3 The ee 4.Model oed o : followg e Af e d d d d d d. Rell logrh boh r we hll receve Af d Af d eer dego lo he h g he. We hll d we hll receve A where M M...:.. We e fco o e A vew of he eered dego we hll receve he roof of he gve ee d we 4. O he eergec heor of olo growh The ew odel of growh of he olo o-clled eergec odel of growh of he olo offered. I how; h ower odel oble o receve fro le of gro of l odel vew of ge rcre d l drbo. To qeo of odelg d forecg of growh of he olo ero wor for ele [4-4] ce ow Mlh odel d d re devoed o logcl odel odel vew of gro d o egro odel vew of ge rcre d l drbo d lo S.P. o odel l e. I odel or e roorol of growh re o ber o he olo Mlh odel or o qre of her ber odel. Thee o oe lo o oe ge d пространственных odel of growh of ber of olo. I or wor roceedg fro oe l gro of he odel decrbg growh of olo vew of

32 3 ge rcre d l drbo he ew odel o-clled b ower wll be receved. Il gro of odel of growh of he olo we hll wre he followg d: N F N N N N B N d r r N F N N N N B N d N N Here N ber olo N N for ce а d N N for ce -b e ge G [ ] F B re fcor deh re d brh re reer < <.. Model vew of ge drbo. e' eer fco / N d < < where а oe o-egve fco wh codo rel of he gve wor we hll forle he followg heore. d. The bc The heore. e fco N N olo of roble ce а. The here wll be fco а d for whch fco olo of he eqo: d d : B e d 3 where B B e F d fco of rvvl re of he olo.

33 33 The roof of he heore. We coder he ce а fro. The fr eqo а we hll cree o - N N F N he we hll receve de. The gve eqo we hll cree o fco d rel we hll egre o rforo we hll receve:. Frher проводя le N d d N d N d d d F N d. Tg o cco bodr codo ddg d brcg eber N d he lef r of l de d lo choog fco he olo d F B of roble d.e. B e F d d we hve he eqo of e eqo Mlh d d. Fro we hll receve B ' e ' ' F d ' d. Hvg eered relcee B B e F d we receve he roof of he heore. Defo. Model we hll e ower odel d he roble rore o h odel ower =. Fco 3 we hll e oel of he olo. b. Polol odel. The deco of ower roble rereeed

34 34 c e co.e. c e co / where fcor c re defed fro codo c = we hve ber ree odel he for of: c 3 d e и - re deco of e B e B e co d d The rer. For defo of olo vew of ge rcre we hll receve he followg forl N d where defed der he forl d fro codo d. c. Model vew of ge d l drbo. A ow he coce of oel of he olo w eered wor [-6] ce of e ge drbo. Dcve fere of he ree wor h drg chge of ower l reer re e o cco. We hll eer fco / N dd 4 where fco. chrcere he fco decrbg ervcebl erdo of he olo d olo of ge o d N N he oe e lo fe б.

35 35 N N d N B N N N N F N r where e -ge l coorde r=r he gve fco decrbg eed of chge of ber o dreco F orl re coeffce B fcor of brh re. Ug or he heore for defo of fco. fro. we hve e eqo d r e B o d F o 5 I 4 we hll = he hvg e ce we hve he eqo of e : d e B d d where r. d. Model of he olo for core vew of ge drbo. e' e h he olo of he core fe o he eqo G d N B N S G G N N E G G G F N N r where F B re he gve re of he order wh elee

36 F f f... f f f f... f f f B b b... b b b b... b b b 36 N l ber of olo. e' eer fco N dd d where. G G - ered vecor of fco d d F o d N dd = e B * r d where B * F * re coge o B F * rce. Hece we hve e eqo :de B e d I d c e co where r. e. oer eere. For he fr ere of coer eere l fco we e he followg d: B b. F F 9. For defo of fcor of we hll wre c where =3 d olved h eqo o he b of l he eqo c c he b of he deco c c eqo wh he hel of rforo offered b. The o 33 we hll receve he deco he ec. o he deco of he reqred eqo The wre lgorh w rogred lgge Borld Delh d ere of cog eere crred o.

