  Reaction-diffusion systems with a Lotka-Volterra-type reaction term, also
known as competition-diffusion systems, have been used to investigate the
dynamics of the competition among $m$ ecological species for a limited resource
necessary to their survival and growth. Notwithstanding their rather simple
mathematical structure, such systems may display quite interesting behaviours.
In particular, while for $m=2$ no coexistence of the two species is usually
possible, if $m \ge 3$ we may observe coexistence of all or a subset of the
species, sensitively depending on the parameter values. Such coexistence can
take the form of very complex spatio-temporal patterns and oscillations.
  Unfortunately, at the moment there are no known tools for a complete
analytical study of such systems for $m \ge 3$. This means that establishing
general criteria for the occurrence of coexistence appears to be very hard. In
this paper we will instead give some criteria for the non-coexistence of
species, motivated by the ecological problem of the invasion of an ecosystem by
an exotic species. We will show that when the environment is very favourable to
the invading species the invasion will always be successful and the native
species will be driven to extinction. On the other hand, if the environment is
not favourable enough, the invasion will always fail.
