Nonlinear Analysis 61 (2005) 491 – 501 www.elsevier.com/locate/na Internal stabilizability for a reaction–diffusion problem modeling a predator–prey system Bedr’Eddine Ainseba a , ∗ , Sebastian Ani¸ ta b a Mathématiques Appliquées de Bordeaux, UMR CNRS 5466, case 26, UniversitéVictor Segalen Bordeaux 2, 33076 Bordeaux Cedex, France b Faculty of Mathematics, University “Al.I. Cuza” and Institute of Mathematics, Romanian Academy, Ia¸ si 6600, Romania Received 11 January 2004; accepted 28 September 2004 Abstract In this work we consider a 2× 2 system of semilinear partial differential equations of parabolic-type describing interactions between a prey population and a predator population, featuring a Holling-type II functional response to predation. We address the question of stabilizing the predator population to zero, upon using a suitable internal control supported on a small subdomain  of the whole spatial domain , and acting on predators.We give necessary and sufficient conditions for this stabilizability result to hold.  2004 Elsevier Ltd. All rights reserved. Keywords: Reaction–Diffusion systems; Stabilization; Non-controllability 1. Introduction and main results We consider a nonlinear model describing the dynamics of a predator–prey system in a spatial bounded domain  ⊂ R n (n  1), with a smooth boundary . We denote by u = u(t,x)  0 the density of predators and by v = v(t,x)  0 the density of prey at time t  0 and position x ∈ . In a predator-free setting, the prey population increases at a natural rate r> 0, and saturates at a level K> 0, referred to as the carrying capacity of the ∗ Corresponding author. E-mail addresses: ainseba@sm.u-bordeaux2.fr (B. Ainseba), anita@uaic.ro (S. Ani¸ ta). 0362-546X/$ - see front matter  2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2004.09.055