常微分方程初步 自治方程与人口模型 (英文)

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1、AutonomousEquationsPopulationDynamicsSection2.5inBoyceandDiPrimaWenhanWangDepartmentofMathematicsUniversityofWashington10-08-201210-10-2012WenhanWangDepartmentofMathematicsUniversityofWashington()FirstOrderODE10-08-201210-10-20121/26AutonomousEquations

2、RecallthatthegeneralformforfirstorderODEisdy=f(t;y):dtIff(t;y)isafunctiononlyiny,thentheODEbecomesdy=f(y):dtSuchequationsarecalledautonomousequations.Suchequationsdonotcontainexplicitlytheindependentvariablet.Therateofchangeofy(t)dependsonlyonyitself.Th

3、ispropertyfitsmanymathematicalmodelsfordynamicalsystems,includingpopulationmodels.WenhanWangDepartmentofMathematicsUniversityofWashington()FirstOrderODE10-08-201210-10-20122/26PropertiesofAutonomousEquationsAutonomousequationsarealwaysseparable:dy1dy=f(

4、y))1+=0:dtf(y)dtRdyThesolutionisgivenby=t+caslongasf(y)6=0.f(y)Whatiff(y)=0?Theseparationofvariablesdoesnotwork,aswearemissingconstantfunctionsassolutions.WenhanWangDepartmentofMathematicsUniversityofWashington()FirstOrderODE10-08-201210-10-20123/26St

5、ableSolutionsZerosoff(y)arecalledcriticalpoints.Supposey1isacriticalpoint,i.e.,f(y1)=0.Thentheconstantfunctiony(t)=y1isasolutiontotheautonomousequation.Suchsolutionsarecalledtheequilibriumsolutionstotheautonomousequation.Tosolveanautonomousequation,wes

6、houldfirstdiscovertheequilibriumsolutions,thenproceedtofindothergeneralsolutionsbyseparationofvariables.WenhanWangDepartmentofMathematicsUniversityofWashington()FirstOrderODE10-08-201210-10-20124/26ExponentialGrowth/DecayAsimplepopulationmodelassumesthat

7、therateofgrowth/decayofpopulationy(t)isproportionaltoy:dy=ry=f(y)dtHereriscalledtheintrinsicrateofgrowth(rateofdecay).r>0meansgrowth,r<0meansdecay.Tofindequilibriumsolutions:setf(y)=ry=0,whichgivesy(t)=0.WenhanWangDepartmentofMathematicsUniversityofWash

8、ington()FirstOrderODE10-08-201210-10-20125/26ExponentialGrowth/DecayOthersolutions:y(t)=yertforaninitialconditiony>0.00Asymptoticbehaviorofsolutions(y0>0):Ifr>0,thenlimy(t)=+1;t!1Ifr<0,thenlimy(t)=0,theequilibriumsolution.t!1WenhanWangD