ProbabilisticModelstoPredicttheMassandShapes

ProbabilisticModelstoPredicttheMassandShapesofLeavesinaTreeControlNumber17164February13,2012AbstractThetaskistocreateamathematicalmodeltodescribeandclassifyleaves.Wehavecomeupwithprobabilisticmodelsthatareabletopredictthemassandshapesofleavesinagiventree.Webeginbyconstructingamodelofatreebasedonphysicalprinciplesofbranchingalgorithmandbifurcationconceptstoapproximatethenumberofleavesinatree.Wethenimproveonthismodeltofurtheremulatethephysicalappearanceofarealtree,whileenablingustovarytheshapeofthetree.Furthermore,researchandfurtheranalysisofphotosyntheticrateandlightirradiancefrommeteorologicaldatabasedata[1]haveenabledustorelatelightintensityandLeafMassperArea,enablingustocalculatethemostlikelymassofaleafinatree.Usingthedataobtainedandourmodel,weareabletopredictthemassofleavesinagiventree.Thismodelisusefultobeusedforswiftapproximationinbotanicalandecologicalresearch.Moreover,wehavealsocreatedacompletelydifferentprobabilisticmodeltoanalyzewhichleafshapeisthebestforaspecifictypeoftreeinagivenregion.Simulationsshowthattheswordleafshapeisthemosteconomicaltobeemployedbyanytreeinanyregion.Sinceweknowthatnotalltreeshavesword-shapedleaves,thisprovesthepointthatplantsdonotalwaysaimtohaveleavesthatmaximizesurfaceareaexposedtosunlight,asthereareotherfactorsthatmustbetakenintoconsideration[2].Lastly,weshowthatourmodelcanbefurthercalibratedandextendedtomoreenvironmentalconditionsintoaccount.Page2of69Control#17164InternationalMCM2012ProblemATableofContents1Introduction......................................................................................................................................32OutlineofOurApproach...............................................................................................................32.1KeyDefinitions...............................................................................................................................................42.2Assumptions....................................................................................................................................................43ConstructingtheModelofaTree...............................................................................................43.1VerticalBifurcationModel........................................................................................................................53.2VerticalTrunk–HorizontalBranchBifurcationModel................................................................63.3FountainTreeModel...................................................................................................................................73.4JustificationofVariables............................................................................................................................84SeasonalandRegionalVariationsinLeafMass....................................................................94.1RelationshipbetweenPhotosyntheticRateandLMA....................................................................94.2RelationshipbetweenLightIrradianceandPhotosyntheticRate.........................................114.3RegionalandSeasonalLeafMassCalculations..............................................................................125PredictingtheMostEfficientLeafShape..............................................................................135.1LightIrradianceProbabilisticModelA.............................................................................................135.2LightIrradianceProbabilisticModelB.............................................................................................155.3JustificationofVariables.........................................................................................................................176Results..............................................................................................................................................216.1LeafMassPredictions..............................................................................................................................216.2LeafShapesClassifications....................................................................................................................237FurtherConsiderations..............................................................................................................258RecommendationsandConclusions......................................................................................259References.......................................................................................................................................2610Appendices......................................................................................................................................2710.1MATLABCode..............................................................................................................................................2710.2LeafShapes..............................................................................................