The long-run behavior of periodic competitive Kolmogorov systems
The paper is concerned with the non-autonomous periodic competitive system \[ \dot{x}_i=x_iN_i(t,x_1,\ldots,x_n). \tag{1} \] Assuming dissipation and irreducibility, the authors prove the existence of a canonically defined countable family of unordered, disjoint invariant sets which attract all persistent trajectories whose \(\omega\)-limit sets are not cycles. Then a more precise characterization of this family is obtained: it is shown that each member of the family (a) lies on an invariant \((n-1)\)-dimensional Lipschitz submanifold; (b) is the lower boundary of the basin of lower repulsion of a certain cycle in the interior of \(C\); (c) consists of finite connected components which are open \((n-1)\)-cells. Applying these results to the case studied by \textit{H. L. Smith} [J. Differ. Equations 64, 165-194 (1986; Zbl 0596.34013)], the authors obtain more information on the long-time behavior of trajectories of the system (1). Finally, they also study the topological entropy of the Poincaré map \(T\) generated by the system (1) and give a more explicit formula compared to that one obtained by Liang and Jiang in a preprint from 2000.
non-autonomous periodic competitive systems
scientific article; zbMATH DE number 1880130
scientific article