Geometric analysis of Oscillations in the Frzilator model

TL;DR

This paper employs geometric singular perturbation theory and blow-up methods to mathematically prove the existence and analyze the structure of stable oscillations in a biochemical oscillator model related to myxobacteria development.

Contribution

It introduces a rigorous mathematical proof of a limit cycle in a biochemical oscillator using advanced geometric methods, expanding understanding of such biological systems.

Findings

Existence of a strongly attracting limit cycle proven mathematically.

The limit cycle corresponds to a relaxation oscillation with specific timescale structure.

The model exhibits stable and robust oscillations within certain parameter ranges.

Abstract

A biochemical oscillator model, describing developmental stage of myxobacteria, is analyzed mathematically. Observations from numerical simulations show that in a certain range of parameters, the corresponding system of ordinary differential equations displays stable and robust oscillations. In this work, we use geometric singular perturbation theory and blow-up method to prove the existence of a strongly attracting limit cycle. This cycle corresponds to a relaxation oscillation of an auxiliary system, whose singular perturbation nature originates from the small Michaelis-Menten constants of the biochemical model. In addition, we give a detailed description of the structure of the limit cycle, and the timescales along it.