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Cobwebbing

XPPAUT does not do cobwebbing, but you can make it do it by using a special ODE file that I have created. You are welcome to look at the file, but I do not include the code here as it is quite baroque. However, it is instructive to use it. Run it my typing xpp cobweb.ode at the prompt. Before running it, click on \fbox{{\tt Graphics}} \fbox{{\tt Add Curve}} (or tap \fbox{{\bf g}} \fbox{{\bf a}}) and fill in the resulting dialog box as
\fbox{
\begin{tabular}{l\vert l\vert l}
{\tt X-axis:} st & {\tt Z-axis:} X & {\tt Line type:} 1\\
\hline
{\tt Y-axis:} map & {\tt Color:} 1 & {}
\end{tabular}}
Click on \fbox{{\tt Ok}}. Repeat the above (that is, add a new curve), filling the dialog box as:
\fbox{
\begin{tabular}{l\vert l\vert l}
{\tt X-axis:} st & {\tt Z-axis:} X & {\tt Line type:} 1\\
\hline
{\tt Y-axis:} st & {\tt Color:} 5 & {}
\end{tabular}}
Click on \fbox{{\tt Ok}}. You have added two graphs to the picture on the screen; a plot of f(x) vs x and a plot of x vs x, which is just a diagonal line.

Now click on \fbox{{\tt Initialconds}} \fbox{{\tt Go}} ( \fbox{{\bf i}} \fbox{{\bf g}}) and you will get a nice cobweb. Change the parameter r to 2.95, erase the screen ( \fbox{{\tt Erase}} or \fbox{{\bf e}}), and rerun the simulation. (Either click on \fbox{{\tt Go}} in the Parameter Window, or tap \fbox{{\bf i}} \fbox{{\bf g}} in the Main Window.) Try r=3.1 and r=3.5.

Ok, now lets use this cobweb to look at the Ricker model

xn+1 = r xn e-xn

To examine this model, you must change three things. Now run the Ricker model with r=5,10,14,18.

For the last cobwebbing example, try the function $f(x)=r\sin(2\pi x)$ with 0<r<1, with a window $[0,1]\times [0,1]$ and scale=1.


next up previous
Next: The predator-prey model revisited Up: No Title Previous: Homework
G. Bard Ermentrout
1999-09-17