Ecological systems at many scales can exhibit multiple stable states and the possibility of regime shifts. These shifts are relatively well understood for a variety of specific systems which are amenable to description by simple mathematical models. Using the insights from these simple models as a bridge to understanding regime shifts in more complex systems raises substantial mathematical and ecological challenges to determine approaches which should help guide both management and adaptation in the face of global change. M athematical approaches in ecology and population biology have a very long history, and are going to become increasingly important in the face of challenges of dealing with global change. These challenges can be met by building on past successes in the use of mathematical models in ecology and including recent advances in understanding of dynamical systems, stochastic models, statistics and ecology. Mathematical tools are particularly important for management and understanding in the face of limited data, as well as for providing insights into how to gather data to update these predictions. One of the most important aspects of theory is to understand and forecast change and particularly the possibility of dramatic change in ecosystems, which clearly maps onto the mathematical notion of bifurcations. More precisely, changing ecological systems are often thought of as ones with a slowly changing parameter or variable, and a faster responding variable or set of variables. The slow variables are thought of as the external drivers, while the faster variables are the quantities of interest. For example, the slow variable could be human caused effects of global change, and the fast variables could be the size of one or more populations. Finally, given the imprecision of the descriptions and the potential role of small population sizes and varying external influences, stochastic models are appropriate. For example, temper
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