Group for Research in Decision Analysis

# Explicit construction of viability kernels for sustainable management of ecosystems with an application to the hake-anchovy Peruvian fisheries

## Michel De Lara

We consider sustainable management issues formulated within the framework of control theory. The problem is one of controlling a discrete-time dynamical system (e.g. population model) in the presence of state and control constraints, representing conflicting economic and ecological issues for instance.The viability kernel is known to play a basic role for the analysis of such problems and the design of viable control feedbacks.

Unfortunately, its computation is not an easy task in general. We show that when the decreasing sequence of viability kernels up to time $t$ is stationary, its limit is the viability kernel. In the case of nonlinear generic ecosystem models, we provide a rather explicit description of the viability kernel for preservation and production constraints. Numerical illustrations are given for the hake-anchovy couple in the Peruvian ecosystem, with particular emphasis on consistency between ecological and economic conflicting objectives.