We wanted to study how volume, rounding, and stochasticity affected the coexistence of the two beetle populations. In order to get a good visual image of this we created a program that showed the long term behavior of every trajectory in a defined window. Our choices were 1) Species 1 goes extinct, 2) Species 2 goes extinct, or 3) Both species survive. We represented extinction of either species by coloring the starting point of a trajectory black, and if both survived, green. The x-axis corresponds to species one and the y-axis, species two.
Here is an example:
The program in its most basic form did one thing. It took every starting point in a defined window and ran it through the model a set number of times. Then it looked at the finishing values and assigned a color to the corresponding starting point. Since the model is six dimensional and we cannot represent six dimensional space, the actual picture represented is a slice of that space. We chose to look only at starting conditions with adult beetles, therefore, at the start both species pupa and larva populations are zero. This restriction also helped plotting the results. Remember, the picture generated is a slice of six dimensional space. Namely the one corresponding to just adults. After getting the program to work we added features. This allowed us to vary the volume, amount of variability, add a lattice, and change the pixel ratio (amount of starting points run).
One aspect of the program to note is the extinction cutoff. As discussed later in the volume section the lattice causes trajectories headed for the axes to get hung up before reaching complete extinction. Therefore, to get an accurate representation of the coexistence basin we added a parameter called the "extinction cutoff." This is the density where a species has had it. There are just to few individuals for the species to exist. There are ecological and mathematical arguments for setting this value but for most of our experiments, after a little debate, we decided on four. The limits on this value are zero, which cannot be reached with the lattice on, and the value where the stable attractor lies in the extinction region, which results in a black picture. Here is an example, continuous model, of increasing the extinction cutoff. Movie
For more information about the program and to run your own plots you can download basinplot. You will need Matlab to run the code but can view the implementation in any text editor. In order to speed up the program the model parameters are hard coded into the program so you can only run that set of parameters without difficult changes.
One note about the movies that appear in the following pages. For the most part they are in a window ranging from densities of 0 to 1024 in both the x and y directions. The exception to this appears in the movies where zooming is employed to get a better look at the specific behavior of an area. Also, the origin is in the lower left hand corner.
In order to generate movies we needed to generate a lot of pictures/frames. To do this in Matlab would have taken a long time, so in the interest off time part of the program was rewritten in C++ and then the data supplied to Matlab to run movies. Also, the more interesting ones were changed to MPEG format and appear in this report. For more information on implementation and a collection of movies see this page.