{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "List Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "List \+ Subitem" 14 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 3 12 0 0 0 0 273 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 270 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 271 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 269 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 41 "ORDINARY DIFFERENTIAL EQUATIONS POWERTOOL" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 270 "" 0 "" {TEXT -1 41 "Unit 7 -- Applicat ion: The Spruce Budworm" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 " " 0 "" {URLLINK 17 "Prof. Douglas B. Meade" 4 "http://www.math.sc.edu/ ~meade/" "" }}{PARA 258 "" 0 "" {URLLINK 17 "Industrial Mathematics In stitute" 4 "http://www.math.sc.edu/~IMI/" "" }}{PARA 259 "" 0 "" {URLLINK 17 "Department of Mathematics" 4 "http://www.math.sc.edu/" " " }}{PARA 260 "" 0 "" {URLLINK 17 "University of South Carolina" 4 "ht tp://www.sc.edu/" "" }}{PARA 261 "" 0 "" {TEXT -1 19 "Columbia, SC 292 08\n" }}{PARA 263 "" 0 "" {TEXT -1 7 "URL: " }{URLLINK 17 "http://ww w.math.sc.edu/~meade/" 4 "http://www.math.sc.edu/~meade/" "" }}{PARA 264 "" 0 "" {TEXT -1 25 "E-mail: meade@math.sc.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT -1 38 "Copyright \251 2001 b y Douglas B. Meade" }}{PARA 266 "" 0 "" {TEXT -1 19 "All rights reserv ed" }}{PARA 268 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT -1 67 "-------------------------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 17 "Outline of Unit 7" }} {EXCHG {PARA 14 "" 0 "" {HYPERLNK 17 "7.A" 1 "" "7.A" }{TEXT -1 22 " B iological Background" }}{PARA 14 "" 0 "" {HYPERLNK 17 "7.B" 1 "" "7.B " }{TEXT -1 21 " Bifurcation Analysis" }}{PARA 14 "" 0 "" {HYPERLNK 17 "7.C" 1 "" "7.C" }{TEXT -1 11 " References" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 14 "Initia lization" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with( DEtools ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with( plots ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with( linalg ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "with( PDEtoo ls ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "7.A" {TEXT -1 25 "7.A Biological Background" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 217 "A classical example of bifurcation in na ture is the interaction between the spruce budworm and balsam fir fore sts in North America. The basic model for the budworm population is a \+ logistic model with a predation term:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f[logistic] := r*B*(1-B /K);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f[predation] := beta*B^2/(a lpha^2+B^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "where, in the absence of predation, " } {XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT -1 34 " is the intrinsic growth rat e and " }{XPPEDIT 18 0 "K" "6#%\"KG" }{TEXT -1 48 " is the carrying ca pacity. All four parameters, " }{XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "K" "6#%\"KG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a lpha" "6#%&alphaG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "beta" "6#%%bet aG" }{TEXT -1 58 ", are positive. The corresponding differential equat ion is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "ode := Diff( B, t ) = f[logistic] - f[predation];" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 158 "Via dimensional analysis, the problem can be reduced to \+ one involving only two parameters. Introduce new, dimensionless, depen dent and independent variables, " }{XPPEDIT 18 0 "y=y(tau)" "6#/%\"yG- F$6#%$tauG" }{TEXT -1 12 ", defined by" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "new_var_eq := B(t)=alph a*y(tau)," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " t=alpha/ beta*tau;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "and new (dimensionless) parameters " } {XPPEDIT 18 0 "R" "6#%\"RG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Q" "6# %\"QG" }{TEXT -1 16 " defined so that" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "new_par_eq := r=beta/al pha*R," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " K=alpha*Q; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "The corresponding differential equation in terms of \+ the new variables and parameters is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "ode2 := dchange( \{new_var s,new_pars\}, eval(ode/beta,B=B(t)), [y,tau,Q,R] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "w hich can be simplified to" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "ode2 := map( collect, ode2, \{R,alp ha\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "The advantage of this ODE is that it invo lves only two parameters." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "7.B" {TEXT -1 24 "7.B Bifurcation Analysi s" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "To begin the bifurcation ana lysis, it would be nice to be able to identify all equilibrium solutio ns for the nondimensionalized ODE" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "ode2;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "While \+ it is easy to see that " }{XPPEDIT 18 0 "y=0" "6#/%\"yG\"\"!" }{TEXT -1 82 " is an equilibrium solution, no other equilibrium solution is i mmediately obvious." