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Leqcia 10 5 maTematikuri modelebis Sesaxeb biologiaSi, qimiaSi, medicinasa da ekologiaSi


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26/06/2016 diferencialuri gantolebebi da maTematikuri fizika

leqcia 10
5 maTematikuri modelebis Sesaxeb biologiaSi, qimiaSi, medicinasa da ekologiaSi
5.1 balansis meTodi
sanam biologiaSi, medicinasa da ekologiaSi zogierTi maTematikuri mo­­delis Seswavlaze gadavidodeT, ganvixiloT maTematikuri modelebis agebis erT-erTi efeq­turi me­Todi, romelic balansis meTodis saxelwodebiT aris cnobili.

davuSvaT, zedapiriT SemosazRvrul moculobaSi mimdinare raime procesi an mov­lena drois t momentSi aditiuri sididis mniSvnelobiT xasiaTdeba. G si­didis adi­­tiuroba Semdegnairad unda gavigoT: Tu moculobas gavyofT or na­wilad da TiToeul nawilSi mimdinare igive procesi an movlena drois t mo­men­tSi xasiaTdeba da sidideebis mniSvnelobebiT, maSin . am sidides SeiZleba hqondes masis, energiis, impulsis, ra­ime regionSi mosaxleobis ra­o­denobis, sawarmoo fondebis da a. S. Sinaarsi. cxadia, rom Tu aRniSnavs raime re­gionSi mosaxleobis raodenobas, maSin war­moadgens am regionis Sesabamis ze­da­pirs, xolo -s qveS am zedapiris Se­mom­sa­zR­vreli wiri igulisxmeba.

Cveni mizania, gavarkvioT, Tu ram SeiZleba gamoiwvios droTa ganmavlobaSi si­di­dis cvlileba raime aTvlis sididis sistemis mimarT fiqsirebul (uc­vlel) mo­cu­lobaSi da rogor SeiZleba am cvlilebis ricxviTi maxa­siaTeb­le­bis gamoTvla.

ganvixiloT konkretuli magaliTi. vTqvaT, raime dasaxlebuli punqtia, ki _ misi sazRvari. maSin am dasaxlebul punqtSi mosaxleobis raodenobis Sec­vlis ori ZiriTadi, erTmaneTisgan gansxvavebuli tipis mozezi arsebobs.

pirveli mizezia dasaxlebuli punqtidan mosaxleobis gamgzavreba anu misi sazRvris gadalaxva Signidan da dasaxlebul punqtSi mosaxleobis Camosvla, anu sazRvris gadalaxva garedan. sxva sityvebiT rom vTqvaT, dasaxlebul punqtSi mosaxleobis raodenobis cvlilebis erTi mizezia am punqtis Semom­sazR­vre­li wiris gamWoli mosaxleobis “nakadebis” arseboba.

mosaxleobis raodenobis cvlilebis meore mizezia dasaxlebul punqtSi ada­mia­ne­bis dabadeba da sikvdili. aqac, Tu sxva sityvebiT davaxasiaTebT am process, SeiZleba vTqvaT, rom -s SigniT mosaxleobis “wyaroebis” cnebis qveS vgulis­xmobT si­di­dis rogorc “gaCenas”, ase “gaqrobasac”. am “gaCenisa” da “gaq­ro­bis” konkretuli mi­ze­ze­bi SeiZleba sxvadasxva iyos, magram am mizezebis detaluri gan­xilva Cvens mizans ar warmoadgens.

amrigad, zedapiriT SemosazRvrul moculobaSi si­di­dis cvlilebas ori sa­xis procesi ganapirobebs: zedapiris gamWoli si­di­dis “nakadi” da TviT mo­cu­lobaSi si­di­dis “wyaroebis” moqmedeba.

rogorc aRvniSneT, sazogadod SesaZlebelia arsebobdes rogorc “nakadebis”, aseve “wyaroebis” sxvadasxva saxeoba. kerZod, zemoT ganxilul magaliTSi mo­sax­leo­bis raodenobis cvlilebis Sesaxeb rogorc “nakadis”, aseve “wyaroebis” or-ori saxeoba gvqonda: “gasvla” da “Semosvla”, “dabadeba” da “sikvdili”. am SemTxvevaSi maTematikuri mo­­delis Sedgenisas gaTvaliswinebuli unda iqnes rogorc nakadebis, aseve wyaroebis yvela saxeobis erToblivi moqmedebis Sedegebi.

gadavideT axla raodenobrivi Tanafardobebis dadgenaze.

Tu vigulisxmebT, rom funqcia uwyveti da warmoebadia, maSin warmoebuli moculobaSi G sididis cvlilebis siCqares, anu moculobaSi drois er­Te­ul­Si G sididis cvlilebas warmoadgens.



si­di­dis im raodenobas, romelic nakadebis yvela saxeobis erToblivi moq­­me­de­bis Sedegad gamWoli moZraobiT gaivlis zedapirs drois er­Te­ul­Si, vuwodoT G si­didis nakadi” da aRvniSnoT igi Q simboloTi, xolo G sididis im raodenobas, ro­melic wyaroebis yvela saxeobis erToblivi moqmedebis Sedegad Cndeba moculobaSi drois er­Te­ul­Si, vuwodoT “G sididis wyaroebis simZlavre” da aRvniSnoT igi F-iT.

axla ukve sirTules aRar warmoadgens, SevadginoT “balansi” _ moculobaSi G sididis Semcvelobis cvlileba drois er­Te­ul­Si gavutoloT G sididis nakadisa da wyaroebis simZlavris jams:



. (5.1.1)

am gantolebas balansis gantolebas vuwodebT. swored is warmoadgens balansis meTodis maTematikur formulirebas.

ganvixiloT ekologiis zogierTi maTematikuri mo­­deli, romelSic diferencialuri gantolebebi iqneba gamoyenebuli. am SemTxvevaSi drois cvlileba uwyvetad xdeba da ma­Te­matikuri mo­­delis saSualebiT drois nebismier momentSi ama Tu im populaciis Se­sa­xeb daskvnis gakeTebis saSualeba gveqneba. SevniSnoT, rom maTematikuri mo­­delebis Sed­ge­nisTvis am SemTxvevaSi arsebiTad Cvens mier aRwerili balansis meTodi iqneba ga­mo­ye­ne­buli.

