Optimal exploitation of distributed population in periodic environment

For a population distributed in a periodic environment (= on a torus) with dynamics provided by an equation of the Kolmogorov-Petrovsky-Piskunov-Fisher type in divergent form, we consider two types of its exploitation – constant and impulse harvesting, which is also distributed. The task is to get the maximum time averaged income from exploitation in kind. It has been proven that such harvesting exists.
For a special case – the case of a circle – for a distributed harvesting carried out by a controlled machine moving cyclically around a circle, the existence of control of this machine, which provide the maximum time averaged income from operation in kind, has also been proven.