數(shù)學(xué)建模課件

MathematicalModeling數(shù)學(xué)建模(英文版)機(jī)械工業(yè)出版社,北京,2003.5經(jīng)典原版書庫(kù),原書名:AFirstCourseinMathematicalModeling(ThirdEdition)byFrankR.Giordano,MauriceD.Weir,WilliamP.Fox1MathematicalModeling數(shù)學(xué)建模(英文版)Chapter1ModelingChangeIntroduction Weoftendescribeaparticularphenomenonmathematically(bymeansofafunctionoranequation,forinstance). Suchamathematicalmodelisanidealizationofthereal-worldphenomenonandneveracompletelyaccuraterepresentation.2Chapter1ModelingChangeInMathematicalModels Weareofteninterestedinpredictingthevalueofavariableatsometimeinthefuture.Amathematicalmodelcanhelpusunderstandabehaviorbetteroraidusinplanningforthefuture.

Let'sthinkofamathematicalmodelasamathematicalconstructdesignedtostudyaparticularreal-worldsystemorbehaviorofinterest.3MathematicalModels Weareoft Themodelallowsustoreachmathematicalconclusionsaboutthe behavior,asillustratedinFigure1.1. Theseconclusionscanbeinterpretedtohelpadecisionmakerplanforthefuture. Inthischapterwedirectourattentiontomodelingchange.4 ThemodelallowsustoreachFigure1.1Aflowofthemodelingprocessbeginningwithanexaminationofreal-worlddataReal-worlddataModelMathematicalconclusionsPredictions/explanationssimplificationAnalysisVerificationInterpretation5Figure1.1AflowofthemodeSimplification Mostmodelssimplifyreality.Generally,modelscanonlyapproximatereal-worldbehavior.Oneverypowerfulsimplifyingrelationshipisproportionality.6Simplification Mostmodelssi

DefinitionTwovariablesyandxareproportional(toeachother)ifoneisalwaysaconstantmultipleoftheother,thatis,ify=kxforsomenonzeroconstantk.Wewritey

x. Thedefinitionmeansthatthegraphofyversusxliesalongastraightlinethroughtheorigin.Thisgraphicalobservationisusefulintestingwhetheragivendatacollectionreasonablyassumesaproportionalityrelationship.7 DefinitionTwovariablesyaExample1

TestingforProportionality

Consideraspring-masssystem(Figure1.2).Weconductanexperimenttomeasurethestretchofthespringasafunctionofthemass(measuredasweight)placedonthespring.Considerthedatacollectedforthisexperiment,displayedinTable1.1.Figure1.2Spring-masssystem8Example1

TestingforPrTable1.1Spring-masssystemMass50100150200250Elongation1.0001.8752.7503.2504.3753003504004505005504.8755.6756.5007.2508.0008.7509Table1.1Spring-masssystem

Ascatterplotgraphofthestretchorelongationofthespringversusthemassorweightplacedonitrevealsanapproximatestraightlinepassingthroughtheorigin.Figure1.3Datafromspring-masssystem10 Ascatterplotgraphofthest Thedataappeartofollowtheproportionalityrulethatelongationeisproportionaltothemassm,orsymbolically,e

m. Wecalculatetheslopeofthelinejoiningthesepointsas Andthemodelisestimatedase=0.0163m.11 Thedataappeartofollowthe Byplottingthelinethemodelrepresentssuperimposedonthescatterplot(Figure1.4),thegraphrevealsthatthesimplifyingproportionalitymodelisreasonable.Figure1.4Datafromspring-masssystem12 ByplottingthelinethemodeModelingChange Apowerfulparadigmtouseinmodelingchangeis

futurevalue=presentvalue+change. Often,wewishtopredictthefutureonwhatweknownowandthechangethathasbeencarefullyobserved.Insuchcases,webeginbystudyingthechangeitselfaccordingtotheformula

change=futurevaluepresentvalue.13ModelingChange Apowerfulpa Ifthebehaviorofinterestistakingplaceoverdiscretetimeperiods,theprecedingconstructleadstoadifferenceequation. Otherwise,ifthebehavioristakingplacecontinuouslywithrespecttotime,thentheconstructleadstoadifferentialequation.14 Ifthebehaviorofinteresti1.1ModelingChangewithDifferenceEquations

