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Mathematical Biosciences and Engineering (MBE)
 

Bogdanov-Takens bifurcations in the enzyme-catalyzed reaction comprising a branched network

Pages: 1499 - 1514, Volume 14, Issue 5/6, October/December 2017      doi:10.3934/mbe.2017078

 
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Qiuyan Zhang - Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China (email)
Lingling Liu - School of Sciences, Southwest Petroleum University, Chengdu, Sichuan 610500, China (email)
Weinian Zhang - Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China (email)

Abstract: There have been some results on bifurcations of codimension one (such as saddle-node, transcritical, pitchfork) and degenerate Hopf bifurcations for an enzyme-catalyzed reaction system comprising a branched network but no further discussion for bifurcations at its cusp. In this paper we give conditions for the existence of a cusp and compute the parameter curves for the Bogdanov-Takens bifurcation, which induces the appearance of homoclinic orbits and periodic orbits, indicating the tendency to steady-states or a rise of periodic oscillations for the concentrations of the substrate and the product.

Keywords:  Enzyme-catalyzed reaction, cusp, Bogdanov-Takens bifurcation, normal form.
Mathematics Subject Classification:  34C23, 92C45.

Received: July 2016;      Accepted: January 2017;      Available Online: May 2017.

 References