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Three-dimensional competitive Lotka-Volterra systems with no periodic orbits. (English) Zbl 0910.34050
Summary: The following conjecture of M. L. Zeeman is proved. If three interacting species modeled by a competitive Lotka-Volterra system can resist each invasion at carrying capacity, then there can be no coexistence among the species. Indeed, two of the species are driven to extinction. It is proved that in the other extreme, if none of the species can resist an invasion from either of the others, then there is a stable coexistence of at least two of the species. In this case, if the system has a fixed point in the interior of a positive cone in ${ℝ}^{3}$, 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 ${ℝ}^{3}$, representing stable coexistence of two of the species.

##### MSC:
 34C25 Periodic solutions of ODE 92D25 Population dynamics (general) 34A26 Geometric methods in differential equations 92D40 Ecology