The dynamics of a fish stock exploited in two fishing zones

This work presents a specific stock-effort dynamical model. The stocks corr espond to two populations of fish moving and growing between two fishery zo nes. They are harvested by two different fleets. The effort represents the number of fishing boats of the two fleets that operate in the two fishing n ones. The bioeconomical model is a set of four ODE's governing the fishing efforts and the stocks in the two fishing areas. Furthermore, the migration of the fish between the two patches is assumed to be faster than the growt h of the harvested stock. The displacement of the fleets is also faster tha n the variation in the number of fishing boats resulting from the investmen t of the fishing income. So, there are two time scales: a fast one correspo nding to the migration between the two patches, and a slow time scale corre sponding to growth. We use aggregation methods that allow us to reduce the dimension of the model and to obtain an aggregated model for the total fish ing effort and fish stock of the two fishing zones. The mathematical analys is of the model is shown. Under some conditions, we obtain a stable equilib rium, which is a desired situation, as it leads to a sustainable harvesting equilibrium, keeping the stock at exploitable densities.