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**Systems of differential equations that are competitive or cooperative. II: Convergence almost everywhere.**
*(English)*
Zbl 0658.34023

[For part I see ibid. 13, 167-179 (1982; Zbl 0494.34017).]

The paper studies the limiting behavior of differential systems \(\dot x=F(x)\) in \({\mathbb{R}}^ n\) which are either cooperative or competitive, i.e. \(\partial F^ i/\partial x^ j(x)\geq 0\) or \(\leq 0\) for \(i\neq j\) and all x. By means of an analysis of monotone or eventually monotone flows in \({\mathbb{R}}^ n\) it is shown that both cooperative and competitive systems cannot have attracting limit cycles. In addition, in the planar case every solution is shown to be eventually monotone implying that every bounded trajectory converges, if the nonnegative quadrant \({\mathbb{R}}^ 2_+\) is positively invariant. For the derivation of the main result of this paper the relative position of \(\omega\)-limit sets of eventually strongly monotone flows is described, leading to the following result: almost all positive semitrajectories with compact closure approach the set of equilibria, if the underlying system is cooperative and has irreducible Jacobian matrices DF(x) for all x. Applications are made to generalizations of positive feedback loops.

The paper studies the limiting behavior of differential systems \(\dot x=F(x)\) in \({\mathbb{R}}^ n\) which are either cooperative or competitive, i.e. \(\partial F^ i/\partial x^ j(x)\geq 0\) or \(\leq 0\) for \(i\neq j\) and all x. By means of an analysis of monotone or eventually monotone flows in \({\mathbb{R}}^ n\) it is shown that both cooperative and competitive systems cannot have attracting limit cycles. In addition, in the planar case every solution is shown to be eventually monotone implying that every bounded trajectory converges, if the nonnegative quadrant \({\mathbb{R}}^ 2_+\) is positively invariant. For the derivation of the main result of this paper the relative position of \(\omega\)-limit sets of eventually strongly monotone flows is described, leading to the following result: almost all positive semitrajectories with compact closure approach the set of equilibria, if the underlying system is cooperative and has irreducible Jacobian matrices DF(x) for all x. Applications are made to generalizations of positive feedback loops.

Reviewer: B.Aulbach

### MSC:

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

37C10 | Dynamics induced by flows and semiflows |