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 1, 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 2.