Cusp catastrophe manifold, explained with an example from community ecology. This manifold represents situations with two control parameters, herbivory and storm intensity in this case, and one response variable. Here the response is the coral or macroalgal dominance of tropical reefs. Dominance emerges from changes that take place within the community as the two control parameters interact. The change is usually abrupt and discontinuous, but the smooth manifold serves as a good explanation of the discontinuity. Prediction using catastrophe manifolds, however, remains a problem.
The manifold was generated with Mathematica 5.2 running on Fedora Linux. The example is a well known example of regime shift in ecological communities. Read more about it here.