matematick e modelov an a syst emov a dynamikaxpelanek/iv109/slidy/mat-sys.pdf · matematick e...
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![Page 1: Matematick e modelov an a syst emov a dynamikaxpelanek/IV109/slidy/mat-sys.pdf · Matematick e modelov an Syst emov a dynamika P r klady Z akladn m ody chov an Syst emov a dynamika](https://reader031.vdocuments.pub/reader031/viewer/2022021810/5c75979809d3f2d3778b8ec0/html5/thumbnails/1.jpg)
Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Matematicke modelovanı a systemova
dynamika
Radek Pelanek
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Modelovanı shora
souhrnne promenne, abstrahovanı od jednotlivcu,lokalnıch vztahu
model = system rovnic
simulace = numericke resenı techto rovnic
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Lovec-korist: matematicky model
dL
dt= pl KL− ul L
dK
dt= pkK − ukKL
(Lotka-Voltera model)
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Lovec-korist: systemovy model
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Matematicke modelovanı
Zakladnı princip:
stav systemu = vektor stavovych promennych
chovanı systemu (zmena) = rovnice nad stavovymipromennymi
Zakladnı delenı:
diskretnı cas
spojity cas
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Diskretnı cas
Diskretnı cas
rekurentnı rovnice
stavova promenna = posloupnost Xt
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Diskretnı cas
Fibonacciho kralıci: model
(velmi zjednoduseny) model mnozenı kralıku
Xt = pocet paru kralıku
kralıci nesmrtelnı
od veku 2 let se mnozı
model:
pocatecnı stav: X1 = X2 = 1rovnice popisujıcı zmenu:
Xt+1 = Xt + Xt−1
![Page 8: Matematick e modelov an a syst emov a dynamikaxpelanek/IV109/slidy/mat-sys.pdf · Matematick e modelov an Syst emov a dynamika P r klady Z akladn m ody chov an Syst emov a dynamika](https://reader031.vdocuments.pub/reader031/viewer/2022021810/5c75979809d3f2d3778b8ec0/html5/thumbnails/8.jpg)
Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Diskretnı cas
Fibonacciho kralıci: chovanı
Model:
Xt+1 = Xt + Xt−1 X1 = X2 = 1
Test: ktere z nasledujıcıho je explicitnım resenım?
Xt =φt + 1
2− 1
Xt =φt − (1− φ)t
√5
Xt =t · (1− φ)
(1 + φ)
ve vsech prıpadech:
φ = (1 +√
5)/2
![Page 9: Matematick e modelov an a syst emov a dynamikaxpelanek/IV109/slidy/mat-sys.pdf · Matematick e modelov an Syst emov a dynamika P r klady Z akladn m ody chov an Syst emov a dynamika](https://reader031.vdocuments.pub/reader031/viewer/2022021810/5c75979809d3f2d3778b8ec0/html5/thumbnails/9.jpg)
Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Diskretnı cas
Fibonacciho kralıci: chovanı
Model:
Xt+1 = Xt + Xt−1 X1 = X2 = 1
Explicitnı resenı:
Xt =φt − (1− φ)t
√5
, kde φ = (1 +√
5)/2
Simulace (= dosazenı):1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Diskretnı cas
Fibonacciho kralıci: poznamky
populace roste nade vsechny meze (exponencialne)
pouze pozitivnı zpetna vazba
chybı korigujıcı negativnı zpetna vazba
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Diskretnı cas
Logisticka rovnice: model
r – mıra reprodukce
K – kapacita prostredı
rovnice:Xt+1 = r · Xt · (1− Xt/K )
Jak se bude model chovat pro K = 1,X1 = 0.2 a ruznehodnoty r?
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Diskretnı cas
Logisticka rovnice: chovanı
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Diskretnı cas
Logisticka rovnice: Feigenbaumuv diagram
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Diskretnı cas
Logisticka rovnice: poznamky
kombinace pozitivnı a negativnı zpetne vazby
velmi jednoduchy system – slozite chovanı (chaos)
nutnost pouzitı vypocetnı simulace
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Spojity cas
Spojity cas
motivace pouzitı spojiteho casu:
nelze cas rozdelit na diskretnı kroky, napr. prıtok a odtokvodyjednodussı matematicke zpracovanı nez diskretnı cas
diferencialnı rovnice
zaklad: dXdt ∼ ”
zmena hodnoty promenne X v case t“
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Spojity cas
Model populace I
zmena velikosti populace = pocet narozenı – pocet umrtı
dX
dt= pX − uX
r = p − u
dX
dt= rX
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Spojity cas
Model populace I: chovanı
Explicitnı resenı diferencialnı rovnice:
X (t) = X (0)ert
exponencialnı rust (pokles) – srovnej Fibonacciho kralıci
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Spojity cas
Model populace II
Podobne jako pro diskretnı logistickou rovnici:
dX
dt= r · X · (1− X
K)
Explicitnı resenı:
X (t) =K
1 + ce−rt, c =
K
X (0)− 1
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Spojity cas
Numericke resenı rovnic
explicitnı obecne resenı – malokdy
numericke resenı:
priblizne resenı pro konkretnı hodnotymırne nepresne, ale pro modelovanı dostatecnenutno vsak pamatovat na nepresnost, robustnost, ...
