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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 355 "          Investigating
 the Solutions of Differential Equations\n\nName:\n\nThis worksheet in
vestigates the qualitative behavior of the solutions\nof first order d
ifferential equations of the form\n                      y' = f(y)\n\n
The methods of direction field and stability analyses are presented us
ing\na threshold model as an example.\n\nDefine the function f:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "restart;\nf := w -> r*(w/theta-1)*(
1-w/K)*w;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eqn := diff(y(
t),t)=f(y(t));" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 125 "We can find \+
the analytic solution for general parameter values but\nthe solution i
s in implicit form (Can you solve for y(t)?)" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "dsolve(eqn,y(t),implicit);" }}}{EXCHG {PARA 257 "" 0 
"" {TEXT -1 180 "Direction field analysis - The drawback is that we mu
st specify the\nparameter values.  If the qualitative behavior changes
 with the\nparameters, we must do a sampling of parameters." }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 28 "r := 1; theta := 1/5; K:= 1;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "with(DEtools):\nDEplot(eqn,y(t),t=0
..3,y=0..2,arrows=slim);" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 125 "We
 can also generate a numerical solution for initial conditions\ny(0)=0
.1 and y(0)=0.5 to study the effects of the threshold." }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 54 "DEplot(eqn,y(t),t=0..3,\{[0,0.1],[0,0.5]\},arr
ows=NONE);" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 436 "Stability Analys
is - First we must locate the equilibrium or rest points\nof the equat
ion, i.e. the constant solutions, which satisfy f(y)=0.  Next\nwe must
 deteremine the stability of each rest point.  This can be\ndone witho
ut specify the parameters.  Let z be a rest point, \nthe test for stab
ility is:\n  i)   stable if f'(z) < 0,    ii)  unstable if f'(z) > 0, \+
 iii) test fails f'(z)=0\n\nWe first must clear the values of the para
meters." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "r := 'r'; theta := 'thet
a'; K := 'K'; " }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 45 "Locate the re
st points and save in a list pts" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 
"pts:= [solve(f(z)=0,z)];" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 58 "Te
st the stability of the rest points.  Differentiate f(z)" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 19 "df := diff(f(z),z);" }}}{EXCHG {PARA 257 "" 
0 "" {TEXT -1 42 "Evaluate f'(z) at each of the rest points." }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 26 "eig1 := subs(z=pts[1],df);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eig2 := subs(z=pts[2],df);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eig3 := subs(z=pts[3],df);" 
}}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 73 "What are the signs of each of
 the eigenvalues?  Remember that\n0< theta<K." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "13" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }