Zitsanzo zosavuta zokhala ndi khalidwe lovuta mwachitsanzo chisokonezo

Kompyuta ndi chida chomwe asayansi akugwiritsa ntchito kwambiri kuwulula zinsinsi zobisika mosamala ndi chilengedwe. Kujambula, pamodzi ndi kuyesa ndi chiphunzitso, kukukhala njira yachitatu yophunzirira dziko.

Zaka zitatu zapitazo, ku yunivesite ya Silesia, tinayambitsa pulogalamu yophatikiza njira zamakompyuta mu maphunziro. Zotsatira zake, zida zambiri zosangalatsa za didactic zidapangidwa, zomwe zimapangitsa kuti zikhale zosavuta komanso zozama kuphunzira mitu yambiri. Python inasankhidwa ngati chida chachikulu, chomwe, pamodzi ndi mphamvu zamalaibulale asayansi omwe alipo, mwina ndi njira yabwino yothetsera "zoyesera zamakompyuta" ndi ma equation, zithunzi kapena deta. Chimodzi mwazinthu zosangalatsa kwambiri zogwirira ntchito ndi Sage [2]. Ndi kuphatikiza kotseguka kwa makina a algebra apakompyuta ndi chilankhulo cha Python, komanso kumakupatsani mwayi kuti muyambe kusewera pogwiritsa ntchito msakatuli ndi imodzi mwazinthu zomwe mungathe kuzipeza kudzera muutumiki wamtambo [3] kapena seva imodzi yamakompyuta yomwe imalumikizana. Baibulo la nkhaniyi lachokera pa [4] .

Chisokonezo w ekologii

M'zaka za 1st ku yunivesite ya Oxford, wasayansi waku Australia Robert May adaphunzira mbali zongopeka za kuchuluka kwa anthu. Anafotokoza mwachidule ntchito yake mu pepala lomwe linatuluka m'magazini ya Nature pansi pa mutu wokopa "Simple Mathematical Models With Very Complex Dynamics" [XNUMX]. Kwa zaka zambiri, nkhaniyi yakhala imodzi mwazinthu zomwe zatchulidwa kwambiri pazachilengedwe. Kodi nchiyani chinachititsa chidwi choterocho m’ntchito imeneyi?

Vuto lachikale la kuchuluka kwa anthu ndi kuwerengera kuchuluka kwa mtsogolo kwa zamoyo zina, kutengera momwe zilili pano. Mwamasamu, zachilengedwe zinkaonedwa kuti ndizosavuta kwambiri zomwe moyo wa mbadwo umodzi wa anthu umakhala nyengo imodzi. Chitsanzo chabwino ndi kuchuluka kwa tizilombo tomwe timatha kusintha nyengo imodzi, monga agulugufe. Nthawi mwachilengedwe imagawika m'zigawo ziwiri zofananira ndi moyo wa anthu. Chifukwa chake, ma equations omwe amafotokoza za chilengedwe chotere mwachilengedwe amakhala ndi zomwe zimatchedwa nthawi yeniyeni, i.e. t = 2…. Robert May adalimbana ndi mphamvu zoterezi, pakati pa zinthu zina. M’lingaliro lake, iye anafewetsa zamoyo kuti zikhale za mtundu umodzi wokha umene chiwerengero chawo chinali chofanana ndi cha anthu a chaka chathachi. Kodi chitsanzochi chinachokera kuti?