37 37 5. Polol Model of Develoe of oe he Wor odo b d wh og Selee - odfco ehod of he ere eghbor Now we coder qeo of corco d vegg ew ehod of clclo e of loe b d wh log elee: odfco ehod of he ere eghbor. The rooo ehod o defe e of loe he wor codo of e d bed o g of o-clled odel of ber ree. B g h ehod for odel he d howed d crred o oe cool eere. I oced h he rooe ehod b o ber ree odel le d verl ehod for defo of vle of oe he Wor odo b d wh og Selee d e rogred o ll coer lgge. The recevg forl re corolled or clclo. A he ehodolog cre clclo e he robbl heor gve d he log-er cl d fcl clclo h o fcl re fll red o deogrh re coeced wh Se hecl d he cl reglre eblhg l relo bewee he rer d he r. The reflec hecl forl he ech of foro edco d eedre of rce fd log-er rce oero. To he lo crr clclo of rff o d of rce: lfe eo fro ccde roer wor cc. The ehodolog cr clclo e he robbl heor gve o deogrh d he log-er cl d fcl clclo. B e of he l rff he coe whch receved b he rer fro e cred reorce of he ccled e of rce e o cco. Ece for h re o cre re red o clclo whch coeced o oe of he wdered roble of ch -cl clclo coeced o defo of or d codo of rce h he of rce e relg e greed receo b rce fr or he e orgo eeced rel. Mg "ree" of deco ecer o drw "r" d he "brche" dlg rcre of roble. "Tree" fro lef o rgh ele dow."brche" dege oble lerve deco whch c be cceed d he oble ocoe rg rel of hee deco. O he crc we e wo d of "brche": The fr - he dhed le coecg qre oble deco he ecod - he coo le coecg crcle of oble ocoe. Sqre "" dege lce where de deco rod "" - occrrece of ocoe. A cceg he

38 38 deco co flece occrrece of ocoe eed o clcle robbl of her occrrece ol. Whe ll deco d her ocoe re ecfed o "ree" ech of vr coed d he ed oer coe dow. All chrge ced b he deco re dow rore "brch". A ow he forec of he fre loe g obervbl verge develoe of he ere eghbor ll defe der he followg crc:??? Y YYO he b of gve o he crc he odel of he ere eghbor e he followg IA A e Tow: Y where... ;.We hll coder ore geerl odel h h odel: / Y Y where > >... ; =3. Hvg / eered dego Y fro h odel we hve Y Y / or

39 d Y 39 >> =34.The fr eqo he eqo of degree wh + ow d h fe ber of deco. For lloco of he ecer deco ecer h we e echo he > Model of he wor develoe of loe: For everhg >... rgh r of he eqo h he l vle h here correod he wor codo of e.e. Y Y M / where M :. The eqo 4 we hll cll Model of he wor develoe of loe 6. Eree Ecooc d Ecooc 5. Model of ecooc cr. We hll coder followg odelg Ecoo offered wor: d A f d d d : e d A f d A A A A A d where = - e of he cl he oe of e = - e of ower hre of he ol coe - Y gog o cl vee - e of coo A=A echologcl level B-B fco of

40 4 bl of ower deered f e d B fco of brh re of ower reorce F - fco of deh re. I ecer o oe h fco = rereeed N where N N N N N F N d he deco of roble: N Here N d he deco of he coeced roble d refer o oel o fco of ower reorce. Defo. e erce fco of roble re gve rge of defo of he reer d lo ow erce reer A Y A oe rodco fco he vecor fco f deered he deco of roble we e odelg ecoo he rore rodco fco f. Th rodco fco A f we hll e odelg o fcre. Defo. We hll ell h he odelg ecoo codo of cr f here re ch co ove ber * * d for whch hve lce of eql 3 wh deco fg codo wold o be d d 4 where A f d. For ele f we hll coder fcre obb-dogl f f where degree of e of ower

41 drg fcre 4. I e o ee h f here r f 3 d d.e. h lce of eql 4. The ee. For oe odelg fcre of e Af he codo 3 ecer d ffce. Rell b defo of rodco f f l d A A A we hve * f f A f f d 5 Fro here f re eeced codo 3 f f d. Slrl g codo f f receve d d we hll. The roof of he gve eql lo follow fro eql 5. Th ecer d ffce codo of cr of odelg ecoo re eqle 4. I ecer o oe h ber re defed fro he deco of he dfferel eqo. For reece of he deco of he eqo d d we hold olve he eqo e d ll over g. Geerll f h eqo h oe l erl roo d ccog ber of cole coeced roo of e o we hve: c e... d herefore e ce co where re Forer coeffce of he gve decooo. For legl of he gve decooo we hold coder cl of fco co ортонормированной. Nber lo for whch e of fco we hll defe fro e e e co d d

42 4 Fro here e o ee h.e.... d roo удовлетворяемой o codo l B B B d B where B B co d. The ee. For he for lw of drbo we hve B c e co The ee 3. e B fco of drbo of he orl lw defed he orl lw of drbo.e. d e he defed fro he deco of he eqo F e where reer of F. The roof. We hll eer relcee т.е d dd re The we hve e d e e d e w o be how we hll oce h f o eer relcee h we hll receve f e f d herefore. Slrl we hll receve

43 43. Th qe erl roo of he eqo e d he fr d Thee vle re ed defo of cr vle.. Polol odel of ecooc cr. We hll coder e.a f l d f l d where we hll e degg f f l d l f l d we hll receve coder roble of o o e of M.e. he ecod ce. d : f. Now we hll of vrble fro here we hve f. Ug codo of of fco f Th o occr o reer of echolog. How o eh lge we reree he bc eqo of odel ree of ber. Ech eber of he of he rgh r or oe of he re decooed he le cooed he e re ede d eber of he rgh r of l r re g decooed he le cooed he e re ede. e f f f... f re he deco he eqo ree of ber he we hll defe e d l fro codo f l d f f l d. f

