Subharmonics in a class of planar periodic predator-prey models

Abstract

This contribution studies the existence of positive subharmonics ofarbitrary order in the planar periodic Volterra predator-prey model. When themodel isnon-degenerate, in the sense that the birth rate of the prey intersects thesupport of the death rate of the predator, as in [8], then the existence of positivesubharmonics can be derived from the Poincaré–Birkhoff theorem version [3].Nevertheless, in thedegeneratecase when these supports do not intersect, then,the Poincaré–Birkhoff theorem fails in general. Still in these degenerate situations,the techniques of [7] provide us with the existence of positive subharmonics ofarbitrary order.