5.2 populaciis malTusis diferencialuri modeli
malTusis models safuZvlad udevs martivi debuleba _ populaciaSi in­di­vide­bis raodenobis cvlilebis siCqare t momentSi individebis rao­de­nobis pir­da­­pir­pro­por­ciulia.

vTqvaT, raime populaciaSi individebis raodenobas warmoadgens t mo­mentSi. Tu A im individebis raodenobaa, romlebic drois erTeulSi iba­de­bian, xolo B _ im in­di­vi­debis raodenoba, romlebic drois erTeulSi kvde­bian, maSin balansis meTodis gaT­va­lis­winebiT SegviZlia davaskvnaT, rom ar­sebobs sakmaod seriozuli safuZveli imisTvis, ra­Ta sididis cvlilebis siCqare ganvsazRvroT Semdegi di­ferencialuri ganto­le­bis saSualebiT



, (5.2.1)

sadac , , xolo , Sesabamisad dabadebisa da sik­vdi­lianobis koeficientebs war­mo­ad­genen. (5.2.1) gantoleba Se­iZleba ase gadaiweros:



. (5.2.2)

Tu , , e. i., Tu da koeficientebi mxolod t drois cvladzea damokidebuli, maSin advilad SevamowmebT, rom (5.2.2) gantolebis amonaxsns aqvs



(5.2.3)

saxe, sadac populaciaSi individebis raodenobas warmoadgens sawyis momen­tSi.


nax. 5.2.1
naxazze (ix. nax. 5.2.1) mocemulia funqciis grafikebi, rodesac , , sadac da mudmivebia (erTmaneTis msgavs mrudeebs sawyisi momen­tis sxva­dasxva mniSvneloba Seesabameba). gantolebis amonaxsns

funqcia warmoadgens.

Tu , maSin populaciaSi individebis raodenoba mudmivia.

Tu , maSin eqsponencialurad, rodesac .

Tu , maSin aseve eqsponencialurad, rodesac .

am garemoebam malTuss saSualeba misca, gamoeTqva eWvi imis Sesaxeb, rom deda­mi­waze mosaxleobis mkveTri zrda moxdeba aqedan gamomdinare yvela SedegiT.

Tumca moyvanili maTematikuri mo­­deli uaRresi gamartivebis gamo arare­a­lis­tu­­ria, miuxedavad amisa, is kargad aRwers baqteriebis koloniebSi individTa rao­de­nobis zrdis dinamikas sakvebi garemos gamofitvamde. amrigad, Cvens mier gan­xi­lu­li maTe­ma­ti­kuri mo­­deli samarTliania garkveul pirobebSi _ drois mcire mo­nak­veTSi. am daskvnas zogierTi tipis baqteriebze Catarebuli realuri eqspe­ri­men­tebis Sedegebi adasturebs.

amasTan dakavSirebiT moviyvanoT maTematikuri mo­­delis erTi saintereso maga­li­Ti. mkvlevarebis mizani iyo, SeeswavlaT dedamiwaze mosaxleobis raodenobis zrdis dinami­ka.

vTqvaT, mosaxleobis zrdis siCqare mamakacebisa da qalebis raodenobis nam­rav­lis proporciulia (es hipoTezaa, romelic bevr praqtikulad saintereso Sem­Txve­vaSi das­turdeba), e. i., dedamiwaze mosaxleobis raodenobis zrdis dinami­ka aR­vweroT Semdegi di­ferencialuri gantolebiT:

, (5.2.4)

sadac dedamiwaze mosaxleobis raodenobaa drois t mo­men­tSi, da _ Se­saba­mi­­sad mamakacebisa da qalebis raodenoba, _ propor­ciu­lo­­­bis garkveuli ko­e­fi­ci­en­ti.

sakmarisad didi sizustiT SeiZleba CavTvaloT, rom

,

maSin (5.2.4) miiRebs



(5.2.5)

saxes.


vTqvaT, momen­tSi mosaxleobis raodenobaa , maSin (5.2.5) gantolebis amo­­nax­sni am sawyisi monacemiT gamoisaxeba Semdegi formulis saSualebiT:

(5.2.6)

sadac


.

marTlac, SemoviRoT



(5.2.7)

aRniSvna. maSin, radganac



,

(5.2.5) gantoleba Semdegi saxiT Caiwereba:



.

saidanac -dan t-mde integrebiT miviRebT, rom



,

e. i., (5.2.7)-is gaTvaliswinebiT,



.

amrigad, momentSi dedamiwaze macxovrebelTa raodenoba usasrulo gaxdeba. “samyaros aRsasrulis” droa. dedamiwis mosaxleobis statistikuri mona­ce­me­bis gaTva­lis­winebiT gamoTvales koeficientis mniSvneloba da daadgines, rom ka­­tas­trofa moxdeba paraskevs, 2026 wlis 13 noembers. magram katastrofa ar mox­­de­ba, radgan ufro zusti meTodebiT dadgenilia, rom am droisTvis dedamiwis mo­sax­leoba 10 miliards miaRwevs da ara -s.