44 44 e. I h ce we e e o e of he cl т.е f d where f f l.e' eer d le he e defed f : f f roble f f. e' receve he eqo d he coderg 6... f f Th eqo ch he bc eqo ree of ber. Solvg we fd d he e f т.е he e cl d geerl force. 7. Model of oer crclo Moer crclo ow ccordg o he qve heor of oe for clve ded deedece bewee q of de rodco o whch rchg ded howed d geerl rce level e lce: M V = P Y Here M he offer of oe V eed of he lo of oe P rce level d Y he q of he de good d ervce. Th eqo rove h he offer of oe defe vole of fcre ol ereo whch r deed fro rce level d q of de rodco: М=к Ру к =. Fro here Р =к V M V d herefore bewee rce level d o vole of fcre here vere reloh. A he vole of fcre defed b vro d de rodco Y = У У.У d he vecor of rce level Р coeced o P= Р Р Р he bc eqo wll be defeddeered he followg d: Y Р =МV where = P У. Bede we hll e h rce level d vole of fcre Р re fco of oe reer r e where -e r-he rel re of ere е-he echge

45 45 re х-he l fcor. The bcll he eqo clr rodc Р у r e defed : e e G P r e r e dred If wll dege hrogh d - ccordgl l d level of he rce he oe of e fro he bc eqo we hll receve eql: M v where r e e R r e drded ol o of fcre. Nrll l d o level of he rce he l d l offer of oe ccordgl wer. The here M M VM VM. Fro. The de of he l offer of oe o level of he rce eqll o he de of he l offer of oe o l rce level. Th de refer o oc of oe. Th coc of vole of fcre o reer r e oc of oe doe o chge. Ug he heore of verge for verge vle rce VM level o r e we hve: ср c d herefore c.the receved rel re fr for verge vle Р M o e he codered e ervl of ervo. e M c M. c where fere bove e e vergg o e of vle of hee e for ele P d. Fg 3.cover ll oble ce whch c re rel

46 46 Deedg o h whch r of fgre o be o М = M У Р hoe who offer oe ecer for oce codc rore olc of chge of vle of vole of fcre d rce level. For ele for co level of vole of fcre У of he rce c vr fro level o l. Slrl we c hold rce level o oe fvorbl ll level Р d vole of fcre o redce or cree fro У o У. I rel he reoble olc der he de he offer of oe defed. A rce level he offer wll rel cree cree of oc of oe d redco wll rel he offer of oe redco. I he fr ce whe he vole of fcre creed d he ecod decree. If he ecoo he begg of ervo codo М h decree of clve ded coeced wh redco fro he offer of oe here ro fro o М o o D whch he vole of fcre of below rel level d he roce of redco of rce occr growh of ecoo o level У. I he e fgre oher cre oberved lo. A vole of fcre eql У ll over g rce level creed o Р h o o Ad d he oohl redced o o I. I rel here ecoo whch fcre o dp d dv becoe l. A MV=Pу V lo we hll eer d d d d dm dv dego v v v v V v M we hve d d d d v d fro here we hll receve d d d d d da d f A f d d A f d d v. da oderg vle d we receve he rce eqo: d M P v P P d d where v d r е. Th eqo he bc eqo of he qve heor of oe. A o d dv dego for v eer v lo v h or rrgee here d d v Pe e MP d MP d v P e e d

47 47 choce of her chge h chge of re of he offer of oe d eed of he referece of oe. Thee re re llowble b gee d he re defed fro he deco of oe of cl roble of o corol. A fro recevg eqo we hll receve he forl: Fro whch coc v A we hve: MP Th forl chrcere e chge rce level coc of oher reer ee fg. 4. Fg4. v Deedece rce level fro he rel re of ere co reer dr co d Fg 5. Deedece rce level fro he rel re of ere co reer

48 If r order of e e: 48 h we receve he eqo rve dervve of - r v. A he olo of h eqo wh codo / r bed o Fg 3. For olo of h eqo we hll e lo l codo d bodr codo ch foro edco rce level / r deedg o reer r h r dr / r r r r dr. Here r d co d r. The receved roble reree ele of roble wh fcol er codo whch re eered d veged wor of he hor [4]. I e o ee h he olo of eqo h ce reree : r r e r r d r r v r e d Y d Fco P we hll defe fro bodr codo of foro edco of he rce h r r r d r e r d f v Y r d where f e ddr. Y r The eqo reoro reree o-for egred he eqo of e. A we hll receve: r r r r e Y r d r r r v e r Y d d where r r r r r r Y d Y r r d d r Y.

