Effective reduction of a three-dimensional circadian oscillator model

Shuang Chen; Jinqiao Duan; Ji Li

doi:10.3934/dcdsb.2020349

Article Contents

2021, Volume 26, Issue 10: 5407-5419. Doi: 10.3934/dcdsb.2020349

This issue Previous Article Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions Next Article Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise

Effective reduction of a three-dimensional circadian oscillator model

1.
School of Mathematics and Statistics & Center for Mathematical Sciences, Huazhong University of Sciences and Technology, Wuhan, Hubei 430074, China
2.
Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA
3.
School of Mathematics and Statistics, Huazhong University of Sciences and Technology, Wuhan, Hubei 430074, China

^* Corresponding author: Shuang Chen
^* Corresponding author: Shuang Chen

Received: July 1, 2020

Early access: November 18, 2020

Published online: November 18, 2020

This work was partly supported by the NSFC grants 11531006, 11771449, 11771161, and the Hubei provincial postdoctoral science and technology activity project

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Abstract

We investigate the dynamics of a three-dimensional system modeling a molecular mechanism for the circadian rhythm in Drosophila. We first prove the existence of a compact attractor in the region with biological meaning. Under the assumption that the dimerization reactions are fast, in this attractor we reduce the three-dimensional system to a simpler two-dimensional system on the persistent normally hyperbolic slow manifold.

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Mathematics Subject Classification: Primary: 34C45, 34A26; Secondary: 34C20.

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Figure 1. The mechanism for the circadian oscillator model (1). Adapted from [33]

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Figure 2. The attraction of the slow manifold. The surface is the critical manifolds $ \mathcal{M}_{0} $, which is the zeroth-order approximation of the slow manifold, and the discrete orbits, respectively, start from $ (10,10,2) $, $ (15,15,2) $, $ (20,20,2) $, $ (10,20,2) $ and $ (20,10,2) $. Here $ k_{a} = 20000 $, $ k_{d} = 100 $ and the remaining parameters in (2) are chosen as in [33,Table 1,p.2414], that is, $ v_{m} = 1 $, $ k_{3} = k_{m} = 0.1 $, $ v_{p} = 0.5 $, $ k_{1} = 10 $, $ k_{2} = 0.03 $, $ P_{c} = 0.1 $ and $ J_{p} = 0.05 $. System (16) with $ \widetilde{k}_{2} = 1 $ has small parameter $ \varepsilon = 0.0003 $

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