Eradicating an Alien Predator Population in a Seasonal Environment

We investigate the problem of eradicating an alien predator population in a seasonal environment. It is expressed as a zero-stabilizability problem with state constraints. The dynamics of the predators is described by a prey–predator system with nonlocal reaction terms and time-periodic rates. First, we are concerned with a control that acts on the predators, in a subset of the relevant habitat. A necessary condition and a sufficient condition for the eradicability (zero-stabilizability) of predators is given in terms of the sign of the principal eigenvalue of an appropriate operator that is not self-adjoint. Next, we investigate the eradicability of predators via controllers of harvesting type which act either on the predator population or on the prey population. Concerning this last approach, the idea is to diminish the prey population to a level at which it cannot sustain the predator population. Some principal eigenvalues will be involved in this investigation as well.