国际建模C题O奖RebalancingHumanInfluencedWord格式文档下载.docx

国际建模C题O奖RebalancingHumanInfluencedWord格式文档下载.docx

《国际建模C题O奖RebalancingHumanInfluencedWord格式文档下载.docx》由会员分享，可在线阅读，更多相关《国际建模C题O奖RebalancingHumanInfluencedWord格式文档下载.docx（19页珍藏版）》请在冰豆网上搜索。

XingyongZhang

Summary

InTask1,weestablishaVolterrapredator-preymodelwiththreebiologicalpopulations,andwespecifythesteady-statenumbersofthethreepopulations.Then,basedontheAnalyticHierarchyProcessandacompetitionmodel,weobtaintheratioofdifferentspeciesinthesecondpopulation,predictthatthesteady-statelevelofwaterqualityisnothigh,andmakethewaterqualitysatisfactorybyadjustingthenumbersofsixspecies.

InTask2,whenmilkfishfarmingsuppressesotheranimalspecies,wesetupalogisticmodel,andpredictthatthewaterqualityatsteady-stateisawful,thesameasinthefishpens—insufficientforthecontinuedhealthygrowthofcoralspecies.Whenotherspeciesarenottotallysuppressed,withanimprovedpredator-preymodelwesimulatethewaterqualityofBolinao（makingitmatchcurrentquality）,obtainpredictednumbersofpopulations,anddiscusschangestothepredator-preymodelaimedatmakingthenumbersofthepopulationsagreemorecloselywithobservations.

InTask3,weestablishapolyculturemodelthatreflectsaninterdependentsetofspecies,introducemusselsandseaweedgrowingonthesidesofthepens,andobtainthenumbersofpopulationsinsteadystateandtheoutputsofourmodel.

InTasks4and5,wedifferentiatethemonetaryvaluesofdifferentkindsediblebiomassanddefinethetotalvalueasthesumofthevaluesofeachspeciesharvested,minusthecostofmilkfishfeed.Undercircumstances

TheUMAPJournal30

（2）（2009）121–139.!

cCopyright2009byCOMAP,Inc.Allrightsreserved.Permissiontomakedigitalorhardcopiesofpartorallofthisworkforpersonalorclassroomuseisgrantedwithoutfeeprovidedthatcopiesarenotmadeordistributedforprofitorcommercialadvantageandthatcopiesbearthisnotice.Abstractingwithcreditispermitted,butcopyrightsforcomponentsofthisworkownedbyothersthanCOMAPmustbehonored.Tocopyotherwise,torepublish,topostonservers,ortoredistributetolistsrequirespriorpermissionfromCOMAP.

ofacceptablewaterquality,webuildanonlinearequilibriumoptimizationmodel,fromwhichweobtainanoptimalstrategyandharvest.

InTask6,weputforwardastrategytoimprovethewaterqualityinBolinao.Withtheratiobetweenfeedcostandnetincomeastheindex,theindexvalueofthemodelissmallerthanthatofBolinaoarea,whichsignifiestheleverageofthestrategy.Also,weanalyzethepolyculturesystemintermsofecology.

Introduction

ToimprovethesituationinBolinao,weneedtoestablishapracticablepolyculturesystemandintroduceitgradually.Soourgoalisprettyclear:

•duction.ModeltheoriginalBolinaocoralreefecosystembeforefishfarmintro-

•ModelthecurrentBolinaomilkfishmonoculture.

•ModeltheremediationofBolinaoviapolyculture.

•Discusstheoutputsandeconomicvaluesofspecies.

•WritemarizingabriefthetorelationshipthedirectorbetweenofthePacificbiodiversityMarineFisheriesandwaterCouncilqualitysum-forcoralgrowth.

Ourapproachis:

•theDeeplycoralanalyzereeffoodweb.dataintheproblem,graduallyestablishingamodelof

•Withavailabledataasevaluationcriteria,confirmthewaterqualitybasedonelementsinthesediment.

•Establishpurposeofmodels,improvingandwaterinterpretquality.theactualsituationwithdata,withthe

•Dofurtherdiscussionbasedonourwork.

Solutions

Task1

Aimingtowardacoralreeffoodwebmodel,weassumethatallthespeciesgrowinthesamefishpen.Wedividethespeciesintothreepopulations:

•onealgaspecies（Population1）;

•oneherbivorousfish,onemolluscspecies,onecrustaceanspecies,andoneechinodermspecies（Population2）;

and

•thesolepredatorspecies,milkfish（Population3）.

TheinterrelationshipsamongthespeciesarepresentedinFigure1.

Onthisbasis,wecanestablishaVolterrapredator-preymodelwiththreepopulations[ShanandTang2007].Letthenumberoftheithpopulationbexi（t）.Ifwedonottakeintoconsiderationtherestrictionsofnaturalresources,thealgaespeciesofPopulation1growinginisolationwillfollowanexponentialgrowthlawwithrelativegrowthrater1,sothatx˙（t）=r1x1.However,speciesofPopulation2feedingonthealgaspecieswilldecreasethegrowthrateofthealgae,sotherevisedmodelofthealgaspeciesis

x˙1（t）=x1（r1−λ1x2）,

wheretheproportionalitycoefficientλ1reflectsthefeedingcapabilityofspeciesinPopulation2forthealgaspecies.

