Systems of differential equations that are competitive or cooperative. V: Convergence in 3-dimensional systems.

*(English)*Zbl 0712.34045[For part III, see Nonlinearity 1, 51-71 (1988; Zbl 0658.34024).]

Within the framework of monotone dynamical systems the author presents new results for existence of equilibria and cycles, convergence of all trajectories, global stability and structural stability of 3-dimensional systems. The overall theme is to show that any nontrivial limit set must lie on the boundary of a bounded set A, having nonempty interior, which is semi-invariant.

In Section 1 these sets A are defined and - among others - statements of the following form are obtained: if an invariant strongly balanced Jordan curve K lies in an order interval [a,b), then K lies on the boundary of a closed 3-ball in [a,b] which is positively or negatively invariant according as the vector field is competitive or cooperative; or: if an order interval [a,b] contains no equilibrium, then the orbit of every point of [a,b] must leave [a,b] in either time direction. In Section 2 three-dimensional systems are studied which are cooperative with negative divergence. It is shown e.g. that there are no cycles, and that in the irreducible case every compact limit set consists of equilibria. Also a global stability result for certain cooperative systems is obtained. Structural stability is considered in Section 3. It is shown that the flow in any attractor basin is structurally stable if equilibria are hyperbolic with stable and unstable manifolds intersecting transversally. The results are applied to positive feedback loops.

Within the framework of monotone dynamical systems the author presents new results for existence of equilibria and cycles, convergence of all trajectories, global stability and structural stability of 3-dimensional systems. The overall theme is to show that any nontrivial limit set must lie on the boundary of a bounded set A, having nonempty interior, which is semi-invariant.

In Section 1 these sets A are defined and - among others - statements of the following form are obtained: if an invariant strongly balanced Jordan curve K lies in an order interval [a,b), then K lies on the boundary of a closed 3-ball in [a,b] which is positively or negatively invariant according as the vector field is competitive or cooperative; or: if an order interval [a,b] contains no equilibrium, then the orbit of every point of [a,b] must leave [a,b] in either time direction. In Section 2 three-dimensional systems are studied which are cooperative with negative divergence. It is shown e.g. that there are no cycles, and that in the irreducible case every compact limit set consists of equilibria. Also a global stability result for certain cooperative systems is obtained. Structural stability is considered in Section 3. It is shown that the flow in any attractor basin is structurally stable if equilibria are hyperbolic with stable and unstable manifolds intersecting transversally. The results are applied to positive feedback loops.

Reviewer: B.Aulbach

##### MSC:

34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |

34D30 | Structural stability and analogous concepts of solutions to ordinary differential equations |

37-XX | Dynamical systems and ergodic theory |

##### Keywords:

cooperative systems; monotone flows; monotone dynamical systems; global stability; structural stability; positive feedback loops
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\textit{M. W. Hirsch}, J. Differ. Equations 80, No. 1, 94--106 (1989; Zbl 0712.34045)

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