A two-patch model for the optimal management of a fishing resource considering a marine protected area

In this work, we propose and analyze a model related with the management optimization of a renewable resource in aquatic environment composed of two different patches. Spatial distribution of each subpopulation is assumed: one is developed in a marine protected area (MPA) or a marine reserve and the other is located in a zone where fishing with open access may be effected.It is generally assumed that there may be migration between both areas, but in this work we will consider that the flux goes.When a fishing ban in the protected area is established it becomes a marine reserve, which can also be assumed as a refuge for the captured species. In this case, the marine reserve is the source and the exploitation area is a sink.The behavior of the renewable resource is modeled by a deterministic continuous time system. To establish the optimal harvesting policy, we will maximize the present value J of a continuous time stream of revenues, given by a cost functional indicating the net economic revenue to the fishermen, the perceived rent. Using Pontragyn’s Maximum Principle we will obtain the Hamiltonian function to determine the optimal policies.