Estimates of size of cycle in a predator-prey system II

TL;DR

This paper provides bounds on predator and prey populations in a predator-prey system's limit cycle, especially when populations are small and oscillations are large, using Lyapunov functions.

Contribution

It generalizes previous results to a broader class of systems and offers simpler proofs for existing estimates.

Findings

Estimates for predator and prey populations during limit cycles.

Applicable to systems with small populations and large oscillations.

Provides new Lyapunov-based proof techniques.

Abstract

We prove estimates for the maximal and minimal predator and prey populations on the unique limit cycle in a standard predator-prey system. Our estimates are valid when the cycle exhibits small predator and prey abundances and large amplitudes. The proofs consist of constructions of several Lyapunov-type functions and derivation of a large number of non-trivial estimates, and should be of independent interest. This study generalizes results proved by the authors in (Differ Equ Dyn Syst 30, 131-159 (2022)) to a wider class of systems and, in addition, it gives simpler proofs of some already known estimates.