Logisic quaion of Human populaion growh gnralizaion o h cas of raciv nvironmn. Srg V. Ershkov Insiu for Tim aur Exploraions M.V. Lomonosov's Moscow Sa Univrsi Lninski gor - Moscow 999 ussia -mail: srgj-rshkov@andx.ru words: Logisic quaion; Human populaion; raciv nvironmn; Al ODE A k poins of nw approach for modling of h populaion dnamics in raciv nvironmn ar prsnd hr: gnralizaion of h Logisic quaion o h cas of raciv nvironmn for modling of populaion dnamics or for h fulfilling of h cological nichs; nw p of asmpoic soluion for such quaion which is sd on human populaion growh; rducion o h Al ODE in gnral cas. Du o a vr spcial characr of Al ODE i's gnral soluion is provd o hav a jumping or h rak-down of h componns for such a soluion. I mans an xisnc of coninuous gnral soluion onl a som dfini rsricd rang of im-paramr or a possiili of suddn gradin caasroph in rgard o h componns of soluion populaion growh a h dfini momn of im-paramr.
According o h rsuls [-] logisic quaion dscris how populaion volvs ovr im. Such an quaion acuall drmins a linar dpndnc of slf-similar ra of voluion procss or dnamics of populaion in rgard o h propr rsidual capaci of non-filld par of nich. Th las is assumd o proporional o h diffrnc of h ponials dfining a propr ra of populaion dnamics as low [4]: d / d - hr is h im-paramr is h populaion oal or a propr lvl of nich sauraion = ; = is h Malhusian paramr h ra of maximum populaion growh; is h carring capaci i.. h maximum susainal populaion a h oal sauraion of a propr nich =. L us assum ha a k poin in modling of such a populaion dnamics procsss - is o ak ino considraion h momn of raciv nvironmn. In his cas h funcion of rsisanc of nvironmn should prsnd in a form low: aciv + raciv - whr aciv is h aciv consan rsisanc of nvironmn or a propr rsisanc of nvironmn du o a sauraion of nich xranous lmns or in modling a procss of an xhausion of main rsourcs which mans h xhausion of h nich in rgard o i s own lmns; raciv - is h raciv non-consan rsisanc of nvironmn i.. h propr rsisanc of nvironmn as a racion in rgard o incrasing of h xranous lmns incorporad ino his populaion or ino h non-filld par of a propr nich whr raciv = raciv. Aciv rsisanc aciv aov dos no dpnd on h amouns of lmns i.. i has a sal consan valu. For xampl in h cas of human populaion dnamics such an aciv rsisanc aciv mans h world-wid accidns which unforunal ak
plac vr ar in h world ~ lik aviaion-accidns or chnical caasrophs caasrophs of naural or ohr characr as a rsul w hav a dcrasing of human populaion millions vr ar. W should also ak ino considraion ha h lvl of sauraion of a propr nich is known o drmind h lvl of dmographic prssur [-]: P = / - whr according o h principl counracion should lik acion w assum: raciv ~ - so h nir nvironmn rsisanc should dfind as low: ~ 0 + /. Summarizing all h assumpions aov w could sa a principl which is o dfin h dnamics of such a procsss of voluion: a slf-similar ra of h voluion procss is o dircl proporional o h rsidual capaci of h non-filld par of a propr nich u also i should simulanousl in invrs proporion o h funcion of rsisanc of h nvironmn spac. Taking ino considraion h univrsal principl aov w should rprsn h logisic quaion of voluion in a form low: d / d / / - hr is h funcion of aciv consan rsisanc of nvironmn 0 u for simplici w will dno i as. Bsids l us rprsn h las quaion as low: d d.
4 - which is provd o a propr Al ordinar diffrnial quaion of h -nd kind [4]. Du o a vr spcial characr of Al quaions i s gnral soluion is known o hav a jumping or h rak-down of h funcion a som momn ₀. I mans h xisnc of coninuous soluion onl a som dfini rsricd rang of paramr or possiili of suddn gradin caasroph [5] a som momn ₀. An appropria chang of varials in.: + = / l us w oain h Al ODE of h -s kind [4] as low: If w assum: = cons = = cons = = cons = quaion. could rducd as low: - which has a propr analical soluion [4]. Bsids l us assum = / ; i mans ha w considr a cas whn h lvl of rsisanc of h nvironmn should dircl proporional o h carring capaci. Such an assumpion should proprl simplif h righ par of quaion. as low: - or:.. d d
5 - h analical ingraion of quaion aov ilds: - or L us rprsn quaion aov in ohr form = / : - so w hav oaind h analical xprssion for. Bsids if w ak ino considraion ha carring capaci is a larg nough for h cas of modling of human populaion ~ 8 illions of prsons h las xprssion could asil simplifid undr h appropria condiion + / as low Fig. : Thus w hav oaind h simpl asmpoic soluion for h final prognosis of Human populaion dnamics. Finall l us spciall no h analical rprsnaion of soluion of quaion. for h cas of sparad varials = / = cons cons / :. d
Fig.. Logisic curv. frncs: [] Vrhuls P.-F. 845. chrchs mahmaiqus sur la loi d'accroissmn d la populaion. ouv. mm. d l'acadmi oal ds Sci. Blls-Lrs d Bruxlls 8-4. [] Vrhuls P.-F. 847. Duxim mmoir sur la loi d'accroissmn d la populaion. Mm. d l'acadmi oal ds Sci. ds Lrs ds Baux-Ars d Blgiqu 0 -. [] Wolfram S. 00. A w ind of Scinc. Champaign IL: Wolfram Mdia p. 98. S also: hp://mahworld.wolfram.com/logisicequaion.hml. [4] Dr. E.amk 97. Hand-ook for ordinar diffrnial quaions. Moscow Scinc. [5] Arnold V.I. 99. Caasroph Thor rd d. Brlin: Springr-Vrlag. 6