Canard Phenomenon in a modified Slow-Fast Leslie-Gower and Holling type scheme model

TL;DR

This paper applies Geometrical Singular Perturbation Theory to a modified predator-prey model, revealing a canard phenomenon near a folded singularity using blow-up techniques, and quantifies the limit-cycle behavior with high precision.

Contribution

It introduces a novel analysis of a folded singularity in a predator-prey model using blow-up methods, extending classical singular perturbation theory to new geometric phenomena.

Findings

Identification of a special folded singularity not previously analyzed

Quantification of the limit-cycle with an error of O(ε)

Demonstration of the canard phenomenon crossing the fold point

Abstract

Geometrical Singular Perturbation Theory has been successful to investigate a broad range of biological problems with different time scales. The aim of this paper is to apply this theory to a predator-prey model of modified Leslie-Gower type for which we consider that prey reproduces mush faster than predators. This naturally leads to introduce a small parameter $\epsilon$ which gives rise to a slow-fast system. This system has a special folded singularity which has not been analyzed in the classical work of Krupa-Szmolyan. We use the blow-up technique to visualize the behavior near this fold point $P$. Outside of this region the dynamics are given by classical singular perturbation theory. This allows to quantify geometrically the attractive limit-cycle with an error of $O(\epsilon$) and shows that it exhibits the \textit{canard} phenomenon while crossing $P$.