Corridors in Shallow Lakes: The Case of Intermediate Pollution

We study optimal management of the standard shallow lake model with special attention to the damage caused by the accumulated phosphorus in the water of the lake. The occurrence of a tipping point leads to a stronger increase in the damage after moving from the low-damage to the high-damage regime, because of the loss of ecosystem services. We modify the standard quadratic damage function by introducing a jump in marginal damage when moving from the low-damage to the high-damage regime. Connecting the lower and the higher marginal damage functions yields a continuous piecewise linear marginal damage function with a strong increase around the jump. In addition to the well-known oligotrophic and eutrophic steady states, we find that a third steady state can occur, separated by two Skiba points. This implies that a corridor or a basin of attraction arises between these two Skiba points where optimal management of the lake stabilizes the amount of phosphorus somewhere in the middle instead of either moving to the low-damage or to the high-damage regime, as is the result in the standard knife-edge case with one Skiba point.