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

## TL;DR

This paper provides simple, parameter-dependent estimates for the maximum and minimum populations in a predator-prey model with logistic prey growth and Holling type II response, applicable to cycles with varying amplitudes.

## Contribution

It introduces a method to estimate predator and prey population bounds in a Rosenzweig-MacArthur system using Lyapunov functions, covering small and large amplitude cycles.

## Key findings

- Derived explicit bounds for predator and prey populations.
- Applicable to cycles with both small and large amplitudes.
- Introduced Lyapunov-based techniques for population estimates.

## Abstract

We consider a Rosenzweig-MacArthur predator-prey system which incorporates logistic growth of the prey in the absence of predators and a Holling type II functional response for interaction between predators and preys. We assume that parameters take values in a range which guarantees that all solutions tend to a unique limit cycle and prove estimates for the maximal and minimal predator and prey population densities of this cycle. Our estimates are simple functions of the model parameters and hold for cases when the cycle exhibits small predator and prey abundances and large amplitudes. The proof consists of constructions of several Lyapunov-type functions and derivation of a large number of non-trivial estimates which are also of independent interest.