Existence of maximum of time averaged harvesting in the KPP-model on sphere with permanent and impulse collection

On a two-dimensional sphere, a distributed renewable resource is considered, the dynamics of which is described by a model of the Kolmogorov–Petrovsky–Piskunov–Fisher type, and the exploitation of this resource, carried out by constant or periodic impulse harvesting. It is shown that after choosing an admissible exploitation strategy, the dynamics of the resource tend to the limiting dynamics corresponding to this strategy, and that there is an admissible harvesting strategy that maximizes the time averaged harvesting of the resource.