Managing systems with non-convex positive feedback

We study here optimal management of dynamic ecological systems that exhibit a destabilizing positive feedback. The prototype example is that of a shallow lake in which phosphorous loading placed by anthropogenic activities ( fertilizers for farming and gardening) is stored in sediments until a critical level is reached after which there is a destabilizing return to the water - the tradeoff between farming interests and lake quality generates an optimal control problem. We show that in such systems, there may be a variety of local optima and associated basins of attraction wherein the optimal path may depend on starting state ( phosphorous stock). We characterize the various possible optimal behaviors and identify the ambiguities that can only be resolved by choice of functional form.