THE LOGISTIC EQUATION: SOLUTIONS AND DEMOGRAPHIC INTERPRETATION

Transcription

1 Oulook on Communiaion HE LOGISIC EQUAIO: SOLUIOS AD DEMOGRAPHIC IERPREAIO Ciprian DARIESCU Prof PhD Fauly of Physis Al I Cuza Univrsiy of Ia[i Romania Corrsponding auhor: marina@uairo Absra h prsn papr dals wih dmographially imporan soluions o h fundamnal logisi quaion Bsids h ommon on h rvrsd h ovr-growh and h supra-xinion logisi modls ar brifly analyzd poining ou hir spifi propris Among h abrran logisi modls h supraxinion on whih migh orrspond o an involuion of populaion soon afr a nular War sams o b h mos dramai Indd ompard o h ohr normal logisi branhs h dmographi rovry nds mor han 8 yars jus for nring h ovr-growh phas nssary in anding h nw quilibrium sag Kywords: Logisi quaion Dmography Logisis of Populaion Dynamis h S-shapd logisi funion was applid for h firs im in h firs half of h 9h nury by Pirr Franois Vrhuls o h sudy of populaion voluion H pu ino vidn afr an xponnial growh a slowing down of h pross whih is h rsul of sauraion followd by sopping a mauriy Suh a rsul rprsnd h basis for h pry-prdaor modl dsribing h populaion dynamis in ologial sysms - indpndnly laborad by Alfrd Jams Loka and Vio Volrra In im h logisi funion aquird mor ompliad forms nowadays bing appliabl in a larg rang of domains suh as: biology mdiin biomahmais nuronal nworks dmography onomy hmisry physis saisis soial and poliial sins h fundamnal quaion of h logisi modl L us hav quaion: whr: d d () is im; - numbr of individuals; ln wih / ln : / = man lif im of / populaion in fr sa; ln : = man im of populaion doubling in fr sa wih : mdium priod of assing or produing rsours : avrag hararisi numbr of individuals from a halhy populaion In prinipl h soluions wih (posiiv) xprsss h numbr of rih popl (from h middl lass upwards) whil hos wih giv h numbr of h poor ons (finanially from h middl lass downwards) h drminaion mhod of h gnral soluion of quaion () indpndn on h pariular signs + or of paramrs an is dsribd in h following Consqunly from: d d d d using noaion: dz z : z d hr rsuls hrough variaion of onsan K 94 volum issu April / Jun p 94-99

2 HE LOGISIC EQUAIO: SOLUIOS AD DEMOGRAPHIC IERPREAIO from h xprssion of soluion: z( ) K( ) h sring of xprssions: dk dk K( ) K d d ha is obaining for z / ( ) h gnral soluion (indpndn on sign): z( ) K ( ) ( ) K Canonially h firs ririon for h sablishmn of h pariular soluions is givn by ondiion Cauhy: ( ) so ha K whih mans: / ( ) ( ) / () h sond ririon is rprsnd by h onr signs + or of h wo onrol paramrs and ; vidnly h following siuaions mniond aording o hir imporan in h liraur of h fild ar possibl: I Common logisi modl ; II Rvrsd logisi modl ; III Ovr-growh logisi modl ; IV Supra-xinion logisi modl Analysis of ah of hs modls will b prformd in h following srss bing laid on hir rally spifi propris I HE COMMO LOGISIC MODEL I orrsponds o an onomially and soially - middl lass populaion wih a naural growh ra whih nvrhlss globally onsums mor han i produs so ha Undr suh irumsans h onr form of h soluion of logisi quaion () boms: ( ) whr rprsns h quilibrium valu As on may obsrv if ha is h populaion is individually prosprous prdominan is h numbr of hos who hav omparaivly wih hos who do no hav all hy nd y h iniial ffiv numbr (i a ) dos no xd h quilibrium valu: (ln) m hn as boms: h whol soluion ( ) (3) bing rgular (ha is laking singulariis) and posiiv ovr h whol ral axis ( ) Indd vn if for whih mans for qui a long im h numbr of prosprous individuals was vry low baus of h low naural birh ra and onsumpion his numbr nrs a quasi-linar growh pross whih will nd whn onsumpion will inras y wihou xding in rmo fuur valu his is h absoluly sabl logisi branh Howvr if maning ha h populaion xds as numbr of individuals (in a rain Inrnaional Journal of Communiaion Rsarh 95

