PERSISTENT UNSTABLE EQUILIBRIA AND THE GRACE PERIOD IN DYNAMIC-MODELSOF ENVIRONMENTAL-CHANGE

Models of the interactions between population, economy and environment often contain nonlinear functional relationships and variables that v ary at different speeds. These properties foster apparent unpredictabi lities in the system behavior. We identify a class of deterministic mo dels on the interaction of demographic, economic and environmental int eractions in which catastrophic changes in environmental quality can t ake place, but involve a delay between passing a critical threshold le vel of pollution and the final collapse of the environment. We denoted this delay as the 'environmental grace period'. We illustrate the use fulness of geometric singular perturbation theory and local bifurcatio n theory to analyse such models. In particular, we show how it is poss ible to obtain analytic expressions for: (1) the level of emissions ab ove which environmental deterioration begins; (2) the time it takes fr om 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, th ey make the outcomes of demographic, economic and environmental intera ctions more predictable and, therefore, potentially more manageable.