%This file creates two figures associated with the
%system x'=f(x,y), y'=g(x,y)
%1. Plots the graphs of x(t) and y(t)
%2. Plots the graph of (x(t),y(t)) in the phase plane.

%verbatimtex
%\input mtplain
%etex

%Generate standard eps
prologues:=2;

beginfig(0);

%Place RHS of x'=f(t,x,y) here
  def fxy(expr t, x, y)=
   (0.4-0.01*y)*x
  enddef;

%Place RHS of y'=g(t,x,y) here
  def gxy(expr t, x, y)=
   (-0.3+0.005*x)*y
  enddef;

%Declare some variables
  path q, trajx, trajy;
  pair L, R, B, T, xt, yt;
  numeric sx[], sy[];

%Initialize clipping window
  a:=0; b:=40;   %left and right of viewing rectangle
  c:=0; d:=150;  %bottom and top of viewing rectangle

%Initialize timespan
  tstart:=a;
  tstop:=b;

%Initialize number of points to be plotted
  N:=500;

%Calculate time increment dt for Euler's method
  dt:=(tstop-tstart)/N;

%Scaling factors for horizontal and vertical axes. Note that this produces
%an image that is 2 inches by 2 inches.
  (b-a)*ux=1.75in;
  (d-c)*uy=1.75in;

%Clipping boundary
  q=(a,c)--(b,c)--(b,d)--(a,d)--cycle;

%Use Runge-Kutta4 to create path (t,x(t))

%Choose initial condition
  t:=tstart;
  x:=40;
  y:=20;
  trajx:=(t,x);
  forever:
   sx1:=fxy(t,x,y);
   sy1:=gxy(t,x,y);
   sx2:=fxy((t+dt/2),(x+dt*sx1/2),(y+dt*sy1/2));
   sy2:=gxy((t+dt/2),(x+dt*sx1/2),(y+dt*sy1/2));
   sx3:=fxy((t+dt/2),(x+dt*sx2/2),(y+dt*sy2/2));
   sy3:=gxy((t+dt/2),(x+dt*sx2/2),(y+dt*sy2/2));
   sx4:=fxy((t+dt),(x+dt*sx3),(y+dt*sy3));
   sy4:=gxy((t+dt),(x+dt*sx3),(y+dt*sy3));
   x:=x+dt*(sx1+2*sx2+2*sx3+sx4)/6;
   y:=y+dt*(sy1+2*sy2+2*sy3+sy4)/6;
   t:=t+dt;
   trajx:=trajx..(t,x);
   exitif ((t>tstop) or (t>b) or (x<c) or (x>d));
  endfor;

%Use Runge-Kutta4 to create path (t,y(t))

%Choose initial condition
  t:=tstart;
  x:=40;
  y:=20;
  trajy:=(t,y);
  forever:
   sx1:=fxy(t,x,y);
   sy1:=gxy(t,x,y);
   sx2:=fxy((t+dt/2),(x+dt*sx1/2),(y+dt*sy1/2));
   sy2:=gxy((t+dt/2),(x+dt*sx1/2),(y+dt*sy1/2));
   sx3:=fxy((t+dt/2),(x+dt*sx2/2),(y+dt*sy2/2));
   sy3:=gxy((t+dt/2),(x+dt*sx2/2),(y+dt*sy2/2));
   sx4:=fxy((t+dt),(x+dt*sx3),(y+dt*sy3));
   sy4:=gxy((t+dt),(x+dt*sx3),(y+dt*sy3));
   x:=x+dt*(sx1+2*sx2+2*sx3+sx4)/6;
   y:=y+dt*(sy1+2*sy2+2*sy3+sy4)/6;
   t:=t+dt;
   trajy:=trajy..(t,y);
   exitif ((t>tstop) or (t>b) or (y<c) or (y>d));
  endfor;

%Draw the paths x(t) and y(t) and clip them to bounding box
  draw trajx xscaled ux yscaled uy withcolor red;
  draw trajy xscaled ux yscaled uy withcolor red dashed evenly;
  clip currentpicture to (q xscaled ux yscaled uy);

