Qualitative Analysis of Spruce Budworm OutbreaksFast Slow Budworms Surface area of Predators Branches Energy ReserveBudworm Growth EquationFinding Equilibrium PointsB=0 is an Equilibrium PointDimensionless ParametersDimensionless VariableDimensionless Means No UnitsGrouping of Parameters in the EquationSub in Dimensionless ParametersPer Capita Death and Growth Rates in Terms of the Dimensionless VariablesPredation and Growth Curves Plotted vs. Budworm DensityR Increases to Point Where G and H CoalesceEquilibria disappear as R increases and Budworm Population jumps to IIf R Decreases Population Remains at IStill at IPopulation Jumps to G as R Continues to DecreaseBifurcation CurvesEquations of Branch Surface Area and Energy ReserveEquilibrium Points of S and EAnalysis of Slow VariablesQualitative Analysis of Spruce Budworm Outbreaksby Ron BassarFast SlowBudworms Surface area of Predators Branches Energy ReserveDivide variables into fast and slow categoriesBudworm Growth Equation222)1(BBKBBrdtdBBBFinding Equilibrium Points0)1(222BBKBBrdtdBBB0)1(0))1((0)1(2222222BBKBrdtdBBBBKBrdtdBBBKBBrdtdBBBBBBBB=0 is an Equilibrium PointDimensionless ParametersBBKQrRDimensionless VariableBBBKQrRB ,,22)1(BBKBrdtdBBBDimensionless Means No UnitsGrouping of Parameters in the Equation01)1(0)1()1(0)1(22222BBBBBBBKrKrBBKBrSub in Dimensionless Parameters01)1(01)1(22QRKrBBPer Capita Death and Growth Rates in Terms of the Dimensionless Variables21)1(QR01)1(2QR0predation curvegrowth curve unstable equilibrium stable equilibrium stable equilibrium R Q G H I B Predation and Growth Curves Plotted vs. Budworm Density0predation curvegrowth curve unstable equilibrium stable equilibrium stable equilibrium Budworm Populationoriginally hereR Q G H I BR Increases to Point Where G and H Coalescepredation curvegrowth curve Stable Equilibrium I Q R B G and H Stable EquilibriumBudworm Population HereEquilibria disappear as R increases and Budworm Population jumps to Ipredation curvegrowth curve Stable Equilibrium I Q R B predation curvegrowth curve Stable Equilibrium Budworm Population is HereI Q R BIf R Decreases Population Remains at I 0predation curvegrowth curve unstable equilibrium stable equilibrium Budworm Population is herestable equilibrium R Q G H I BStill at Ipredation curvegrowth curve Q B R G Stable EquilibriumH and I Stable EquilibriumBudworm Populationis here predation curvegrowth curve Q B R G Stable EquilibriumH and I Stable EquilibriumBudworm Populationis here predation curvegrowth curve Q B R G Stable EquilibriumH and I Stable EquilibriumBudworm Populationis herePopulation Jumps to G as R Continues to Decreasepredation curvegrowth curve Q B R G Stable EquilibriumBudworm Population is hereBifurcation Curves 00.5H=I G=H 3 equilibrium points 1 equilibrium point 1 equilibrium point R QEquations of Branch Surface Area and Energy ReserveSBPKEErdtd EEKS KSrd td SEESES)1()1(Equilibrium Points of S and E)(00,0EKErPBKSifdtdESEKKSifdtdSEEEESESE=0 S=0 C D separatix separatix saddle sink separatix Analysis of Slow
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