{VERSION 2 3 "SUN SPARC SOLARIS" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 10 255 0 0 1 2 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "" 1 12 0 0 0 0 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 }{CSTYLE "" -1 258 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 260 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 265 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Helvetica" 1 10 0 0 0 0 1 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 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Headi ng 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 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 15 2 } {PSTYLE "" 3 256 1 {CSTYLE "" -1 -1 "" 1 12 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 "" 3 257 1 {CSTYLE "" -1 -1 "" 1 12 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 "" 3 258 1 {CSTYLE "" -1 -1 "" 1 14 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 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 257 10 "BDH2-6.mws" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 246 "This worksheet is designed to be used in conjunction with Section 2.6 of the text. In particular, note how the nullclines are plotted a nd how this information is used to determine the different types of be havior that a system of ODEs can possess." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 15 "Getting \+ started" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 140 "Every Maple worksheet \+ should begin by re-initializing the Maple \"kernel\" and loading the a dditional packages that we are most likely to use." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with( plots ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with( DEtools ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 256 18 "Defining the model" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 178 "All differential equation models \+ begin with a differential equation. If the problem is an intial value \+ problem, an initial condition is also needed. Replace the question mar ks ( " }{TEXT 0 1 "?" }{TEXT -1 67 " ) in the following input regions \+ to define the relevant ODE model." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "ode1 := diff( x(t), t ) = 2*x(t)*(1-x(t)/2) - x(t)* y(t) ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "ode2 := diff( y(t), t ) = 3*y(t)*(1-y(t)/3) - 2*x(t)*y(t) ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "MODEL := \{ ode1, ode2 \} :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "VAR := \{ x(t), y(t) \} ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 258 "" 0 "" {TEXT 263 10 "Nullclin es" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 185 "It is difficult to give a g eneral outline for the determination of nullclines. The key is to set \+ each RHS equal to zero and find all curves on which the resulting equa tion is satisfied." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 264 12 "x-nullcli nes" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Here we illustrate first fo r the " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 59 "-nullclines of the competing species example from the text:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 37 "xNULLeqn := factor( rhs(ode1) ) = 0 ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Thus, the " }{XPPEDIT 18 0 "x" "I\"xG6\" " }{TEXT -1 38 "-nullcline consists of the two lines " }{XPPEDIT 18 0 "x=0" "/%\"xG\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "x+y=2" "/, &%\"xG\"\"\"%\"yGF%\"\"#" }{TEXT -1 39 ". Here is how these might be p lotted: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "xNULL1 := plot ( [ 0, y, y=-2..4 ], color=GREEN ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "xNULL2 := plot( 2-x, x=-2..4, color=GREEN ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "xNULLplot := display( [ xNULL1, xNULL2 ], axes=BOXED \+ ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "xNULLplot;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 265 12 "y-nullclines" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "And, now, the \+ same general steps for the " }{XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 12 "-nullclines:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "yNULLeq n := factor( rhs(ode2) ) = 0 ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Thus, the " }{XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 38 "-nullcline co nsists of the two lines " }{XPPEDIT 18 0 "y=0" "/%\"yG\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "2*x+y=3" "/,&*&\"\"#\"\"\"%\"xGF&F&%\"y GF&\"\"$" }{TEXT -1 39 ". Here is how these might be plotted: " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "yNULL1 := plot( 0, x=-2. .4, color=BLUE ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "yNULL2 := plot ( 3-2*x, x=-2..4, color=BLUE ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 " yNULLplot := display( [ yNULL1, yNULL2 ], axes=BOXED ):" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 10 "yNULLplot;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "The complete nullcline plot is the union of the plots of the " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 6 "- and " }{XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 12 "-nullclines." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "NULL plot:=display( [ xNULLplot, yNULLplot ], view=[ -1 .. 3, -1 .. 4 ]," } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 " title=`Nullclin es for Competing Species` ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "NULL plot;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 267 8 "Question" }}{EXCHG {PARA 15 "" 0 "" {TEXT -1 52 "How many equilbrium solutions exist for \+ this system?" }}{PARA 15 "" 0 "" {TEXT -1 35 "What are the equilibrium solutions?" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 266 4 "Hint" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Recall that the equilibrium solutions are located at the equilibrium solutions of this system." }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 259 21 "Equilibrium Solutions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "EQUILeqns := \{ rhs( ode1 ) = 0, rhs( ode2 ) = 0 \} ;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "EQUILsoln := solve( EQUILeqn s, VAR ) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 258 54 "Direction Field, Phase Portrait, and Sol ution Curves (" }{HYPERLNK 17 "DEplot" 2 "DEplot" "" }{TEXT -1 1 ")" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "DOMAIN := t = 0 .. 5 ;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "RANGE := x = -1 .. 3, y = -2 .. 4 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 260 15 "Direction Field" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "DFplot := DEplot( MODEL, VAR, DOMAIN, RANGE, arr ows = MEDIUM," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 " \+ title = `Direction Field for Competing Species` ):" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 7 "DFplot;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "display( [ NULLplot, D Fplot ], title=`Nullclines and Direction Field` );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 262 14 " Phase Portrait" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "IC := [ x (0) = 0.4, y(0) = 0.2 ]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " \+ [ x(0) = 0.4, y(0) = 0.3 ]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " \+ [ x(0) = 0.4, y(0) = 0.4 ]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " \+ [ x(0) = 0.4, y(0) = 0.6 ]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " [ x(0) = 2, y(0) = 1 ]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " [ x(0) = 2, y(0) = 2 ]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " [ x(0) = 2, y(0) = 3 ]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " [ x(0) = 2, y(0) = 4 ] ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "PPplot := DEplot( MODEL, VAR, DOMAIN, [ IC ], RANGE, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 " scene=[ x, y \+ ], arrows=NONE," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 " \+ title = `Phase Portrait for Competing Species` ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "PPplot;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "display( [ DFplot, P Pplot ], title=`Direction Field and Phase Portrait` );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "display( [ NULLplot, DFplot, PPplot ], title=`Nullclines, Direction Field, and Phase Portrait` );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT 261 15 "Solution Curves" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Select the initial condition of your choice" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "P1:= DEplot( MODEL, VAR, DOMAIN, [ IC[1] ]," }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 " scene=[ t, x ], arrows =NONE, linecolor=BLUE ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "P2:= DE plot( MODEL, VAR, DOMAIN, [ IC[1] ]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 " scene=[ t, y ], arrows=NONE, linecolor=GREEN ):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "display( [ P1, P2 ] , title = `x and y vs. t` ); # combined solution curves" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "#display( array([P1,P2]), title = `Soluti on Curve` ); # side-by-side solution curves" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 257 "" 0 "" {TEXT 268 20 "Conc luding Questions" }}{EXCHG {PARA 15 "" 0 "" {TEXT -1 131 "Can you find one initial condition with the property that leads to the equilibrium solution that corresponds to mutual coexistence?" }}{PARA 15 "" 0 "" {TEXT -1 58 "Can you find another initial condition with this property ?" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }