Stabilization of a Predator-Prey System with Nonlocal Terms

Faculty of Mathematics, “Alexandru Ioan Cuza” University of Iaşi, Iaşi 700506, Romania “Octav Mayer” Institute of Mathematics of the Romanian Academy, Iaşi 700506, Romania

^⋆ Corresponding author. E-mail: sanita@uaic.ro

Abstract

We investigate the zero-stabilizability for the prey population in a predator-prey system via a control which acts in a subregion ω of the habitat Ω, and on the predators only. The dynamics of both interacting populations is described by a reaction-diffusion system with nonlocal terms describing migrations. A necessary condition and a sufficient condition for the zero-stabilizability of the prey population are derived in terms of the sign of the principal eigenvalues to certain non-selfadjoint operators. In case of stabilizability, a constant stabilizing control is indicated. The rate of stabilization corresponding to such a stabilizing control is dictated by the principal eigenvalue of a certain operator. A large principal eigenvalue leads to a fast stabilization to zero of the prey population. A method to approximate all these principal eigenvalues is presented. Some final comments concerning the relationship between the stabilization rate and the properties of ω and Ω are given as well.