An Introduction to Viability Theory and Management of Renewable Resources

Abstract

The main purpose of viability theory is to explain the evolution of the state of a control system, governed by nondeterministic dynamics and subjected to viability constraints, to reveal the concealed feedbacks which allow the system to be regulated and provide selection mechanisms for implementing them. It assumes implicitly an “opportunistic” and “conservative ” behavior of the system: a behavior which enables the system to keep viable solutions as long as its potential for exploration (or its lack of determinism) — described by the availability of several evolutions — makes possible its regulation. We shall illustrate the main concepts and results of viability theory by revisiting the Verhulst type models in population dynamics, by providing the class of all Malthusian feedbacks (mapping states to growth rates) that guarantee the viability of the evolutions, and adapting these models to the management of renewable resources. Other examples of viability constraints are provided by architectures of networks imposing constraints described by connectionist tensors operating on coalitions of actors linked by the network. The question raises how to modify a given dynamical system governing the evolution of the signals, the connectionist tensors and the coalitions in such a way that the architecture remains viable. 1