# A model of regulatory dynamics with threshold-type state-dependent delay

## TL;DR

This paper introduces a mathematical model for intracellular regulatory dynamics incorporating threshold-type state-dependent delays, analyzing their impact on stability and oscillations using bifurcation theory and numerical simulations.

## Contribution

It extends classic delay differential equations by including state-dependent delays and analyzes their effects on stability and oscillatory behavior.

## Key findings

- State-dependent delays can induce both supercritical and subcritical Hopf bifurcations.
- The model predicts conditions for the emergence of periodic oscillations.
- Numerical simulations confirm theoretical bifurcation analysis.

## Abstract

We model intracellular regulatory dynamics with threshold-type state-dependent delay and investigate the effect of the state-dependent diffusion time. A general model which is an extension of the classic differential equation models with constant or zero time delays is developed to study the stability of steady state, the occurrence and stability of periodic oscillations in regulatory dynamics. Using the method of multiple time scales, we compute the normal form of the general model and show that the state-dependent diffusion time may give arise to both supercritical and subcritical Hopf bifurcations. Numerical simulations of the prototype model of genetic regulatory dynamics are also given to illustrate the general results.