Slow-fast dynamics in a planar parasite--host model with an extinction singularity

TL;DR

This paper analyzes a complex parasite-host model with slow-fast dynamics near extinction, using advanced mathematical techniques to understand its rich behavior including homoclinic orbits and bifurcations.

Contribution

The study applies Geometric Singular Perturbation Theory and blow-up methods to desingularize and analyze a non-standard parasite-host model with an extinction singularity.

Findings

Identification of homoclinic orbits and canard-like transitions.

Demonstration of topological changes due to infinitesimal parameter variations.

Numerical validation of analytical results.

Abstract

We study a slow-fast parasite--host model featuring a singularity at the extinction state. Using techniques from Geometric Singular Perturbation Theory (GSPT), and in particular the so-called blow-up method, we desingularize that point and reconstruct the local and global dynamics. The system we consider is in non-standard GSPT form and is characterized by a rich dynamical behavior: families of slow-fast homoclinic orbits, canard-like transitions generated by trajectories that remain close to a repelling critical manifold, and topological changes produced by infinitesimal variations of the infection rate, including the creation and destruction of an endemic equilibrium. We conclude with a numerical exploration of the model, to illustrate our analytical results.