Persistent Unstable Equilibria in Wonderland

Models of the interactions between population, economy, and environment often contain nonlinear functional relationships and variables that vary a t different speeds. These properties foster apparent unpredictabilities in system behavior. Using a simple deterministic model of demographic, economic and environmental interactions, we illustrate the usefulness of geometric singular perturbation theory and local bifurcation theory. 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 demographic, economic, and environmental interactions more predictable, and, therefore, potentially more manageable.