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.