A classification theorem for three dimensional competitive systems
type: Item
instance of: scholarly article
title: A classification theorem for three dimensional competitive systems
zbMATH Open document ID: 0638.34022
DOI: 10.1016/0022-0396(87)90154-9
author: Paul Waltman
author: Hal L. Smith
published in: Journal of Differential Equations
publication date: 1987-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 4040336
zbMATH Keywords: first order differential equation
zbMATH Keywords: global dynamics
zbMATH Keywords: Kolmogorov-type competitive system
zbMATH Keywords: basins of attraction
zbMATH Keywords: phase portraits
MaRDI profile type: Publication
full work available at URL: 0022-0396(87)90154-9
OpenAlex ID: W2016297725
cites work: Persistence in dynamical systems
cites work: Uniformly Persistent Systems
cites work: Persistence in a model of three competitive populations
cites work: Systems of Differential Equations Which Are Competitive or Cooperative: I. Limit Sets
cites work: Systems of Differential Equations that are Competitive or Cooperative II: Convergence Almost Everywhere
cites work: The dynamical systems approach to differential equations
cites work: On $\omega $-Limits for Competition Between Three Species
cites work: Periodic orbits of competitive and cooperative systems
cites work: Systems of Ordinary Differential Equations Which Generate an Order Preserving Flow. A Survey of Results
cites work: On the differential equations of species in competition
description: scientific article; zbMATH DE number 4040336