Periodic Solutions of Periodic Competitive and Cooperative Systems

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Abstract
The periodic solutions, their basins of attraction and invariant manifolds are considered for periodic systems of differential equations which are cooperative or competitive following Hirsch. Competitive and cooperative mappings are introduced which possess the essential features of the Poincaré map for such systems. The geometrical properties of these mappings and the discrete dynamical system they generate are the objects of study. The main tools in this study are the Perron–Frobenius theory of positive matrices and invariant manifold theory. A complete description of the “phase portrait” of the discrete dynamical system generated by an orientation preserving planar cooperative map is obtained.

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