Catastrophe theory applied to ecological systems

Abstract

Catastrophe theory is a new field in mathematical topology that allows the formulation of comprehensive qualitative models of systems which have previously eluded rigorous mathematical formulation. Because the models have a topological foundation, many seem ingly dissimilar phenomena can be related to a common underlying topological structure. The properties of that structure can then be studied in a convenient form and the conclusions related back to the original problem. This paper provides an introduction to catastrophe theory and defines the principal condi tions required for its application. The basic prop erties of bimodality, discontinuity (catastrophe), hysteresis, and divergence are defined and illustra ted using the simplest structures of the theory. In this paper the spruce budworm/forest ecosystem of eastern Canada illustrates how catastrophe theory may be applied in ecology. With only a minimum of general descriptive information about the budworm system, a qualitative catastrophe theory model is hypothesized. This model is rich in its ability to provide predictions and insight about the global behavior of the system. To further check and verify the assumptions of this qualitative model, an exist ing detailed simulation model is analyzed from the perspective of catastrophe theory and is found to exhibit a basic underlying structure that is very similar to the catastrophe theory model. A detailed examination of the simulation model shows that catas trophe theory provides a consistent framework for analyzing and interpreting the behavior of this eco system.