美国数学建模一棵树的叶子TheLeavesofaTree.docx

美国数学建模一棵树的叶子TheLeavesofaTree.docx

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美国数学建模一棵树的叶子TheLeavesofaTree

TheLeavesofaTree

"Howmuchdotheleavesonatreeweigh?

"Howmightoneestimatetheactualweightoftheleaves（orforthatmatteranyotherpartsofthetree）?

Howmightoneclassifyleaves?

Buildamathematicalmodeltodescribeandclassifyleaves.Considerandanswerthefollowing:

•Whydoleaveshavethevariousshapesthattheyhave?

•Dotheshapes“minimize”overlappingindividualshadowsthatarecast,soastomaximizeexposure?

Doesthedistributionofleaveswithinthe“volume”ofthetreeanditsbrancheseffecttheshape?

•Speakingofprofiles,isleafshape（generalcharacteristics）relatedtotreeprofile/branchingstructure?

•Howwouldyouestimatetheleafmassofatree?

Isthereacorrelationbetweentheleafmassandthesizecharacteristicsofthetree（height,mass,volumedefinedbytheprofile）?

Inadditiontoyouronepagesummarysheetprepareaonepagelettertoaneditorofascientificjournaloutliningyourkeyfindings.

Nowadaystheheavymetalpollutionissocommonthatpeoplepaymoreandmoreattentiontoit.Theaimofthispaperistocalculatethemaximumofmethylmercuryinhumanbodyduringtheirlifetimeandthemaximumnumberoffishtheaverageadultcansafelyeatpermonth.FromCityOfficialsresearch[1],wegetinformationthatthemeanvalueofmethylmercuryinbasssamplesoftheNeversinkReservoiris1300ug/kgandtheaverageweightofbasspeopleconsumepermonthis0.7kg.Accordingtothedifferentconsumingtimeineverymonth,weconstructadiscretedynamicalsystemmodelfortheamountofmethylmercurythatwillbebioaccumulatedintheaverageadultbody.Inidealconditions,weassumepeopleconsumebassatfixedtermpermonth.Basedonit,weconstructfixed-ingestionmodelandwereachtheconclusionthatthemaximumamountofmethylmercurytheaverageadulthumanwillbioaccumulateintheirlifetimeis3505ug.Asmethylmercuryingestedisnotonlycomingfrombassbutalsofromotherfood,hence,wemakefurtherrevisetoourmodelsothatthemodelisclosertotheactualsituation.Asaresult,wefigureoutthemaximumamountofmethylmercurytheaverageadulthumanwillbioaccumulateintheirlifetimeis3679ug.Asamatteroffact,althoughweassumepeopleconsumeonefishpermonth,theconsumingtimehasgreatrandomness.Takingtherandomnessintoconsideration,weconstructarandom-ingestionmodelatthebasisofthefirstmodel.Throughcomputersimulations,weobtainthemaximumofmethylmercuryinhumanbodyis4261ug.Wealsocalculatethemaximumamountis4420ugafterrandom-ingestionmodelisrevised.Asitisknowntous,differentcountriesanddistrictshavedifferentcriterionsformercurytoxicity.Inourcase,weadoptLD50asthetoxiccriterions（LD50isthedosageatwhich50%ofthehumansexposedtoaparticularchemicalwilldie.TheLD50formethylmercuryis50mg/kg.）.Wespeculatemercurytoxicityhaseffectontheabilityofeliminatingmercury,therefore,wesetupvariable-eliminationmodelatthebasisofthefirstmodel.Accordingtothefirstmodel,theamountofmethylmercuryinhumanbodyis50ug/kg,farlessthan50mg/kg,sowereachtheconclusionthatthefishconsumptionrestrictionsputforwardbythereservoiradvisoriescanprotecttheaverageadult.Iftheamountofmethylmercuryingestedincreases,theamountofbioaccumulationwillgoupcorrespondingly.If50mg/kgisthemaximumamountofmethylmercuryinhumanbody,wecanobtainthemaximumnumberoffishthatpeopleconsumesafelypermonthis997.

Keywords:

methylmercurydiscretedynamicalsystemmodelvariable-eliminationmodel

discreteuniformrandomdistributionmodelrandom-ingestionmodel

Introduction

Withthedevelopmentofindustry,thedegreeofenvironmentalpollutionisalsoincreasing.Humanactivitiesareresponsibleformostofthemercuryemittedintotheenvironment.Mercury,abyproductofcoal,comesfromacidrainfromthesmokestackemissionsofold,coal-firedpowerplantsintheMidwestandSouth.Itsparticlesriseonthesmokestackplumesandhitcharideonprevailingwinds,whichoftenblownortheast.AftercollidingwiththeCatskillmountainrange,theparticlesdroptotheearth.Onceintheecosystem,micro-organismsinthesoilandreservoirsedimentbreakdownthemercuryandproduceaverytoxicchemicalformknownasmethylmercury.Ithasgreateffectonhumanhealth.