AssumethatthedeathrateofthespeciesinPopulationIIisr2whenexistinginisolation;

thenx˙2（t）=−r2x2,sobasedonthefoodwebweconcludethat

x˙2（t）=x2（−r2+λ2x1）,

wheretheproportionalitycoefficientλ2reflectsthesupportcapabilityofthealgaspeciesforPopulation2—whichinturnprovidefoodforthemilkfish.ThemilkfishreducethegrowthrateofthespeciesinPopulation2,sowemustsubtracttheirfeedingeffecttoget

x˙2（t）=x2（−r2+λ2x1−µ

x3）.

Likewise,themodelforthemilkfishis

x˙3（t）=x3（−r3+λ3x2）.

Altogether,wehaveaninterdependentandmutually-restrictingmathematicalmodelofthethreepopulations:

Sincethissystemofdifferentialequationshasnoanalyticsolution,weneedtouseMatlabtogetitsnumericalsolution.

Ecologistspointoutthataperiodicsolutioncannotbeobservedinmostbalancedecosystems;

inabalancedecosystem,thereisanequilibrium.Inaddition,someecologiststhinkthatthelong-existingandperiodicallychangingbalancedecosystemsinnaturetendtowardastableequilibrium;

thatis,ifthesystemdivergesfromtheformerperiodiccyclebecauseofdisturbance,aninternalcontrolmechanismwillrestoreit.However,theperiodically-changingstatedescribedbytheVolterramodelisnon-structuredstability,andevensubtleadjustmentstotheparameterswillchangetheperiodicsolution.

Soweimprovethemodelbylettingthealgaspeciesfollowlogisticgrowthifinisolation:

whereN1isthemaximumpopulationofthealgaspeciesallowedbytheenvironmentalresources.ThealgaspeciesprovidesfoodforthespeciesofPopulation2,sothemodelforthealgaespeciesis

whereN2isthemaximumcapacityofthespeciesinPopulation2andσ1referstothequantityofthealgae（comparedtoN1）eatenbytheunitquantityspeciesinPopulation2（comparedtoN2）.

Withoutthealgae,thespeciesinPopulation2willperish;

letitsdeathrateber2,sothatinisolationwewillhave:

x˙2（t）=−r2x2.

ThealgaeprovidefoodforPopulation2,soweshouldaddthateffect;

andthegrowthofthespeciesinPopulation2isalsoinfluencedbyinternalblockingaction;

soweget

whereσ2isanalogoustoσ1.Analogously,wecangetafullmodelofthespeciesinPopulation2via

.

WithoutthespeciesinPopulation2,milkfishwilldisappear;

wesettheirdeathrateasr3.ThespeciesinPopulation2providefoodforthemilkfish,andthegrowthofmilkfishisalsorestrictedbyinternalblockingaction.Herethemodelis

Summarizing,wehavesimultaneousequationsconstitutinganinterdependentmathematicalmodelforthethreepopulations:

Weobtainthevaluesofsomeparametersinthemodel,andthroughnonlineardatafittingoftheoriginaldataofthelocalthreepopulations[ShanandTang2007;

Sumagaysay-Chavoso1998;

ChenandChou2001],wegettheirnaturalgrowthrates:

σ1=0.6,σ2=0.5,σ3=0.5,σ4=2;

N1=150×

103,N2=30×

103,N3=2.2×

103.

Accordingtothevolumeoflocalfishpensandrelevantmaterials,wegettheoriginalnumbersofthethreepopulations:

x1（0）=121.5×

103,x2（0）=27×

103,x3（0）=2×

ThenweuseMatlabtoimplementthemodel,withtheresultsofFigure2,whereweseethatcanseethatwiththepassageoftime,thexi（t）tendtothesteady-statevalues69,027,27,015,and1,760.

Thenumber27,015ofthespeciesinPopulation2ismadeupofherbivorousfish,molluscs,crustaceans,andechinoderms.NowweconfirmthenumbersofallthespeciesinPopulation2,whichstayatthesametrophiclevel,coexistingandmutuallycompeting.

Figure2.Numericalsolutionsforxi（t）.

WeapplyexpertsystemandgroupdecisiontheorytodeterminetheweightsofthespeciesinPopulation2.Wehaveamulti-attributedecisionproblem,wheretheaimistoselecttheoptimalsolutionfrommanyalternativesortosorttheavailablealternatives.

AssumethatthefinitesolutionsetisY={y1,...,yn},theattributesetisC={c1,...,cq},andthedecisionexpertsetisE={e1,...,em}.LetS={s1,...,sg}beapredefinedsetconsistingofodd-chainelements.ExpertekselectsoneelementfromSasthevalueofsolutionyiunderattributecj;

letitbedenotedas

andlet

denotethejudgmentmatrixofexpertekonallthesolutionsforalltheattributes.Theattributeweightvectorinevaluatinginformationgivenbyexpertekis

istheweightofattributecjselectedbyexpertekfromsetS,wj∈S.

ThistheorycanbeactualizedthroughtheAnalyticalHierarchyProcess（AHP）,firstputforwardbyAmericanoperationalresearcherT.L.Saatyinthe1970s.AHPisamethodfordecision-makinganalysisthatcombinesqualitativeandquantitativemethods.Usingthismethod,decision-makerscanseparatecomplexproblemsintoseverallevelsandfactors,andcompareandfindtheweightsfordifferentsolutions,andprovidethebasisfortheoptimumsolution.

AHPfirstclassifiestheproblemintodifferentlevelsbasedonthenatureandthepurposeoftheproblem,constructingamultilevelstructuremodelrankedasthelowestlevel（programfordecisionmaking,measuresetc.）,comparedwiththehighestlevel（thehighestpurpose）.Ba