Optimal states of distributed exploited populations with periodic impulse selection

The dynamics of a population distributed on a torus is described by an equation of the Kolmogorov–Petrovsky–Piskunov–Fisher type in the divergence form. The population is exploited by periodic sampling of a constant distributed measurable ratio of its density. We prove that there exists a sampling ratio maximizing the time-averaged income in kind, i.e., a ratio that provides an optimal stationary exploitation in the long run.