Amamodeli alula anokuziphatha okuyinkimbinkimbi i.e. isiphithiphithi

Ikhompiyutha iyithuluzi elisetshenziswa kakhulu ososayensi ukuze bembule izimfihlo ezifihlwe ngokucophelela yindalo. Ukumodela, kanye nokuhlola kanye nethiyori, kuba indlela yesithathu yokufunda umhlaba.

Eminyakeni emithathu edlule, eNyuvesi yaseSilesia, sethula uhlelo lokuhlanganisa izindlela zamakhompiyutha emfundweni. Ngenxa yalokho, kuye kwadalwa izinto eziningi ezithokozisayo ze-didactic, okwenza kube lula futhi kujule ukufunda izihloko eziningi. I-Python yakhethwa njengethuluzi eliyinhloko, okuthi, kanye namandla emitapo yolwazi yesayensi etholakalayo, cishe iyisixazululo esingcono kakhulu "sokuhlolwa kwekhompyutha" ngezibalo, izithombe noma idatha. Okunye okuthakazelisa kakhulu ukusetshenziswa kwebhentshi eliphelele yiSage [2]. Kuwukuhlanganiswa okuvulekile kwesistimu ye-algebra yekhompyutha nolimi lwePython, futhi ikuvumela ukuthi uqale ukudlala ngokushesha usebenzisa isiphequluli sewebhu kanye nenye yezinketho zokufinyelela okungenzeka ngesevisi yefu [3] noma iseva eyodwa yekhompyutha lapho ukuxhumana inguqulo yalesi sihloko isekelwe [4] .

Isiphithiphithi ku-ecology

Eminyakeni ye-1st e-Oxford University, usosayensi wase-Australia uRobert May wafunda izici zethiyori ze-demographic dynamics. Ufingqe umsebenzi wakhe ephepheni elivele ephephabhukwini Imvelo ngaphansi kwesihloko esivusa inkanuko esithi "Amamodeli Ezibalo Alula Anamandla Ayinkimbinkimbi Kakhulu" [XNUMX]. Ngokuhamba kweminyaka, le ndatshana ibe ngomunye wemisebenzi ecashunwe kakhulu kusayensi yemvelo. Yini eyabangela isithakazelo esinjalo kulo msebenzi?

Inkinga yakudala yokuguquguquka kwabantu ukubala inani lesikhathi esizayo lohlobo oluthile, uma kubhekwa isimo salo samanje. Ngokwezibalo, izimiso zemvelo zazibhekwa njengezilula kakhulu lapho impilo yesizukulwane esisodwa sabantu ithatha isizini eyodwa. Isibonelo esihle isibalo sezinambuzane ezibhekana nokushintshashintsha okuphelele ngesizini eyodwa, njengezimvemvane. Isikhathi ngokwemvelo sihlukaniswa sibe yizikhathi ezihlukene2 ezihambisana nemijikelezo yempilo yabantu. Ngakho-ke, izibalo ezichaza i-ecosystem enjalo ngokwemvelo zinalokhu okubizwa isikhathi esinqunyiwe, i.e. t = 1,2,3…. URobert May wabhekana nokuguquguquka okunjalo, phakathi kwezinye izinto. Ekucabangeni kwakhe, wenza i-ecosystem yaba lula ukuze ibe uhlobo olulodwa olunenani labantu bangonyaka odlule. Uvelaphi lo modeli?

