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\centerline{\bf Math 221 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Project
with Maple \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Due on April 5, 2001}
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\noindent
Name:$\underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ }$
   \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
Score:$\underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$
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\hrule\smallskip \noindent {\bf Suggestions:} To get started, go
to my web site http://www.math.unl.edu/$\sim$bdeng to get the ode
sample file in Maple.
%credits. No text book, notes, formula sheets allowed.
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\item{\bf 1} (a) Use Maple to find the exact solution to the
logistic equation $${{d P(t)}\over{d t}}=rP(t)(1-{{P(t)}\over
K})$$ with $r>0$ the intrinsic growth rate and $K>0$ the carrying
capacity.
\itemitem{(b)} Plot the slope field for the equation
with $r=0.5$ and $K=5$ together with solutions  over the time
interval $[0,20]$ with distinct initial conditions
$P(0)=1,2,3,4,5,6,7,8$.
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\item{\bf 1} (a) Consider the equations of two competing species
(Eq.(3) of page 493 of the hand-out supplement.) $$\eqalign{{{d
x}\over{d t}}&=x(1-x-y)\cr {{d y}\over{d
t}}&=y(0.75-y-0.5x).\cr}$$

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\itemitem{(i)} Plot the direction field similar to Figure 9.4.1. of
the supplement.
\itemitem{(ii)} Plot the direction field together with several
solutions in one plot. Such a plot is referred to as a phase
portrait of the equations. With the exception of the direction
field, this plot should look like Figure 9.4.2 of the supplement.
\itemitem{(iii)} Plot the solutions curves $x(t),y(t)$ verses the
time $t$ over the interval $[0,20]$ with initial conditions
$x(0)=0.2,y(0)=0.05$. You need to put these two curves in one plot
similar to Figure 9.5.3.

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\itemitem{(b)} Consider equations of two competing species
(Eq.(21) of page 496 of the supplement.) $$\eqalign{{{d x}\over{d
t}}&=x(1-x-y)\cr {{d y}\over{d t}}&=y(0.5-0.25y-0.75x).\cr}$$
Generate three plots exactly the same as those of (a)(i,ii,iii)
above.
\itemitem{(c)} Consider equations of one prey and one predator
(Eq.(2) of page 505 of the supplement.) $$\eqalign{{{d x}\over{d
t}}&=x(1-0.5y)\cr {{d y}\over{d t}}&=y(-0.75+0.25x).\cr}$$
Generate three plots exactly the same as those of (a)(i,ii,iii)
above.
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\centerline{\bf END}
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