SUMMER 2005

This figure is an example of a basin of attraction, with the two colors denoting initial conditions whose orbits will eventually have different properties.

Creating such graphs requires an enormous number of computations, and the graphs can be misleading if the algorithm used to generate the picture is not sufficient.

This graph, though beautiful, is in fact inaccurate because the underlying algorithm did not perform enough computations.

Jared and Allison focused on an even simpler competition model with one species (J and A) stage-structured and the other (Z) not. For a certain choice of parameter values, this system of three equations showed fully 5 separate attractors - 3 periodic and 2 chaotic.

We investigated the basin of attraction for each of these 5 attractors. While the computation involved was tremendous, the resulting graphs were quite striking. In this graph, initial conditions (J,A,Z) in blue lead to the 96-cycle in the graph above, points in red lead to the white chaotic attractor, and points in black lead to the 24-cycle. The union of the JZ- and AZ- planes forms an invariant set, with some points leading to chaos (green in the graph above) and some leading to the 48-cycle.

We also were able to complete some mathematical analysis of the underlying system of equations. We were able to generate 'backwards orbits' and prove that no point has more than 4 pre-image points.