Dynamics of the periodic type-K competitive Kolmogorov systems

For time-periodic dissipative and irreducible type-K competitive Kolmogorov systems, it is proved that there is a canonically defined countable family F of unordered, disjoint invariant sets with the property that, for every persistent trajectory whose (omega)limit set is not a cycle, there exists a unique trajectory in some element of F such that these two trajectories are asymptotic and the corresponding points in these two trajectories are K-related.