Three-Dimensional Competitive Lotka--Volterra Systems with no Periodic Orbits

The following conjecture of M. L. Zeeman is proved. If three interacting species modeled by a competitive Lotka--Volterra system can each resist invasion at carrying capacity, then there can be no coexistence of the species. Indeed, two of the species are driven to extinction. It is also proved that in the other extreme, if none of the species can resist invasion from either of the others, then there is stable coexistence of at least two of the species. In this case, if the system has a fixed point in the interior of the positive cone in R3 , then that fixed point is globally asymptotically stable, representing stable coexistence of all three species. Otherwise, there is a globally asymptotically stable fixed point in one of the coordinate planes of R3 , representing stable coexistence of two of the species.