rogorc aRvniSneT, (5.2.2) gantoleba kargad aRwers zogierT populaciaSi in­di­vid­Ta raodenobis zrdis dinamikas drois mcire monakveTSi. amis Semdeg am pro­ces­ze gav­le­nas axdens sakvebi resursebis nakleboba, rac maTematikur mo­­dels Se­sam­Cnevad cvlis. am SemTxvevaSi SeiZleba populaciaSi garkveul doneze individTa raodenobis stabi­li­za­cia moxdes, an es raodenoba SeiZleba regularul an arare­gu­larul fluqtuaciebs ga­nicdides, an es raodenoba SeiZleba Semcirdes.

iseTi populaciis yofaqceva, romelSic individTa raodenoba garkveul mdgrad do­­ne­ze stabi­li­zirdeba, xSirad aRiwereba Semdegi logisturi gantolebis saSua­le­biT:



, (5.2.8)

sadac populaciaSi individTa raodenobas warmoadgens t mo­mentSi, xolo , garkveuli mudmivebia.

(5.2.8) gantoleba warmoadgens umartives diferencialur gantolebas, ro­mel­sac Sem­de­gi ori mniSvnelovani Tviseba gaaCnia:

1) x-s mcire mniSvnelobebisTvis (5.2.8) gantolebis amonaxsni uaxlovdeba (5.2.2) gan­tolebis amonaxsns da aqvs eqsponencialuri xasiaTi.

2) t-s zrdasTan erTad monotonurad uaxlovdeba garkveul mudmiv mniS­vne­­lo­bas, rac populaciaSi individTa raodenobis stabi­li­za­cias asaxavs.

ganvixiloT koSis amocana (5.2.8) gantolebisTvis



, (5.2.9)

sadac populaciaSi individTa raodenobas warmoadgens momentSi.

advilad SeiZleba SevamowmoT, rom (5.2.8), (5.2.9) amocanis amonaxsns aqvs Sem­de­gi sa­xe:

. (5.2.10)
t

0

nax. 5.2.2



aqedan Cans, rom, rodesac , maSin populaciaSi individTa raodenoba . amasTan SesaZlebelia ori SemTxveva: da . gansxvaveba am or Sem­Txve­vas Soris kargad Cans nax. 5.2.2-ze (aq ganxilulia SemTxveva, rodesac ).

SevniSnoT, rom (5.2.8) gantoleba da, maSasadame, (5.2.10) funqcia sakmaod kargad aR­­wers zogierTi tipis baqteriis populaciaSi individebis raodenobis cvlis di­na­mikas saw­yis etapze (mag., kulturaSi safuaris ujredebis cvlilebis dinamikas).



5.3 “mtacebeli _ msxverplis” maTematikuri modeli
5.2 paragrafSi arsebiT cvladad miviReT ama Tu im populaciaSi individebis ra­o­­de­no­ba da SevecadeT, Segveqmna romelime erTi calkeuli populaciis ganvi­Ta­rebis mate­ma­ti­kuri modeli im pirobiT, rom populacia izolirebulia.

axla gadavdgaT Semdegi nabiji da ufro metad davuaxlovdeT realur si­tu­a­cias.

ganvixiloT ori saxeobis urTierTqmedeba. SeviswavloT ori “izoli­re­bu­li” po­­­pu­la­ci­is ganviTarebis dinamika sxvadasxva faqtorebis gaTvaliswine­biT.

sxvadasxva saxeobis or populacias Soris urTierTqmedebis meqanizmebi Se­­iZleba sam ka­tegoriad davyoT:

a) konkurecia, rodesac erTi saxeobis ganviTareba me­oris ganviTarebaze dam­Trgun­vel zegavle­nas ax­dens.

b) komensalizmi, rodesac erTi saxeoba meoris ganviTarebis stimulire­bas ax­dens.

g) mtacebloba, rodesac erTi saxeoba (“mtacebeli”) meore saxeobiT (“msxver­pliT”) ik­vebeba da, maSasadame, misi raodenobis Semcirebas iwvevs, xo­lo “msxver­pli” xels uw­yobs “mtaceblebis” raodenobis zrdas.

Cvens mizans ar Seadgens saxeobaTa urTierTqmedebis am meqanizmebis deta­lu­ri gan­­xil­va. SevecdebiT, maTematikuri modelis saSualebiT aRvweroT ise­Ti ori sa­xe­­obis po­pu­laciis ganviTarebis dinamika, romlebic erTmaneT­Tan “mtacebeli _ msxver­plis” prin­ci­piT urTierTqmedeben.

maTematikuri modelis Sedgenisas vigulisxmebT, rom msxverpls yo­vel­Tvis aqvs sa­­Su­a­leba, ipovos sak­vebi, xolo yoveli Sexvedrisas mtacebe­li aucileblad klavs msxverpls, romelic misi erTaderTi sakvebia. cxa­dia, rom am daSvebis Se­de­gad miviRebT sakmaod “idealizebul” models, rom­lis gamoyenebac mxolod zo­gi­erT SemTxvevaSi Se­iZ­le­ba, Tumca am mode­lis saSualebiT bevri saintereso, praq­tikisTvis mniSvne­lo­va­ni das­kvnis gakeTebaa SesaZlebeli.

