QUANTITATIVE-ANALYSIS OF THE HOPF-BIFURCATION IN THE GOODWIN N-DIMENSIONAL METABOLIC CONTROL-SYSTEM

We study, from a quantitative point of view, the Hopf bifurcation in an ODE model of feedback control type introduced by Goodwin (1963) to describe the dynamics of end-product inhibition of gene activity. We formally prove that the exchange of linear stability of the positive equilibrium in the n-dimensional Goodwin system with equal reaction constants coexists with a Hopf bifurcation of nontrivial periodic solutions emanating from this equilibrium, without any further restriction on the dimension n greater-than-or-equal-to 3 or on the Hill coefficient-rho. The direction of the bifurcation, and the stability and the period of the bifurcating orbits are estimated by means of the algorithm proposed by Hassard et al. (1981).