##
**Pedro Gajardo**

## Departamento de Matemática, Universidad Técnica Federico Santa María, Valparaíso - Chile

###
Friday June 2, 2017

GREThA - room F340 : 14:00-15:30

##
*"Methods for a sustainable rebuilding of overexploited natural resources: a viability approach"*

Abstract:

What is the ability of a natural resource to satisfy some constraints (objectives), usually in conflict, from now on? (e.g., to sustain a minimal level of biomass and to get a minimum profit due to harvesting). The viability theory has addressed this problem for more than 15 years. In general, given desired constraints, the idea is to find the states of the resource, for which it can be assured there exists a future viable trajectorie, that is, a way to manage the resource from now in order to satisfy the desired contraints. The set of these (initial) states is called the viability kernel, a key concept in the mentioned theory.

In this talk, we analyze a different problem but strongly related: given the current state of the resource (i.e. the initial condition), what are the constraints (parametrized by thresholds) that can be satisfied from now on? We call the set of these constraints the set of sustainable thresholds. Mathematically, it can be seen as the inverse mapping of the viability kernel. If for the current state of the resource the set of sustainable thresholds is not so large, the resource should be recovered, that is, to manage the resource in order to enlarge the set of sustainable thresholds. This process can be carried out optimizing a criteria (e.g., minimizing time or some cost).

During the talk, we will present the formalism for defining the set of sustainable thresholds and the recovery problem, together with examples concerning some Chilean fisheries. These examples are implemented in a web platform, designed in order to be a visualization tool for the stakeholders, that usually must take decisions about biological and productive objectives.