Emergence of global behaviour in a host–parasitoid model with density-dependent dispersal in a chain of patches Tri Nguyen Huu a,b, *, Pierre Auger b,c , Chrisphe Lett b,c , Marcos Marva ´ d a Laboratoire de Microbiologie, Ge ´ochimie et Ecologie Marines, UMR 6117, OSU, Case 901, Campus de Luminy, 13288 Marseille Cedex 9, France b Institut des Syste `mes Complexes, Ecole Normale Supe ´rieure de Lyon, 46 alle ´e d’Italie, 69364 Lyon Cedex 7, France c 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 d Departamento de Matema ´ ticas, Universidad de Alcala ´ , 28771 Alcala ´ de Henares, Spain 1. Introduction The study of spatial dynamics of host–parasitoid associations has received a lot of attention (see the review by Briggs and Hoopes, 2004). Most models consider a set of spatial patches connected by dispersal events. These models usually combine two sub-models, one describing local host–parasitoid inter- actions on each patch and the other describing dispersal among patches. In early works, Hassell et al. (1991a) and Comins et al. (1992) considered a spatial environment which is a two-dimensional network of patches connected by dispersal. The local dynamics was represented by the classical Nichol- son–Bailey model which is unstable. Regarding the dispersal sub-model, they defined host (resp. parasitoid) mobility as the proportion of hosts (resp. parasitoids) moving from one patch to its eight closest neighbouring patches between two time steps. In this model, it was assumed that dispersal is ‘‘constant’’, i.e. proportions of migrants do not depend on local host and parasitoid densities and are simply constant parameters. Furthermore, dispersal was considered as iso- tropic, i.e. migrant individuals were uniformly distributed on neighbouring patches. These models were mostly developed to study the dynamics and the persistence of the host– parasitoid system (Adler and Nuernberger, 1994; Allen, 1975; Reeve, 1988; Rohani et al., 1994; Rohani and Ruxton, 1999) and the spatial structures that may emerge such as spiral waves, ecological complexity 5 (2008) 9–21 article info Article history: Received 11 April 2007 Received in revised form 18 July 2007 Accepted 19 July 2007 Published on line 14 September 2007 Keywords: Emergence Host–parasitoid interactions Spatial model Density dependent dispersal Fast dispersal Aggregation of variables abstract We present a time discrete spatial host–parasitoid model. The environment is a chain of patches connected by dispersal events. Dispersal of parasitoids is host-density dependent. When the host density is small (resp. high), the proportion of migrant parasitoids is close to unity (resp. to zero). We assume fast patch to patch dispersal with respect to local inter- actions. Local host–parasitoid interactions are described by the classical Nicholson–Bailey model. By using time scales separation methods (or aggregation methods), we obtain a reduced model that governs the total host and parasitoid densities (obtained by addition over all patches). The aggregated model describes the time evolution of the total number of hosts and parasitoids of the system of patches. This global model is useful to make predictions of emerging behaviour regarding the dynamics of the complete system. We study the effects of number of patches and host density-dependent parasitoid dispersal on the overall stability of the host–parasitoid system. We finally compare our stability results with the CV 2 > 1 rule. # 2007 Elsevier B.V. All rights reserved. * Corresponding author. Tel.: +33 4 72 72 87 52; fax: +33 4 72 72 80 80. E-mail address: tri.nguyen-huu@ens-lyon.fr (T. Nguyen Huu). available at www.sciencedirect.com journal homepage: http://www.elsevier.com/locate/ecocom 1476-945X/$ – see front matter # 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ecocom.2007.07.003