{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 257 "" 0 1 0 0 19 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 19 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 19 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 19 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 117 60 193 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 262 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 0 1 0 0 24 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 19 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Nor mal" -1 0 1 {CSTYLE "" -1 -1 "Times" 0 14 0 0 0 1 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 0 12 128 0 128 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE " " -1 -1 "Courier" 0 11 0 128 128 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 20 "Predator-Prey Models" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 19 "Part 1. \+ Background " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Enter the \"with\" commands to load packages." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(plots): with (DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Answer th e questions in Part 1 here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 32 "Part 2. The Lotka-Volter ra Model" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Answer questions 1-3 here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Enter sample values of a , b, c, p:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "a:=1; b:=0.03; c:=0.4; p:=0.01;" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "The next group of \+ commands will draw the direction field. We name the Lotka-Volterra eq uations " }{TEXT 264 3 "LV1" }{TEXT -1 5 " and " }{TEXT 263 3 "LV2" } {TEXT -1 27 ". The inputs start with a " }{TEXT 261 4 "list" }{TEXT -1 54 " of the differential equations. Next comes the list " }{TEXT 262 5 "[x,y]" }{TEXT -1 108 " identifying the dependent variables, th en ranges for the variables, and finally any optional parameters. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "LV 1 := diff(x(t),t)=a*x-b*x*y;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "LV2 := diff(y(t),t)=-c*y+p*x*y;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "DEp lot([LV1,LV2], [x,y], t=0..12, x=0..140, y=0..80, title=`Lotka-Volterr a Model`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Answer qu estion 4 here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 150 "The following command modifies DEplot to add some trajectories determined by a list of starting points, each g iven by x and y coordinates at time 0. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "DEplot([LV1,LV2], [x,y], t= 0..12, x=0..140, y=0..80, title=`Lotka-Volterra Model`, [[x(0)=15,y(0) =15], [x(0)=20,y(0)=20], [x(0)=25,y(0)=25], [x(0)=30,y(0)=30]], stepsi ze=0.2, linecolor=blue);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Answer question 5 here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Enter formulas to calcula te the coordinates of the equilibrium point." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "xs:=???; ys:=???;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "An swer the rest of the questions in Part 2 here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 " " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 34 "P art 3. Graphical Representations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 60 "The following command creates a plot of t he prey population." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "preyplot:=DEplot([LV1, LV2], [x,y], t=0..20, x=0. .140, y=0..140, title=`Population for 2 Periods`, [[x(0)=15,y(0)=15]], stepsize=0.2, scene=[t,x], linecolor=blue):%;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 76 "Now we plot the predator population the s ame way and overlay the two graphs." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "predplot:=DEplot([LV1, LV2], [x,y ], t=0..20, x=0..140, y=0..140, [[x(0)=15,y(0)=15]], stepsize=0.2, sce ne=[t,y], linecolor=green): display([predplot,preyplot]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Answer questions 3 and 4 here. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 236 "The following command will construct a 3-dimensio nal plot of the curve [x(t),y(t),t]. If you click on the plot, you w ill see buttons and menus at the top of the screen for controlling vie wpoint, axes, redrawing [R], and other options." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "spaceplot := DEplo t3d([LV1,LV2], [x,y], t=0..30, x=0..140, y=0..80, [[x(0)=15,y(0)=15]], scene=[x,y,t], stepsize=0.2, axes=normal, linecolor=blue): %;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "To make additional copies of the 3D plot, " }{TEXT 265 7 "display" }{TEXT -1 30 " the plot alre ady constructed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "display(spaceplot);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "display (spaceplot);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "display(spaceplot);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 31 "Part 4. Varying the Parameter s" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 168 "Ent er your adjustments to the parameters here, and recompute the differen tial equations. Then use your choice of representation to examine the change in the solutions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 7 "a:=???;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "LV1 := diff(x(t),t)=a*x-b*x*y;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " LV2 := diff(y(t),t)=-c*y+p*x*y;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 40 "Part 5. \+ The Effect of Hunting Predators" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 264 "The following commands reset the paramet ers and define the modified differential equations with constant hunti ng of predators. Note that the names of the equations have been chang ed. Use your choice of graphical representation to determine the effe ct of hunting." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "a:=1; b:=.03; c:=.4; p:=.01;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "DE1 := diff(x(t),t)=a*x-b*x*y;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "DE2 := diff(y(t),t)=-c*y+p*x*y-5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 16 "Part 6. Summary" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 36 "Answer the questions in part 6 here." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} }{MARK "0 6 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }