In this paper, we are concerned with the existence of subharmonic solutions for the degenerate periodic systems of Lotka-Volterra type with impulsive effects. In our degenerate model, the variation of the predator and prey populations may vanish on a time interval, which imitates the (real) possibility that the predation is seasonally absent. Our proof is based on the Poincaré-Birkhoff theorem. By using phase plane analysis, we can find the large gap in the rotation numbers between the "small" solutions and the "large" solutions, which guarantees a suitable twist property. By applying the Poincaré-Birkhoff theorem, we then obtain the existence of subharmonic solutions. Our main theorem extends the associated results by J. López-Gómez et al.

We investigate the resonance problem and prove the existence of multiple periodic solutions to a second order parameter-dependent equation $x^{″}+f(t,x)=sp(t)$. We weaken the usual requirement on the sublinearity of perturbations when $|x|$ becomes large; and develop a more general method to investigate the rotational characterizations of the Landesman-Lazer conditions. Furthermore, $f$ does not satisfy the common sign condition, and even the global existence of the solution is not guaranteed.