Local Bifurcations of the Enzyme-Catalyzed Reaction Comprising a Branched Network

Abstract

An enzyme-catalyzed reaction system with four parameters a, b, c, κ is discussed. The system can be reduced to a quartic polynomial differential system with four parameters, which leads to difficulties in the computation of semi-algebraic systems of large degree polynomials. Those systems have to be discussed on subsets of special biological sense, none of which is closed under operations of the polynomial ring. In this paper, we overcome those difficulties to determine the exact number of equilibria and their qualitative properties. Moreover, we obtain parameter conditions for all codimension-1 bifurcations such as saddle-node, transcritical, pitchfork and Hopf bifurcations. We compute varieties of Lyapunov quantities under the limitations of biological requirements and prove that the weak focus is of at most order 2. We further obtain parameter conditions for exact number of limit cycles arising from Hopf bifurcations.