Tu am situacias ganvixilavT, cxadi gaxdeba, rom mtaceblebis raodenoba ma­nam ima­­­tebs, sanam maT sakmarisad aqvT sakvebi, e. i. sanam msxverpli sak­ma­ri­si rao­de­no­bi­Taa. bo­­los da bolos dadgeba momenti, rodesac mtaceb­le­bis zegavleniT msxver­plis ra­o­de­no­ba sakmarisad Semcirdeba, am dros mta­ceb­lebs sakvebi ar eyofaT da da­iwyeba maTi ra­o­denobis Semcireba. es iqamde mig­viyvans, rom mtaceblebis rao­de­no­bis Semcirebis gamo da­iwyeba msxver­plis raodenobis mateba. es kvlav miscems sti­muls mtaceblebis rao­de­no­bis zrdas da a. S. cikli kvlav ganmeordeba. “mta­ce­beli _ msxverplis” tipis ur­Ti­erT­qmedeba sakmaod xSirad gvxvdeba sxvadasxva praqtikuli amocanis Ses­wavlisas. prob­le­mis aqtualurobis gamo misi Seswavla bo­lo periodSi ro­gorc ekologiis, ise sxva dar­gis mecnierebis, maT Soris ma­Te­ma­tikosTa, yuradRebis centrSi moeqca.

aRvniSnoT da -Ti Sesabamisad mtaceblisa da msxverplis rao­de­no­ba t mo­mentSi. imisTvis, rom Ca­movayaliboT maTematikuri modeli, romelic gar­kve­ul miax­lo­ebaSi po­pulaciaSi individTa raodenobis cvlilebis dinamikas aR­wers, gavakeToT ram­­de­ni­me daSveba, rac amocanas gaamartives. jer erTi, davuSvaT, rom rom im Sem­TxvevaTa raodenoba, rodesac mtacebeli msxverpls klavs, damoki­de­­bu­lia maT Sex­ved­raTa sixSireze. CavTvaloT, rom es sidide xy namravlis pro­por­ciulia. meore, ugu­le­belvyoT is dro, romelic mtacebels msxver­plis Se­saW­me­lad sWirdeba. rac Seexeba bu­nebrivi Sobadobisa da sikvdilis pi­robebSi popu­la­ciis raodenobis cvlilebas, is (5.2.8) saxis logisturi gan­tolebebis saSua­le­biT aRvweroT.

(5.2.2) saxis logisturi gantolebis saSualebiT miviRebT, rom orive po­­pu­la­ci­aSi individTa raodenobis cvlileba aRiwereba Semdegi pirveli rigis dife­ren­ci­a­lur gan­to­lebaTa sistemis saSualebiT:



, (5.3.1)

(5.3.2)

sadac a, b, c da d gar­kve­uli da­de­bi­Ti mudmivebia.

(5.3.1), (5.3.2) gantolebebi pirvelad gamoyvanili iqna 1925 w. da cno­bi­lia lot­­ka-volteras gantolebebis saxelwodebiT.

aq gaTvaliswinebulia is garemoeba, rom msxverplis ararsebobis SemTxvevaSi mta­­­ce­bel­Ta raodenoba bunebrivi sikvdilis gamo iklebs (maT sakvebi ar aqvT) da ami­tom in­di­vidTa raodenobis cvlilebis siCqare x sididis proporciulia pro­por­­ciulobis uar­yofiTi koeficientiT. msxverplis sakmao raodenobiT arsebobis Sem­TxvevaSi , mta­ce­bel­Ta raodenoba izrdeba. analogiurad, mtaceblebis ar­arsebobis Sem­Txve­vaSi msxverplTa raodenoba imatebs da am populaciaSi in­di­vidTa raodenobis cvli­lebis siCqare y sididis proporciulia proporciulobis dadebiTi koeficientiT (aq, iseve rogorc adre, vgulisxmobT, rom msxverplis sakvebis raodenoba SemosazRvruli ar aris). amave dros, Tu arseboben sakmao raodenobis mtaceblebi , maSin msxver­plis raodenoba iklebs.

amocana jer-jerobiT mTlianad dasmuli ar aris, unda iyos cnobili sawyis mo­men­tSi TiToeul populaciaSi individebis raodenoba. amrigad, (5.3.1), (5.3.2) ganto­le­bebs unda daematos e. w. sawyisi pirobebi

, , (5.3.3)

sadac da mocemuli dadebiTi ricxvebia.

(5.3.1)-(5.3.3) amocana warmoadgens koSis amocanas pirveli rigis diferencialur gan­­to­lebaTa sistemisTvis. SevniSnoT, rom gansxvavebiT zemoT ganxiluli koSis amo­ca­nis­gan ar arse­bobs am amocanis amoxsnis analizuri warmodgena. am amocanis amo­naxsni Se­iZ­le­ba miRe­bu­li iqnes mxolod ricxviTi meTodebis gamoyenebiT. rac Se­exeba (5.3.1)-(5.3.3) koSis amo­ca­nis amonaxsnis arsebobas da erTaderTobas, ga­mom­di­nareobs Teorema 3.7.5-dan, radgan (5.3.1), (5.3.2) sistemis marjvena mxareebi y-is mimarT lipSicis pirobas akma­yo­fi­lebs.

SevecadoT, gamovikvlioT (5.3.1)-(5.3.3) amocana da davadginoT kavSiri da funqciebs Soris. am mizniT SemoviRoT Semdegi aRniSvnebi:



, , , .

am aRniSvnebis Semdeg (5.3.1)-(5.3.2) diferencialuri gantolebebi miiReben Sem­deg sa­xes:



, , (5.3.4)

sadac .

am diferencialur gantolebaTa sistemas daemateba

, (5.3.5)

sawyisi pirobebi, sadac



, , .

dasmuli amocanebis fizikuri Sinaarsidan gamomdinare, SemdgomSi Cven ganvi­xi­lavT (5.3.4), (5.3.5) amocanis dadebiT amonaxsnebs.

(5.3.4) gantolebaTa sistema Semdegi saxiT gadavweroT:

, . (5.3.6)

(5.3.6) sistemis meore gantoleba gavamravloT -ze da Semdeg es gantolebebi Sev­­kri­boT:



. (5.3.7)

amis Semdeg (5.3.6) gantolebaTa sistema Semdegi saxiT gadavweroT:



,

an

, . (5.3.8)

(5.3.8) sistemis meore gantoleba gavamravloT -ze da mivumatoT pirvels, mivi­RebT:

.

es ukanaskneli gamovakloT (5.3.7) gantolebas, ris Semdegadac gveqneba:



.

vaintegroT es gantoleba -dan -mde, miviRebT:



, (5.3.9)

sadac


.