DefinitionForasequenceofnumbersA={a0,a1,a2,…},thenthfirstdifferencesarean=an+1an,n=0,1,2,…

NotefromFigure1.5thatthedifferencerepresentstheriseorfallbetweenconsecutivevaluesofthesequence.151.1ModelingChangewithDiffeFigure1.5Thefirstdifferenceofasequenceistheriseinthegraphduringonetimeperiod16Figure1.5ThefirstdifferenExample1ASavingsCertificate Considerthevalueofasavingscertificateinitiallyworth$1000thataccumulatesinterestpaideachmonthat1%permonth.Thefollowingsequenceofnumbersrepresentsthevalueofthecertificatemonthbymonth:A={1000,1010,1020.10,1030.30,…}.17Example1ASavingsCertifi ThefirstdifferenceofAareasfollows: Thisexpressioncanberewrittenasthedifferenceequation:whichgivesthedynamicalsystemmodel:18 ThefirstdifferenceofAare Equation(1.1)representsaninfinitesetofalgebraicequations,calledadynamicalsystem. Dynamicalsystemsallowustodescribethechangefromoneperiodtothenext. Thedifferenceequationformulacomputesthenexttermknowingtheimmediatelypreviousterminthesequence,butitdoesnotcomputethevalueofaspecifictermdirectly(e.g.,thesavingsafter100periods).19 Equation(1.1)representsan Tomodifyourexample,ifweweretowithdraw$50fromtheaccounteachmonth,thechangeduringaperiodwouldbetheinterestearnedduringthatperiodminusthemonthlywithdrawal,or20 Tomodifyourexample,ifwe Inmostexamples,mathematicallydescribingthechangeisnotgoingtobeaspreciseaprocedureasillustratedhere.Oftenitisnecessarytoplotthechange,observeapattern,andthendescribethechangeinmathematicalterms.Thatis,wewillbetryingtofindchange=an=somefunctionf.21 Inmostexamples,mathematica Thechangemaybeafunctionofprevioustermsinthesequence,oritmayalsoinvolvesomeexternalterms.Thus,wewillbemodelingchangeindiscreteintervalsthisway:

change=an=an+1

an

=f(termsinthesequence,externalterms). Modelingchangeinthiswaybecomestheartofdeterminingorapproximatingafunctionfthatrepresentsthechange.22 ThechangemaybeafunctionExample2MortgagingaHome Sixyearsagoyourparentspurchasedahomebyfinancing$80000for20years,payingmonthlypaymentsof$880.87withamonthlyinterestof1%. Theyhavemade72paymentsandwishtoknowhowmuchtheyoweonthemortgage,whichtheyareconsideringpayingoffwithaninheritancetheyreceived.23Example2MortgagingaHome Thechangeintheamountowedeachperiodincreasesbytheamountofinterestanddecreasesbytheamountofthepayment: Solvingforbn+1andincorporatingtheinitialconditiongivesthedynamicalsystemmodel24 ThechangeintheamountowedThus,yieldingthesequenceB={80000,79919.13,79837.45,…}. ThesequenceisgraphedinFigure1.6.ThefigureisplottedwithMatlab,b72=71532,b241=025Thus,yieldingthesequence25Figure1.6ThesequenceandgraphforExample226Figure1.6Thesequenceand Inthissectionwehavediscussedbehaviorsintheworldthatcanbemodeledexactlybydifferenceequations.Inthenextsection,weusedifferenceequationtoapproximateobservedchange.Aftercollectingdataforthechangeanddiscerningpatternsofthebehavior,wewillusetheconceptofproportionalitytotestandfitmodelsthatwepropose.27 Inthissectionwehavediscu1.2ApproximatingChangewithDifferenceEquations

