Viability Kernel for Ecosystem Management Models

AbstractWe consider sustainable management issues formulated within the framework ofcontrol theory. The problem is one of controlling a discrete–time dynamical system(e. g. population model) in the presence of state and control constraints, representingconflicting economic and ecological issues for instance. The viability kernel is knownto play a basic role for the analysis of such problems and the design of viable controlfeedbacks, but its computation is not an easy task in general. We study the viabilityof nonlinear generic ecosystem models under preservation and production constraints.Under simple conditions on the growth rates at the boundary constraints, we providean explicit description of the viability kernel. A numerical illustration is given for thehake–anchovy couple in the Peruvian upwelling ecosystem.Key words: control theory; state constraints; viability; predator–prey; ecosystemmanagement; Peruvian upwelling ecosystem. ∗ IMCA, Instituto de Matema´tica y Ciencias Afines, UniversidadNacional de Ingenier´ia, Calle los Bio´logos245, La Molina, Lima 12, Peru´. eocana@imca.edu.pe, fax +511349-9838