Viability Analysis of a Stock-Capital Fishery Model

Abstract

This work is based on C. Clark, F.Clarke and G. Munro (C.C.M) fishery model, [1], where the dynamic is the evolution of both stock and capital controlled by the investment and the part of the capital dedicated to the harvest. The contribution of the present work consists of substituting the profit maximization objective to a more of a sustainability approach that could be seen as constraints on the states, which corresponds more to the concerns of decision makers and legal authorities. In this new formulation, we give conditions on the parameters of the model which guarentee the non-emptiness of the viability kernel, that means the determination of the biggest set of intial conditions on which an investment and exploitaion sustainable policies could be defined. At the center of this result, we find the viability theory introducted by Aubin [2], which consist of studying the existence of controls yielding admissible states. Thus, we build the viability kernel corresponding to the studied model, under the conditions cited above.