A classification theorem for three dimensional competitive systems
The authors obtain a complete characterization of the global dynamics of the three dimensional Kolmogorov-type competitive system \(\dot x_ i=x_ if_ i(x)\), \(1\leq i\leq 3\), in the interior of the positive octant under the assumptions: 1) there is at most one positive equilibrium, 2) the system is uniformly persistent and dissipative, 3) all equilibrium points and closed orbits are hyperbolic. The main result asserts that, in addition to the positive equilibrium (which must exist), there is a (possibly empty) finite set of limit cycles and that the omega limit set of every positive trajectory is one of these. The geometry of the stable and unstable manifolds and basins of attraction are described. There are countably many inequivalent phase portraits, all of which can be realized. A recent result of \textit{M. W. Hirsch} [Systems of differential equations which are competitive or cooperative: III: Competing species, nonlinearity 1, 51-71 (1988)] implies that our equilibrium and all limit cycles lie on an invariant two dimensional manifold in the positive octant.
scientific article; zbMATH DE number 4040336
scientific article