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "equil_eq := eval( rhs(ode2), y(tau)=y ) = 0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "factor( equil_eq ); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 251 "What is apparent is that, in addition to the trivia l equilibrium, there can be up to three additional equilibria. While M aple is capable of finding explicit formulas for the remaining equilib ria, these expressions are not likely to be terribly useful." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "s olve( equil_eq, \{y\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "The bifurcation analysis pres ented in " }{HYPERLNK 17 "Unit 6, Section B" 1 "unit06.mws" "" }{TEXT -1 71 " can be used to identify bifurcation points in terms of the par ameters " }{XPPEDIT 18 0 "Q" "6#%\"QG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "R" "6#%\"RG" }{TEXT -1 34 ". The two necessary conditions are" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "bif_eq1 := equil_eq;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "bif_eq2 := diff( eval(rhs(ode2),y(tau)=y), y ) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "The parametric solutions to these two equations are" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "bif _sol := solve( \{bif_eq1,bif_eq2\}, \{R,Q\} );" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "From t he fact that the original unknowns and parameters are positive, " } {XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Q" "6#%\" QG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "R" "6#%\"RG" }{TEXT -1 108 " s hould also be positive. Thus, it is observed that there are physically realistic equilibria only when y>1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "p1 := plot( eval([Q,R,y=1 .001..100],bif_sol), labels=['Q','R']," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " view=[0..200,0..1], numpoints=200 ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "p2 := textplot([[15,0.1,`Region I`],[50,0.7,` Region I`]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " [100 ,0.25,`Region II`]]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display([p 1,p2]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 169 "In Region I there is only one non-trivia l equilibria while in Region II there are three non-trivial equilibria . At every point of the boundary between Regions I and II, " }{TEXT 260 4 "i.e." }{TEXT -1 123 ", the red curve, there are two non-trivial equilibria. To understand this conclusion, note that the equilibria m ust satisfy" }}{PARA 271 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R*(1-y /Q) = y/(1+y^2)" "6#/*&%\"RG\"\"\",&F&F&*&%\"yGF&%\"QG!\"\"F+F&*&F)F&, &F&F&*$F)\"\"#F&F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 242 "Th e function on the left-hand side is linear while the one on the right- hand side does not depend on the parameters. From a graph of these fun ctions it is possible to visually see how the number of equilibria cha nge with the parameter values." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "P := proc(R,Q,T)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " if nargs=2 then T := `` end if;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 " plot( [R*(1-y/Q),y/(1+y^2)], y=0..10, t itle=T )" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "end proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "display( array([P(0.5,5,`Region I`) ,P(0.5,7.3,`Boundary`),P(0.5,10,`Region II`)]) );" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "To s ee the transition from three, to two, to one equilibrium, the followin g animation is helpful:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "display( seq( P(0.5, 10-q/10, spri ntf(\"P= 0.5, Q=%4.1f\",10.-q/10)), q=0..50), view=0..0.5, insequence= true );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "For additional information about this pr oblem -- both biological and mathematical -- please consult the refere nces." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "7.C" {TEXT -1 14 "7.C References" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "1. Ludwig, Jones, and Holling, ``Qualitative Analysis of Insect Breakout Systems: The Spruce Budworm and Forest,'' " } {TEXT 256 17 "J. Animal Ecology" }{TEXT -1 9 " (1978), " }{TEXT 257 2 "47" }{TEXT -1 10 ", 315-332." }}{PARA 0 "" 0 "" {TEXT -1 15 "2. Murra y, J., " }{TEXT 258 20 "Mathematical Biology" }{TEXT -1 24 ", Springer -Verlag, 1993." }}{PARA 0 "" 0 "" {TEXT -1 17 "3. Strogatz, S., " } {TEXT 259 28 "Nonlinear Dynamics and Chaos" }{TEXT -1 23 ", Addison-We sley, 1994." }}{PARA 0 "" 0 "" {TEXT -1 44 "4. McKelvey, S., Spruce Bu dworm Model, URL: " }{URLLINK 17 "http://www.stolaf.edu/people/mckelve y/envision.dir/spruce.html" 4 "http://www.stolaf.edu/people/mckelvey/e nvision.dir/spruce.html" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "[Back to " }{HYPERLNK 17 "ODE Powertool Table of Contents" 1 "unit00.mws" "" } {TEXT -1 1 "]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "0 1 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }