Optimal Cyclic Harvesting of a Distributed Renewable Resource with Diffusion

We study the problem of optimizing the harvesting of a renewable resource distributed on a circle. The dynamics of the resource restoration process is described by a Kolmogorov–Petrovskii–Piskunov–Fisher type equation in divergence form, and the harvesting of the resource is performed by a machine that moves cyclically along the circle. The objective functional is an average quantity depending on the position of this machine, the difficulty of detecting or harvesting the resource from this position, and the distance of the resource from this position. We prove that there exists an optimal motion of the harvesting machine that maximizes the average time profit in the natural form in the long run when the initial distribution of the resource is not less than the limit value in the absence of harvesting.