Regional control in optimal harvesting of population dynamics

Here we investigate the regional control for some optimal harvesting problems related to population dynamics; namely we consider the problem of maximizing the profit for spatially structured harvesting problems with respect to both the harvesting effort and the selected subregion ω (of the whole domain Ω) where the effort is localized. For a fixed subregion ω we state necessary optimality conditions and use them to get the structure of the optimal effort and to reformulate the maximization problem with respect to the subregion ω, where the harvesting effort is localized, in a more convenient way. We derive an iterative algorithm to increase at each iteration the profit by changing the subregion where the effort is localized. Some numerical tests are given to illustrate the effectiveness of the results for a particular optimal harvesting problem. Final comments are given as well concerning further directions to extend the results and methods presented here.