AnnouncementsTopics:

-  sections3.1+3.2(DTDSs),3.3(modellingwithDTDSs)*Readthesesectionsandstudysolvedexamplesinyourtextbook!

Homework:-  reviewlecturenotesthoroughly-  workonpracticeproblemsfromthetextbookandassignmentsfromthecoursepackasassignedonthecoursewebpage(underthe“SCHEDULE+HOMEWORK”link)

DynamicalSystems

•  Discrete-timedynamicalsystemsdescribeasequenceofmeasurementsmadeatequallyspacedintervals

•  Continuous-timedynamicalsystems,usuallyknownasdifferentialequations,describemeasurementsthatarecollectedcontinuously

Discrete-TimeDynamicalSystems

Adiscrete-timedynamicalsystemconsistsofaninitialvalueandarulethattransformsthesystemfromthepresentstatetoastateonestepintothefuture.

Discrete-TimeDynamicalSystemsandUpdatingFunctions

Letrepresentthemeasurementofsomequantity.Therelationbetweentheinitialmeasurementandthefinalmeasurementisgivenbythediscrete-timedynamicalsystemTheupdatingfunctionacceptstheinitialvalueasinputandreturnsthefinalvalueasoutput.Note:representspresenttimeandrepresentsonetime-stepintothefuture

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Solutions

Definition:Thesequenceofvaluesoffor0,1,2,…isthesolutionofthediscrete-timedynamicalsystemstartingfromtheinitialconditionThegraphofasolutionisadiscretesetofpointswiththetimeonthehorizontalaxisandthemeasurementontheverticalaxis.

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Example:ADiscrete-TimeDynamicalSystemfor

aBacterialPopulation

Colony InitialPopulationbt(millions)

FinalPopulationbt+1(millions)

1 0.47 0.94

2 3.30 6.60

3 0.73 1.46

4 2.80 5.60

5 1.50 3.00

6 0.62 1.24

Data:

Example:ADiscrete-TimeDynamicalSystemfor

aTreeGrowth

Tree InitialHeight,ht(m)

FinalHeight,ht+1(m)

1 23.1 23.9

2 18.7 19.5

3 20.6 21.4

4 16.0 16.8

5 32.5 33.3

6 19.8 20.6

Data:

Example:ADiscrete-TimeDynamicalSystemfor

AbsorptionofPainMedicationApatientisonmethadone,amedicationusedtorelievechronic,severepain(forinstance,aftercertaintypesofsurgery).Itisknownthateveryday,thepatient’sbodyabsorbshalfofthemethadone.Inordertomaintainanappropriatelevelofthedrug,anewdosagecontaining1unitofmethadoneisadministeredattheendofeachday.

BasicSolutions

BasicExponentialDiscrete-timeDynamicalSystemIfwithinitialcondition,thenBasicAdditiveDiscrete-timeDynamicalSystemIfwithinitialcondition,then

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ht+1 = ht + a

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ht = h0 + at.

CobwebbingCobwebbingisagraphicaltechniqueusedtodeterminethebehaviourofsolutionstoaDTDSwithoutcalculations.Thistechniqueallowsustosketchthegraphofthesolution(asetofdiscretepoints)directlyfromthegraphoftheupdatingfunction.

CobwebbingAlgorithm:1.  Graphtheupdatingfunctionandthediagonal.

2.  Plottheinitialvaluem0onthehorizontalaxis.Fromthispoint,moveverticallytotheupdatingfunctiontoobtainthenextvalueofthemeasurement.Thecoordinatesofthispointare(m0,m1).

3.  Movehorizontallytothepoint(m1,m1)onthediagonal.Plotthevaluem1onthehorizontalaxis.Thisisthenextvalueofthesolution.

4.  Fromthepoint(m1,m1)onthediagonal,moveverticallytotheupdatingfunctiontoobtainthepoint(m1,m2)andthenhorizontallytothepoint(m2,m2)onthediagonal.Plotthepointm2onthehorizontalaxis.

5.  Continuealternating(or“cobwebbing”)betweentheupdatingfunctionandthediagonaltoobtainasetofsolutionpointsplottedalongthehorizontalaxis.

Cobwebbing

Example:Startingwiththeinitialcondition,sketchthegraphofthesolutiontothesystembycobwebbing3steps. €

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Cobwebbing

Example:ConsidertheDTDSforthemethadoneconcentrationinapatient’sblood:Cobwebfor3stepsstartingfrom(i)  (ii)  (iii) 

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Equilibria

Definition:ApointiscalledanequilibriumoftheDTDSifGeometrically,theequilibriacorrespondtopointswheretheupdatingfunctionintersectsthediagonal.

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SolvingforEquilibria

Algorithm:1.  Writetheequationfortheequilibrium.2.  Solvefor3.  Thinkabouttheresults.

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SolvingforEquilibria

Examples:Findtheequilibria,iftheyexist,foreachofthefollowingsystems.(a) (b)

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Cobwebbing

Example:ConsidertheDTDSforapopulationofcodfishwhereisthenumberofcodfishinmillionsandistime.Supposethatinitiallythereare1millioncodfish.Determinetheequilibriaandthebehaviourofthepopulationovertimebycobwebbing.