Izibalo ezicacile ezilula kakhulu ezichaza ukuvela kwesibalo sabantu imodeli yomugqa:

lapho i-Ni iyinala ngesizini ye-i-th, futhi u-Ni + 1 uchaza inani labantu ngesizini elandelayo. Kulula ukubona ukuthi i-equation enjalo ingaholela ezimweni ezintathu. Uma i- = 1, ukuziphendukela kwemvelo ngeke kushintshe usayizi wesibalo sabantu, futhi <1 iholela ekuqothulweni, futhi icala elithi a > 1 lisho ukukhula kwabantu okungenamkhawulo. Lokhu kuzoholela ekungalinganini kwemvelo. Njengoba yonke into emvelweni inomkhawulo, kunengqondo ukulungisa lesi sibalo ukuze sibhekane nenani elilinganiselwe lezinsiza. Ake ucabange ukuthi izinambuzane zidla okusanhlamvu, okuthi minyaka yonke kufane ncamashi. Uma izinambuzane ziyingcosana uma ziqhathaniswa nenani lokudla ezingakwazi ukuzala, zingazalana ngamandla okuzala aphelele, ngokwezibalo ezinqunywa ngokuqhubekayo a > 1. Nokho, njengoba inani lezinambuzane likhula, ukudla kuyoba yindlala futhi amandla okuzala azokwehla. Esimweni esibucayi, umuntu angacabanga ukuthi izinambuzane eziningi zizalwa kangangokuthi zidla konke okusanhlamvu ngaphambi kokuba zibe nesikhathi sokuzala, futhi inani labantu liyafa. Imodeli ecabangela lo mphumela wokufinyelela okulinganiselwe kokudla yaqala ukuphakanyiswa ngu-Verhulst ngo-1838. Kulo modeli, izinga lokukhula alishintshi, kodwa lincike esimweni sabantu:

Ubudlelwano phakathi kwezinga lokukhula kuka-a kanye ne-Ni kufanele kube nalokhu okulandelayo: uma inani labantu likhuphuka, izinga lokukhula kufanele lehle ngoba ukufinyelela kokudla kunzima. Yebo, miningi imisebenzi enalesi sakhiwo: lena imisebenzi esuka phezulu. U-Verhulst uphakamise ubudlelwano obulandelayo:

lapho u-a>0 kanye no-K>0 ongaguquki bebonisa izinsiza zokudla futhi zibizwa ngokuthi umthamo wendawo ezungezile. Ushintsho ku-K lulithinta kanjani izinga lokukhula kwabantu? Uma i-K inyuka, i-Ni/K iyehla. Ngokulandelayo, lokhu kuholela eqinisweni lokuthi i-1-Ni / K ikhula, okusho ukuthi iyakhula. Lokhu kusho ukuthi izinga lokukhula liyakhula futhi inani labantu likhula ngokushesha. Ngakho-ke masilungise imodeli yangaphambilini (1) ngokuthatha ukuthi izinga lokukhula liyashintsha njengesibalo (3). Bese sithola i-equation

Le zibalo zingabhalwa njengezibalo eziphindaphindayo

lapho u-xi = Ni / K kanye no-xi + 1 = Ni + 1 / K besho inani labantu elincishisiwe ngesikhathi i futhi ngokuhamba kwesikhathi i + 1. Izibalo (5) zibizwa ngokuthi i-logistic equation.

Kungase kubonakale sengathi ngokuguqulwa okuncane kangaka, imodeli yethu kulula ukuyihlaziya. Ake sikuhlole. Cabangela isibalo (5) sepharamitha a = 0.5 eqala kubantu bokuqala x0 = 0.45. Amanani abantu abalandelanayo angatholwa kusetshenziswa i-recursive equation (5):

x₁= izembe₀(1st₀)

x₂= izembe₁(1st₁)

x₃= izembe₂(1st₂)

Ukwenza izibalo ku-(6), singasebenzisa uhlelo olulandelayo (lubhalwe nge-Python futhi lungaqhutshwa, phakathi kwezinye izinto, endaweni yesikhulumi se-Sage. Sincoma ukuthi ufunde incwadi http://icse.us.edu .pl/e-book . ), silingisa imodeli yethu:

a =0.5 x = 0.45 ngoba mina ebangeni (10):      x \u1d a * x * (XNUMX-x)      phrinta x

Sibala amanani alandelanayo ka-xi futhi siqaphela ukuthi avame ukuba uziro. Ngokuzama ikhodi engenhla, kulula futhi ukubona ukuthi lokhu kuyiqiniso kungakhathaliseki inani lokuqala lika-x0. Lokhu kusho ukuthi inani labantu lihlala lifa.