(5.3.9) saSualebas iZleva, avagoT misi Sesabamisi wirebi H-is sxvadasxva mniS­vne­lo­bis­Tvis: ,, (ix. nax. 5.3.1).

0

nax. 5.3.1 nax. 5.3.2


rogorc vxedavT, sibrtyeze miviRebT Caketil wirTa ojaxs. davuSvaT, rom sawyisi monacemebi mocemulia A wertilis traeqtoriaze, romelic mniSvnelobas Seesabameba. vTqvaT, A wertili Seesabameba sawyis mniS­vne­lo­bebs, amitom da . (5.3.4) sistemis pirveli gantoleba gviCvenebs, rom saw­yis etapze U cvla­di iklebs. analogiuri Tviseba gaaCnia V cvladsac. Semdeg, rodesac U miiRebs mniS­vnelobas , maSin . Semdeg, -s cvlilebis gar­kve­ul SualedSi, V cvladi zrdas iwyebs, xolo U cvladi klebas agrZelebs. rodesac toloba gveqneba, ma­Sin da am momentidan daiwyebs zrdas U cvla­dic da a. S. radgan wer­ti­li Caketil traeqtoriaze moZraobs, es imas niSnavs, rom gantolebaTa sistemis da amonaxsni periodul fun­qci­ebs warmoadgens, amasTan rxeva orive populaciaSi ( da funqciebis sa­SualebiT aRiwereba populaciebSi individTa raodenoba) sxvadasxva fazaSi xde­ba. moviyvanoT da funqciebis tipiuri grafiki im SemTxvevaSi, rodesac , , (ix. nax. 5.3.2).

SevniSnoT, rom radgan mtaceblebis sakvebis raodenoba SemosazRvrulia (es ra­o­­de­no­ba msxverplis ra­o­de­no­biT ganisazRvrvreba), amitom mtaceblebis popu­la­ci­a­Si indivi­de­bis ra­o­de­no­bis aRwerisTvis mizanSewonilia (5.2.8) logisturi gan­to­le­bis gaTva­lis­wi­neba. maSin ori populaciis individTa ra­o­de­no­bis cvlileba “mtacebeli _ msxver­plis” tipis urTierTqmedebis SemTxvevaSi aRiwereba Semdeg diferencialur ganto­leba­Ta sistemis saSualebiT:



, ,

sadac a, b, c, d, e gar­kve­uli da­de­bi­Ti mudmivebia, xo­lo da kvlav Sesa­ba­­mi­sad mtaceblisa da msxver­plis raodenobas gamosaxavs t mo­­­mentSi.

cxadia, rom Cvens mier moyvanili maTematikuri modelebi ori populaciis ur­Ti­erT­qme­debis meqanizms garkveul, sakmarisad mkacri SezRudvebis pirobebSi aRwe­ren. ufro zus­ti maTematikuri mo­­delis Sesaqmnelad mravali sxva faqtori (mag., po­pulaciis in­di­vidTa asaki, gamravlebis sezonuroba, is, rom mtacebeli yoveli Sexvedrisas ver axer­­xebs msxverplis daWeras da a. S.) unda iqnas gaTva­lis­wi­ne­bu­li. cxadia, am faq­to­rebis gaTvaliswineba maTematikuri models sakmaod gaarTu­lebs, magram amis safasurad ufro kargad aRwers realur situacias.
5.4. epidemiologiis umartivesi maTematikuri modelebi
istoriidan cnobilia faqtebi, rodesac sxvadasxva epidemiuri daava­de­bis­gan didi raodenobiT adamianebi iRupebodnen. qolera, Savi Wiri, gri­pi da sxva xSirad iwvevs adamianTa mniSvnelovani raodenobis daavadebas. imis­Tvis, rom rom efeqturad vebrZoloT am daavadebaTa gavrcelebas, saWiroa, wi­naswar ganisa­zR­vros, Tu ra Sedegi moyveba daavadebis sawinaaRmdego Ro­nis­Ziebebs, e. i., saWiroa moxdes sxvadasxva RonisZiebis Catarebis Sedegad avad­myofTa raodenobis dinamikis prog­no­zi­re­ba. aqedan gamomdinare mivdi­varT iseTi maTematikuri mo­­delis Seqmnis aucileblobamde, romelic daava­de­bis gavrcelebis garkveuli prognozirebis saSualebas iZleva.

simartivisTvis jer ganvixiloT situacia, rodesac araferi ar keTdeba ama Tu im epi­demiis gavrcelebis winaaRmdeg, e. i., movaxdinoT epi­demiis gav­rce­lebis bunebrivi pro­cesis prog­no­zi­re­ba.

cxadia, rom maTematikuri mo­­deli epidemiis gavrcelebaze sxvadasxva faq­toris gav­le­nas unda iTvaliswinebdes. ase, magaliTad, gaTvaliswinebuli un­da iqnes is kanonebi, romlis mixedviTac xdeba ama Tu im virusis gam­rav­le­ba, calkeuli adamianis imuniteti ama Tu im daavadebis mimarT, infeqciis matarebeli adamianebis Sexvedris albaToba jan­mrTel adamianebTan, da mra­va­li sxva faqtori. sxva sityvebiT rom vTqvaT, epidemiis ase Tu ise sru­li modeli unda Seicavdes im kvlevis Sedegebs, romelsac mecnierebis sul co­ta oTxi dargi mainc awarmoebs, kerZod, mikrobiologia, medicina, far­ma­ko­logia da so­cialuri fsiqologia.

