## Section: New Results

### Global behavior of $N$ competing species with strong diffusion: diffusion leads to exclusion

In [19] , we study the following problem.
For a large class of models involving several species competing for
a single resource in a *homogeneous* environment,
it is known that the competitive exclusion principle holds:
only one species survives eventually.
Various works indicate though that coexistence of many species
is possible when the competition occurs in a *heterogeneous* environment.
We propose here a spatially heterogeneous system modeling several species
competing for a single resource, and migrating in the spatial domain. For this model,
it is known, at least in particular cases, that if migrations are *slow* enough, then coexistence
occurs.
In this paper we show at variance that if the spatial migrations are *fast* enough,
then our system can be approximated by a spatially homogeneous system, called aggregated model, which can be explicitly computed, and
we show that if the competitive exclusion principle holds for the aggregated model,
then it holds as well for the original, spatially heterogeneous model. In other words,
we show the persistence of the competitive exclusion principle
in the spatially heterogeneous situation
when migrations are fast. As a consequence, for fast migrations
only one species may survive, namely the best competitor *in average*.
We last study which is the best competitor *in average* on some examples, and draw some
ecological consequences.