Following the well-extablished mathematical approach to persistence and its developments contained in [19] we give a rigorous theoretical explanation to the numerical results obtained in [4] on a certain prey-predator model with functional response of Holling type II equipped with an infectious disease in the predator population.The proof relies on some repelling conditions that can be applied in an iterative way on a suitable decomposition of the boundary. A full stability analysis is developed, showing how the "invasion condition" for the disease is derived. Some counterexamples and possible further investigations are discussed.

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In this paper we prove the existence of multiple periodic solutions (harmonic and subharmonic) for a class of planar Hamiltonian systems which include the case of the second order scalar ODE x ′′ + a(t)g(x) = 0 with g satisfying a one-sided condition of sublinear type. We consider the classical approach based on the Poincaré-Birkhoff fixed point theorem as well as some refinements on the side of the theory of bend-twist maps and topological horseshoes. The case of complex dynamics is investigated, too.

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