Optimal Placement of Marine Protected Areas to Avoid the Extinction of the Fish Stock

In this paper, we propose to study a fishery model with variable price. We assume that the price evolves much faster than the rest of the system. Under certain assumptions, this makes it possible to consider the fishery system as a slow–fast system, on two time scales, and to study it with a reduced model of dimension two. Two main cases can occur. The first one which we called catastrophic equilibrium corresponds to over-exploitation leading to fish extinction and a booming price. The second case corresponds to a sustainable fishery equilibrium which is stable. The possible effects of the creation of marine protected areas (MPAs), sites where fishing is prohibited, on the fish stock and fishery are evaluated. We show that MPAs can have a positive effect on the restoration of depleted fish stocks by destabilizing the catastrophic equilibrium and keeping only one positive equilibrium which will be globally asymptotically stable. This problem is addressed by proposing a model with MPA for the fish dynamics. Fish are assumed to move between MPA and fishing area and are subject to harvesting through fishing. We show that to avoid the extinction of the stock and stabilize the fishery in the long term, it is necessary to define a fishing zone such that the ratio of its carrying capacity to its surface is small enough. We further show that with a judicious choice of the surface area of the MPA, it is possible to optimize the total capture.