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Spojity cas
Zakladnı myslenka
(podrobneji viz predmety na PrF:”Numericke metody“)
numericke metody – zalozeny na diskretizaci
cas – intervaly delky ∆t
v bodech tn = t + n ·∆t pocıtame hodnoty yn
zbytek aproximujeme (napr. prımkou)
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Spojity cas
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Spojity cas
Metody aproximace
hodnotu yn+1 aproximujeme s vyuzitım hodnoty yn:
Eulerova metoda: pouzitı diferencnıch rovnic,yn+1 = yn + ∆t · f (yn, t)
Runge-Kutta metody (2. radu, 4. radu): sofistikovanejsımetody aproximace; vıce operacı, ale o hodne presnejsı
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Spojity cas
Presnost a vypocetnı narocnost
zmensujıcı se ∆t:
metody konvergujı k presnemu resenı
simulace vypocetne (a tedy i casove) narocnejsı
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Spojity cas
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Spojity cas
Vyber metody: doporucenı
Runge-Kutta metoda – nevhodna pro modely sdiskretnımi prvky, na ciste spojitych lepsı nez Eulerova
Eulerova metoda – nepresna u modelu svysokofrekvencnımi oscilacemi
volba diskretnıho kroku δt (v softwaru Stella znacenyDT):
maximalne polovina minimalnıho intervalu vyskytujıcıhose v modeluvyzkouset simulaci pro ruzne hodnoty δt
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Spojity cas
Nepresnosti numerickych metod a typy modelu
”presne“ modely, ucel predpovedi – stabilita a presnost
numerickych metod zasadnı
”hrube“ modely, ucel pochopenı/vhled – nepresnosti
modelovanı vesmes vyznamnejsı nez nepresnostinumerickych metod
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Systemova dynamika
”graficky front-end“ pro matematicke modelovanı
1 graficke vyjadrenı zakladnıch vztahu
2 automaticke vygenerovanı diferencialnıch rovnic
3 doplnenı zbyvajıcıch rovnic a hodnot parametru
4 simulace (numericke resenı rovnic)
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Prıklad
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Zakladnı prvky
Systemovy model: zakladnı prvky
1 zasobarny
2 toky
3 parametry
4 vztahy
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Zakladnı prvky
Proc?
proc nepsat rovnou rovnice?
proc rozdelenı na uvedene 4 kategorie?
prehlednost – snadnejsı navrh, ladenı, komunikace
v modelovanı omezenı muze byt vyhodou
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Zakladnı prvky
Zakladnı prvky: prıklady
zasobarna tok parametr
populace narozenı, umrtı porodnost, umrtnost,mıra emigrace
penıze na uctu uroky urokova mıra
teplota ohrıvanı tepelna kapacita
podıl na trhu novı zakaznıci naklady na reklamu,ucinnost reklamy, kva-lita vyrobku
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Zakladnı prvky
Zasobarny
= systemove promenne, reservoirs, stocks= podstatna jmena v modelu
komponenty systemu, kde se necoakumuluje
lze cıselne vyjadrit, v case stoupa aklesa
nereprezentuje (vetsinou)geografickou lokalitu
system zmrazeny v urcitemokamziku – zasobarna manenulovou hodnotu
velikostpopulace
penıze nauctu
teplota
podıl natrhu
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Zakladnı prvky
Toky
= processes, flows= slovesa v modelu
aktivity, ktere urcujı hodnotuzasobaren v case
urcujı zda obsah zasobarnynarusta/klesa
jednosmerne i obousmerne
system zmrazeny v urcitemokamziku – toky majı nulovouhodnotu
narozenı,umrtı,emigrace
uroky
ohrıvanı,ochlazenı
novızakaznıci
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Zakladnı prvky
Parametry
= convertors, auxilaries, system constants
tempo s jakym dochazı ke zmeneobsahu zasobarny vlivem toku
casto vnejsı (exogenous) promennesystemu – chovanı nemodelujeme
hodnoty – pozorovanı, uvaha,odhad
porodnost,umrtnost
urokovamıra
tepelnakapacita
naklady nareklamu,ucinnostreklamy
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Zakladnı prvky
Vztahy
= interrelationships
zavislosti mezi jednotlivymi castmi systemu
co s cım souvisı, co na cem zavisı
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Prıklady
Lisky a kralıci
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Prıklady
Specifikace modelu
pocatecnı hodnoty zasobaren (K a L)
hodnoty parametru (pl , pk , ul , uk)
rovnice pro velikost toku:
prıbytek lisek = plKL,prıbytek kralıku = pkK ,ubytek lisek = ulL,ubytek kralıku = ukKL.