radgan Cveni mizani mxolod sailustrcio modelis Sedgenaa, amitom ma­Te­matikuri mo­­­delis Sedgenisas bevr faqtors ar gaviTvaliswinebT. amis mi­u­xedavad, aseTi uxeSi mo­delis saSualebiTac ki SeiZleba epidemiis gav­rce­le­bis meqanizmis aRwera mis gar­kveul etapze.

amrigad, ganvixiloT adamianebis jgufi, romelic individisgan Sedge­ba. vTqvaT, momentSi am jgufSi moxvda avadmyofi adamiani (infeqciis wya­ro). vigu­lis­xmoT, rom am jgufisgan arc erTi avadmyofis CamoSoreba (mag., karantinis saSua­le­biT) ar xdeba, aseve ar aris arc gamojan­mrTe­le­bi­sa da arc sikvdilis SemTxvevebi. ase­Ti daSvebebi sruliad bunebrivia epi­de­miis dawyebidan drois mcire intervalis gan­mav­lo­baSi. CavTvaloT agreT­ve, rom nebismieri adamiani infeqciis wyarod iTvleba maSinve, rodesac is da­avaddeba.

aRvniSnoT t mo­mentSi daavadebuli adamianebis raodenoba simbolo­Ti, xolo jer­jerobiT janmrTeli adamianebis raodenoba _ simbolo­Ti. cxadia, rom Cveni daSve­bebis pirobebSi t-s nebismieri mniSvnelobisTvis sa­marTliania



(5.4.1)

toloba. rodesac , maSin .

cxadia, rom daavadebuli adamianebis raodenobis cvlilebis siCqare da­mo­kidebulia avadmyofi da janmrTeli adamianebis Sexvedraze, e. i., SeiZleba CavTvaloT, rom es siC­qare namravlis proporciulia. am daSvebis sa­fuZvelze SegviZlia davweroT

an, Tu (5.4.1) tolobas gaviTvaliswinebT,



, (5.4.2)

sadac garkveuli mudmivia. miRebuli diferencialuri gantolebis­Tvis ganvixi­loT koSis amocana



. (5.4.3)

(5.4.2), (5.4.3) warmoadgens epi­demiis gavrcelebis umartives models, rom­lis sa­Su­a­le­biTac drois nebismier t mo­men­tSi daavadebuli adamianebis ra­odenobis gansazRvra SeiZleba.

amovxsnaT es amocana. am mizniT SemoviRoT

aRniSvna, saidanac miviRebT, rom



.

Tu am ukanasknel tolobas gaviTvaliswinebT, (5.4.2) gantoleba SeiZleba Sem­degi saxiT CavweroT:



, (5.4.4)

radgan , amitom .

advilad SeiZleba Semowmeba, rom sawyisi pirobebis gaTvalis­wi­nebiT (5.4.4) gan­tolebis amonaxsns warmoadgens Semdegi funqcia:

,

saidanac miviRebT, rom



. (5.4.5)

gavaanalizoT miRebuli formula. t-s zrdasTan erTad wiladis mniS­vne­li izrdeba, e. i., izrdeba. es Seesabameba Cvens varauds, rom avad­myof­Ta raodenoba SeiZleba mxo­lod gaizardos.

sainteresoa, gamovikvlioT, Tu rogor icvleba daavadebuli adamianebis ra­odenobis zrdis siCqare. am sakiTxis Sesaswavlad unda gamovikvlioT sidide.

(5.4.5)-is gaTvaliswinebiT miviRebT, rom



.

aqedan gamomdinareobs, rom



, rodesac ;

Tu , maSin ;

Tu , maSin .

amrigad, funqcia, romelic avadmyofTa raodenobis zrdis siCqares aR­wers, momentamde izrdeba, xolo Semdeg iklebs. es Sedegi, uxeSi maTematikuri mo­­­delis miuxedavad, sakmaod kargad eTanxmeba eqsperimentul monacemebs, gansakuTrebiT epidemiis sawyis etapze.

axla ganvixiloT epidemiis sxva maTematikuri mo­­­deli, romelSic zo­gi­er­Ti iseTi faqtoris gaTvaliswineba xdeba, romelsac epidemiis Cvens mier gan­xiluli umartivesi (da rogorc aRvniSneT, uxeSi) modeli ar iTva­lis­wi­nebda.

davuSvaT, rom raime populacia, romelic N individisgan Sedgeba, sam jgu­fad iyofa. pirvel jgufs mivakuTvnoT individebi, romlebic mocemul mo­mentSi janmrTelebi arian, magram ar aqvT imuniteti garkveuli da­a­va­de­bis mimarT, e. i., arsebobs imis garkveuli albaToba, rom isini daavaddebian. aseTi individebis raodenoba t mo­mentSi aRvniSnoT simboloTi. meore jgufs mivakuTvnoT individebi, romlebic aRebul momentSi avad arian da, maSasadame, infeqciis gamavrceleblad iTvlebian. maTi raodenoba t mo­mentSi aRvniSnoT simboloTi. da bolos, mesame jgufs mivakuTvnoT indivi­de­bi, romlebic ar arian avad da amave dros aqvT imuniteti am daavadebis mi­marT. maTi raodenoba aR­vniS­noT simboloTi. e. i.,



. (5.4.6)

davuSvaT, rom im SemTxvevaSi, rodesac daavadebulTa ricxvi gar­kve­ul fiqsirebul ricxvs gada­aWarbebs, e. i. , iwyeba epidemiis pro­cesi, e. i., maT Se­uZ­li­aT daaavadon is individebi, romelTac am daava­de­bis mimarT imu­niteti ar aqvT. es ki imas niS­navs, rom garkveul momentamde SesaZlebelia daavadebul individTa izolacia (mag., ka­rantinis saSualebiT) da rom sawyis etapze daavadeba garkveuli, mcirericxovani jgu­­fis SigniT mimdinareobs. daavadebis am etaps maTematikur mo­­­delSi ar aRvwerT. Sem­deg Cav­TvaloT, rom da imunitetis armqone janmrTeli individebis jguf­Si da­a­vadebis gamo individTa raodenobis cvlilebis siCqare am jguf­Si myofi individebis ra­odenobis proporciulia. am daSvebebis Sedegad miviRebT



(5.4.7)

diferencialur gantolebas.

