2012年美国数学建模——一棵树的叶子The-Leaves-of-a-Tree

TheLeavesofaTreeHowmuchdotheleavesonatreeweigh?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-eliminationmodeldiscreteuniformrandomdistributionmodelrandom-ingestionmodelIntroductionWiththedevelopmentofindustry,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.ProblemOnediscretedynamicalsystemmodelThemeanvalu