3 Ciprian DARIESCU givn momn hr sld as ) h quilibrium valu hn h sond rm from h dnominaor of (3) rvrss h sign whih boms ngaiv and h soluion gs h form: ( ) wih < (4) whih vidns h sond imporan hararisi of h ommon logisi modl namly is slf-rgulaion apaiy Evn if in fr sa bing no inflund by h ass o h limid subsisn rsours h populaion would hav a naural growh whn boming supra-riial ( ) h xagg- rad onsumpion ra xds h naural growh on so ha as a mar of fa h ffiv numbr of individuals ha is bgins o dras: d d for Jus on h lin for ha is in rmo fuur on may obain: ( ) lim maning r-sablishmn of h quilibrium valu (on h suprior branh) Mor han ha in his as ( ) h modl shows on mor hing namly ha h soluion anno b indfinily prolongd in h pas namly for ( ] baus a (singular) riial momn xiss: ln whn h dnominaor of soluion (4) is vanishd (whih is no h as of h numraor) and onsqunly xprssd in sinifi rms h soluion xplods : ( ) and ( ) Suh asp of unboundd disoninuiy of h soluion is obviously a shoroming of h modl as h onrol paramrs and wr akn as onsan Aually ohr dynami non-linar ffs also xis vn if no onsidrd hr suh as volunary birh onrol ha is h valu and sign of paramr and spially h propnsiy of h rih popl of invsing for highr produiviy ha is for hanging paramr from a onsumpion on ino a produion on Anyway supra-riial logisi modls wih onsan paramrs announ auhni soial and onomi splis II HE REVERSE LOGISIC MODEL his as rfrs o a suffiinly maur populaion also possssing h orrsponding wlfar for invsing in produiviy ha is bu unforunaly suffiinly agd for providing any naural inras any mor so ha aually Consqunly h logisi quaion () boms: d wih (5) d wo onlusions bing alrady possibl on h global bhavior of soluion ( ) hus if giving in (5) as a faor on obains: d d whih shows ha for all ass wih ( ) h sign of h driva is ngaiv so ha furhr on for h numbr of individuals rgisrs a oninuous dras h onr form of h soluion bing givn by xprssion: 96 volum issu April / Jun p 94-99

4 HE LOGISIC EQUAIO: SOLUIOS AD DEMOGRAPHIC IERPREAIO ( ) wih Indd for hr rsuls h limi: lim ( ) whih mans ha a populaion wih a ngaiv birh growh and a lowr numbr han h quilibrium valu will b xinguishd baus a lowr and lowr fraion of i onribus o is subsisn As a firs rovry sp from suh an xinion his yp of populaion nds an infusion of young disiplind (and qualifid) oupls from h ousid for mainaining produiviy and for ransforming paramr ino a posiiv on hus nring a loal ovrgrowh phas Furhr on whn h quilibrium valu is xdd onsumpion > should b innsifid for avoiding supra-produion h ommon logisi modl wih ( ) is hus obaind sn as nding owards h quilibrium valu As alrady known his was h as of Swizrland and of h norhrn ounris in h sond half of h XX-h nury Whn mnioning h rm of supraproduion in h as ( ) i may b obsrvd ha: d d whih mans nring a pross of populaion inras dsribd by soluion: ( ) wih whih announs h infinisimal riial momn: ln / prior o whih ha is for on has: lim ( ) From a praial prspiv suh a rsul is unapabl rprsning an insuffiiny of h rvrsd logisi modl wih onsan paramrs as wih h inras of populaion h onsumpion ra will nssarily inras Consqunly paramr boms ngaiv xprssing onsumpion whih mans nring a loal supra-xinion phas unil obviously followd by a naural inras phas for aaining h rgion of absolu sabiliy of h ommon logisi modl As rrospivly obsrvd i was his h xa sragy of h USA afr World War II namly supra-produion unil h 55 is simulaion of onsumpion unil abou 969 and massiv apan of immigrans bwn 97 and 98 afr whih h govrnmnal program Visa Lory was launhd III HE OVER-GROWH MODEL oghr wih h supra-xinion on hy form h abrran logisi modls ha is as long as paramrs and ar onsan having h sam sign wih rfrn o h anoni soluions wih h possibiliy of slfadjusmn (produd in h as of h auhni logisi modl) xiss no mor ha is why saring wih posiiv h orrsponding soluions anno b indfinily prolongd any longr ovr h whol ral im axis soio-onomi splis similar o h (finanial) rashs and/or rvoluions hus ourring as an inrpraion Uilizaion of suh modls assums prisly ha if paramrs and may b qui prisly drmind by (global) saisial Inrnaional Journal of Communiaion Rsarh 97