Semdeg davuSvaT, rom im individTa raodenobis cvlilebis siCqare, rom­le­bic daa­va­de­bulTa jgufs miekuTvnebian, magram gamojanmrTeldnen, pro­porciulia am jgufSi in­di­videbis raodenobisa da -s tolia. Tu ga­viT­va­liswinebT agreTve, rom imunitetis ar­mqone individi bolos da bolos avad­deba da TviTon xdeba infeqciis gamavrcelebeli (e. i., aseTi individi me­o­re jgufSi gadadis), amitom infeqciis gamavrcelebelTa raode­no­bis cvlilebis siCqare damokidebulia drois erTeulSi daavadebul da gamojan­mrTe­le­bul individTa raodenobis sxvaobaze. amrigad,

(5.4.8)

(5.4.7) da (5.4.8) gantolebebSi proporciulobis da ko­e­fi­ci­entebs Se­sa­bamisad vuwodoT daavadebisa da gamojan­mrTe­le­bis koefi­ci­en­te­bi.

da bolos, Tu CavTvliT, rom gamojan­mrTe­le­buli individebi imunitets iZe­nen, maSin jan­mrTe­li individebis (romlebsac am daavadebis mimarT imu­ni­te­ti gaaCniaT) raode­no­bis cvlileba SeiZleba Semdegi gantolebiT aRvwe­roT:

. (5.4.9)

miviReT (5.4.7)-(5.4.9) diferencialur gantolebaTa sistema. ganvixiloT ko­Sis amo­ca­na am gantolebaTa sistemisTvis (TiToeul jgufSi individTa ra­odenoba sawyis mo­men­tSi):



, , . (5.4.10)

(5.4.7)-(5.4.10) amocana epidemiis gavrcelebis maTematikur mo­­­dels war­mo­adgens gar­kve­ul SezRudvebSi, romelTa Sesaxebac zemoT gvqonda la­pa­ra­ki. cxadia, rom am gan­to­le­bebidan SegviZlia nebismieri SevcvaloT (5.4.6) gan­tolebiT. kerZod, Tu amovxsniT (5.4.7) da (5.4.8) gantolebebs, Seg­viZlia miviRoT (5.4.6)-dan.

msjelobis simartivisTvis davuSvaT, rom sawyis etapze aRebul po­pu­la­ci­aSi ar ari­an individebi, romlebsac garkveuli daavadebis mimarT imu­ni­te­ti aqvT, e. i., . am daSvebis Sedegad imunitets is individebi iZenen, rom­lebic avadmyofobis Semdeg ga­mo­janmrTeldebian. aseve davuSvaT, rom da­a­vadebisa da gamojan­mrTe­le­bis koeficientebi erTmaneTis tolia: (SevniSnoT, rom Tu , es msjelobaSi raime garTulebas ar gamo­iw­vevs).

ganvixiloT ori SemTxveva:

I. vTqvaT, . Cveni daSvebis Tanaxmad am dros populaciaSi Sema­val indi­vi­debs daavadeba ar gadaecemaT. am SemTxvevaSi



,

ris safuZvelzec (5.4.6) tolobis da



pirobis gaTvaliswinebiT miviRebT:



.

es modeli im situacias aRwers, rodesac yvela daavadebuli individi ga­mojanmrTe­le­bulia. am SemTxvevaSi (5.4.8) miiRebs



saxes, saidanac miviRebT, rom



,

.

nax. 5.4.1-ze grafikulad aris gamoxatuli individebis raodenobis cvlileba samive jgufSi.

nax. 5.4.1

II. vTqvaT, . am SemTxvevaSi iarsebebs intervali, sadac Sesru­le­bu­li iqneba utoloba. aqedan gamomdinareobs, rom Tu , maSin avad­myo­fo­ba gavrceldeba im individebze, romlebsac am da­a­va­debis mimarT imuniteti ar aqvT. maSin (5.4.7) gantolebidan gamom­di­na­re­obs, rom



, Tu . (5.4.11)

CavsvaT -s gamosaxuleba (5.4.8) gantolebaSi, maSin miviRebT, rom



.

am gantolebis orive mxare gavamravloT -ze:



.

am tolobis gaTvaliswinebiT SegviZlia davweroT, rom



,

sadac C nebismieri mudmivia. ukanaskneli tolobidan miviRebT, rom



.

