Cusps and butterflies: multiple stable states in marine systems as catastrophes

Abstract

Ecologists usually view smooth threshold-like shifts and sudden discontinuous jumps in stable states as an either–or proposition. This need not be the case, and using only graphs and no equations, it will be shown how it is possible to have a single model containing one, two or three stable points. This is not a new idea and the basics, known as catastrophe theory, were developed in the 1960s, and are well known to engineers and physicists. Systems with two stable points, which are known as cusp catastrophes, and those with three points, which are known as butterfly catastrophes, will be introduced without equations. Coral reefs and temperate intertidal rocky shores are discussed as possible examples of cusp and butterfly catastrophes. It has also been well known since the 1960s that there are nine hallmarks of catastrophes, and the relative merit of these hallmarks for use by experimentalists will be discussed. The hallmarks can be placed into three groups: the shape of the equilibrium surface (modality and inaccessibility), the behaviour of the equilibrium points as conditions change (discontinuous jumps, hysteresis, divergence and one-jump paths) and transient behaviour near cusps and folds (critical slowing down, anomalous variances and non-linear responses). There are two caveats. First, hysteresis and divergence may not occur in systems with noise. Second, unusual transient behaviour such as critical slowing down is not unique to systems with catastrophes and can be found in systems with smooth threshold-like shifts. We suggest that the two-state system of rockweeds and mussels in the Gulf of Maine is an example of a cusp catastrophe, and the three-state systems of corals, seaweeds and algal turfs may be an example of a butterfly catastrophe. In closing, we speculate why ecologists have overlooked and then reinvented catastrophe theory and rediscovered its hallmarks.