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Ecole Nationale de Commerce et de Gestion, B.P. 1255, 90000 Tanger, Morocco; Laboratoire de Biom´etrie et Biologie Evolutive (UMR 5558), CNRS, Univ. Lyon1, 43 bd 11 nov, 69622 Villeurbanne Cedex, France; 3 Ecole Nationale des Sciences Appliqu´ees de Safi (E.N.S.A.S.) B.P. 89, route Dar si Aissa, 46000 Safi, Morocco; 4 Laboratoire S.I.A.N.O., Department de maths et info., Universit´e Ibn Tofail B.P. 133, 14000 K´enitra, Morocco; E-mail: [email protected] 2

Received 22 October 2005; accepted in final form 22 October 2005

ABSTRACT We study the effects of density dependent migrations on the stability of a predator-prey model in a patchy environment which is composed with two sites connected by migration. The two patches are different. On the first patch, preys can find resource but can be captured by predators. The second patch is a refuge for the prey and thus predators do not have access to this patch. We assume a repulsive effect of predator on prey on the resource patch. Therefore, when the predator density is large on that patch, preys are more likely to leave it to return to the refuge. We consider two models. In the first model, preys leave the refuge to go to the resource patch at constant migration rates. In the second model, preys are assumed to be in competition for the resource and leave the refuge to the resource patch according to the prey density. We assume two different time scales, a fast time scale for migration and a slow time scale for population growth, mortality and predation. We take advantage of the two time scales to apply aggregation of variables methods and to obtain a reduced model governing the total prey and predator densities. In the case of the first model, we show that the repulsive effect of predator on prey has a stabilizing effect on the predator-prey community. In the case of the second model, we show that there exists a window for the prey proportion on the resource patch to ensure stability.

Key Words: predator prey model, refuge, resource, time scales, aggregated model, stability, Dulac’s criterion.

1. INTRODUCTION There was a lot of interest for the study of the effect of a prey refuge on the stability of predator-prey communities (one can see Chiorino et al., 1999; Gonz´alez-Olivares and Ramos-Jilibert, 2003; Hausrath, 1994; Kar, 2004; K˜rivan, 1998; Srinivasu and Gayatri, 2005; Yakubu, 1997). In the case of a refuge for preys, the environment is heterogeneous and is usually represented by a set of discrete patches connected by migrations, see Chiorino et al. (1999), Ives (1992), Mchich et al. (2002, 2004) and Reeve (1988). The simplest situation corresponds to a set of two patches. The mathematical model is thus composed of two parts, a part which describes local predator-prey interactions on each patch and a part which describes the migration from patch to patch, for example from Acta Biotheoretica (2005) 53: 331–340

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the refuge to another patch where preys can go. It is also realistic to assume that prey as well a