Persistent unstable equilibria and the grace period in dynamic models of environmental change

Abstract

Models of the interactions between population, economy and environment often contain nonlinear functional relationships and variables that vary at different speeds. These properties foster apparent unpredictabilities in the system behavior. We identify a class of deterministic models on the interaction of demographic, economic and environmental interactions in which catastrophic changes in environmental quality can take place, but involve a delay between passing a critical threshold level of pollution and the final collapse of the environment. We denoted this delay as the 'environmental grace period'. We illustrate the usefulness of geometric singular perturbation theory and local bifurcation theory to analyse such models. In particular, we show how it is possible to obtain analytic expressions for: (1) the level of emissions above which environmental deterioration begins; (2) the time it takes from reaching the critical level of emissions to the beginning of rapid environmental deterioration and (3) the level of emissions at the time that rapid deterioration begins. Because our results are analytic, they make the outcomes of demo- graphic, economic and environmental interactions more predictable and, therefore, poten- tially more manageable.