Inmostexamples,describingthechangemathematicallywillnotbeaspreciseaprocedureasinthesavingscertificateandmortgageexamplespresentedintheprevioussection.Typically,wemustplotthechange,observeapattern,andthenapproximatethechangeinmathematicalterms.281.2ApproximatingChangewithExample1GrowthofaYeastCulture Thedatainthetablebellowwascollectedfromanexperimentmeasuringthegrowthofayeastculture.TheGraph1.7representstheassumptionthatthechangeinpopulationisproportionaltothecurrentsizeofthepopulation.Thatis,pn=pn+1

pn=kpn,wherepnrepresentsthesizeofthepopulationbiomassafternhours,andkisapositiveconstant.Thevalueofkdependsonthetimemeasurement.Inthisexamplek

0.5.29Example1GrowthofaYeastTimeinhoursn01234567Observedyeastbiomasspn9.618.329.047.271.1119.1174.6257.3Changeinbiomasspn+1

pn8.710.718.223.948.055.582.7

30Timeinhoursn01234567ObserveFigure1.7Growthofayeastcultureversusbiomass31Figure1.7Growthofayeast Usingtheestimatek=0.5fortheslopeoftheline,wehypothesizetheproportionalitymodelpn=pn+1

pn=0.5pn,yieldingthepredictionpn+1=1.5pn. Thismodelpredictsapopulationthatincreasesforever,whichisquestionable.32 Usingtheestimatek=0.5fo

ModelRefinement:ModelingBirths,Deaths,andResources Ifbothbirthsanddeathsduringaperiodareproportionaltothepopulation,thenthechangeinpopulationshouldbeproportionaltothepopulation,aswasillustratedinExample1.However,certainresources(e.g.,food)cansupportonlyamaximumpopulationlevelratherthanonethatincreasesindefinitely.Asthesemaximumlevelsareapproached,growthshouldslow.33 ModelRefinement:ModelingBiExample2GrowthofaYeastCultureRevisited FindingaModelThedatainFigure1.8showwhatactuallyhappenstotheyeastculturegrowinginarestrictedareaastimeincreasesbeyondtheeightobservationsgiveninFigure1.734Example2GrowthofaYeastCTimeinhoursn01234567Observedyeastbiomasspn9.618.329.047.271.1119.1174.6257.3Changeinbiomasspn+1

pn8.710.718.223.948.055.582.793.489101112131415161718350.7441.0513.3559.7594.8629.4640.8651.1655.9659.6661.190.372.346.435.134.611.42.2

35Timeinhoursn01234567ObserveFigure1.8Yeastbiomassapproachesalimitingpopulationlevel36Figure1.8Yeastbiomassappr Fromthethirdrowofthedatatablenotethatthechangeinpopulationperhourbecomessmallerastheresourcesbecomemorelimitedorconstrained.Fromthegraphofpopulationversustime,thepopulationappearstobeapproachingalimitingvalueorcarryingcapacity.Basedonourgraphweestimatethecarryingcapacitytobe665.37 Fromthethirdrowofthedat Because665

pngetssmalleraspnapproaches665,weproposethemodelpn=pn+1

pn=k(665

pn)pn,whichcausesthechangeptobecomeincreasinglysmallaspnapproaches665. Mathematically,thishypothesizedmodelstatesthatthechangepisproportionaltotheproduct(665

pn)pn. Totestthemodel,plot(pn+1

pn)versus(665

pn)pntoseeifthereisareasonableproportionality.Thenestimatetheproportionalityconstantk.38 Because665pngetssmallerpn+1

pn8.710.718.223.948.055.5pn(665

pn)6291.8411834.6118444.0029160.1642226.2965016.6911.42.222406.6415507.369050.295968.693561.8434.641754.9682.793.490.372.346.435.185623.84104901.21110225.0198784.0077867.6158936.41Dataofpn+1

pnversus(665

pn)pn39pn+1pn8.710.718.223.948.055Figure1.9Testingtheconstrainedgrowthmodel40Figure1.9Testingtheconstr ExaminingFigure1.9,weseethattheplotdoesreasonablyapproximateastraightlineprojectedthroughtheorigin.Weestimatetheslopeofthelineapproximatingthedatatobeaboutk0.00082,whichgivesthemodelpn+1

pn=0.00082(665

pn)pn.(1.2)SolvingtheModelNumerically

SolvingEquation(1.2)forpn+1givespn+1=pn+0.00082(665

pn)pn,(1.3)whichgivesadynamicalsystemmodelwiththeinitialvaluep0=9.6.41 ExaminingFigure1.9,wesee ThisnumericalsolutionofmodelpredictionsispresentedinFigure1.10.Thepredictionsandobservationsareplottedtogetherversustimeonthesamegraph.Notethatthemodelcapturesfairlywellthetrendoftheobserveddata.42 ThisnumericalsolutionofmoFigure1.10Modelpredictionsandobservations43Figure1.10ModelpredictionsExample3SpreadofaContagiousDisease Supposethereare400studentsinacollegedormitoryandthatoneormorestudentshaveaseverecaseoftheflu.Letinrepresentthenumberofinfectedstudentsafterntimeperiods. Assumesomeinteractionbetweenthoseinfectedandthosenotinfectedisrequiredtopassonthedisease.44Example3SpreadofaContagi Ifallaresusceptibletothedisease,then400

inrepresentsthosesusceptiblebutnotyetinfected.Ifthoseinfectedremaincontagious,wecanmodelthechangeofthoseinfectedasaproportionalitytotheproductofthoseinfectedbythosesusceptiblebutnotyetinfected,orin=in+1

in=kin(400

in).(1.4)45 Ifallaresusceptibletothe Inthismodeltheproductin(400

in)representsthenumberofpossibleinteractionsbetweenthoseinfectedandthosenotinfectedattimen. Afractionkoftheseinteractionswouldcauseadditionalinfections,representedbyin.46 Inthismodeltheproductin( Equation(1.4)hasthesameformasEquation(1.2),butintheabsenceofanydatawecannotdetermineavaluefortheproportionalityconstantk.Nevertheless,agraphofthepredictionsdeterminedbyEquation(1.4)wouldhavethesameSshapeastheyeastpopulationinFigure1.10.47 Equation(1.4)hasthesamefExample4

DecayofDigoxinintheBloodstream Digoxinisusedinthetreatmentofheartdisease.Doctormustprescribeanamountofmedicinethatkeepstheconcentrationofdigoxininthebloodstreamaboveaneffectivelevelwithoutexceedingasafelevel(thereisvariationamongpatients).48Example448 Foraninitialdosageof0.5mginthebloodstream,tablebelowshowstheamountofdigoxinanremaininginthebloodstreamofaparticularpatientafterndays,togetherwiththechangeaneachday.n0123an0.5000.3450.2380.164an0.1550.1070.0740.051456780.1130.0780.0540.0370.0260.0350.0240.0170.011

49 Foraninitialdosageof0.5 AscatterplotofanversusanfromthetableisshowninFigure1.11.Thegraphshowsthatthechangeanduringatimeintervalisapproximatelyproportionaltotheamountofdigoxinanpresentinthebloodstreamatthebeginningofthetimeinterval. Theslopeoftheproportionalitylinethroughtheoriginisapproximatelyk