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Solution:

StabilityofEquilibria

Anequilibriumisstableifsolutionsthatstartneartheequilibriummoveclosertotheequilibrium.

Anequilibriumisunstableifsolutionsthatstartneartheequilibriummoveawayfromtheequilibrium.

MODELLINGWITHDTDSs

BacterialPopulationGrowth:Theparameteriscalledpercapitaproduction.Itrepresentsthenumberofnewbacteriaproducedperbacterium.

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BacterialPopulationGrowthinGeneral

Solution:Assumption:risconstantReality:rmaydependonthesizeofthepopulation (resourcesarelimited)

smallpopulationslesscompetitionhigherrlargepopulationsmorecompetitionlowerr

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lowrate

small pop’n

highrate

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MODELLINGWITHDTDSs

ModelforLimitedBacterialPopulationGrowth:Example:

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-2000 -1500 -1000 -500 0 500 1000 1500 2000

-16

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MODELLINGWITHDTDSs

ModelforLimitedBacterialPopulationGrowth:Example:Determineequilibriaandbehaviourofnearbysolutionsbycobwebbing.

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⎝ ⎜

⎞

⎠ ⎟ ⋅ bt

0 250 500 750 1000 1250 1500 1750 2000 2250

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500

750

1000

1250

elimination of chemicals

*** filtration by kidneys (kidneys break down constant amount per hour … caffeine) *** breaking down the chemicals using enzymes from the liver (amount of chemical broken down depends on the amount present … alcohol)

SubstanceAbsorption(Elimination)andReplacement(Consumption)Models

AbsorptionofCaffeine:Ourbodieseliminatecaffeineataconstantrateof13%perhour.DTDS:*Similarto“methadone”example

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ct+1 = 0.87ct + d

amountofcaffeine(mg)1hourlater

amountofcaffeinenow

amountof“new”caffeineconsumedattimet+1

SubstanceAbsorption(Elimination)andReplacement(Consumption)Models

EliminationofAlcohol:Theamountofalcoholthatisbrokendownbytheliverdependsontheamountofalcoholpresentinthebody.Thelargertheamount,thesmallertheproportionofalcoholbeingeliminated.*Similartothelimitedgrowthpopulationmodel

r( )

a t

a t

large amount

lowrate

small amount

highrate

SubstanceAbsorption(Elimination)andReplacement(Consumption)Models

EliminationofAlcohol:DTDS:€

at+1 = at − r(at )at + d

amountofalcohol(g)1hourlater

amountofalcoholnow

amountof“new”alcoholconsumedattimet+1

rateofelimination

SubstanceAbsorption(Elimination)andReplacement(Consumption)Models

EliminationofAlcohol:Example:RateofElimination:DTDS:

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r(at ) =10.14.2 + at

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at+1 = at −10.14.2 + at

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⎠ ⎟ at + d-15 -10 -5 0 5 10 15 20

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SubstanceAbsorption(Elimination)andReplacement(Consumption)Models

EliminationofAlcohol:Example:RateofElimination:DTDS:

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r(at ) =10.14.2 + at

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at+1 = at −10.14.2 + at

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⎝ ⎜

⎞

⎠ ⎟ at + d

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at

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r(at )

0 8 16 24 32 40 48 56 64 72

1

2

3

4

5

definition

one drink = 14 grams of alcohol * 5 ounces of wine, or * 12 ounces of beer, or * 1.5 ounces of 80 proof (vodka, rum, gin, etc.)

SubstanceAbsorption(Elimination)andReplacement(Consumption)Models

EliminationofAlcohol:Example:Astandarddrinkcontains14gofalcohol.Comparewhathappensovertimeforthefollowingsituations:(a) Youconsumetwodrinksrightawayandcontinuetohavehalfofadrinkeveryhour(b) Youconsumeonedrinkeveryhour

SubstanceAbsorption(Elimination)andReplacement(Consumption)Models

EliminationofAlcohol:(a) Youconsumetwodrinksrightawayandcontinuetohavehalfofadrinkeveryhour

0 8 16 24 32 40 48 56 64 72

8

16

24

32

40

€

f (at ) = at −10.14.2 + at

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⎝ ⎜

⎞

⎠ ⎟ at + 7, a0 = 28

SubstanceAbsorption(Elimination)andReplacement(Consumption)Models

EliminationofAlcohol:(b)Youconsumeonedrinkeveryhour

0 8 16 24 32 40 48 56 64 72

8

16

24

32

40

€

f (at ) = at −10.14.2 + at

⎛

⎝ ⎜

⎞

⎠ ⎟ at +14, a0 = 0

so … how much alcohol is in the body

** 2 rapid drinks, then 1/2 drink every hour … decreases, stabilizes at 9.5 grams ** one drink every hour … increases, after 5 hours reaches 41 grams. keeps increasing, no limit