Math. Model. Nat. Phenom.
Volume 10, Number 6, 2015Nonlocal reaction-diffusion equations
|Page(s)||6 - 16|
|Published online||02 October 2015|
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
⋆ Corresponding author. E-mail: email@example.com
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.
Mathematics Subject Classification: 35B35 / 35B09 / 35B51 / 92D25 / 35K57
Key words: zero-stabilizability / state constraints / comparison result / principal eigenvalue / population dynamics / reaction-diffusion system
© EDP Sciences, 2015
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