Minimal complexity of subharmonics in a class of planar periodic predator-prey models

Abstract

This contribution analyzes the existence of $nT$-periodic coexistence states, for $n\geq1$, in two classes of non-autonomous predator-prey Volterra systems with periodic coefficients. In the first place, when the model is non-degenerate it is shown that the Poincaré–Birkhoff twist theorem can be applied to get the existence of subharmonics of arbitrary order. In the second place, it will be analyzed a degenerate predator-prey model introduced in [9] and [5] and, then, deeply studied in [7]. By analyzing the iterates of the Poincaré map of the system, it is shown that it admits nontrivial $nT$-periodic coexistence states for every $n\geq2$.