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Math 325: Differential	Equations    Quiz 5    11/22/96
:[font = input; initialization; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450]
*)
<<Graphics`PlotField`
(*
:[font = subsubsection; inactive; preserveAspect; rightWrapOffset = 538]
Consider the non-linear initial value problem:
    x'[t] == x[t](3 - x[t] - y[t])
    y'[t] == x[t](4 -2y[t] - x[t])
a) Find all the critical points of this system.
b) Classify each critical point by type and stability.
c) Describe what happens to the solution that passes through the point {1,1} in the phase plane as t->Infinity.
;[s]
12:0,0;47,1;117,2;120,0;165,2;168,0;220,2;223,0;291,1;296,0;319,1;330,0;332,-1;
3:6,13,9,Times,1,12,0,0,0;3,13,10,Courier,1,12,0,0,0;3,13,9,Times,3,12,0,0,0;
:[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 538; startGroup]
Solution
:[font = input; preserveAspect; rightWrapOffset = 538]
F[x_, y_] = x(3 - x - y);
G[x_, y_] = y(4 -2y - x);
:[font = text; inactive; preserveAspect; rightWrapOffset = 538]
a) Find the critical points:
;[s]
2:0,1;3,0;29,-1;
2:1,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0;
:[font = input; preserveAspect; rightWrapOffset = 538; startGroup]
cp = Solve[{F[x,y] == 0, G[x,y] == 0}]
:[font = output; output; inactive; preserveAspect; rightWrapOffset = 538; endGroup]
{{x -> 0, y -> 0}, {x -> 2, y -> 1}, {x -> 3, y -> 0}, 
 
  {y -> 2, x -> 0}}
;[o]
{{x -> 0, y -> 0}, {x -> 2, y -> 1}, {x -> 3, y -> 0}, 
 
  {y -> 2, x -> 0}}
:[font = text; inactive; preserveAspect; rightWrapOffset = 538]
The critical points are thus {0,0}, {2,1}, {3,0}, and {0,2}.

b) To analyze them we need to examine the matrix of partial derivatives evaluated at each.
;[s]
11:0,0;29,1;34,0;36,1;41,0;43,1;48,0;54,1;59,0;62,2;65,0;153,-1;
3:6,13,9,Times,0,12,0,0,0;4,13,10,Courier,1,12,0,0,0;1,13,9,Times,2,12,0,0,0;
:[font = input; preserveAspect; rightWrapOffset = 538; startGroup]
A = {{D[F[x,y],x], D[F[x,y],y]},
     {D[G[x,y],x], D[G[x,y],y]}};
A//MatrixForm
:[font = output; output; inactive; preserveAspect; rightWrapOffset = 538; endGroup]
MatrixForm[{{3 - 2*x - y, -x}, {-y, 4 - x - 4*y}}]
;[o]
3 - 2 x - y   -x

-y            4 - x - 4 y
:[font = text; inactive; preserveAspect; rightWrapOffset = 538]
At the critical point {0,0}:
;[s]
3:0,0;22,1;27,0;29,-1;
2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0;
:[font = input; preserveAspect; rightWrapOffset = 538; startGroup]
A1 = A /. cp[[1]];
A1//MatrixForm
:[font = output; output; inactive; preserveAspect; rightWrapOffset = 538; endGroup]
MatrixForm[{{3, 0}, {0, 4}}]
;[o]
3   0

0   4
:[font = text; inactive; preserveAspect; rightWrapOffset = 538]
The eigenvalues are both positive so the point is an unstable node.
At the point {2,1}:
;[s]
3:0,0;81,1;86,0;88,-1;
2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0;
:[font = input; preserveAspect; rightWrapOffset = 538; startGroup]
A2 = A /. cp[[2]];
A2//MatrixForm
:[font = output; output; inactive; preserveAspect; rightWrapOffset = 538; endGroup]
MatrixForm[{{-2, -2}, {-1, -2}}]
;[o]
-2   -2

-1   -2
:[font = input; preserveAspect; rightWrapOffset = 538; startGroup]
Eigenvalues[A2]//N
:[font = output; output; inactive; preserveAspect; rightWrapOffset = 538; endGroup]
{-3.414213562373095049, -0.5857864376269049512}
;[o]
{-3.41421, -0.585786}
:[font = text; inactive; preserveAspect; rightWrapOffset = 538]
The eigenvalues are both negative, so the critical point is an asymptotically stable node.
At the point {3,0}:
;[s]
3:0,0;104,1;109,0;111,-1;
2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0;
:[font = input; preserveAspect; rightWrapOffset = 538; startGroup]
A3 = A /. cp[[3]];
A3//MatrixForm
:[font = output; output; inactive; preserveAspect; rightWrapOffset = 538; endGroup]
MatrixForm[{{-3, -3}, {0, 1}}]
;[o]
-3   -3

0    1
:[font = text; inactive; preserveAspect; rightWrapOffset = 538]
The eigenvalues have opposite signs, so the critical point is an unstable saddle point.
At the point {0,2}:
;[s]
3:0,0;101,1;106,0;108,-1;
2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0;
:[font = input; preserveAspect; rightWrapOffset = 538; startGroup]
A4 = A /. cp[[4]];
A4//MatrixForm
:[font = output; output; inactive; preserveAspect; rightWrapOffset = 538; endGroup; endGroup]
MatrixForm[{{1, 0}, {-2, -4}}]
;[o]
1    0

-2   -4
:[font = text; inactive; preserveAspect; rightWrapOffset = 538]
The eigenvalues again have opposite signs, and the critical point is another unstable saddle point.
:[font = text; inactive; preserveAspect]
c) This system is a competing species model. At the point {1,1} both derivatives, x'[t] and y'[t], are positive. The solution will increase in both coordinates and approach the asymptotically stable critcial point {2,1}.
;[s]
10:0,0;3,1;58,2;63,1;82,2;87,1;92,2;97,1;214,2;219,1;221,-1;
3:1,13,9,Times,2,12,0,0,0;5,13,9,Times,0,12,0,0,0;4,13,10,Courier,1,12,0,0,0;
:[font = input; Cclosed; preserveAspect; startGroup]
v = PlotVectorField[{F[x,y],G[x,y]},{x,0,3.5},{y,0,2.5},
ScaleFunction->(Log[50#+1]&), Frame->True]
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 272; pictureHeight = 194]
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