AnnouncementsTopics:

-  section3.2(cobwebbing,equilibria),3.3(modellingwithDTDSs);6.7+6.8(stabilityofdynamicalsystems)

*Readthesesectionsandstudysolvedexamplesinyourtextbook!

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

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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xt+1 =axt1+ xt

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:rwilldependonthesizeofthepopulation (resourcesarelimited)

smallpopulationslesscompetitionhigherrlargepopulationsmorecompetitionlowerr

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MODELLINGWITHDTDSs

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MODELLINGWITHDTDSs

ModelforLimitedBacterialPopulationGrowth:Example:Determineequilibriaandbehaviourofnearbysolutionsbycobwebbing.

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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

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amountof“new”caffeineconsumedattimet+1

SubstanceAbsorption(Elimination)andReplacement(Consumption)Models

EliminationofAlcohol:Theamountofalcoholthatisbrokendownbytheliverdependsontheamountofalcoholpresentinthebody.Thelargertheamount,thesmallertheproportionofalcoholbeingeliminated.*Similartothelimitedgrowthpopulationmodel

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SubstanceAbsorption(Elimination)andReplacement(Consumption)Models

EliminationofAlcohol:DTDS:€

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amountofalcohol(g)1hourlater

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amountof“new”alcoholconsumedattimet+1

rateofelimination

SubstanceAbsorption(Elimination)andReplacement(Consumption)Models

EliminationofAlcohol:Example:RateofElimination:DTDS:

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SubstanceAbsorption(Elimination)andReplacement(Consumption)Models

EliminationofAlcohol:Example:RateofElimination:DTDS:

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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.)

effects depend on many factors (body mass, constitution, gender, age, etc.)

10 g … slight impairment 40 g … driving while intoxicated (DWI) limit; blunted feelings, disinhibition; impaired reasoning 60 g … emotional swings; staggering, slurred speech, reaction time slows down 100 g … loss of understanding, impaired sensations, memory blackout 150 g … serious breathing problems, irregular heart rate, unconsciousness 200 g … possible coma, death

SubstanceAbsorption(Elimination)andReplacement(Consumption)Models

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

SubstanceAbsorption(Elimination)andReplacement(Consumption)Models

EliminationofAlcohol:(a) Youconsumetwodrinksrightawayandcontinuetohavehalfofadrinkeveryhour

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SubstanceAbsorption(Elimination)andReplacement(Consumption)Models

EliminationofAlcohol:(b)Youconsumeonedrinkeveryhour

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f (at ) = at −10.14.2 + at

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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

CheckingStabilityofEquilibria

Todeterminestability,wecanuse:1.  Cobwebbing2.  “GraphicalCriteria”(ifthetheupdating

functionisincreasingattheequilibrium)3.  “SlopeCriteria”i.e.theStabilityTheorem

(providedtheslopeattheequilibriumisn’texactly-1or1)

StabilityTheoremforDTDSs

•  Anequilibriumisstableiftheabsolutevalueofthederivativeoftheupdatingfunctionis<1attheequilibrium,i.e.,

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StabilityTheoremforDTDSs

•  Anequilibriumisstableiftheabsolutevalueofthederivativeoftheupdatingfunctionis<1attheequilibrium,i.e.,

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StabilityTheoremforDTDSs

•  Anequilibriumisunstableiftheabsolutevalueofthederivativeoftheupdatingfunctionis>1attheequilibrium,i.e.,

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StabilityTheoremforDTDSs

•  Iftheslopeoftheupdatingfunctionisexactly1or-1attheequilibrium,i.e.,

thentheequilibriumcouldbestable,unstable,orhalf-stable.

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StabilityTheoremforDTDSs

Example:DTDSforalimitedpopulation

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StabilityTheoremforDTDSs

Example:DTDSforalimitedpopulation

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StabilityTheoremforDTDSs

Example:logisticdynamicalsystem

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xt+1 = rxt (1− xt )

StabilityTheoremforDTDSs

Example:logisticdynamicalsystem

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xt+1 =1.5xt (1− xt )

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GraphicalCriterionforStabilityofEquilibriaforaDTDSwithanIncreasingUpdatingFunction

•  Anequilibriumisstableifthegraphofthe(increasing)updatingfunctioncrossesthediagonalfromabovetobelow.

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GraphicalCriterionforStabilityofEquilibriaforaDTDSwithanIncreasingUpdatingFunction

•  Anequilibriumisunstableifthegraphofthe(increasing)updatingfunctioncrossesthediagonalfrombelowtoabove.

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GraphicalCriterionforStabilityofEquilibriaforaDTDSwithanIncreasingUpdatingFunction

Example:DTDSforalimitedpopulation

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