Esigabeni sesibili sokuhlaziya, sinyusa inani lepharamitha a kunoma yiliphi inani kububanzi ae (1,3). Kuvela ukuthi ukulandelana kuka-xi kuya enanini elithile x * > 0. Uma sichaza lokhu ngokombono we-ecology, singasho ukuthi ubukhulu besibalo sabantu bunqunywa ezingeni elithile, elingaguquki kusukela enkathini kuya kwesinye. . Kubalulekile ukuqaphela ukuthi inani lika-x * alincikile esimweni sokuqala esingu-x0. Lona umphumela wokuphokophela kwe-ecosystem ukuzinzisa - inani labantu lilungisa usayizi walo ukuze likwazi ukuzondla ngokwalo. Ngokwezibalo, kuthiwa uhlelo luvame ukuya endaweni ezinzile, okungukuthi. ukwanelisa ukulingana x = f(x) (lokhu kusho ukuthi ngomzuzu olandelayo isimo siyafana nesesikhathini esidlule). Nge-Sage, singakubona ngeso lengqondo lokhu kuguquguquka ngokuhlela abantu ngokuhamba kwesikhathi.

Umthelela wokuzinzisa onjalo ubulindelwe ngabacwaningi, futhi i-logistic equation (5) ibingeke idonse ukunaka okukhulu ukube bekungekona ukumangala. Kwavela ukuthi kumanani athile wepharamitha, imodeli (5) iziphatha ngendlela engalindelekile. Okokuqala, kunezifunda ze-periodic kanye ne-multiperiodic. Okwesibili, ngesinyathelo ngasinye sesikhathi, inani labantu liyashintsha ngokungalingani, njengokunyakaza okungahleliwe. Okwesithathu, kunokuzwela okukhulu ezimeni zokuqala: izimo ezimbili ezicishe zingabonakali ziholela ekuguqukeni kwabantu okuhluke ngokuphelele. Zonke lezi zici ziyisici sokuziphatha okufana nokunyakaza okungahleliwe ngokuphelele futhi okubizwa ngokuthi isiphithiphithi esinqunyiwe.

Masihlole lesi sakhiwo!

Okokuqala, ake sibeke inani lepharamitha a = 3.2 futhi sibheke ukuvela. Kungase kubonakale kumangala ukuthi ngalesi sikhathi inani labantu alifinyeleli inani elilodwa, kodwa amabili, okwenzeka ngokulandelana njalo ngesizini yesibili. Nokho kuvele ukuthi izinkinga azigcinanga lapho. Nge = 4, isistimu ayisabikezeli. Ake sibheke umfanekiso (2) noma sizozikhiqizela izinombolo ngokulandelana sisebenzisa ikhompuyutha. Imiphumela ibonakala ingahleliwe futhi ihluke kakhulu kubantu abaqalayo abahluke kancane. Nokho, umfundi oqaphile kufanele aphikise. Isistimu echazwe nge-deterministic equation1, ngisho nelula kakhulu, ingaziphatha kanjani ngokungalindelekile? Hhayi-ke, mhlawumbe.

Isici salolu hlelo ukuzwela kwayo okuphawulekayo ezimweni zokuqala. Kwanele ukuqala ngezimo ezimbili zokuqala ezihluka ngesigidi esisodwa, futhi ngezinyathelo ezimbalwa nje sizothola amanani abantu abahluke ngokuphelele. Ake sihlole kukhompuyutha:

a = 4.0

x = 0.123 y = 0.123 + 0.000001 I-PCC = [] ngoba mina ebangeni (25): x = a*x*(1-x) u = a *u * (1-u) phrinta x, y