49 49 Fro h forl co Y v we hve r v r followg fgre. v r r Y. The receved forl erreed he Fg.6.Deedece rce level fro he rel re of ere co reer dr r co. d e' oce h fgre re decl hogh ree dffere bodr codo. Slrl we coder he ce whe d r e. 8. Model of erco of core wh dffere ecooc level of develoe I ow h eoch of ble d ed growh of ecooc he ehod o he b of cl roceg vro ecooc reer elded orl rel. B he o h hrl chged wh he begg of ecooc cr. The forec o he b of he l of log-er edece bece ble. Br world fcl d ecooc cr whch roch bece for eecede h hrl red he qeo bo oore of ecooc forecg bo bl of cece deqel o decrbe cole ocl rocee d o redc her develoe. Ad he deco ch ce wh he hel of oher ehod. Mehod of oler dc chge cocer o hee ehod ecooc ergerc ed o he decro of oe eqlbr rocee o he l lw derco old d foro of ew ocl d ecooc rcre. Th we coder he e of oler dc for reerch of ecooc d ocoolcl rocee cle he ere cor d he world whole he bc eo

50 5 hvg gve he l: lw of foro of ed ocl d ecooc rcre; lw of re cre d he ro fro oe rcre o oher. For develoe of ecoo of he core here re core wh HP d he core wh P wh followg he crc of develoe: Fg 7.The core wh hgh rodco HP d low rodco P. d hecl odel of e d d b d d b... ; where... ; - gro ol rodc er c cooerg core wh hgh rodco HP d low rodco P - he fcor decrbg e of erco l codo > h - core - leder - cchg core.

51 5 Ele. We hll coder ce whe we hve e oe cchg cor wh wo leder he he wre odel hgher loo le d d d d d d d d d 3 b d d d 3 c d d d 3 where = < = < = α β γ bcd re reer of he eqo. I e o ee h he feld of vle : β d +γ d 3 d /4 α d +γ d 3 d /4b α d + β d d 3 /4c he gve e h wo eqlbr codo of e fg.. Ug dfferel e of h ele oble o how h ll of olo f o codo for ol ol coe +b +c d where - e he lef r =.e. he Gro Nol Prodc GNP core he l oe of e. I ecer o oe h he lef r reree GNP ll core he oe of e d he rgh r egrl eerg of e «eerg of ecoo». For he deco of he receved roble we hll e dvge e Borld DEPI c odelg gve o

52 5 Fg.8. Rel of coer eere =. Fg. 9. Rel of coer eere =5. fg. 8 he we hll receve rel were cchg cor ver rogl lg behd he core of leder. e' oere ow he gve e wh he hel of reer fro ervl. We hll e for ele reer =.5 he we hve fg 9.Th rel of ccefl gee wh he hel of reer we hll receve ro of he cor fro cr e roel 47 o hghw of dvced rodco. 9. MODE OF POPUATION TURBUENE AND ATASTROPHE IN ETREME EONOMIS Th e del wh d of rblece odel of olo wh he e-ce drbo of ge-reled chge he reer dffo coeffce defed re of oler eqo. oder odel olo wh he e-ce-ge drbo : N N N B N d N N N N N N F N D where N N he e of he olo he o ge e F F deh re B B brh re N N he olo

53 53 e he l e. O he be of or relcee D D d F e U N ' ' 4 ed of we ob he roble:. d U B U D Soe h. M D D where... : M Ug he de of he ehod of ero of vrble for he roble forle wo cle of oble olo:. l of le olo. D. l of eoel olo D whch D defe he rereeo. D D The defo. A olo "rblece" he frewor of he odel or we cll h e olo whch cer vle of he vecor M... he vle / 3 e vle.e. M. 4

54 54 The followg heore: Theore.The Eqo / M Z d he eqo Z re eqvle. Now we coder he eqo / M 5 Theore. e codo. he here vle of M for whch he eqo 5 d he eqo D 6 eqvle. Ideed der he codo of. 5 r fored o he eqo of he e h correod o he o of he fcol 3. Th eqo 6 Z he eqo of he "rblece".e ddreed b he fcol 3 e he vle. Solve he eqo 6 he cl of oble le olo:. Theore 3. The olo of eqo 6 he cl of oble le olo c be rereeed. U U 7 Proof. For lc he roof roceed he ce where = d D =. oder he cl of le olo. I e o ee h follow. Delverg h o codo we ob 7. Forl 7 correod o he o roble 3 h of he ecod dervve whch correod o he "olo of rblece." Now coder he ce where he olo he

55 55 roce of rblece whch o he feld reched he ber of olo. Fro U we hve: Hece. oe he r of ecod dervve. Th f ; d he he o fco defed b 7 reche. Rer. I 7 ll he coeffce deed o.e fc he re fco d he re deered b he l codo. Rereeo 7 c be wre he followg olol: 3 7 ' where. 3 Fro h olol follow h he ce of rblece olo behvor "forgoe" he l codo.

