Delayed stability switches in singularly perturbed predator-prey models

TL;DR

This paper proves the existence of canard solutions in certain predator-prey models with singular perturbations, highlighting delayed stability switches during bifurcations, and classifies models that admit such solutions.

Contribution

It provides an elementary proof of canard solutions in predator-prey systems with transcritical bifurcations, extending one-dimensional theory to two dimensions and classifying applicable models.

Findings

Identified conditions for canard solutions in predator-prey models.

Extended one-dimensional canard theory to two-dimensional systems.

Classified quadratic vector field models allowing canard solutions.

Abstract

In this paper we provide an elementary proof of the existence of canard solutions for a class of singularly perturbed predator-prey planar systems in which there occurs a transcritical bifurcation of quasi steady states. The proof uses a one-dimensional theory of canard solutions developed by V. F. Butuzov, N. N. Nefedov and K. R. Schneider, and an appropriate monotonicity assumption on the vector field to extend it to the two-dimensional case. The result is applied to identify all possible predator-prey models with quadratic vector fields allowing for the existence of canard solutions.