Controlling an Alien Predator Population by Regional Controls

Abstract

We investigate the problem of minimizing the total cost of the damages produced by an alien predator population and of the regional control paid to reduce this population. The dynamics of the predators is described by a prey-predator system with either local or nonlocal reaction terms. A sufficient condition for the zero-stabilizability (eradicability) of predators is given in terms of the sign of the principal eigenvalue of an appropriate operator that is not self-adjoint, and a stabilizing feedback control with a very simple structure is indicated. The minimization related to such a feedback control is treated for a closely related minimization problem viewed as a regional control problem. The level set method is a key ingredient. An iterative algorithm to decrease the total cost is obtained and numerical results show the effectiveness of the theoretical results. A spatially structured SIR problem may be described by the same system; in this case the above mentioned minimization problem is related to the problem of eradication of an epidemic by regional controls.