5 Ciprian DARIESCU prossing hn h riial momns may b aniipad so ha orrsponding masurs for avoiding hm shall b akn For xampl in h as of h ovr-grow modl wih h orrsponding soluion aks xprssion: ( ) wih whih shows ha by vanishing h dnominaor h momn of risis is givn by xprssion: ln + If in h iniial momn h populaion is muh sub-riial ha is hn: + so ha: ( sub) ln( / ) ln Consqunly i is suffiin im a las a h lvl of a gnraion for modifying prfrnially paramr ino a onsumpion on so ha h sysm will nr h usual slf-adjusing logisi phas and h oal numbr of individuals will asympoially nd owards h quilibrium valu If howvr a h sam momn h soiy was onsidrably supra-riial ha is hn ln + onsqunly baus ( supra ) and hrfor vn if aking yars and h risis will b vry soon insalld namly: ( supra) yar As h mniond daa ar similar o hos of Japan on may spula (a las ha is wihou any laim of aniipaion) ha if his ounry will winss ovrgrowh i would fa in no mor han on-wo yars a supra-produion risis his xplains why for avoiding a risis Japan hangs is hnologial lins a inrvals of abou yars h mos frqunly usd word in hir ommrial advrismns bing aarashii maning nw whih aims a a priodial inras of h inrnal onsumpion IV HE SUPRA-EXICIO MODEL Considring paramrs and onsan and ngaiv orrsponds o h involuion of a populaion (immdialy) afr a nular global onfli whn h numbr of survivors is low (omparaivly wih h normal hararisi sandard quilibrium valu ( ) / ) whil h ln is high baus of h half im priod / of h populaion whih boms muh lowr as a rsul of radiaions han h avrag normal lif im of h (iniially) halhy individuals Consqunly srily mahmaially ha is wihou any r-sablishmn masurs h soluion of h supra-xinion modl aks h form: ( ) vidning ha for boms: and i 98 volum issu April / Jun p 94-99

6 HE LOGISIC EQUAIO: SOLUIOS AD DEMOGRAPHIC IERPREAIO ( ) orrsponding as is larg o a rapid xponnial xinion h surviving on inludd o avoid suh an asp as long as h survivors ar sill abl o work a firs masur o b akn is o mov hm in undrground loaions In his way as a rsul of h rrsrial srning o radiaions offiin is brough o zro whih mans saisial supprssion of moraliy and nssarily rvrsion of h sign of h onsumpion ra ino a posiiv on of miro-limari agrarian produion so ha h rmaining olliviy o nr saisially spaking upon h only asnding branh availabl o i in suh momns namly h soluion of h dgnrad logisi quaion: dp P P P( ) d P Hr P rprsns h populaion apabl of supporing h ommuniy as boh mainaining of produiviy and inras of birh ra On known ha afr a nular aalysm boh P and ar low hir produ wih is muh subuniary P whih lads o a quasi-linar voluion phas P( ) P ( P ) d P ons d A (normal) biologial sal and onsidring h subsqun (soio-onomi) slf-supporing abiliy of h nw-born ons an avrag priod of 8yars is hr involvd rsuling from 6 afr whih h naural inras boms posiiv whih is h bginning of an ovr-growh phas nssary for h rapid ( nw) aainmn of a nw quilibrium lvl afr whih by simulaing onsumpion o nr h usual logisi branh whih is absoluly sabl o any subsqun fluuaions h alrady obsrvabl onlusion is ha rovry afr a larg-sal nular aasroph is h mos diffiul on among all normal logisi branhs whih xplains why h nular arming during h Cold War was mor a mna and no a raliy announing a ral aggrssion Rfrns Vrhuls P F Corrspondan mahémaiqu physiqu (838) p3- Loka A J Analyial hory of Biologial Populaions Plnum Prss w York Rihards F J A flxibl growh funion for mpirial us J Exp Bo (959) p Bakr G and J Gollub Chaoi dynamis Cambridg Univrsiy Prss 99 5 iolis G Inroduion o nonlinar sin Cambridg Univrsiy Prss 995 Inrnaional Journal of Communiaion Rsarh 99