56 56 Theore 4. e he rereeo 7 ' 3 ow q d olo for oe d 3 4 he ever roce of olo rblece decrbed fco 7' e eergec ldg ol wo o d. U / / where he roof follow fro he defo of der: de where.. ool eere r. I e fgre we how rel of cool eere wh ecoocl cr reer d we defed he rcre of he cr. Fg..The rcre of he cr Polo rblece. Now here re oe coer lo wh he followg odel d: = 5; =.; =.; 3 = -.; = ; c = ; = -:.: ; = -:.: ; c = c ^ -c ^ * -4 * 3 ^ ^ / / ^ / ; c * c = 3 ^ ; c ^ + c ^ = c ^. For hee d he followg re oe of he rel of ercl eere.

57 3= =. Fg.. Deedece of he olo rblece 3=-.4 e for dffere vle reer 3=.4 Fg.. Deedece of he olo rblece e for dffere vle reer Fro he gve rel vble h chge reer here "rblece" olo ber. Eere wh cl e dffere ce. Now we coder oe rel of cool eere wh cl e e eree rege e odel d =.7==.=. f=de=-.5be= l=7 l=.l=.

58 58 l=. l=. b l=.3c l=.4 d Fg..Deedece of he cl e o e for dffere vle reer ce of -d The oclo A e of Ecooc dffere erod b hrogh he develoe cre. I brod ee roce h hree he eece of he Eree Ecooc oe orgo. re ofe coe o he orgo' gee ddel. However rcce he eergece of he cr how o: lo of coe fro he le of good wor ervce redco of oher dcor of fcl d ecooc cve of he orgo. M coe re o ble o overcoe ch cre develo d leve he re. The ver obl of cr defed r develoe of orgo h lw here. Therefore dvble

59 59 o coder he olog of cre he develoe of he orgo he ce d er of her occrrece he crer ecfc o he cr. A cole obec cor orgo he dffere erod b - hrogh he develoe cre. I brod ee roce h hree he eece of he orgo. re ofe coe o he orgo' gee ddel. However rcce he eergece of he cr how o: lo of coe fro he le of good wor ervce redco of oher dcor of fcl d ecooc cve of he orgo. M coe re o ble o overcoe ch cre develo d leve he re. The obl of cr defed r develoe of orgo h lw here. Therefore dvble o coder he olog of cre he develoe of he orgo he ce d er of her occrrece he crer ecfc o he cr. Aed. Polol odel foro ecr I oble ed roeco of he foro or d N Y 3 Z Y where Z=+Y d ll deco of he eqo rereeed = re. P N olol of degree - o d cl crve hgher degree whch re ver well ed roeco of he foro. For ele we hll wre cl of ch crve ecl ce wh he hel of dl: N... ; I.e. crve P Y P 3 Y YZ 3 N 4 Y YZ Z d Y 3 P 4 Y YZ YZ. d lo e Z=+Y whch belog o cl ellc crve. Pr l o e "o" whch c "be " bo oher lr o of ellc crve. «The» wo o r oo "l" o ellc crve. Ece for he o lg o ellc crve «he ero o» codered lo. The of wo o A wh coorde A codered h YA d B wh coorde B YB eql O f A =

60 6 B YA = - YB od. The ero o doe o l o ellc crve b everhele rce clclo; c be codered defel reoved fro crve. Se of o of ellc crve ogeher wh ero o d wh he eered oero of ddo we hll e he "gro". For ech ellc crve he ber of o gro cerl b gre eogh. The or role lgorh of he gre wh e of ellc crve led wh "lle" o. Po Q refer o o of freqec re f for oe o P e eeced eql: P = Q + Q + Q +. + Q = Q. If for oe o P here ch ber h P = h ber ed he order of o P. Mlle o of ellc crve re loge of degree of ber le feld. The roble of clclo of freqec re of o eqvle o roble of clclo of he dcree logrh. Follow oced h relbl of he dgl gre bed o cole of clclo of "freqec re" of o of ellc crve d. Thogh eqvlece of roble dcree logrh d roble of clclo of freqec re lo roved he ecod h he bg cole. For h reo corco of lgorh of he gre gro of o of ellc crve ered oble o do who horer e coro wh le feld ece of he greer bl. A cofdel e coder oe rdo ber. The oe e coder coorde of oe o o ellc crve P whch defed P = Q where Q - ecl ge he choe o of ellc crve ed «be o. oorde of o Q ogeher wh fcor of he eqo ecfg crve re reer of he crc of he gre d he hold be ow o ll rc of echge ege. Fro here follow h oe "oder" croe c "be hfed" o ellc crve. Aed. Algebrc rereeo of ree ber of oe dfferel odel The Model of Tree Nber Defo of ber ree d rereeo. e N oe rl ber. We hll ell h he ber N for ree of ber f here wll be rl ber d eger... for whch N... d r oe or ll rereeed

61 6... d oe of lo c be bed... d l decooo e lce whch eber of he rgh r 3 co be re bed he fl cooed -h degree of oe eger o-clled b b b of ree. ocel Model of Nber Tree geerl ce gve fg.. N Fg.. ocel Model of Nber Tree geerl ce So l level he ree co of he ch 3.be elee c eer o ech level ree herefore fro ech level we e ol hoe elee whch o представимы 3. The rel he ber N qel rereeed N q q... q 4 where q - ber of occrrece of bc elee... q ree of ber.