Equation yosavuta kwambiri yofotokozera za kusinthika kwa chiwerengero cha anthu ndi mzere wamzere:

kumene Ni ndi kuchuluka mu nyengo ya i-th, ndipo Ni + 1 akufotokoza chiwerengero cha anthu mu nyengo yotsatira. Ndikosavuta kuwona kuti equation yotere imatha kubweretsa zochitika zitatu. Pamene = 1, chisinthiko sichidzasintha kukula kwa chiwerengero cha anthu, ndipo <1 imayambitsa kutha, ndipo mlandu a> 1 amatanthauza kuwonjezeka kwa chiwerengero cha anthu mopanda malire. Izi zipangitsa kusalinganika kwachilengedwe. Popeza kuti chilichonse m’chilengedwe chili ndi malire, n’zomveka kusintha chigawochi kuti chikhale chochepa cha zinthu. Tangoganizani kuti tizirombo timadya tirigu, zomwe chaka chilichonse zimakhala zofanana. Ngati tizilombo tating'ono poyerekezera ndi kuchuluka kwa chakudya chomwe tingathe kubereka, tikhoza kuberekana ndi mphamvu zonse zobereka, masamu amatsimikiziridwa ndi nthawi zonse a > 1. Komabe, pamene chiwerengero cha tizilombo chikuwonjezeka, chakudya chidzakhala chochepa ndipo mphamvu yobereka idzachepa. Pazifukwa zovuta, munthu akhoza kuganiza kuti tizilombo tochuluka timabadwa kuti timadya mbewu zonse asanakhale ndi nthawi yobereka, ndipo anthu amafa. Chitsanzo chomwe chimaganizira zotsatira za kuchepa kwa chakudya chochepa chinaperekedwa koyamba ndi Verhulst mu 1838. Muchitsanzo ichi, kukula kwake sikukhazikika, koma kumadalira chikhalidwe cha anthu:

Ubale pakati pa chiwerengero cha kukula kwa a ndi Ni uyenera kukhala ndi katundu wotsatira: ngati chiwerengero cha anthu chikuwonjezeka, chiwerengero cha kukula chiyenera kuchepa chifukwa kupeza chakudya kumakhala kovuta. Zoonadi, pali ntchito zambiri ndi katunduyu: izi ndi ntchito zapamwamba. Verhulst adapereka mgwirizano wotsatirawu:

pomwe a> 0 ndi K> 0 nthawi zonse amawonetsa chakudya ndipo amatchedwa mphamvu ya chilengedwe. Kodi kusintha kwa K kumakhudza bwanji kuchuluka kwa anthu? Ngati K iwonjezeka, Ni/K imachepa. Komanso, izi zimabweretsa kuti 1-Ni / K imakula, zomwe zikutanthauza kuti imakula. Izi zikutanthauza kuti chiwonjezeko chikuwonjezeka ndipo chiwerengero cha anthu chikukula mofulumira. Chifukwa chake tiyeni tisinthe mtundu wam'mbuyomu (1) poganiza kuti kukula kumasintha monga momwe zilili (3). Kenako timapeza equation

Equation iyi ikhoza kulembedwa ngati equation yobwerezabwereza

pamene xi = Ni / K ndi xi + 1 = Ni + 1 / K amatanthauza kuchuluka kwa anthu mu nthawi i ndipo m'kupita kwa nthawi i + 1. Equation (5) imatchedwa equation ya logistic.

Zingawoneke kuti ndi kusinthidwa kwakung'ono koteroko, chitsanzo chathu ndi chosavuta kusanthula. Tiyeni tifufuze. Ganizirani za equation (5) ya parameter a = 0.5 kuyambira pa chiwerengero choyambirira x0 = 0.45. Ziwerengero zotsatizana zitha kupezeka pogwiritsa ntchito recursive equation (5):

x₁= ax₀(1st₀)

x₂= ax₁(1st₁)

x₃= ax₂(1st₂)

Kuti tithandizire kuwerengera mu (6), titha kugwiritsa ntchito pulogalamu yotsatirayi (yalembedwa mu Python ndipo imatha kuthamanga, mwa zina, pa nsanja ya Sage. Tikukulimbikitsani kuti muwerenge bukuli http://icse.us.edu .pl/e-book . ), kutengera chitsanzo chathu:

= 0.5 x = 0.45 pa kwa ine (10):      x \u1d a * x * (XNUMX-x)      sindikiza x

Timawerengera motsatizana zikhalidwe za xi ndikuwona kuti zimakhala ziro. Poyesa nambala yomwe ili pamwambapa, ndizosavuta kuwona kuti izi ndi zoona mosasamala kanthu za mtengo woyamba wa x0. Izi zikutanthauza kuti chiwerengero cha anthu chikufa nthawi zonse.

