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Cell["\<\
Qualitative Behavior of Non-linear Systems:  A Competing Species \
Example\
\>", "Section"],

Cell["\<\
Suppose that x(t) and y(t) are the populations (counted in , say, \
millions) of two different species living in the same environment at time \
t.Assuming that the two species compete for food, water, and space,one might \
imagine that the populations are governed
roughly by the system of ODE's 

   x' = x(1 - x- y)
   y' = y(2-4x-y)
\t
A fundamental idea for studying the qualitative behavior of solutions to \
nonlinear systems such as this one is to first determine the constant \
solutions, then consider the behavior of solutions near these constant \
solutions, and finally describe the entire phase portrait in terms of its \
behavior relative to the constant solutions.  We proceed by
first setting up the system and finding  its critical points with the help of \
a direction field.\
\>", "Text",
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\>"],
  ImageRangeCache->{{{0, 287}, {287, 0}} -> {-0.0550006, 0.419801, 
  0.00199642, 0.00220656}}]
}, Open  ]],

Cell["\<\
Why was the above choice of ranges for x and y a reasonable one?  \
Think in terms of the situation we are trying to model.
\t\t
Critical points can be approximately detected with a direction field
by looking for those places in which the vector field seems to be turning \
quite a lot.  In this picture there appear to be three
such places--roughly speaking, the points (.4,.5), (0,1.75),
and (.9,0).  We can improve these guesses with a little help:\
\>", "Text",
  FontSize->24],

Cell[BoxData[{
    \(FindRoot[{f[x, y]\  == \ 0, \ g[x, y]\  == \ 0}, \ {x,  .4}, {y,  .5}]\ 
    \), 
    \(FindRoot[{f[x, y]\  == \ 0, \ g[x, y]\  == \ 0}, \ {x,  .9}, {y, 0}]\ 
    \), 
    \(FindRoot[{f[x, y]\  == \ 0, \ g[x, y]\  == \ 0}, \ {x, 0}, {y, 1.75}]\ 
    \n\)}], "Input",
  FontSize->24],

Cell["\<\
That is, (1/3, 2/3), (1,0), and (0,2) are the critical points.  \
Note that we missed the critical point (x,y) = (0,0).
\t
Now we want to examine the behavior of solutions near each critical
point in turn.  First we pick on (1/3,2/3).  The direction field nears this \
point looks like:\
\>", "Text",
  FontSize->24],

Cell[BoxData[
    \(\(PlotVectorField[{f[x, y], g[x, y]}, {x, 1/3 -  .01, 1/3\  +  .01}, {
        y, 2/3 -  .01, 2/3 +  .01}, \n\tAxesOrigin -> {1/3, 2/3}, \n\t
      Axes -> \ True, \n\tScaleFunction \[Rule] \((1&)\), Ticks \[Rule] None, 
      Frame \[Rule] True, AspectRatio \[Rule] 1, \ PlotPoints\  -> \ 20]; 
    \)\)], "Input",
  FontSize->24],

Cell["\<\
This looks a bit like the vector field of a linear homogeneous
constant coefficient system with one positive and one negative
eigenvalue.  Now we look at a phase portrait.  Since the system
is non-linear, we must use NDSolve to generate solution curves.
We define a function to produce these solution given (in order)

     starting time     t1
     stopping time     t2
            x(0)                  a
            y(0)                  b

Note that this will be most meaningful if t1<0<t2.  Also notice
that the semi-colon is important in the following definition.
Without it, Mathematica would try to evaluate the NDSolve command
and then crash when it didn't have definite values for t1,t2,a, and
b.\
\>", "Text",
  FontSize->24],

Cell[BoxData[
    \(\(soln[t1_, t2_, a_, b_]\  := 
      \ {x[t], y[t]} /. 
        First[\n\t\t\t
          NDSolve[\ {\(x'\)[t] == f[x[t], y[t]], \(y'\)[t] == g[x[t], y[t]], 
              \n\t\t\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x[0]\  == \ a, \ 
              y[0]\  == \ b}, \n
            \t\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {x[t], y[t]}, \n
            \t\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {t, t1, t2}]\n\t\t\t]
          \n\)\)], "Input",
  FontSize->24],

Cell[BoxData[
    \(\(\n\(Show[\n\n
      ParametricPlot[\n\t\ \ 
        Evaluate[
          Table[\n\t\t\t
            soln[\(-4\), 4, 1/3 + s, 2/3 +  .01], {s, \(- .01\),  .01,  .001}
              \n\t\t]], \n\t\ \ \ {t, \(-4\), 4}\n], \n
      ParametricPlot[\n\t\ \ 
        Evaluate[
          Table[\n\t\t\t
            soln[\(-4\), 4, 1/3 + s, 2/3 -  .01], {s, \(- .01\),  .01,  .001}
              \n\t\t]], \n\t\ \ \ {t, \(-4\), 4}\n], \n\n\t
      AspectRatio -> Automatic, \n\tAxesOrigin -> {1/3, 2/3}, \n\t
      PlotRange -> {{1/3 -  .01, 1/3 +  .01}, {2/3 -  .01, 2/3 +  .01}}, \n\t
      Ticks \[Rule] None, \n\tFrame \[Rule] True\t\ \ \ \n]; \)\t\t\t\)\)], 
  "Input",
  FontSize->24],

Cell[BoxData[{
    FormBox[
      RowBox[{
      "Again", ",", " ", 
        \(this\ looks\ very\ much\ like\ the\ picture\ of\ linear\ homogeneous
          \ system\ of\ saddle\ type . \ \ In\ particular\), ",", " ", 
        \(solutions\ that\ begin\ close\ to\ \((1/3, 2/3)\)\ tend\ to\ move\ 
          away\ from\ this\ point\ when\ t\ gets\ large . \ \ For\ this\ 
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This looks exactly like the direction field picture for the orginal \
system near (1/3,2/3).  \
\>", "Text",
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The phase portrait also matches with that of the original system.\
\
\>", "Text",
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Now we turn to the other critical points of our example, examining \
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\>", "Text",
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we see that the linear system associated to the point (0,0) has a \
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the orginal system.  This can be seen by looking at a direction field near \
(0,0):\
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This time we have a sink in the associated linear system.  Thus \
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Again, a sink.  The point (0,2) is also a stable critical \
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Now we know more or less how solutions of our system behave near \
each critical point.  It remains to understand the global behavior of the
system.  To this end we plot some trajectories on a larger scale.  We will \
take our initial conditions in the first quadrant along the line 2x + y = \
1.\
\>", "Text",
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