62 6 We hll coder roble for ber N d vecor... fro he eqo N M where M... ;. The e M reree ered crvler herod d r l - ered herod. Ug heore fro wor [] we hll receve: N N Th h eqo o ee 4 d he ree of ber rore b h eqo rereeed o rce. lo o ree. Now we coder qeo of he be rereeo oe chrcer of obec elee wh he hel of oe roere elee gve obec d lco oe ce where drbo rocee of growh rocee he d wve rocee dffo rocee oe reer vle e lce he l rege. Eree rege of ch hcl rocee re rg he ce whe he vle of her reer re chg oe gve e. The c be cclo of he wrh rcle wve eerg oe re where he codered hcl rocee re rg. Sere cool eere lo crred o wh odel d for oe coderg he rfer rocee der cole codo d eree rege. eнd - oe orled ce d he e of M gve.e' e h for obec elee H d roere of obec or P h d lo w of dl H of roere of he codered obec re deered. For ele we c defe her or he rore ce.e. P T / d / d T T.

63 63 Defo. We hll ell h obec elee H w bed wh he hel of oe roere elee he be f for oe elee M frl r / P h / h... h M For oe d.. The Theore. Th he obec elee l ge hrogh roere h hd lce bo r: H h P h h he be bed w ecer d eogh Ele. Drbo roble of reorce. A ele oble o rel roble of rereeo of oer re or word codg ec. he ber: P h h. he P h / we hll receve he eqo eqo for oe > > h ccog ber of eger deco h h... h d P. Ele. Growh of l. e coder odel of growh of l wh r for ele roo r leve bd. e fr r e e of rodco bo whch c be e for develoe of ll oher r of l. e re bo of r ow of e. Develoe of l we hll e o he followg odel: f l

64 64 I M... where... re gve ber. f f he lw of foro of ew bo l - e of hoohe reer. O he be of or heore Bell fco correode o ol corol roble f e eqo M.e. or... l Ele 3. Dfferel eqo wh eree roere. M rocee drbo rocee of he d wve dffo rocee belog o o-clled odel eqo wh eree roere. I he geerl ce ch eqo be rereeed he for of: M or o here re gve oeror whch chrcere he codered hcl rocee. e gve re E G wh border Г: Г Г d he ow h he eerre ode de re G d wve roce f o e eqo[]: o G o G. Here re coeffce of he codcv or wve veloc chrcerg vero of evroe. We hll e he followg o. e he evroe of he codcv hoe h = where o o

65 M The eqo of he codcv correod ee ce whe =.e. he echge occr der he l lw of Newo. Therefore choce of e of reer... fro M =. The we hve he eqo wh eree roere M d o G M Ad herefore o he b of he bc heore we hll receve he oler eqo of e o G o G 3 The rgh r of l eqo correod o q of he d wve re G fored rel of o of he revo eqo o e M. For he olo of l eqo for ele he fr eqo 3 ecer o e cl of oble olo or le e c c or eoe e c c where c c re olo of he coordo eqo: c c. The e. e re G recglr wh he de: l l Ad he fr eqo 3 gve lrl for he ecod d lo re gve l d bodr codo: G l. The he olo rereeed he followg d: l l c e D l l where D re coeffce Forer of fco d reer c c re he olo of he coordo eqo c c c 3... l l c.

66 66 The e. e G E recglr wh he de l l d e l d bodr codo re gve ecod eqo 3 rereeed A co c B c The he olo of he where c = fro l codo. l l l l A B re defed Ele 4. The fco. e here bch of rcle wh he cer le d oe eerg. I ow h ch bch of rcle decrbed b wve fco.e. lde of robbl. Th wve fll o he cree crc d frher fro hee crc leve hercl wve d o he followg cree hee wve re refereced. Sg wve fro he o d boo crc of he cree orged wve where for oe rcle d for rcle ccordgl hre of wve levg crc geerl g wve. We hll e h where he rore robble reer of he evroe of drbo for ele. The coderg egred roch wve fco.e. P P* h h o he b of rcle of o rereeo we hve: erp h * * erp erp h h where er= > d <. The fr forl chrcered robbl of rcle deeco d he ecod erferece cre. Ug - rforo h ' h h ' erp P ' er P we re el eblhed coeco b codo for vro rcle. Here oe olo of he eqo. Ele 5. Soe roere of coder o-clled fco: fco d lco. e'

67 67 where х - ove ber - rl ber. Proere. Fco o * h l d l vle eql.e. where d. Rell d d d d d

68 68 h fro codo * d d we hve *. I e o ee h * d d d * d d. Hece he o * * qe o of of fco : * d qe o of of fco : *. Bede d. Proere. If he deco of he eqo oe he o qe o eree of he fco. Rell le he deco of he eqo. A hvg eered dego we hll receve e of he eqo be relve :

