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August  2018, 15(4): 863-882. doi: 10.3934/mbe.2018039

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

 Department of Mathematical Sciences, The University of Texas at Dallas, 800 W. Campbell Road, FO. 35, Richardson, TX, 75080, USA

* Corresponding author

Received  March 10, 2017 Accepted  December 16, 2017 Published  March 2018

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 classical 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 lead to both supercritical and subcritical Hopf bifurcations. Numerical simulations of the prototype model of Hes1 regulatory dynamics are given to illustrate the general results.

Citation: Qingwen Hu. A model of regulatory dynamics with threshold-type state-dependent delay. Mathematical Biosciences & Engineering, 2018, 15 (4) : 863-882. doi: 10.3934/mbe.2018039
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##### References:
Hes1 regulatory system in a cell: a. inhibition of mRNA transcription in nucleus from protein diffused from cytoplasm, b. translation of mRNA for protein synthesis in cytoplasm
(a) Equilibrium $(r^*, \, \xi^*) = (11.97050076, \, 2992.625189)$ is stable with $\epsilon = \epsilon_0-\delta$, $c = 0.01 < c_0$ with $\delta = 0.1$; (b) periodic solution appears at $\epsilon = \epsilon_0+\delta$, $c = 0.01 < c_0$
(a) Equilibrium $(r^*, \, \xi^*) = (11.97050076, \, 2992.625189)$ is asymptotically stable with $\epsilon = \epsilon_0-\delta < \epsilon_0$, $c = c_0+0.001$ (see the solid curve); when initial value is far enough from the equilibrium $(r^*, \, \xi^*) = (11.97050076, \, 2992.625189)$, solution may converge to another equilibrium (see the dashed curve). Subcritical bifurcation occurs at $\epsilon_0$ with $0 < c < c_0$. (b) If $\epsilon>\epsilon_0$ and $c = c_0+0.001$, the equilibrium $(r^*, \, \xi^*)$ is unstable
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