Maple Worksheet to go with the notes from Chapter 9 on Competing Species, Predator-Prey, and the Lorenz Equations. with(DEtools): with(plots): Compete:=[diff(x(t),t)=x(t)*(1-x(t)-y(t)),diff(y(t),t)=y(t)*(3/4-y(t)-(1/2)*x(t))]; A1:=DEplot(Compete,[x(t),y(t)],t=0..10,x=0..1.5,y=0..1,dirgrid=[30,30],[[x(0)=1,y(0)=1],[x(0)=1,y(0)=1/2],[x(0)=1,y(0)=2],[x(0)=0.2,y(0)=0.1],[x(0)=0.1,y(0)=0.5]],stepsize=0.01): B1:=plot(-x+1,x=0..1.5,y=0..1,color=black,thickness=2): B2:=plot(-(1/2)*x+3/4,x=0..1.5,y=0..1,color=black): display({A1,B1,B2}); Second Competing Species Model: Compete2:=[diff(x(t),t)=x(t)*(1-x(t)-y(t)),diff(y(t),t)=y(t)*(1/2-(1/4)*y(t)-(3/4)*x(t))]; A2:=DEplot(Compete2,[x(t),y(t)],t=0..10,x=-1/10..1.25,y=-1/10..2.5,dirgrid=[30,30],[[x(0)=1,y(0)=1],[x(0)=1,y(0)=1/2],[x(0)=1,y(0)=2],[x(0)=0.2,y(0)=0.1],[x(0)=0.1,y(0)=0.5]],stepsize=0.01): b1:=plot(-x+1,x=-1/10..1.25,y=-1/10..2.5,color=black,thickness=2): b2:=plot(-3*x+2,x=-1/10..1.25,y=-1/10..2.5,color=black): display({A2,b1,b2}); Next is the Predator-Prey model: PredPrey:=[diff(x(t),t)=x(t)*(1-(1/2)*y(t)),diff(y(t),t)=y(t)*((1/4)*x(t)-(3/4))]; DEplot(PredPrey,[x(t),y(t)],t=0..12,[[x(0)=1.2,y(0)=1.2],[x(0)=1,y(0)=.7],[x(0)=3,y(0)=1.5]],stepsize=.2,method=rkf45); Ax:=DEplot(PredPrey,[x(t),y(t)],t=0..20,[[x(0)=1.2,y(0)=1.2]],scene=[t,x],stepsize=.2,method=rkf45,linecolor=red,thickness=3): Ay:=DEplot(PredPrey,[x(t),y(t)],t=0..20,[[x(0)=1.2,y(0)=1.2]],scene=[t,y],stepsize=.2,method=rkf45,linecolor=blue,thickness=1): display({Ax,Ay}); Lastly, the Lorenz Equations. Lorenz:=[diff(x(t),t)=-10*(x(t)-y(t)),diff(y(t),t)=28*x(t)-y(t)-x(t)*z(t),diff(z(t),t)=-(8/3)*z(t)+x(t)*y(t)]; DEplot3d(Lorenz,[x(t),y(t),z(t)],t=0..50,[[x(0)=0.1,y(0)=-0.1,z(0)=27]],stepsize=0.01,linecolor=black,thickness=1); DEplot(Lorenz,[x(t),y(t),z(t)],t=0..50,[[x(0)=0.1,y(0)=-0.1,z(0)=27]],stepsize=0.01,scene=[t,x],linecolor=black,thickness=1);