0.107/0.345

0.310.50 AscatterplotofanversusaFigure1.11Aplotofanversusanfromthedatasuggestsastraightlinethroughtheorigin.51Figure1.11Aplotofanver TheModelFromFigure1.11,an=an+1

an=0.31an,an+1

=0.69an. Adifferenceequationmodelforthedecayofdigoxininthebloodstreamgivenaninitialdosageof0.5mgisan+1

=0.69an,a0=0.5.52 TheModelFromFigure1.11,5Project: YouwishtobuyanewcarandnarrowyourchoicestoaSaturn,Cavalier,andHyundai.Eachcompanyoffersyouitsbestdeal:Saturn:$13,990,$1000down,3.5%interestforupto60months;Cavalier:$13,550,$1500down,4.5%interestforupto60months;Hyundai:$12,400,$500down,6.5%interestforupto48months. Youareabletospendatmost$475amonthonacarpayment.Useadynamicalsystemtodeterminewhichcartobuy.53Project:53MathematicalModeling數(shù)學(xué)建模(英文版)機(jī)械工業(yè)出版社,北京,2003.5經(jīng)典原版書庫(kù),原書名:AFirstCourseinMathematicalModeling(ThirdEdition)byFrankR.Giordano,MauriceD.Weir,WilliamP.Fox54MathematicalModeling數(shù)學(xué)建模(英文版)Chapter1ModelingChangeIntroduction Weoftendescribeaparticularphenomenonmathematically(bymeansofafunctionoranequation,forinstance). Suchamathematicalmodelisanidealizationofthereal-worldphenomenonandneveracompletelyaccuraterepresentation.55Chapter1ModelingChangeInMathematicalModels Weareofteninterestedinpredictingthevalueofavariableatsometimeinthefuture.Amathematicalmodelcanhelpusunderstandabehaviorbetteroraidusinplanningforthefuture.

Let'sthinkofamathematicalmodelasamathematicalconstructdesignedtostudyaparticularreal-worldsystemorbehaviorofinterest.56MathematicalModels Weareoft Themodelallowsustoreachmathematicalconclusionsaboutthe behavior,asillustratedinFigure1.1. Theseconclusionscanbeinterpretedtohelpadecisionmakerplanforthefuture. Inthischapterwedirectourattentiontomodelingchange.57 ThemodelallowsustoreachFigure1.1Aflowofthemodelingprocessbeginningwithanexaminationofreal-worlddataReal-worlddataModelMathematicalconclusionsPredictions/explanationssimplificationAnalysisVerificationInterpretation58Figure1.1AflowofthemodeSimplification Mostmodelssimplifyreality.Generally,modelscanonlyapproximatereal-worldbehavior.Oneverypowerfulsimplifyingrelationshipisproportionality.59Simplification Mostmodelssi

DefinitionTwovariablesyandxareproportional(toeachother)ifoneisalwaysaconstantmultipleoftheother,thatis,ify=kxforsomenonzeroconstantk.Wewritey

x. Thedefinitionmeansthatthegraphofyversusxliesalongastraightlinethroughtheorigin.Thisgraphicalobservationisusefulintestingwhetheragivendatacollectionreasonablyassumesaproportionalityrelationship.60 DefinitionTwovariablesyaExample1

TestingforProportionality

Consideraspring-masssystem(Figure1.2).Weconductanexperimenttomeasurethestretchofthespringasafunctionofthemass(measuredasweight)placedonthespring.Considerthedatacollectedforthisexperiment,displayedinTable1.1.Figure1.2Spring-masssystem61Example1

TestingforPrTable1.1Spring-masssystemMass50100150200250Elongation1.0001.8752.7503.2504.3753003504004505005504.8755.6756.5007.2508.0008.75062Table1.1Spring-masssystem