69 69 e relve h he qe deco ove he deer of e eql o ero. de. Fro here d fro - eqo of e we hll receve vle d fco. Proere 3. All deco of he eqo re rereeed he followg rercl forl: * where d ove ber d. Rell g reeo we hve hoogeeo e of he lgebrc eqo be relve : or A he deer of e eql o ero h he o rvl deco. B vre of for of e deco re rereeed. co. Fro here d hvg eered dego

70 7 ll deco of he eqo we hll co *. The rer. For deco * re fr eo.. d lo for he e lce forl: d d. orollr. A > deco * re o he whole ove ber. I ecer o oe h = deco * c be eger d o eger. For ele ll deco of e * re eger ber d he re rereeed : odd d eve =34.. Soe ele of Nber Tree Now we hll coder ow ele of Nber Tree for dffere ber.. e 5 N he we hve Fg.. Model of Nber Tree for 5 N 5 or 3 Roo evel

71 7 - e q o of elee of he gve o of ree d ber o edge w decooo. Fro here follow h rereeo 4 e he followg d: Now we hll coder ber N 5. 5 Roo 3 evel evel Fg.3. Model of Nber Tree for N N Roo or 3 evel 36 4 evel Fg.4. Model of Nber Tree for N N Roo evel Fg.5. Model of Nber Tree for N d herefore evel

72 5. N or 3 Roo evel 3 evel evel3 Fg.6. Model of Nber Tree for N

73 6. N Fg.7. Model of Nber Tree for N Roo 5 4 evel 4 evel evel 3 evel4

74 74

75 75 7. Slrl for N 35 we hll receve rereeo 35 5 Roo evel evel evel3 6 3 evel Fg.8. Model of Nber Tree for N 35 evel

76 76 For ce whe o lw oble o corc grcefl ele. We hll rel oe ele. e N 35 3 he Fg.9 Fro he gve ele follow h we hve o receved ree b ol h ветку. We hll e ch "ree" o growg ree. For growh of ch ree ecer o e "cg" fro oher for ele ree d o er he o o growg ree. For ele Roo or 3 evel evel Fg Ecoocl erreo rcre of ree of olo. The ree of deco reree oe of w of lg of e of he d o cle or cegore. The roo of ree lcl co ll clfed d d leve - he cer cle fer erforce of clfco. Ierede of ree reree e o of deco g o choce or erforce of eg rocedre wh rbe of elee of he d whch erve for he frher dvo of he d h. Ull] he ree of deco deered rcre whch co fro U - leve ech of whch reree he cer

77 77 cl; U of ccece of deco специфицирующих he cer e rocedre whch hold be eeced relo o oe of vle of rbe; he of ccece of deco lef wh brche whch q correod o q o of oblerobble ocoe of eg rocedre. I oble o coder ree of deco d fro oher o of vew: erede of ree correod o rbe of clfed obec d rche - o oble lerve vle of hee rbe. The ele of ree bed o fgre. O h ree erede reree rbe ervo hd wd. eve of ree re red b oe of wo cle P or Н. I oble o co h P correod o cl of ove coe colo d Н - o cl egve. For ele P c reree cl " o leve o wl " d Н- he cl " o hoe ". Thogh obvo h he ree of deco he w of rereeo whch dc fro dcg rle he ree c core he cer rle of clfco whch gve for ech obec hvg he rore e of rbe bed b e of erede of ree he deco o wh fro cle o rbe rele h obec e of cle bed b e of vle of leve of ree. I he gve ele he rle wll crr obec o cl P or Н. Referece []. Aev A. A. 9. Moder fcl d ecooc cr vew of he heor of ovve-echologcl rogre of ecoo d gee of ovve roce. Se oorg: Globl d regol Progre. D. A. Hlr d er oroev. С []. I. S. Mlov.. Noler Dc of he Noler World. [3]. Av S. A. V. M. Bhhber I. S. Eov. D. Mehl Aled c: clfco d decree redco of deo. - M.: The Fce d Sc. [4]. Zhrvlev J. I. V.V. Rov Se'o O.V. 6. "Recogo". Mhecl Mehod. Progre. Prccl Alco. - M.: The Phe. ISBN [5]. Zgoro N Aled ehod of he l of he d d owledge. - Novobr: HIMIT he Sber Brch of he R Acde of Scece ISBN [6]. He T. R. Tbhr J. Fred.. The Elee of Scl erg. - Srger. ISBN

