Effects of density-dependent migrations on stability of a two-patch predator–prey model Abderrahim El Abdllaoui a , Pierre Auger a, * , Bob W. Kooi b , Rafael Bravo de la Parra c , Rachid Mchich d a IRD, Institut de Recherche pour le De ´veloppement, U. R. GEODES, Centre de Recherche d’Ile de France, 32 Avenue Henri Varagnat, 93143 Bondy cedex, France b Department of Theoretical Biology, Institute of Ecological Science, Vrije Universiteit, De Boelelaan 1087, 1081 HV Amsterdam, The Netherlands c Departamento de Matema ´ ticas, Universidad de Alcala ´ , 28871 Alcala ´ de Henares, Madrid, Spain d Ecole Nationale de Commerce et de Gestion, B.P. 1255, 90000 Tanger, Morocco Received 20 November 2006; received in revised form 5 March 2007; accepted 9 March 2007 Available online 18 March 2007 Abstract We consider a predator–prey model in a two-patch environment and assume that migration between patches is faster than prey growth, predator mortality and predator–prey interactions. Prey (resp. predator) migration rates are considered to be predator (resp. prey) density-dependent. Prey leave a patch at a migra- tion rate proportional to the local predator density. Predators leave a patch at a migration rate inversely proportional to local prey population density. Taking advantage of the two different time scales, we use aggregation methods to obtain a reduced (aggregated) model governing the total prey and predator densi- ties. First, we show that for a large class of density-dependent migration rules for predators and prey there exists a unique and stable equilibrium for migration. Second, a numerical bifurcation analysis is presented. We show that bifurcation diagrams obtained from the complete and aggregated models are consistent with each other for reasonable values of the ratio between the two time scales, fast for migration and slow for local demography. Our results show that, under some particular conditions, the density dependence of 0025-5564/$ - see front matter Ó 2007 Published by Elsevier Inc. doi:10.1016/j.mbs.2007.03.002 * Corresponding author. E-mail addresses: elabdll@bondy.ird.fr (A.E. Abdllaoui), pierre.auger@bondy.ird.fr (P. Auger), kooi@bio.vu.nl (B.W. Kooi), rafael.bravo@uah.es (R.B. de la Parra), racmchich@yahoo.com (R. Mchich). www.elsevier.com/locate/mbs Available online at www.sciencedirect.com Mathematical Biosciences 210 (2007) 335–354