Dynamics of a fishery on two fishing zones with fish stock dependent migrations: aggregation and control

This work presents a specific stock-effort dynamic model. The stock corresponds to two fish populations growing and moving between two fishing zones, on which they are harvested by two different fleets. The effort represents the number of fishing vessels of the two fleets which operate on the two fishing zones. The bioeconomical model is a set of four ordinary differential equations governing the stocks and the fishing efforts in the two fishing areas. Fish migration, as well as vessels displacements, between the two zones are assumed to take place at a faster time scale than the variation of the stocks and the changes of fleets sizes, respectively. The vessels movements between the two fishing areas are assumed to be stock dependent, i.e. the larger the stock density is in a zone the more vessels tends to remain in it. The global fish stock and the total number of vessels keep constant at the fast time scale. This property enables, via aggregation methods, the reduction of the system dimension in order to proceed to its qualitative analysis. Under some assumptions, we obtain either a stable equilibrium or a stable limit cycle which involves large cyclic variations of the total fish stock and fishing effort. Finally, we introduce a control parameter to maintain the system at a sustainable equilibrium and thus avoiding the important fluctuations founded otherwise.