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Prıklady
Automaticky vygenerovane rovnice
zmena hodnoty zasobarny = vstupnı toky – vystupnı toky
dL/dt = pl KL− ul L
dK/dt = pkK − ukKL
(Jde o Lotka-Voltera model.)
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Prıklady
Caste problemy
toky mezi zasobarnami vs.”mimo model“
konstanty ve spatnem radu (0, 05 vs. 5)
prılis rychle toky
prekombinovane”skryte“ rovnice
magicke nepojmenovane konstanty
nesmyslne jednotky, napr. tok”lide na druhou“
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Epidemie
Epidemie
model epidemie SIRS (susceptible – ill – resistant –susceptible)
predpokladejme uzavreny system (ryby v rybnıku)
stavy: zdrava, nemocna, odolna
parametry epidemie: infekcnost, umrtnost, doba nemoci,doba odolnosti
(vıce o epidemiıch pozdeji)
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Epidemie
Pozn. Sick fish, Resistant fish –”fronta“ = rozsırenı zasobarny
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Epidemie
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Polya process
Polya process
model:
pytel s cernymi a bılymi kamenytahame kameny – pravdepodobnost, ze vytahneme cernyje prımo umerna podılu dosud vytazenych cernychkamenu
otazky:
Jaky bude pomer vytazenych cernych/bılych vdlouhodobem horizontu?Co situace modeluje?
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Polya process
J. Sterman, Business Dynamics
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Polya process
Chovanı
Pocatecnı nahodne tahy stanovı pomer, ktereho se systemnadale drzı (lze dokazat tez analyticky).
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Polya process
Variace
pravdepodobnost vytazenı je nelinearne zavisla na pomerukamenu ⇒ pomer konverguje k 0 nebo 1
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Polya process
Polya process: komentare
lock-in: system se zamkne do urcite konfigurace, aniz by ktomu byl specificky duvod
system rızeny pozitivnı zpetnou vazbou
o osudu rozhodujı nahodne vychylky na pocatku
existence radu nenı dıky nahode, je zarucena pozitivnızpetnou vazbou
prıklady?
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Polya process
Polya process: prıklady
typicky prıklad: dve firmy soutezı o dominanci na trhu sestejnym produktem
videokazety: VHS X Betamax
Wintel
Facebook vs MySpace
QWERTY
Silicon Valey
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Demografie
Modelovanı demografie
demografie – studium lidskych populacı
typicka aplikace”modelovanı shora“
relativne dobra predvıdatelnost vyvoje
ne uplne intuitivnı, modely uzitecne
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Demografie
Demografie: Kvızova otazka
populacnı dynamika
zeme s vysokou porodnostı a nızkou umrtnostı (tj. prudkyrust populace)
porodnost prudce klesne na cca 2 deti/zenu
jak bude vypadat vyvoj velikosti populace?
kdy se ustalı?
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Demografie
Vekove pyramidy – kvız
Brazılie, CR, Japonsko, Nigerie, Rusko, USA
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Demografie
Vekove pyramidy – kvız
http://esa.un.org/unpd/wpp/Graphs/DemographicProfiles/
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Demografie
Vekova pyramida
Joe McFalls (2007), Population: A Lively Introduction
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Demografie
Vekova pyramida: Nemecko
Joe McFalls (2007), Population: A Lively Introduction
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Demografie
Vekova pyramida: CR
Wikipedia: Vekova pyramida
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Demografie
Modelovanı demografie: Rozklad zasobaren
rozklad zasobarny na podzasobarny, kterymi elementysekvencne prochazı
populace: vekove skupiny
zamestnanci: postavenı ve firme, akademicke tituly
CFC, pesticidy
finance: solventnost klientu
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Demografie
J. Sterman, Business Dynamics
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Demografie
Modelovanı demografie: zakladnı parametry
porodnost (distribuce podle veku zeny)
umrtnost (distribuce podle veku)
migrace
I jednoduchy model prinası zajımavy vhled (viz kvızovaotazka), prıklady:
http://www.learner.org/courses/envsci/
interactives/demographics/
Modelovanı zakladnıch demografickych procesu, BP JanBleha
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Demografie
Demograficky prechod
Joe McFalls (2007), Population: A Lively Introduction
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Demografie
Demografie – dopad, kontext
dopad mj. na:
ekonomika
zdravotnictvı
skolstvı
dulezite faktory mj.:
pomer pracujıcıch k celkove populaci, demografickadividenda
pomer skupiny 15-25 v populaci – socialnı nepokoje
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Svet sedmikrasek
Hypoteza Gaia
Hypoteza Gaia (James Lovelock)
Ziva hmota na planete Zemi funguje jako jeden organismusudrzujıcı si vhodne podmınky pro zivot.