Ascatterplotgraphofthestretchorelongationofthespringversusthemassorweightplacedonitrevealsanapproximatestraightlinepassingthroughtheorigin.Figure1.3Datafromspring-masssystem63 Ascatterplotgraphofthest Thedataappeartofollowtheproportionalityrulethatelongationeisproportionaltothemassm,orsymbolically,e

m. Wecalculatetheslopeofthelinejoiningthesepointsas Andthemodelisestimatedase=0.0163m.64 Thedataappeartofollowthe Byplottingthelinethemodelrepresentssuperimposedonthescatterplot(Figure1.4),thegraphrevealsthatthesimplifyingproportionalitymodelisreasonable.Figure1.4Datafromspring-masssystem65 ByplottingthelinethemodeModelingChange Apowerfulparadigmtouseinmodelingchangeis

futurevalue=presentvalue+change. Often,wewishtopredictthefutureonwhatweknownowandthechangethathasbeencarefullyobserved.Insuchcases,webeginbystudyingthechangeitselfaccordingtotheformula

change=futurevaluepresentvalue.66ModelingChange Apowerfulpa Ifthebehaviorofinterestistakingplaceoverdiscretetimeperiods,theprecedingconstructleadstoadifferenceequation. Otherwise,ifthebehavioristakingplacecontinuouslywithrespecttotime,thentheconstructleadstoadifferentialequation.67 Ifthebehaviorofinteresti1.1ModelingChangewithDifferenceEquations

DefinitionForasequenceofnumbersA={a0,a1,a2,…},thenthfirstdifferencesarean=an+1an,n=0,1,2,…

NotefromFigure1.5thatthedifferencerepresentstheriseorfallbetweenconsecutivevaluesofthesequence.681.1ModelingChangewithDiffeFigure1.5Thefirstdifferenceofasequenceistheriseinthegraphduringonetimeperiod69Figure1.5ThefirstdifferenExample1ASavingsCertificate Considerthevalueofasavingscertificateinitiallyworth$1000thataccumulatesinterestpaideachmonthat1%permonth.Thefollowingsequenceofnumbersrepresentsthevalueofthecertificatemonthbymonth:A={1000,1010,1020.10,1030.30,…}.70Example1ASavingsCertifi ThefirstdifferenceofAareasfollows: Thisexpressioncanberewrittenasthedifferenceequation:whichgivesthedynamicalsystemmodel:71 ThefirstdifferenceofAare Equation(1.1)representsaninfinitesetofalgebraicequations,calledadynamicalsystem. Dynamicalsystemsallowustodescribethechangefromoneperiodtothenext. Thedifferenceequationformulacomputesthenexttermknowingtheimmediatelypreviousterminthesequence,butitdoesnotcomputethevalueofaspecifictermdirectly(e.g.,thesavingsafter100periods).72 Equation(1.1)representsan Tomodifyourexample,ifweweretowithdraw$50fromtheaccounteachmonth,thechangeduringaperiodwouldbetheinterestearnedduringthatperiodminusthemonthlywithdrawal,or73 Tomodifyourexample,ifwe Inmostexamples,mathematicallydescribingthechangeisnotgoingtobeaspreciseaprocedureasillustratedhere.Oftenitisnecessarytoplotthechange,observeapattern,andthendescribethechangeinmathematicalterms.Thatis,wewillbetryingtofindchange=an=somefunctionf.74 Inmostexamples,mathematica Thechangemaybeafunctionofprevioustermsinthesequence,oritmayalsoinvolvesomeexternalterms.Thus,wewillbemodelingchangeindiscreteintervalsthisway:

