STABLE PERIODIC-ORBITS FOR A CLASS OF 3-DIMENSIONAL COMPETITIVE-SYSTEMS

It is shown that for a dissipative, three dimensional, competitive, an d irreducible system of ordinary differential equations having a uniqu e equilibrium point, at which point the Jacobian matrix has negative d eterminant, either the equilibrium point is stable or there exists an orbitally stable periodic orbit. If in addition, the system is analyti c then there exists an orbitally asymptotically stable periodic orbit when the equilibrium is unstable. The additional assumption of analyti city can be replaced by the assumption that the equilibrium point and every periodic orbit are hyperbolic. In this case, the Morse-Smale con ditions hold and the flow is structurally stable. (C) 1994 Academic Pr ess, Inc.