The Long-run Behavior of Periodic Competitive Kolmogorov Systems †

Persistent trajectories of the Poincaré map T generated by the ndimensional periodic system ẋi = xiNi(t, x1, · · · , xn), xi ≥ 0, are studied under the assumptions that the system is dissipative, competitive and strongly competitive in C = {x : xi > 0}. The main result is that there is a canonically defined countable family of disjoint, totally invariant sets which attract all persistent trajectories whose ω-limit sets are not cycles. Each totally invariant set is the union of finite disjoint open (n − 1) cells which are Lipschitz submanifolds and are transverse to positive rays. This result extends to time-periodic Kolmogorov systems a well known result of M. W. Hirsch for autonomous systems.