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Reaction-diffusion systems with a Lotka-Volterra-type reaction term, alsoknown as competition-diffusion systems, have been used to investigate thedynamics of the competition among $m$ ecological species for a limited resourcenecessary to their survival and growth. Notwithstanding their rather simplemathematical structure, such systems may display quite interesting behaviours.In particular, while for $m=2$ no coexistence of the two species is usuallypossible, if $m \ge 3$ we may observe coexistence of all or a subset of thespecies, sensitively depending on the parameter values. Such coexistence cantake the form of very complex spatio-temporal patterns and oscillations.  Unfortunately, at the moment there are no known tools for a completeanalytical study of such systems for $m \ge 3$. This means that establishinggeneral criteria for the occurrence of coexistence appears to be very hard. Inthis paper we will instead give some criteria for the non-coexistence ofspecies, motivated by the ecological problem of the invasion of an ecosystem byan exotic species. We will show that when the environment is very favourable tothe invading species the invasion will always be successful and the nativespecies will be driven to extinction. On the other hand, if the environment isnot favourable enough, the invasion will always fail.

Ecological invasion in competition-diffusion systems when the exotic species is either very strong or very weak