Ill-behaviour of transformation-frontiers with environmental quality versus pollution

If one looks at environmental quality N as a renewable natural resource (with its own natural dynamics) which is changed by emissions Z, then one gets a better understanding of the shape of the damage function. In the case of a logistic growth dynamics one obtains a well-shaped transformation frontier N(Z), even in a two-species natural dynamics. For a more general natural interaction in the framework of a prey-predator model one obtains the possibilities of a cusp catastrophe which makes the transformation frontier N(Z) ill-shaped for the standard arguments of economists and their most favoured environmental policy instruments.