Pa gawo lachiwiri la kusanthula, timawonjezera mtengo wa parameter a ku mtengo uliwonse mumtundu ae (1,3). Zikuoneka kuti ndiye kuti kutsatizana xi kumapita ku ndalama zina x * > 0. Kutanthauzira izi kuchokera ku lingaliro la chilengedwe, tikhoza kunena kuti kukula kwa chiwerengero cha anthu kumakhazikitsidwa pamlingo wina, womwe sumasintha nyengo ndi nyengo. . Ndizofunikira kudziwa kuti mtengo wa x * sudalira mtundu woyamba x0. Izi ndi zotsatira za kuyesetsa kwa chilengedwe kuti zikhazikike - anthu amasintha kukula kwake kuti athe kudzidyetsa okha. Masamu, zimanenedwa kuti dongosololi limakhala lokhazikika lokhazikika, i.e. kukhutiritsa kufanana x = f (x) (izi zikutanthauza kuti panthawi yotsatira boma liri lofanana ndi nthawi yapitayi). Ndi Sage, titha kuwona chisinthikochi mwachidwi popanga chiwembu cha anthu pakapita nthawi.

Kukhazikika kotereku kumayembekezeredwa ndi ochita kafukufuku, ndipo kugwirizana kwa zinthu (5) sikukanakopa chidwi chachikulu ngati sikunali kudabwa. Zinapezeka kuti pazinthu zina za parameter, chitsanzo (5) chimachita mosadziwika bwino. Choyamba, pali periodic ndi multiperiodic states. Kachiwiri, ndi sitepe iliyonse, chiwerengero cha anthu chimasintha mosagwirizana, monga kuyenda mwachisawawa. Chachitatu, pali chidwi chachikulu pazikhalidwe zoyambira: maiko awiri osadziwika bwino amatsogolera ku chisinthiko chosiyana. Zonsezi ndi khalidwe la khalidwe lomwe limafanana ndi kayendetsedwe kachisawawa ndipo amatchedwa chisokonezo cha deterministic.

Tiyeni tifufuze malowa!

Choyamba, tiyeni tiyike mtengo wa parameter a = 3.2 ndikuyang'ana chisinthiko. Zingawoneke zodabwitsa kuti nthawi ino chiwerengero cha anthu sichimafika pamtengo umodzi, koma ziwiri, zomwe zimachitika motsatira nyengo yachiwiri iliyonse. Komabe, zinapezeka kuti mavutowo sanathere pamenepo. Ndi = 4, dongosololi silinadziwikenso. Tiyeni tione chithunzi (2) kapena tidzapanga manambala angapo pogwiritsa ntchito kompyuta. Zotsatira zimawoneka ngati zachisawawa komanso zosiyana kwa anthu oyambira osiyana pang'ono. Komabe, wowerenga mwachidwi ayenera kutsutsa. Kodi dongosolo lofotokozedwa ndi deterministic equation1, ngakhale losavuta kwambiri, lingachite bwanji mosayembekezereka? Chabwino, mwina.

Mbali ya dongosololi ndi chidwi chake chodabwitsa pamikhalidwe yoyambira. Ndikokwanira kuyamba ndi mikhalidwe iwiri yoyambira yomwe imasiyana ndi miliyoni imodzi, ndipo munjira zochepa chabe tipeza zikhalidwe zosiyana kwambiri za anthu. Tiyeni tione pa kompyuta:

ndi = 4.0

x = 0.123 pa y=0.123+0.000001 PKC = [] kwa ine (25): x = a*x*(1-x) y = a*y*(1-y) sindikiza x,y

