Methods of aggregation of variables in population dynamics

In ecology, we are faced with modelling complex systems involving many vari ables corresponding to interacting populations structured in different comp artmental classes, ages and spatial patches. Models that incorporate such a variety of aspects would lead to systems of equations with many variables and parameters. Mathematical analysis of these models would, in general, be impossible. in many real cases, the dynamics of the system corresponds to two or more time scales. For example, individual decisions can be rapid in comparison to growth of the populations. In that case, it is possible to pe rform aggregation methods that allow one to build a reduced model that gove rns the dynamics of a lower dimensional system, at a slow time scale. in th is article, we present a review of aggregation methods for time continuous systems as well as for discrete models. We also present applications in pop ulation dynamics. A first example concerns a continuous time model of a sin gle population distributed on a system of two connected patches (a logistic source and a sink), by fast migration. It is shown that under a certain co ndition, the total equilibrium population can be larger than the carrying c apacity of the logistic source. A second example concerns a discrete model of a population distributed on two patches, still a source and a sink, conn ected by fast migration. The use of aggregation methods permits us to concl ude that density-dependent migration can stabilize the total population. (C ) 2000 Academie des science/Editions scientifiques et medicales Elsevier SA S.