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

This work is supported by NSFC grant 11831012 and 11771168

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  • 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.

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


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  • Figure 1.  Reaction scheme with branched sink

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

    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)

    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)

    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) $

    Figure 6.  Two limit cycles bifurcate from Hopf bifurcation

    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 $
     | Show Table
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  • [1] B. AgudaL. Frisch and L. Olsen, Experimental evidence for the coexistence of oscillatory and steady states in the peroxidase-oxidase reaction, J. Amer. Chem. Soc., 112 (1990), 6652-6656.  doi: 10.1021/ja00174a030.
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