78 78 [7]. Vg V.N. E. J. Golov A.A Zgor M. V. Fo. 4. Ahec d Plble oclo Iellecl Se // Uder V. N. Vg' edo D. A. Poelov. - M.: Fl.74. [8]. Gelov V.A. A. A. Bhlov V. B. Brov E. D. Vlov.. Iellecl of e of or of ccece of deco ererr o wh e of he foro o codo of rl //-М.:Edorl of evroe. 34. [9]. Aod A. E. Pl e-bed reog: fodol e ehodologcl vro d e roche // AI oco. IOS Pre. Vol. 7:. P []. Bhov A.I. 5. I.A. Iellecl' Boo foro echologe: Sde. The gr //-M.:MGTU. N.E. B. 34с. []. Vrhv P.R. A. P. Ereeev. 6. The ehod of lble reog o he b of loge d recede for ellecl e of or of ccece of deco // New of he rfcl Iellec []. Wrw P.R. 6. Relo of ehod of lble reog o he b of recede for ellecl e or of ccece of deco // Wor of he eh ol coferece o II wh erol rco - I 3 т. Т.. - M.: Fl [3]. Ereeev A. P. Vrhv. 7. Mehod d Tool for Reog b Alog Iellge Deco Sor Se // Proc. of he Ierol oferece o Deedbl of oer Se. Slr Poreb Pold 4-6 Je 7 IEEE P [4]. M. Y. 9. Model ecooc. TNU [3]. M. Y. 5. Soe Hohecl Model of Rel Sce d he Pheoe Occrrg he. The Blle of Nol Uver 3 ere of rl cece. c ISSN [5]. M. Y. 5. Abo he Eqo wh Eree Proere d her Alco. Jorl of Ve Nol Uver ere of Mhec ISSN [6]. M. Y. 9. Modelg of Nber Tree d Alco. Proceedg of Ierol oferece o Mhec d Iforo Secr Sohg Eg. 3-5 Noveber 9. h:// / eer/.

79 79 [7]. M. Y. 4. Abo Oe l of he Modelg Eqo wh Eree Proer. Jorl of Ve Nol Uver ere Mhec.8-35 ISSN [8]. Y M.. The Boo Abrc IM Hderbd Id [9]. Y M The Boo Abrc IM 998 Berl Ger. []. Y M.. The Boo Abrc IM Beg he []. Y M The Boo Abrc Edbrgh.33. []. Y M.. Abo olo of he eqo Z. Ve Nol Uver 4 с ISSN [3]. Y M. 8. Ivego of Soe Noler Sglr Model Ecoe d New ocered Mhecl Proble. J. Ecologcl Modelg Vole 6 Ie 4 Ag [4]. Y M.. Irodco o Model of Ecooc. Dhbe -37. [5]. Y M.. F.M. Y d M.M. Y. 9. Abo The Be Model Prodco d Globl Ecoo oeced o The. erl A Jorl of Iforo Techolog. AJIT P [6]. Y M.. F.M. Y d M..Y. 9. Power odel of growh of he olo. Ve Nol Uver Sc. Jorl 49 P ISSN [7]. Y M.. Dfferel Eqo wh Eree Proere. Ve Nol Uver Sc. Jorl.77 P.3-. ISSN [8]. Y M.. J. Gev S. Odev.. Mhecl Qere of ee Polo Nber. Ve Nol Uver Sc. Jorl 385 P.3-5. ISSN [9]. Y M. 3. Polol rereeo of dfferel eqo wh eree roere. Moder roble of fco heor d dfferel eqo. Mer.ehd.ch.of.ov. 85 verr d. AN RT.G. Mhlov. Dhbe.6-6. [3]. Y M.. ool l of he codco ele - grod e d oe correodg odel eree rege. I he

80 8 boo «He-Phc reerche d eree Eerg d e he corol d ql gee rodco d ervce Merl for Ierol He-Phcl School 6 Yer elebro of frov M.M. 8-3 Ocober.-8. [3]. Y M.. Theor of dfferel eqo wh Eree Proere d Alco Phc Ecolog d Ecooc. Ierol Scece oferece «Proble of Dfferel Eqo» for 8 er celebro of Jrev A. Dhbe с8-. [3]. Y M.. Abo Eree d Algebrc Rereeo of Procee Decrbg Nber Tree Model. 3 Ierol Pre Mhec oferece. Abrc -3 Seeber. Ilbd.45. [33]. Y M.. Rereeo of Soe Phcl Procee Eree Rege. Merl of Nol oferece Moder Proble of Phc for 7-er elebro of T Profeor Boboev T.B. Dhbe с [34]. Y M.. Model of Develoe of oe he Wor odo b d wh og Selee - Modfco Mehod of he Nere Neghbor. Ierol ogre Acre Mrch 7- e- Tow SA.3. [35]. Y M.. The forl: ree of ber for boh lco roeco of he foro d he l cole e. Ve of Nol Uver Secl relee devoed o er of edco d echcl owledge с.-3. ISSN [36]. Y M.. Soe Qeo of Ecooc r d her Mgee. Ve of Nol Uver 67 he ecl e devoed o he verr of Ideedece D of Reblc T c.3-. ISSN [37]. Y M.. Abo Reglo of Uble Srcre of Regol Reerve oeced wh Model of Proeco of Rre d Derg Te. Ve of Nol Uver 67 he ecl e devoed o he h verr of Ideedece D of Reblc T с.-7. ISSN [38]. Y M.. Dfferel Eqo wh Eree Proere. Ve T Nol Uver Nre Sc. 77 c.3-. ISSN [39]. Y M.. The Model of er-cor relo. Ve of T Se Nol Uver N ISSN