3-DIMENSIONAL COMPETITIVE LOTKA-VOLTERRA SYSTEMS WITH NO PERIODIC-ORBITS

The following conjecture of M. L. Zeeman is proved. If three interacti ng species modeled by a competitive Lotka-Volterra system can each res ist 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 r esist invasion from either of the others, then there is stable coexist ence 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 R-3, then that fi xed point is globally asymptotically stable, representing stable coexi stence of all three species. Otherwise, there is a globally asymptotic ally stable fixed point in one of the coordinate planes of R-3, repres enting stable coexistence of two of the species.