Nansi imodeli elula ye-deterministic evolution. Kodwa lokhu kuzimisela kuyakhohlisa, kumane kuwukunquma kwezibalo. Ngokombono ongokoqobo, uhlelo luziphatha ngendlela engalindelekile ngoba asikwazi neze ukusetha izimo zokuqala ngokwezibalo. Eqinisweni, yonke into inqunywa ngokunemba okuthile: ithuluzi ngalinye lokulinganisa linokunemba okuthile, futhi lokhu kungabangela ukungaqiniseki okungokoqobo ezinhlelweni ezinqumayo ezinempahla yesiphithiphithi. Isibonelo amamodeli wokubikezela isimo sezulu, ahlala ebonisa indawo yesiphithiphithi. Yingakho izibikezelo zezulu zesikhathi eside zizimbi kakhulu.

Ukuhlaziywa kwezinhlelo ze-chaotic kunzima kakhulu. Kodwa-ke, singakwazi ukuxazulula izimfihlakalo eziningi zesiphithiphithi kalula ngosizo lwemifanekiso yekhompyutha. Ake sidwebe lokho okubizwa ngokuthi umdwebo we-bifurcation, lapho sibeka khona amanani epharamitha eduze kwe-abscissa axis, kanye namaphoyinti azinzile wemephu ye-logistic eduze kwe-axis ehlangene. Sithola amaphuzu azinzile ngokulingisa inani elikhulu lamasistimu kanyekanye nokuhlela amanani ngemva kwezikhathi eziningi zesampula. Njengoba ungase uqagele, lokhu kudinga izibalo eziningi. Ake sizame "ngokucophelela" ukucubungula amanani alandelayo:

ngenisa i-numpy njenge-np Nx = 300 Nge = 500 х = isb. indawo yomugqa (0,1, Nx) х = х + isb. eros ((Na, Nx)) h = np.transpose (h) a = np.linspace (1,4, Na) a = a + np.zeros ((Nx, Na)) ngoba mina ebangeni (100): x=a*x*(1-x) pt = [[a_,x_] kwe_,x_ in zip(a.flatten(),x.flatten())] iphuzu (pt, usayizi = 1, figsize = (7,5))

Kufanele sithole okuthile okufana nomfanekiso (3). Ungawuhumusha kanjani lo mdwebo? Isibonelo, ngevelu yepharamitha a = 3.3, sinamaphuzu agxilile angu-2 (usayizi wesibalo sabantu uyafana njalo ngesizini yesibili). Nokho, kupharamitha a = 3.5 sinamaphuzu angu-4 angashintshi (njalo ngesizini yesine inani labantu linenombolo efanayo), futhi kupharamitha a = 3.56 sinamaphuzu angu-8 angashintshi (njalo ngesizini yesishiyagalombili inani labantu linenombolo efanayo). Kodwa kupharamitha engu-a≈3.57, sinamaphuzu angashintshiwe amaningi (usayizi wenani labantu awuphindi futhi uyashintsha ngezindlela ezingalindelekile). Kodwa-ke, ngohlelo lwekhompiyutha, singashintsha ububanzi bepharamitha a futhi sihlole isakhiwo sejometri esingapheli salo mdwebo ngezandla zethu.

Lona nje ithiphu leqhwa. Izinkulungwane zamaphepha esayensi abhaliwe mayelana nalesi sibalo, kodwa namanje sifihla izimfihlo zayo. Ngosizo lokulingisa kwekhompiyutha, ungakwazi, ngaphandle kokuthi usebenzise izibalo eziphakeme, udlale iphayona lomhlaba lokuguquguquka okungaqondile. Sikumema ukuthi ufunde inguqulo eku-inthanethi equkethe imininingwane yezinto eziningi ezithokozisayo ze-logistic equation nezindlela ezithokozisayo zokuzibona ngeso lengqondo.

¹ Umthetho wokunquma umthetho lapho ikusasa linqunywa ngokuhlukile isimo sokuqala. I-antonym umthetho we-probabilistic. ² Kumathematika, elithi "discrete" lisho ukuthola amanani kusethi ethile ebalekayo. Okuphambene nalokho "okuqhubekayo".