Systems of Delay Differential Equations: Analysis of a model with feedback

TL;DR

This paper analyzes a delay differential equation system modeling hormonal feedback mechanisms, using topological degree theory to identify stable equilibria and periodic solutions in biological feedback systems.

Contribution

It introduces a mathematical framework for analyzing delay differential systems with feedback, applying topological degree to establish stability and periodicity results.

Findings

Existence of stable equilibria in the model

Conditions for periodic solutions with periodic parameters

Application of topological degree to biological feedback systems

Abstract

Self-regulatory models are common in nature, as described e.g. in (\cite{mur}), (\cite{ha}) and (\cite{Gb}).\\ Let us consider a system made up of a number of glands as a motivation. Each gland secretes a hormone that allows secretion in the {next} gland, which successively generates another hormone to stimulate the next one and so on. In the end, a final hormone is released which, by increasing its concentration, will inhibit the secretion of previous hormones that allowed the production process. This generates the decay of this hormone to a minimum threshold that re-activates the cycle again.\\ This behavior can be seen in other biochemical processes, such as enzymatic or bacterial models.\\ Topological degree is a useful tool to find stable equilibria in a wide variety of models with constant parameters and, furthermore, allows to deduce the existence of periodic solutions when the constant parameters are replaced by periodic functions.