Three mutually competitive systems are modeled by a Lotka-Volterra
system. If each of the species can resist invasion at carrying capacity,
or if none of the species can resist invasion from either of the others, 
then 
there are no periodic orbits.  In the former case, there can be no 
coexistence; two of the species are driven to extinction.  In the latter 
case, 
there is coexistence of at least two species.
This is joint work with Mary Lou Zeeman