Formalizing the Panarchy Adaptive Cycle with the Cusp Catastrophe

The panarchy adaptive cycle, a general model for change in natural and human systems, can be formalized by the cusp catastrophe of René Thom's topological theory. Both the adaptive cycle and the cusp catastrophe have been used to model ecological, economic, and social systems in which slow and small continuous changes in two control variables produce fast and large discontinuous changes in system behavior. The panarchy adaptive cycle, the more recent of the two models, has been used so far only for qualitative descriptions of typical dynamics of such systems. The cusp catastrophe, while also often employed qualitatively, is a mathematical model capable of being used rigorously. If the control variables from the adaptive cycle are taken as parameters in the equation for the cusp catastrophe, a cycle very similar to the adaptive cycle can be constructed. Formalizing the panarchy adaptive cycle with the cusp catastrophe may provide direction for more rigorous applications of the adaptive cycle, thereby augmenting its usefulness in guiding sustainability efforts.