Math 20 Lab #4 Another bifurcation diagram

Save your work regularly--you never know when Mathematica may crash.  

Email your completed file to tleise@amherst.edu by Friday 4pm.

Make sure your bifurcation diagram show the correct solid versus dashed lines.  For example, the previous lab looked at the DE (dx/dt)= r-x-(e^(-x)), whose bifurcation diagram can be constructed with the following Show command.

Show[ParametricPlot[{x+Exp[-x],x},{x,0,3},PlotStyle->{Blue,Thickness[.008]}],ParametricPlot[{x+Exp[-x],x},{x,-2,0},PlotStyle->{Red,Dashing[{0.02,0.02}],Thickness[.008]}],PlotRange->All,AxesOrigin->{0,0}]

Be sure to plot the stable branches with a solid line and the unstable branches with a dashed line (or you can use colors like blue for stable and red for unstable).

Neat way to visualize the graphical analysis in Mathematica (hit the little plus sign to see what the value of r as you move the slider):

f[x_,r_]:=1-x-Exp[-r x]
Manipulate[Plot[f[x,r],{x,-5,2},PlotRange->{{-5,2},{-1,1}}],{r,-2,2}]

Exercise:  Consider the differential equation (dx/dt)= 1-x-(e^(-rx)).  Plot the bifurcation diagram.  
What kind of bifurcation is occurring?  What is the critical value of r?