Environmental Management of Shallow Lakes Facing a Possible Catastrophe

Abstract

This contribution examines the environmental management of shallow lakes, taking into account thewell-documented potential for a catastrophy. Our model incorporates a constraint to ensure a minimumlevel of economic activity while accounting for a limited budget allocated to abatement efforts. In this setupfour different long run outcomes are possible. If abatement is expensive and the budget is low, a catastropheoccurs in finite time. Here we must distinguish between a global and a local solution pattern, where in thelatter case a catastrophe is avoided if the lake is initially sufficiently clean. If the budget is slightly larger,a unique long run steady state results, with heavy pollution but no catastrophe. With a large budget andnot too expensive abatement, the outcome is also history dependent with the lake only becoming heavilypolluted when the initial pollution level exceeds a threshold value. Alternatively, in such a situation it ispossible to clean the lake, but this is too expensive from an economic perspective. When the budget isslightly reduced, a fourth outcome arises. Here, history dependence persists, but with a crucial distinction:if pollution surpasses the threshold, the available budget is insufficient to reverse the degradation underany circumstances. Here the threshold level corresponds to a Stalling equilibrium, where pollution does notincrease any further only if the maximum budget is spend on abatement.

Keywords: optimal control theory, shallow lake, environmental catastrophe, Skiba points, abnormal solution, Stalling equilibrium