On the set of robust sustainable thresholds

In natural resource management, or more generally in the study of sustainability issues, the objective often consists in maintaining the state of a given system within a desirable configuration, typically established in terms of standards or thresholds. For instance, in fisheries management, the procedure of designing policies may include keeping the spawning stock biomass over a precautionary threshold and also ensuring minimal catches. Whenever the evolution of some natural resources, under the action of controls and un- certainties, is represented by a dynamical system in discrete-time, the aim of this paper is to characterize the set of robust sustainable thresholds. That is, the thresholds for which there exists a trajectory satisfying, for all possible uncertainty scenarios, prescribed constraints parametrized by such thresholds. This set provides useful information to users and decision-makers, illustrat- ing the trade-offs between constraints. Using optimal control, maximin and level-set approaches, we characterize the weak Pareto front of the set of robust sustainable thresholds and derive a numerical method for computing the en- tire set, as we show with a numerical example relying on renewable resource management.