Catastrophe Theory

Abstract

Catastrophe theory is a mathematical framework that deals with discontinuous transitions between the states of a system, given smooth variation of the underlying parameters. The term catastrophe, derived from the French in this usage, refers to the abrupt nature of the transitions, and does not necessarily bear negative connotations. The main thesis of the theory is that the parameter space of the system is a low-dimensional projection of the state parameters and state relationships of the system, which are summarized as higher-dimensional, smooth manifolds. Apparent discontinuities or singularities in the parameter space of the system are explained as folds and cusps on the manifolds, thereby allowing the application of smooth differentiable models to discontinuous phenomena. The theory is well defined for systems up to five input or control parameters, and one or two output or response variables. Low-dimension catastrophe manifolds serve as good models and explanations of discontinuous transitions between alternative stable states in biological populations and ecological communities. In addition to the transitions, the models also explain the divergence between systems of slightly different initial conditions as those states evolve, as well as hysteresis in the reversal of state transitions. The prediction, using ‘catastrophe theory’, of future states of ecological systems is more problematic given the often semiquantitative nature of ecological mathematical models, as well as the complexity of those systems. Nevertheless, the broad applicability of catastrophe manifolds to dynamic systems suggests that they should be useful in predicting the potential behaviors of ecological systems.