Local bifurcations of an enzyme-catalyzed reaction system with cubic rate law

Nonlinear Dyn (2018) 94:521–539 https://doi.org/10.1007/s11071-018-4375-y ORIGINAL PAPER Local bifurcations of an enzyme-catalyzed reaction system with cubic rate law Juan Su · Bing Xu Received: 17 December 2017 / Accepted: 17 May 2018 / Published online: 31 May 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 Abstract In this paper, we study the dynamics of reaction scheme is given in Fig. 1. The change of reac- a system arising from enzyme-catalyzed reaction. tants in this scheme can be described by the following Parameter conditions for the existence and qualita- system (see [5,21]) tive properties of equilibria are given. Various kinds of s˙ = α − V (s, p) − V (s), 1 3 bifurcations including saddle-node bifurcation, (1.1) p ˙ = β(V (s, p) − V (p)), 1 2 Bogdanov–Takens bifurcation and Hopf bifurcation are where s, p ≥ 0 denotes the concentrations of substrate investigated. The order of weak focus is proved to be at most 2, and the parameter conditions of exact order S and product P, respectively, V is the rate law func- tion of S converting into P such that V (0, p) = 0, are obtained. Numerical simulations are employed to illustrate the results obtained. ∂V /∂s > 0 and ∂V /∂p > 0for s > 0 and p > 0, 1 1 V (p) ≥ 0 and V (s) ≥ 0 represent the sink rate func- 2 3 Keywords Enzyme-catalyzed reaction · Saddle-node tions of P and S respectively, and α, β> 0 are kinetics constants. Moreover, system (1.1) with V (s) ≡ 0 cor- bifurcation · Bogdanov–Takens bifurcation · Hopf bifurcation · Resultant responds to the removal of branched sink of S in Fig. 1, and has been discussed extensively in [6,10,11,18,19] and the references therein. Mathematics Subject Classification 34C23 · 92C45 System (1.1) with V (s) ≡ 0 includes several spe- cial cases studied in previous works. 1...