A simple geometrical condition for the existence of periodic solutions of planar periodic systems. Applications to some biological models

Using invariant regions for proving the existence of periodic solutions of periodic ordinary differential equations is a common tool. However, describing such a region is, in general, far from trivial. In this paper we provide sufficient conditions for the existence of an invariant region for certain planar systems. Our method locates the solution, in the sense that the region we determine evolves with time around the solution in the phase plane. Also, unlike other approaches, the construction does not depend on upper or lower bounds with respect to time of the functions involved in the system. The criterion is formulated for a general planar periodic ODEs system, and therefore it can be applied in very different contexts. In particular, we use the criterion to improve on previously known results on Holling's type II predator-prey periodic model, and on the classic periodic competition model.