IntroductionDivide variables into fast and slow categoriesDecide on the form of the fast variablesAnalysis of fast variablesDecide upon the form of the slow variablesAnalysis of slow variablesCombining of fast and slow systemsCoupling of equations0predation curvegrowth curve unstable equilibrium stable equilibrium stable equilibrium R Q G H I B IntroductionDivide variables . . .Decide on the form . . .Analysis of fast . . .Decide upon the . . .Analysis of slow . . .Combining of fast . . .Coupling of equationsHome PageTitle PageContentsJJ IIJ IPage 1 of 20Go BackFull ScreenCloseQuitQualitative Analysis of Spruce Budworm Outbreaksand DeclinesRon BassarMay 17, 2000AbstractThis paper shadows the qualitative analysis of spruce budworm out-break and decline phenomena in the balsam fir forests of the NortheasternUnited States and Canada by [Ludwig et. al., (1978)]. Qualitative theoryof differential equations and catastrophe theory is employed to model onecomplete cycle. This paper does not attempt to justify the analysis throughreplacing parameter values with real values. For a treatment of this subjectsee [Ludwig et. al., (1978)].0predation curvegrowth curve unstable equilibrium stable equilibrium stable equilibrium R Q G H I B IntroductionDivide variables . . .Decide on the form . . .Analysis of fast . . .Decide upon the . . .Analysis of slow . . .Combining of fast . . .Coupling of equationsHome PageTitle PageContentsJJ IIJ IPage 2 of 20Go BackFull ScreenCloseQuit1. IntroductionIn quantitative sciences, mathematics is used as a tool to describe data, conveyinformation and make predictions about future occurrences of systems. Popu-lation ecology is one such field. Models of specifics are created by identifyingand separating variables, establishing formulas, and comparing the model withdata obtained from the field. These models aid practioners in making informedmanagement decisions without changing the system they model.A classic application of the modeling of natural systems is found in ”Qualita-tive Analysis of Insect Outbreak Systems: The Spruce Budworm and Forest” byLudwig et. al.[Ludwig et. al., (1978)]. The paper se eks to model the outbreakof the Spruce Budworm in the Northeastern United States and Canada by usingthe Qualitative theory of Differential Equations and Catastrophe Theory. Thenin an effort to justify the model, the authors replace the parameters with valuesprovided by an informed entomologist.The purpose of this paper is to explore the methodologies of the model;specifically as it relates to the qualitative theory of differential e quations. Thispaper will shadow the step by step analysis of the qualitative section in the orig-inal paper by [Ludwig et. al., (1978)]. The following analysis will not considerthe final testing of the model through replacement of the parameters with realvalues. For a treatment of this subject, see [Ludwig et. al., (1978)].0predation curvegrowth curve unstable equilibrium stable equilibrium stable equilibrium R Q G H I B IntroductionDivide variables . . .Decide on the form . . .Analysis of fast . . .Decide upon the . . .Analysis of slow . . .Combining of fast . . .Coupling of equationsHome PageTitle PageContentsJJ IIJ IPage 3 of 20Go BackFull ScreenCloseQuit2. Divide variables into fast and slow categoriesEach variable in the following analysis has an associated time interval overwhich change will occur. Some variables, such as budworm density, can changedramatically in a few years. Therefore, an appropriate time interval for bud-worm density is the order of months. A major influence on budworm densityis predation–particularly by birds (warblers,etc.). Due to reproductive strate-gies avian predators cannot alter their own numbers at a rate comparable tobudworms. However, avian predators can rapidly alter their feeding behavior.Consequently, avian predation is assigned a fast variable.Forests are assigned a slow variable since they cannot alter their numbers in ashort amount of time. An appropriate time scale is on the order of tens of years.The forest will further by divided into two variables; one variable describing theenergy reserve of the forest and the other the total surface area of branches.0predation curvegrowth curve unstable equilibrium stable equilibrium stable equilibrium R Q G H I B IntroductionDivide variables . . .Decide on the form . . .Analysis of fast . . .Decide upon the . . .Analysis of slow . . .Combining of fast . . .Coupling of equationsHome PageTitle PageContentsJJ IIJ IPage 4 of 20Go BackFull ScreenCloseQuit3. Decide on the form of the fast variablesIn the case of the budworm the main limiting features are the food supply andpredation. Therefore, [Ludwig et. al., (1978)] chose the Logistic form,dBdt= rBB1 −BKB, (1)where B represents the budworm density and KBis the carrying capacity whichis dependent upon the amount and quality of foliage available.The effect of predation is given by the function g(B); which must posses sev-eral characteristics. There is an upper limit to the rate of budworm mortalitydue to predation. This upp er limit is a function of variables such as predatorsearch strategies, territorial behavior, and other habitat characters. In otherwords, predators can only eat so many budworms even when budworm popula-tions are high. This effect is termed saturation. Conversely, there is a decreasein the effect of predation at low budworm densities. This is a common effectwhen predators have many alternative food sources. When budworm densitiesare low, avian predators encounter them only incidentally. In contrast, as theirdensity increases, predators encounter them more often, develop a search image,and will seek them out. Therefore, g(B) should approach an upper limit β asB → ∞ and vanish quadratically as B → 0. β may depend on slow variablesand will be considered later. A form for g(B) that meets the criteria above is,g(B) = βB2α2+ B2, (2)where α determines the scale at which saturation begins to take affect.0predation curvegrowth curve unstable equilibrium stable equilibrium stable equilibrium R Q G H I B IntroductionDivide variables . . .Decide on the form . . .Analysis of fast . . .Decide upon the . . .Analysis of slow . . .Combining of fast . . .Coupling of equationsHome PageTitle PageContentsJJ IIJ IPage 5 of 20Go BackFull ScreenCloseQuitEquation 2 is subtracted from the right-hand side of Equation 1 since theinteraction has a negative effect on Budworm