Tu gaviTvaliswinebT sawyis pirobebs, sabolood miviRebT:



, Tu . (5.4.12)

cxadia, rom upirveles yovlisa unda davadginoT T-s mniSvneloba, ag­reT­ve _ drois is momenti, rodesac daavadebul individTa raodenoba maq­simaluria.

pirvel kiTxvaze pasuxis gacema mniSvnelovania imdenad, ramdenadac T mo­mentSi daa­va­debis Semdgomi gavrceleba Sewydeba. Tu ganvixilavT (5.4.12) gantolebas, maSin Cvens mier zemoT gamoTqmuli mosazrebis gamo samar­Tli­a­nia

. (5.4.13)

toloba.


miviReT T-s mimarT arawrfivi gantoleba, romlis amonaxsnic SeiZleba vipovoT, ma­ga­liTad, biseqciis meTodiT.

imisTvis, rom pasuxi gavceT meore kiTxvas, e. i., davadginoT -is mniS­vneloba, gan­vixiloT -s (5.4.12) gamosaxuleba. radgan -s maq­simalur mniSvnelobas Se­esabameba, amitom unda ganvixiloT



gantoleba, saidanac miviRebT, rom



. (5.4.14)

rodesac , infeqciis gavrceleba Sewydeba da (5.4.8) gantolebis amo­naxsni sawyisi pirobiT, roca , iqneba:



.

naxazze (ix. nax. 5.4.2) grafikulad aris gadmocemuli individTa rao­de­no­bis cvlis dinamika TiToeul jgufSi.


tmax

nax. 5.4.2



cxadia, rom SeiZleba analogiurad iqnes ganxiluli SemTxvevebi, rode­sac , .

SevniSnoT, rom Cvens mier ganxiluli epidemiis gavrcelebis maTemati­ku­ri mo­­­deli ram­denadme gamartivebulad aRwers realur situacias, radgan igi zogierT mniS­vne­lo­van faqtors ar iTvaliswinebs. Tumca rogorc eqs­pe­ri­mentul monacemebTan Sedareba gvi­Cvenebs, es gamartivebuli mo­­­delic sak­ma­od karg warmodgenas iZleva infeqciuri daavadebebis gavrcelebis meqa­niz­mze.



5.5 naxSirbadoqsidis daJangvis maqsimaluri siCqaris gansazRvra
vTqvaT, saWiroa, da­vad­gi­noT Jangbadis optimaluri koncentracia, romlis dro­sac airTa narevSi (naxSir­Jan­gi da Jangbadi) Semavali naxSirbadis (II) oqsidis da­Jan­gvis siCqare maqsimaluria. Se­­sa­ba­mi­si reaqciis gantoleba ase iwereba:

. (5.5.1)

CavTvaloT, rom mocemul pirobebSi procesi Seuqcevadia. maSin, cnobilia, rom moq­med masaTa kanonis Tanaxmad, (5.5.1) reaqciis siCqare aRiwereba



(5.5.2)

formuliT, sadac x CO-s koncentraciaa, y _ -is, xolo k reaqciis siCqaris mud­mi­vaa (igi damoki­de­bu­li araa moreagire komponentTa koncentraciaze).

gamovsaxoT reaqciis komponentebis koncentracia moculobiT procentebSi. ma­Sin Seg­­viZlia vTqvaT, rom

.

y-is am mniSvnelobis (5.5.2) formulaSi SetaniT miviRebT:

.

am funqciis pirveli rigis warmoebuli Semdegi sididis tolia:



. (5.5.3)

amovxsnaT



gantoleba. Tu gaviTvaliswinebT, rom , miviRebT x-is or mniSvnelobas: , . Amocanis qimiuri azridan gamomdinare saintereso es ukanas­kne­lia. amis mkac­rad dasasabuTeblad gavixsenoT, rom funqciis maqsimums Seesa­ba­me­ba pi­roba , mi­nimums _ . (5.5.3)-is gawarmoebiT miviRebT



.

cxadia, roca , maSin



,

e. i., am dros siCqare minimaluria, Tu , maSin



,

e. i., am dros siCqare maqsimaluria.

Tu , maSin Jangbadis koncentracia

.

amgvarad, naxSirbadis (II) oqsidis daJangvis siCqare maqsimaluria, roca daJangvis Sem­cve­loba airTa narevis 33,3%-s Seadgens.



5.6 maqsimaluri ganaTebulobis dadgena fotoqimiuri procesebisTvis
mTeli rigi qimiu­ri reaqciebis warmarTvisTvis iyeneben sinaTles (fotoqimia). xSirad dRis wesrigSi dgeba amocana _ ra simaRleze unda ganlagdes sareaqcio sistemidan (WurWlidan) sinaTlis wyaro, rom misi ganaTebuloba maqsimaluri iyos.

nax. 5.6.1-ze moyvanilia amocanis pirobis Sesabamisi sqema (igulisxmeba, rom ganaTebis are sxivebis perpendikularuli araa). cnobilia formula, romlis mixed­vi­Tac gana­Te­bu­loba



, (5.6.4)

sadac k proporciulobis koeficientia, r manZili sinaTlis wyarosa da obieqts Soris, _ sinaTlis sxivis dacemis kuTxe.


nax. 5.6.1-dan cxadia, rom

da

.

Tu SevitanT parametrebis am mniSvnelobas (5.6.4) formulaSi, gveqneba:

. (5.6.5)

x

nax. 5.6.2


davafiqsiroT a da x vcvaloT 0-dan -mde. x-is mniSvnelobaTa Soris un­da vipovoT iseTi, romlisTvisac udidesia. maqsimumis wertilis sapovnelad, Teorema 5.6.2-is Tanaxmad, unda ganvixiloT

,

e. i.,


gantoleba. radgan da , amitom -ze gamravlebis Semdeg mivi­RebT



, e. i.,

kvadratul gantolebas, romlis dadebiTi fesvia:



. (5.6.6)

SevniSnoT, rom, rogorc es (5.6.5)-dan gamomdinareobs,



da ,

xolo intervalze dadebiTia.

vinaidan (5.6.6)-iT gansazRvruli x0 x-is erTaderTi mniSvnelobaa, sadac warmoebuli 0-is tolia, cxa­dia, rom maqsimumis wertilia (ix. nax. 5.6.2).

rodesac


,

maSin


.

maSasadame, ganaTebuloba udidesia, rodesac dacemis kuTxis tangensi -is tolia. es ki maSin xdeba, roca



.



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