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Svet sedmikrasek
Svet sedmikrasek (Daisy world)
Ucel modelu
Podpora teorie Gaia.
Zakladnı myslenka modelu
Hypoteticky svet obıhajıcı slunce, jehoz teplota roste a ktery jeschopen castecne regulovat svou teplotu.
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Svet sedmikrasek
Svet sedmikrasek
cerne a bıle sedmikrasky
rust zavisly na teplote, rustova krivka = parabola
cerne absorbujı svetlo
bıle svetlo odrazı
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Svet sedmikrasek
Svet sedmikrasek
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Svet sedmikrasek
Svet sedmikrasek: regulacnı mechanismus
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Svet sedmikrasek
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Svet sedmikrasek
Chovanı modelu
Chovanı: prekvapive stabilnı, dosahuje homeostasis (schopnostudrzovat rovnovahu pomocı regulacnıch mechanismu)
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Zakladnı mody chovanı
dobre dılo (viz napr. dum):
malokdy uzasne nove zakladnı dılyspıs dobra kombinace osvedcenych dılu
modelovanı – zakladnı mody chovanı
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Zakladnı mody
1 linearnı vyvoj
2 exponencialnı vyvoj
3 logisticky vyvoj
4 prestrel a kolaps
5 oscilace
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Linearnı vyvoj
Linearnı vyvoj
charakteristika zmena konstantnı rychlostızpetna vazba zadnadiff. rovnice dR/dt = kexplicitnı resenı R(t) = R0 + ktprıklad fixnı cerpanı neobnovitelneho zdroje
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Exponencialnı vyvoj
Exponencialnı vyvoj
charakteristika rychlost zmen umerna velikosti zasobarnyzpetna vazba pozitivnı zpetna vazbadiff. rovnice dR/dt = k · R(t)explicitnı resenı R(t) = R0 · ekt
prıklad populacnı rust pri neomezenych zdrojıch
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Logisticky vyvoj
Logisticky vyvoj
charakteristika nejdrıve exponencialnı rust, nasledovanypriblizovanım k rovnovaze (kapacita C )
zpetna vazba kombinace pozitivnı a negativnı zpetne vazby
diff. rovnice dR/dt = k(t) · R(t), kde k(t) = k0 · (1− R(t)C )
explicitnı resenı R(t) = C1+Ae−k0 t
, kde A = C−R0R0
prıklad populacnı rust s fixnımi zdroji, epidemie(vylecitelna nemoc), sırenı informacı
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Prestrel a kolaps
Prestrel a kolaps
charakteristika dve zasobarny, jeden neobnovitelny, druhy na nemzavisı a spotrebovava jej
zpetna vazba kombinace pozitivnı a negativnı zpetne vazbydiff. rovnice -prıklad populacnı rust s neobnovitelnymi zdroji, epidemie
(nevylecitelna nemoc)
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Oscilace
Oscilace
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Oscilace
Oscilace (pokracovanı)
charakteristika dve vzajemne zavisle zasobarny (Consument C , Re-source R)
zpetna vazba negativnı zpetna vazba (se zpozdenım)diff. rovnice dC/dt = kG R(t)− kD
dR/dt = kW − kQC (t)rovnovaha C = kW
kQ, R = kD
kG
prıklad dravec-korist, konzument a obnovitelny zdroj, regu-lace teploty
Vysvetlivky: kG : rust konzumenta, kD : umrtı konzumenta, kW :rust zdroje, kQ : konzumace zdroje
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Matematicke modelovanı Systemova dynamika Prıklady Zakladnı mody chovanı
Oscilace
Shrnutı
pohled shora: sumarnı promenne, rovnice popisujıcı zmenu
matematicke modelovanı: diskretnı, spojite
numericke resenı diferencialnı rovnic
systemova dynamika: graficka nadstavba
prıklady: lovec a korist, epidemie, Svet sedmikrasek, cernea bıle kulicky
zakladnı mody chovanı