%Label graph x(t) and initial condition
  len:= 0.65*(length trajx);
  xt:=point len of trajx;
  label.urt(btex $\scriptstyle x(t)$ etex, (xt xscaled ux yscaled uy));

%Label graph y(t) and initial condition
  len:= 0.5*(length trajy);
  yt:=point len of trajy;
  label.lrt(btex $\scriptstyle y(t)$ etex, (yt xscaled ux yscaled uy));


%Initialize left and right endpoints on time-axis
  L=(a*ux,0);R=(b*ux,0);

%Draw and label t-axis
  drawarrow L--R;
  label.rt(btex $\scriptstyle t$ etex,(b*ux,0));

%Initialize bottom and top endpoints on time-axis
  B=(0,c*uy);T=(0,d*uy);

%Draw and label vertical axis
  drawarrow B--T;
  label.lft(btex $\scriptstyle 0$ etex, B);
  label.lft(btex $\scriptstyle 150$ etex, T);

endfig;


beginfig(2);

%Make some variables local
  save ux, uy;

%Place RHS of x'=f(t,x,y) here
  def fxy(expr t, x, y)=
   (0.4-0.01*y)*x
  enddef;

%Place RHS of y'=g(t,x,y) here
  def gxy(expr t, x, y)=
   (-0.3+0.005*x)*y
  enddef;

%Declare some variables
  path q, trajxy;
  pair L, R, B, T;

%Initialize clipping window
  a:=0; b:=150;   %left and right of viewing rectangle
  c:=0; d:=100;  %bottom and top of viewing rectangle

%Initialize timespan
  tstart:=a;
  tstop:=b;

%Initialize number of points to be plotted
  N:=500;

%Calculate time increment dt for Euler's method
  dt:=(tstop-tstart)/N;

%Scaling factors for horizontal and vertical axes. Note that this produces
%an image that is 2 inches by 2 inches.
  (b-a)*ux=1.75in;
  (d-c)*uy=1.75in;

%Clipping boundary
  q=(a,c)--(b,c)--(b,d)--(a,d)--cycle;

%Use Runge-Kutta4 to create path (x(t),y(t))

%Choose initial condition
  t:=tstart;
  x:=40;
  y:=20;
  trajxy:=(x,y);
  forever:
   sx1:=fxy(t,x,y);
   sy1:=gxy(t,x,y);
   sx2:=fxy((t+dt/2),(x+dt*sx1/2),(y+dt*sy1/2));
   sy2:=gxy((t+dt/2),(x+dt*sx1/2),(y+dt*sy1/2));
   sx3:=fxy((t+dt/2),(x+dt*sx2/2),(y+dt*sy2/2));
   sy3:=gxy((t+dt/2),(x+dt*sx2/2),(y+dt*sy2/2));
   sx4:=fxy((t+dt),(x+dt*sx3),(y+dt*sy3));
   sy4:=gxy((t+dt),(x+dt*sx3),(y+dt*sy3));
   x:=x+dt*(sx1+2*sx2+2*sx3+sx4)/6;
   y:=y+dt*(sy1+2*sy2+2*sy3+sy4)/6;
   t:=t+dt;
   trajxy:=trajxy..(x,y);
   exitif ((t>tstop) or (t>b) or (x<a) or (x>b) or (y<c) or (y>d));
  endfor;


%Draw the paths x(t) and y(t) and clip them to bounding box
  draw trajxy xscaled ux yscaled uy withcolor red;
  clip currentpicture to (q xscaled ux yscaled uy);

%Initialize left and right endpoints on x-axis
  L=(a*ux,0);R=(b*ux,0);

%Draw and label x-axis
  drawarrow L--R;
  label.rt(btex $\scriptstyle x$ etex,(b*ux,0));
  label.bot(btex $\scriptstyle 0$ etex,L);
  label.bot(btex $\scriptstyle 150$ etex,R);

%Initialize bottom and top endpoints on y-axis
  B=(0,c*uy);T=(0,d*uy);

%Draw and label vertical axis
  drawarrow B--T;
  label.rt(btex $\scriptstyle y$ etex,(0,d*uy));
  label.lft(btex $\scriptstyle 0$ etex, B);
  label.lft(btex $\scriptstyle 100$ etex, T);

endfig;

end;
