Viable states for monotone harvest models

Abstract

This paper deals with the control of discrete-time dynamical, monotone both in the state and in the control, in the presence of state and control monotone constraints. A state x is said to belong to the viability kernel if there exists a trajectory, of states and controls, starting from x and satisfying the constraints. Under monotonicity assumptions, we present upper and lower estimates of the viability kernel. Our motivation comes from harvest models, where some monospecies age class models, as well as specific multi-species models (with so-called technical interactions), exhibit monotonicity properties both in the state and in the control. In this context, constraints represent production and preservation requirements to be satisfied for all time, which also possess monotonicity properties. Our results help delineating domains where a viable management is possible. Numerical applications are given for two Chilean fisheries. We obtain upper bounds for production which are interesting for managers in that they only depend on the model’s parameters, and not on the current stocks.