Dynamics of a fishery with nonlinear harvesting: control, price variation, and MSY

In this paper, we construct and analyse a new fishing mathematical model, which describes the time evolution of a fish stock, which is harvested by a fishing fleet, described by its fishing effort.  We consider that the price, which is given by the difference between supply and demand, is varying with respect to time.  For the harvesting function, we use the Holling II function.  On the other hand, we consider two different time scales: a fast one for the price variation and a slow one for fish stock and fishing effort variations.  We use an "aggregation of variables" method to get the aggregated model that governs fish biomass and fishing effort in the slow time.  By analyzing this reduced model, and under some conditions, we prove that three interesting equilibria can occur.  Furthermore, we show how one can control the model to avoid the undesirable situations and to reach the stable equilibrium.  Another interesting aspect given in this manuscript is the possibility of the implementation of Marine Protected Areas (MPAs).  We show how that MPAs permits us to contribute significantly to the rehabilitation of depleted fish populations.  This is achieved by disrupting the state of "Fish Extinction" equilibrium, and establishing a stable one.