change=an=an+1

an

=f(termsinthesequence,externalterms). Modelingchangeinthiswaybecomestheartofdeterminingorapproximatingafunctionfthatrepresentsthechange.75 ThechangemaybeafunctionExample2MortgagingaHome Sixyearsagoyourparentspurchasedahomebyfinancing$80000for20years,payingmonthlypaymentsof$880.87withamonthlyinterestof1%. Theyhavemade72paymentsandwishtoknowhowmuchtheyoweonthemortgage,whichtheyareconsideringpayingoffwithaninheritancetheyreceived.76Example2MortgagingaHome Thechangeintheamountowedeachperiodincreasesbytheamountofinterestanddecreasesbytheamountofthepayment: Solvingforbn+1andincorporatingtheinitialconditiongivesthedynamicalsystemmodel77 ThechangeintheamountowedThus,yieldingthesequenceB={80000,79919.13,79837.45,…}. ThesequenceisgraphedinFigure1.6.ThefigureisplottedwithMatlab,b72=71532,b241=078Thus,yieldingthesequence25Figure1.6ThesequenceandgraphforExample279Figure1.6Thesequenceand Inthissectionwehavediscussedbehaviorsintheworldthatcanbemodeledexactlybydifferenceequations.Inthenextsection,weusedifferenceequationtoapproximateobservedchange.Aftercollectingdataforthechangeanddiscerningpatternsofthebehavior,wewillusetheconceptofproportionalitytotestandfitmodelsthatwepropose.80 Inthissectionwehavediscu1.2ApproximatingChangewithDifferenceEquations

Inmostexamples,describingthechangemathematicallywillnotbeaspreciseaprocedureasinthesavingscertificateandmortgageexamplespresentedintheprevioussection.Typically,wemustplotthechange,observeapattern,andthenapproximatethechangeinmathematicalterms.811.2ApproximatingChangewithExample1GrowthofaYeastCulture Thedatainthetablebellowwascollectedfromanexperimentmeasuringthegrowthofayeastculture.TheGraph1.7representstheassumptionthatthechangeinpopulationisproportionaltothecurrentsizeofthepopulation.Thatis,pn=pn+1

pn=kpn,wherepnrepresentsthesizeofthepopulationbiomassafternhours,andkisapositiveconstant.Thevalueofkdependsonthetimemeasurement.Inthisexamplek

0.5.82Example1GrowthofaYeastTimeinhoursn01234567Observedyeastbiomasspn9.618.329.047.271.1119.1174.6257.3Changeinbiomasspn+1

pn8.710.718.223.948.055.582.7

83Timeinhoursn01234567ObserveFigure1.7Growthofayeastcultureversusbiomass84Figure1.7Growthofayeast Usingtheestimatek=0.5fortheslopeoftheline,wehypothesizetheproportionalitymodelpn=pn+1

pn=0.5pn,yieldingthepredictionpn+1=1.5pn. Thismodelpredictsapopulationthatincreasesforever,whichisquestionable.85 Usingtheestimatek=0.5fo

ModelRefinement:ModelingBirths,Deaths,andResources Ifbothbirthsanddeathsduringaperiodareproportionaltothepopulation,thenthechangeinpopulationshouldbeproportionaltothepopulation,aswasillustratedinExample1.However,certainresources(e.g.,food)cansupportonlyamaximumpopulationlevelratherthanonethatincreasesindefinitely.Asthesemaximumlevelsareapproached,growthshouldslow.86 ModelRefinement:ModelingBiExample2GrowthofaYeastCultureRevisited FindingaModelThedatainFigure1.8showwhatactuallyhappenstotheyeastculturegrowinginarestrictedareaastimeincreasesbeyondtheeightobservationsgiveninFigure1.787Example2GrowthofaYeastCTimeinhoursn01234567Observedyeastbiomasspn9.618.329.047.271.1119.1174.6257.3Changeinbiomasspn+1

pn8.710.718.223.948.055.582.793.489101112131415161718350.7441.0513.3559.7594.8629.4640.8651.1655.9659.6661.190.372.346.435.134.611.42.2

88Timeinhoursn01234567ObserveFigure1.8Yeastbiomassapproachesalimitingpopulationlevel89Figure1.8Yeastbiomassappr Fromthethirdrowofthedatatablenotethatthechangeinpopulationperhourbecomessmallerastheresourcesbecomemorelimitedorconstrained.Fromthegraphofpopulationversustime,thepopulationappearstobeappro