PublicofficialsareworriedabouttheelevatedlevelsoftoxicmercurypollutioninreservoirsprovidingdrinkingwatertotheNewYorkCity.Theyhaveaskedforourassistanceinanalyzingtheseverityoftheproblem.Asaresultofthebioaccumulationofmethylmercury,ifthereservoirispolluted,wecanmakesurethattheamountofmethylmercuryinfishisalsoincreasing.IfeachpersonadherestothefishconsumptionrestrictionsaspublishedintheNeversinkReservoiradvisoryandconsumesnomorethanonefishpermonth,throughanalyzing,weconstructadiscretedynamicalsystemmodeloftimefortheamountofmethylmercurythatwillbioaccumulateintheaverageadultperson.Thenwecanobtainthemaximumamountofmethylmercurytheaverageadulthumanwillbioaccumulateintheirlifetime.Atthesametime,wecanalsogetthetimethatpeoplehavetakentoachievethemaximumamountofmethylmercury.Asweknow,differentcountriesanddistrictshavedifferentcriterionsforthemercurytoxicity.Inourcase,weadoptthecriterionofKellerArmyCommunityHospital.Ifthemaximumamountofmethylmercuryinhumanbodyisfarlessthanthesafecriterion,wecanreachtheconclusionthatthereservoirisnotpollutedbymercuryorthepolluteddegreeisverylow,otherwisewecansaythereservoirisgreatpollutedbymercury.Finally,thedegreeofpollutionisdeterminedbytheamountofmethylmercuryinhumanbody.

ProblemOne

discretedynamicalsystemmodel

ThemeanvalueofmethylmercuryinbasssamplesoftheNeversinkReservoiris1300ug/kgandtheaverageweightofbassis0.7kg.Accordingtothesubject,peopleconsumenomorethanonefishpermonth.Forthesafetyofpeople,wemustconsiderthebioaccumulationofmethylmercuryundertheworstconditionthatpeopleabsorbthemaximumamountofmethylmercury.Therefore,weassumethatpeopleconsumeonefishpermonth.

Assumptions

●Theamountofmethylmercuryinfishisabsorbedcompletelyandinstantlybypeople.

●Theeliminationofmercuryisproportionaltotheamountremaining.

●Peopleabsorbfixedamountofmethylmercuryatfixedtermpermonth.

●Weassumethehalf-lifeofmethylmercuryinhumanbodyis69.3days.

Solutions

Let

denotetheproportionofeliminatingmethylmercurypermonth,

denotetheaccumulationproportion.Asweknow,methylmercurydecaysabout50percentevery65to75days,ifnofurthermethylmercuryisingestedduringthattime.Consequently,

Throughcalculating,weget

Let’sdefinethefollowingvariables：

denotestheamountofmethylmercuryatinitialtime,

denotesthenumberofmonth,

denotestheamountofmethylmercuryinhumanbodyatthemomentpeoplehavejustingestedthemethylmercuryinthemonth

denotestheamountofmethylmercurythatpeopleingestpermonthand

.

Moreover,weassume

Though,

weget

Withtheremainingamountofmethylmercuryincreasing,theeliminationofmethylmercuryisalsogoingup.Weknowtheamountofingestedmethylmercurypermouthisaconstant.Therefore,withtimegoingby,therewillbeabalancebetweenabsorptionandelimination.Wecanobtainthesteady-statevalueofremainingmethylmercuryas

approachesinfinity.

Thevalueof

isshownbyfigure1.

Figure1.merthylmercurycompletelycomingfromfishandingestedatfixedtermpermonth

Ifthedifferenceoftheremainingmethylmercurybetweenthemonth

and

islessthanfivepercentoftheamountofmethylmercurythatpeopleingestpermonth,thatis,

Thenwecanget

Atthesametime,wecanworkoutthetimethatpeoplehavetakentoachieve3380ugis11months.

Fromourmodel,wereachtheconclusionthatthemaximumamountofmethylmercurytheaverageadulthumanwillbioaccumulateintheirlifetimeis3505ug.

Ifpeopleingestmethylmercuryeveryhalfofamonth,however,thesumofmethylmercuryingestedpermonthisconstant,consequently,

Asaresult,weobtainthemaximunamountofmethylmercuryinhumanbodyis3270ug.Whenthedifferenceiswithin5%,wegetthetimepeoplehavetakentoachieveitis11months.

Similarly,ifpeopleingestmethylmercuryperday,wegetthemaximumamountis3050ug,andthetimeis10months.

RevisingModel

Asamatteroffact,theamountofmethylmercuryinhumanbodyisnotcompletelycomingfromfish.AccordingtotheresearchofHongKongSARFoodandEnvironmentalHygieneDepartment[1],undernormalcondition,about76percentofmethylmercurycomesfromfishand24percentcomesfromotherseafood.Inordertomakeourmodelmoreandmoreinlinewiththeactualsituation,itisnecessaryforustoreviseit.TheU.S.environmentalProtectionAgency（USEPA）setthesafemonthlydoseformethylmercuryat3microgramperkilogram（ug/kg）ofbodyweight.IfweadoptUSEPAcriterion,wecancalculatetheamountofmethylmercurythattheaverageadultingestfromseafoodis50.4ugpermonth.

Assumptions

●Theamountofmethylmercuryintheseafoodisabsorbedcompletelyandinstantlybypeople.

●Theeliminationofmethylmercuryisproportionaltotheamountremaining.

●Peopleingestfixedamountofmethylmercuryfromotherseafoodeveryday.

●Weassumethehalf-lifeofmethylmercuryinhumanbodyis69.3days.

Solutions

Let

denotetheamountofmethylmercuryatinitialtime,

denotethenumberofdays,

denotethere