OPTIMAL PHOSPHORUS LOADING FOR A POTENTIALLY EUTROPHIC LAKE

We are concerned with the management of ecological and economic systems with threshold responses and with several time scales. Although optimal control of such systems is seldom attainable, the form of such optimal controls provides important insights for more practical schemes. Here we optimize the expected discounted net benefits of phosphorus (P) loadings for a potentially eutrophic lake. The benefits accrue to agricultural interests from activities that result in loading, and costs accrue to other interests from the resulting deterioration of water quality. We extend the 1999 results of S. R. Carpenter, D. Ludwig, and W. A. Brock to account for dependence of P recycling upon the concentration of P in sediments. We obtain optimal policies using methods of dynamic programming with two state variables. We find a strong interaction between economic and ecological parameters in determining the optimal policy: the economic discount rate determines whether the time horizon is long or short, and this in turn strongly influences the magnitude of the optimal loadings. Simple policies that neglect dynamics of P in the sediments are inadequate unless the time horizon is short and the dynamics are slow. A stochastic model is essential if there are substantial random fluctuations in loadings. Uncertainty in the determination of the critical P density that triggers recycling cannot be neglected. Our results may be interpreted as a quantitative precautionary principle that takes account of both economic and ecological aspects of the management of the lake. Our results may also be used to illustrate economic ideas such as sustainable development, natural capital, option values, and income.