A discrete model with density dependent fast migration

The aim of this work is to develop an approximate aggregation method for ce rtain non-linear discrete models. Approximate aggregation consists in descr ibing the dynamics of a general system involving many coupled variables by means of the dynamics of a reduced system with a few global variables. We p resent discrete models with two different time scales, the slow one conside red to be linear and the fast one non-linear because of its transition matr ix depends on the global variables. In our discrete model the time unit is chosen to be the one associated to the slow dynamics, and then we approxima te the effect of fast dynamics by using a sufficiently large power of its c orresponding transition matrix. In a previous work the same system is treat ed in the case of fast dynamics considered to be linear, conservative in th e global variables and inducing a stable frequency distribution of the stat e variables. A similar non-linear model has also been studied which uses as time unit the one associated to the fast dynamics and has the non-linearit y in the slow part of the system. In the present work we transform the syst em to make the global variables explicit, and we justify the quick derivati on of the aggregated system. The local asymptotic behaviour of the aggregat ed system entails that of the general system under certain conditions, for instance, if the aggregated system has a stable hyperbolic fixed point then the general system has one too. The method is applied to aggregate a murti regional Leslie model with density dependent migration rates. (C) 1999 Else vier Science Inc. All rights reserved.