Marine fisheries are very important to the economy and livelihood of coastal communities, providing food security and job opportunities. The preservation of long-term prosperity and sustainability of marine fisheries is of political and social significance as well as economical and ecological importance. Due to new technologies allowing to catch more fish, various fish stocks like tuna, swordfish, shark, cod, halibut, etc. have declined by up to 90% in the last decades. Therefore, some states introduced policy instruments including landing fees, total allowable catches (TAC's) and marine protect areas (MPA's). In this context, important social and economical questions arise about the optimal amount of TAC's as well as the design of those MPA's: What is the optimal size and location of MPA's with regard to suficient recovery of the fish stock as well as suficient amount of fisheries  yield.

To address to these questions mathematically, the fishing strategy can be formulated as an optimal control problem. The fish stock dynamics are modeled by a time-dependent, non-linear PDE including reproduction and growth rate. The space-time distributed control describes the fishing intensity and is assumed to be bilinear with the biomass. The cost functional takes into account the benefit of the harvest, fishing costs and  the fish stock density at final time which guarantees sustainability. It turns out that the optimization problem complemented with additional control constraints results in a non-standard and non-linear optimal control problem. This talk is dedicated to the analysis of this problem in terms of solvability and optimality conditions. We also show first numerical examples.