Bifurcation analysis of an enzyme-catalyzed reaction system with branched sink

Juan Su; Bing Xu; Lan Zou

doi:10.3934/dcdsb.2019167

Article Contents

2019, Volume 24, Issue 12: 6783-6815. Doi: 10.3934/dcdsb.2019167

This issue Previous Article Remarks on basic reproduction ratios for periodic abstract functional differential equations Next Article Flocking of Cucker-Smale model with intrinsic dynamics

Bifurcation analysis of an enzyme-catalyzed reaction system with branched sink

1.
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
2.
Department of Mathematics, Chengdu Normal University, Chengdu, Sichuan 611130, China

^* Corresponding author: Lan Zou, Email: lanzou@163.com
^* Corresponding author: Lan Zou, Email: lanzou@163.com

Received: August 31, 2018

Revised: January 31, 2019

Early access: July 2019

Published: September 2019

This work is supported by NSFC grant 11831012 and 11771168

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Abstract

In this paper, we study the local bifurcations of an enzyme-catalyzed reaction system with positive parameters $ \alpha $, $ \beta $, $ \gamma $ and integer $ n\geq 2 $. This system is orbitally equivalent to a polynomial differential system with order $ n+2 $. Although not all coordinates of equilibria can be computed because of the high degree of polynomial, parameter conditions for the coexistence of equilibria and their qualitative properties are obtained. Furthermore, it is proved that this system has various bifurcations, including saddle-node bifurcation, transcritical bifurcation, pitchfork bifurcation and Hopf bifurcation. Based on Lyapunov quantities, the order of weak focus is proved to be at most 3. Furthermore, parameter conditions of the exact order of weak focus are obtained. Finally, numerical simulations are employed to illustrate our results.

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Mathematics Subject Classification: Primary: 34C23; Secondary: 92C45.

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Figure 1. Reaction scheme with branched sink

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Figure 2. Partition of parameter quadrant for $ (\alpha, \gamma)\in \mathbb{R}_+^2 $

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Figure 3. Phase portraits of system (7) with $ (n, \alpha, \beta, \gamma) = (4, 0.55, 50, 0.1) $ in (A) and $ (n, \alpha, \beta, \gamma) = (4, 0.57, 50, 0.1) $ in (B)

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Figure 4. Phase portraits of system (7) with $ (n, \alpha, \beta, \gamma) = (4, 0.67247, 10, 0.2) $ in (A) and $ (n, \alpha, \beta, \gamma) = (4, 0.672, 10, 0.2) $ in (B)

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Figure 5. Oscillation of substrate and product in the reaction. Solutions of system (43) with initial value $ (x(0), y(0)) = (0.569, 0.255) $

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Figure 6. Two limit cycles bifurcate from Hopf bifurcation

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Table 1. Parameter conditions of equilibria for system (7)

Possibility of parameters	Equilibria
$ (\alpha, \gamma)\in \mathcal{D}_0\cup \mathcal{D}_4 \cup\mathcal{L}_1 \cup\mathcal{L}_3 \cup \mathcal{P}_0 $	$ E_b $
$ (\alpha, \gamma)\in \mathcal{L}_4 $	$ E_b $ $ E_0 $
$ (\alpha, \gamma)\in \mathcal{D}_{1}\cup \mathcal{L}_{2} $	$ E_b $ $ E_1 $
$ (\alpha, \gamma)\in \mathcal{D}_{2} $	$ E_b $ $ E_2 $
$ (\alpha, \gamma)\in \mathcal{D}_{3} $	$ E_b $ $ E_1 $ $ E_2 $

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