Nayi chitsanzo chosavuta cha chisinthiko chotsimikizika. Koma determinism iyi ndi yachinyengo, ndi masamu determinism. Kuchokera pamalingaliro othandiza, dongosololi limachita zinthu mosayembekezereka chifukwa sitingathe kukhazikitsa zomwe zidayambira masamu ndendende. M'malo mwake, chilichonse chimatsimikiziridwa ndi kulondola kwina: chida chilichonse choyezera chimakhala ndi kulondola kwina, ndipo izi zingayambitse kusadziwikiratu m'machitidwe otsimikiza omwe ali ndi chisokonezo. Chitsanzo ndi zitsanzo zowonetsera nyengo, zomwe nthawi zonse zimasonyeza katundu wachisokonezo. Ichi ndichifukwa chake zoneneratu zanyengo zanthawi yayitali ndizoyipa kwambiri.

Kusanthula machitidwe achisokonezo ndizovuta kwambiri. Komabe, titha kuthetsa zinsinsi zambiri za chisokonezo mosavuta mothandizidwa ndi zoyeserera zamakompyuta. Tiyeni tijambule zomwe zimatchedwa bifurcation, pomwe timayika mfundo za parameter motsatira abscissa axis, ndi mfundo zokhazikika za mapu opangira zinthu motsatira njira yolumikizirana. Timapeza mfundo zokhazikika potengera makina ambiri nthawi imodzi ndikukonza chiwembu pambuyo pa zitsanzo zambiri. Monga momwe mungaganizire, izi zimafuna mawerengedwe ambiri. Tiyeni tiyese "mosamala" kukonza mfundo zotsatirazi:

import numpy monga np Nx = 300 kuti = 500 x = np.linspace(0,1, nx) x = x + np.zero ((Na, Nx)) x = np.transpose(x) a=np.linspace(1,4,Na) a=a+np.zeros((Nx,Na)) kwa ine (100): x=a*x*(1-x) pt = [[a_,x_] kwa_,x_ mu zip(a.flatten(),x.flatten())] dontho(pt, size=1, figsize=(7,5))

Tiyenera kupeza zofanana ndi chithunzichi (3). Kodi mungatanthauzire bwanji chojambulachi? Mwachitsanzo, ndi mtengo wa parameter a = 3.3, tili ndi mfundo zokhazikika za 2 (chiwerengero cha anthu ndi chofanana nyengo yachiwiri iliyonse). Komabe, kwa chizindikiro a = 3.5 tili ndi mfundo 4 zokhazikika (nthawi iliyonse yachinayi chiwerengero cha anthu chimakhala ndi kukula kofanana), ndipo kwa chizindikiro a = 3.56 tili ndi mfundo 8 zokhazikika (nyengo yachisanu ndi chitatu iliyonse chiwerengero cha anthu chimakhala chofanana). Koma pagawo a≈3.57, tili ndi mfundo zambiri zokhazikika (chiwerengero cha anthu sichibwerezabwereza ndikusintha m'njira zosayembekezereka). Komabe, ndi pulogalamu ya pakompyuta, tikhoza kusintha kukula kwa parameter a ndikuyang'ana mawonekedwe a geometric opanda malire a chithunzichi ndi manja athu.

Iyi ndi nsonga chabe ya madzi oundana. Mapepala masauzande ambiri asayansi alembedwa za equation iyi, koma amabisabe zinsinsi zake. Mothandizidwa ndi kayeseleledwe ka makompyuta, mutha, popanda ngakhale kutengera masamu apamwamba, kusewera mpainiya wadziko lopanda mizere. Tikukupemphani kuti muwerenge mtundu wapaintaneti womwe uli ndi zambiri zamakhalidwe osangalatsa a equation ndi njira zosangalatsa zowonera.

¹ Lamulo la deterministic ndi lamulo lomwe tsogolo limatsimikiziridwa mwapadera ndi dziko loyamba. Mawu otsutsana ndi lamulo la probabilistic. ² Mu masamu, "discrete" amatanthauza kupeza mfundo kuchokera kumagulu ena owerengeka. Chosiyana ndi "chopitiriza".