229ModelinginDiscreteDynamicalSystemsRodneyX.SturdivantScenario1:TanksareDiscreteConsiderapurearmorbattle(tankvs.tank)betweenCountryXandCountryY.CountryXattackswith200tankswhileCountryYdefendswithonly65tanks.Aftereachfive-minuteperiod,bothsideslosesometanksbaseduponthenumberofoppositiontanks,thereliabilityofthetheirweapons,andthesurvivabilityofthetanks.OnceatankfromCountryXsightsanenemytank,ithasa20%chanceofstrikingthetarget.Likewise,CountryYtanksare30%effective.AtankfromCountryXhasa40%chanceofbecominga“kill”ifitishit;CountryYtankssurvivehits65%ofthetime.Problem1:Modelthissituationusingadiscretedynamicalmodel.Whatassumptionsdidyoumake?Solution1:Theassumptionsarecriticaltothismodel!Manydifferentapproachesarepossibledependingontheassumptionsmade.Forexample:assumeonlyonetankfromcountryitargetsagiventankfromcountryjatatime.Assumenofratricide.Assumealltanksonlyacquireonetargetineach5-minuteperiod.Assumea“kill”meansthetanknolongerparticipatesinthebattleinanyway.Further,assumethatifcountryXhasmoretanksthancountryY,theydon’tfiremoretanksthantargetsavailable.Thislastassumptioniscritical.Takethefirst5-minuteperiod:CountryXhas200tanks.Ifall200acquiretargets,theymayfireatthesametankoratankalreadykilled.Weassumeeachofthe200waitstheirturntoacquirealivetarget!Isthisrealistic?Probablynot,butwithoutthisassumptionmodelingthissituationisverydifficult.Let’sexamineapossiblemodel:LetX(n)=thenumberofcountryXtanksremainingaftern5-minuteperiodsY(n)=thenumber“Y“X(0)=200Y(0)=65X(n+1)=X(n)–0.4(0.3)Y(n)Y(n+1)=Y(n)–0.35(0.2)X(n)Problem2:Ifacountrysurrenderswhentheyhave5orfewertanks,whichcountrywillwin?Howlongwillthebattletake?Howmanytanksdoesthelosingcountryneedtostartwithtochangetheoutcome?230Solution2:Problem3:Inordertobriefyourresults,youmustprepareforquestionsabouttheassumptionsyoumadeandtheirimpact.Chooseoneassumptionanddiscusshowiteffectsyourresults.Solution3:IteratingthepreviousmodelinMathcadshowsthatcountryXwinsafter5five-minuteintervals:Next,wecan“playwith”theinitialcondi...
